Theory of The Chemostat Cambridge Studies in Mathematical Biology
Theory of The Chemostat Cambridge Studies in Mathematical Biology
Theory of The Chemostat Cambridge Studies in Mathematical Biology
The Theory
of the Chemostat
Dynamics of Microbial Competition
CAMBRIDGE STUDIES
IN MATHEMATICAL BIOLOGY: 13
Editors
C. CANNINGS
Department of Probability and Statistics, University of Sheffield, UK
F. C. HOPPENSTEADT
College of Natural Sciences, Michigan State University, East Lansing, USA
L. A. SEGEL
Weizmann Institute of Science, Rehovot, Israel
CAMBRIDGE STUDIES
IN MATHEMATICAL BIOLOGY
Brian Charlesworth Evolution in age-structured populations
2 Stephen Childress Mechanics of swimming and flying
3 C. Cannings and E. A. Thompson Genealogical and genetic structure
4 Frank C. Hoppensteadt Mathematical methods of population biology
5 G. Dunn and B. S. Everitt An introduction to mathematical taxonomy
6 Frank C. Hoppensteadt An introduction to the mathematics of neurons
7 Jane Cronin Mathematical aspects of Hodgkin-Huxley neural theory
8 Henry C. Tuckwell Introduction to theoretical neurobiology
Volume 1 Linear cable theory and dendritic structures
Volume 2 Non-linear and stochastic theories
9 N. MacDonald Biological delay systems
10 A. G. Pakes and R. A. Maller Mathematical ecology of plant species
competition
11 Eric Renshaw Modelling biological populations in space and time
12 Lee A. Segel Biological kinetics
1
HAL L. SMITH
PA UL WALTMAN
Emory University
CAMBRIDGE
UNIVERSITY PRESS
1994
576'.15 - dc20
94-19083
CIP
A catalog record for this book is available from the British Library.
ISBN 0-521-47027-7 Hardback
To
Contents
page xi
Preface
1
7
13
14
19
20
28
28
2. Liapunov Theory
28
30
vii
34
37
42
43
43
44
47
51
53
59
65
68
69
71
Contents
viii
78
78
79
81
86
91
93
96
99
101
129
129
159
159
162
164
169
Variable-Yield Models
182
1. Introduction
2. The Single-Population Growth Model
3. The Competition Model
4. The Conservation Principle
5. Global Behavior of the Reduced System
6. Competitive Exclusion
7. Discussion
182
183
188
192
195
101
103
106
111
114
119
121
125
126
133
137
140
151
157
171
180
203
206
Contents
10 New Directions
11
231
231
231
Open Questions
248
C Monotone Systems
D Persistence
E
208
208
209
214
219
222
225
226
228
1. Introduction
2. The Unstirred Chemostat
3. Delays in the Chemostat
4. A Model of Plasmid-Bearing, Plasmid-Free
Competition
Appendices
B
ix
F A Convergence Theorem
References
Author index
Subject index
238
243
253
255
261
268
277
282
294
299
309
311
Preface
xii
Preface
Lienard and van der Pol, whose work led to significant advances in the
understanding of nonlinear differential equations. The equations of celestial mechanics motivated the original work of Poincare and the foundation of the "geometric" theory of nonlinear differential equations. Although models in physics are often accepted in spite of their restrictive
(and perhaps unrealistic) hypotheses, this has not been the case with equations that arise in biology. For example, the equations of motion for the
simple pendulum assume a perfect (frictionless) bearing, movement in a
vacuum, and the constancy of gravity. None of these conditions is exactly
satisfied, yet physicists and engineers study the pendulum and related equations for what they can learn about oscillations. In biology there are relatively few accepted mathematical models. The chemostat is one - and, in
microbial ecology, perhaps the only - such model that does seem to have
wide acceptance. This is true, at least in part, because the parameters are
measurable and the experiments confirm that the mathematics and the
biology are in agreement. The equations of population biology, in general, and those of the chemostat-like models, in particular, provide an
interesting class of nonlinear systems of differential equations that are
worthy of study for their mathematical properties. The study of these
equations has already lead to the discovery of new mathematics. The theory of cooperative and competitive dynamical systems has been motivated, at least in part, by problems in population biology. The theory of
persistence in dynamical systems grew directly from such considerations.
The chemostat models an open system, and although (strictly speak-
trient uptake term that is most accepted comes from a model, usually
Preface
xiii
credited to Monod, that appears in enzyme kinetics as the MichaelisMenten model. Yet the mathematics frequently allows for more general
functions than this particular nonlinearity. We have resolved this issue
using the following principle: We retain the specifics of the Monod model
as long as significantly stronger results (in the sense of describing behavior in the complete parameter space) can be obtained with its use. When
the problems reach a level of complexity where stronger results cannot be
obtained using the Monod model, we make use of more general classes
of nonlinearities that subsume the Monod nonlinearity. The problems in
xiv
Preface
to avoid using more abstract mathematical results, which might have al-
that this is the proper language for problems from population biology.
Most of the deterministic models in population biology make use of ordinary differential equations, functional differential equations, or reactiondiffusion equations; the language of dynamical systems is appropriate for
all three. This language is used throughout the book. Chemostats have a
certain conservation property - indeed, the presence of this property may
be fundamental to the definition of a chemostat - which requires some
mathematics to treat properly. A simpler proof of the result for a threedimensional system of chemostat equations is given in Chapter 1, and
Appendix F treats a special case of a very general theorem of Thieme.
When the limit sets are equilibria, this theorem is sufficient to make rigorous the reduction in complexity that follows from the conservation property. In Chapter 2 it is shown that the mathematical results for the simple
chemostat hold in much greater generality. Specifically, they do not depend
Preface
xv
xvi
Preface
mentioned. Many of these appear in the text proper, but are highlighted
here to call attention to mathematical questions whose answers would
help to complete the theory.
The choice of material reflects the authors' personal mathematical interests, so it is hoped that we have used both the engineering and biological literature appropriately. Although some of the results contained in
this monograph are new, we have not aimed for the most general possible
results, especially when achieving them would complicate and obscure
the basic ideas. However, when sharper results than those presented here
are known, we have tried to point out the appropriate references.
The material was used as a basis for a topics course at the Georgia Institute of Technology and one at Emory University. Both authors wish to
thank their students for many comments and discussions. Several colleagues read the entire manuscript, and we especially wish to express our
sincere thanks for this effort: S. B. Hsu, Y. Kuang, W. Rivera, and J. Wu.
Several others read portions of the manuscript as it was developed; we wish
1. Introduction
Competition modeling is one of the more challenging aspects of mathematical biology. Competition is clearly important in nature, yet there are
so many ways for populations to compete that the modeling is difficult to
carry out in any generality. On the other hand, the mathematical idea
seems quite simple: when one population increases, the growth rate of the
others should diminish (or at least not increase), a concept that is quite
easily expressed by partial derivatives of the specific growth rates. If an
ecosystem is modeled by a system of ordinary differential equations - for
example, by
y; =yifi(y),
where i = 1, 2, ..., n, f,, is a continuously differentiable function defined
on lib", and y = (yi, yz, ..., y") - then competition is expressed by the condition
of 0
<
ay;
when i # j. Dynamical systems with such properties have been studied extensively; see Hirsch [Hit; Hi2] and Smith [S3].
Such models easily reflect the direct impact of one population upon the
The name "chemostat" seems to have originated with Novick and Szilard
[NS].
In this monograph the basic literature on competition in the chemostat
is collected and explained from a common viewpoint. The subject is by
Figure 2.1. A schematic of the simple chemostat. (From [W2], Copyright 1990,
Rocky Mountain Mathematics Consortium. Reproduced by permission.)
S'(t) = S(O)D-S(t)D.
The formulation of the consumption term, based on experimental evidence, goes back at least to Monod [Mol; Mo2]. The term takes the form
mSx
a+S'
where x is the concentration of the organism (units are mass/13), m is the
maximal growth rate (units are 1/t), and a is the Michaelis-Menten (or
half-saturation) constant with units of concentration. The form (and the
terminology) of the consumption term is that of enzyme kinetics, where
S would be a substrate. Both a and m can be measured experimentally.
Since it is generally accepted by microbial ecologists, and since it contains
parameters that can be measured, the Michaelis-Menten (or Monod) formulation is most often used as the uptake function, but the mathematical
results are valid for much more general functions. Simple monotonicity
in S, with a limit as S tends to infinity, is usually sufficient. Trying to
squeeze the greatest mathematical generality from the theorems could
interfere with our presentation, so the emphasis here is on the Monod
evidence and does not rest on any physiological basis. The uptake of
nutrient is a very complex phenomenon from the standpoint of molecular
biology. Indeed, the transport of the nutrient through the cell wall is itself
a very complex phenomenon. Dawes and Sutherland [DSu] give a descriptive (i.e. nonmathematical) introduction to microbial physiology and
its complexities. Koch [Ko] considers the uptake and the factors affecting
growth in considerable detail. The Monod and other similar formulations give an aggregate description of the nutrient uptake; to do otherwise would make the modeling problem very difficult. One can, however,
take into account that the uptake by "larger" cells is more than that of
"smaller" ones [Cu2].
The differential equation for S takes the form
S'=(S1-S)D- Sx
(2.1)
-y
while that of the corresponding equation for the microorganism, assuming growth is proportional to consumption, is
x'=x(
+S
-D),
(2.2)
where -y is a "yield" constant reflecting the conversion of nutrient to organism. The constant -y can be determined (in batch culture) by measuring
mass of the organism formed
mass of the substrate used
and hence is dimensionless. (We will scale it out in the simple chemostat,
but it is important for multiple-nutrient problems.) That 1' is a constant
is a hypothesis; this hypothesis will be reconsidered in Chapter 8. The
assumption that reproduction is proportional to nutrient uptake is a vast
simplification. The cell cycle is a very complex phenomenon, and entire
books have been devoted to its description (see e.g. Murray and Hunt
[MH]). Incorporating the essentials of the cell cycle into the chemostat
model would be an interesting problem. From the mathematical point of
view, this introduces a delay between nutrient uptake and cell division.
Comments on the delay models can be found in Chapter 10, and their
proper incorporation in microbial models is very much an open problem.
The appropriate initial conditions are S(0) >- 0 and x(0) > 0. The number of parameters in the system is excessive, so some scaling is in order.
First of all, note that 5(0) and D (the input concentration and the washout
rate) are under the control of the experimenter. The S(O) term has units of
x
S' =(I- S )D- MS/S(O)
- 1- SMO> a/S(0)+S/S(O) S(0)y'
S(O)
x'y
S 0)
xMS/S(O)
C S(0) y
a/S(0)+S/S()
By measuring S, a, and x/y in units of 5(0) and time in units of D-1, one
obtains the following nondimensional differential equations (note that m
and a have changed their meanings):
S'=15
x'=x1
S(0)?\\\0,
mSx
a+S
-1
(2.3)
S
x(0)>0.
This sort of scaling will occur frequently in the problems that follow.
The constants m and a can be regarded as the "natural" parameters of the
organism in this particular environment. We have standardized the environment, scaling out the factors that can be changed by the experimenter;
hence the use of natural parameters expressed in (2.3). This unclutters
the mathematics from the "real" world and focuses attention on the selection of the parameters a and m. This, of course, is in marked contrast to
the point of view of a person who wishes to perform an experiment. There
the parameters a and m are given; they come from the organism selected.
An experimenter wishes to tune the chemostat to make the organisms
grow. Thus, particularly in the engineering literature, one finds an emphasis on presenting results in the form of "operating diagrams," graphs
that show where to operate the chemostat. Since the emphasis here is
Dynamical Systems
theoretical, the scaling just described will be used whenever possible; results in terms of the original parameters can easily be obtained by reinterpreting the parameters.
3. Dynamical Systems
Although the system of equations (2.3) is simple enough to handle directly, we pause here to introduce some mathematical material that will
be important in the remainder of the book. The reader who is not interested in mathematical tidiness may just note the definitions and go on to
the next section. [CL] and [H2] are standard references for the material
presented here. The focus throughout the book will be on the "dynamical
systems" point of view. Dynamical systems are used primarily as a language, not because we need many deep results from that subject. The
language, however, does seem natural for the problems considered. The
dynamical system will be defined in terms of R', but the natural (and
most efficient) formulation is that of a metric space. In a later chapter we
will use the space C[0, 1], the space of continuous functions on the interval [0, 1] with the usual sup norm, and the definition will be expanded
at that time.
The most basic concept is that of a dynamical (or a semidynamical) system. Let ir: M x 118 -> M be a function of two variables, where M is 118"
and R denotes the real numbers. (We use M for the first variable or state
space to suggest that the results are true in greater generality.) The function it is said to be a continuous dynamical system if 7r is continuous and
has the following properties:
Y,=f(y),
(3.1)
nonpositive t are considered, the set is called the negative orbit or nega-
tive trajectory through the point and is denoted by y-(x). The union
of the positive and negative orbits is simply called the orbit or trajectory
through the point, denoted by y(x). For emphasis, the latter is sometimes
called the full orbit. For biological systems one wants to determine the
eventual behavior - the asymptotic properties - of trajectories. Biological
models require that trajectories remain positive (concentrations or populations are positive numbers) and that trajectories do not tend to infinity
with increasing time. If a set S is such that all trajectories that begin (have
their initial condition) in S remain in S for all positive time, then S is said
to be positively invariant. (If trajectories remain in S for both positive
and negative time, S is said to be invariant.) Hence the basic condition
for positivity (of the dependent variables) can be stated as "the positive
cone is positively invariant for the dynamical system generated by (3.1)."
The dynamical system is said to be dissipative if all positive trajectories
eventually lie in a bounded set. This is sufficient to ensure that all solutions of (3.1) exist for all positive time.
Let ( t D be a sequence of real numbers which tends to infinity as n tends
to infinity. (Such a sequence is sometimes called an extensive sequence.)
If P = lr(x, converges to a point P, then P is said to be an omega limit
point of x. (More correctly, P is an omega limit point of the positive trajectory y+(x); both references will be used, but since there is a unique
trajectory through each point x, the abuse of terminology will cause no
confusion when dealing with systems of the form (3.1).) The set of all
such omega limit points is called the omega limit set of x, denoted w(x).
If the system is dissipative, the omega limit set is a non-empty, compact,
connected, invariant set. Moreover, the orbit y+(x) is asymptotic to the
omega limit set of x in the sense that the distance from 7r(x, t) and w(x)
tends to zero as t tends to infinity.
Now let It, I be a sequence of real numbers which tends to negative infinity as n tends to infinity. If P = lr(x, converges to a point P, then
P is said to be an alpha limit point of x. The set of all such alpha limit
points is called the alpha limit set of x, denoted a(x). It enjoys similar
properties if the trajectory lies in a compact set for t < 0.
A particularly important class of solutions are the constant ones, which
are called steady states, rest points, or equilibrium points. In terms of
(3.1), such a solution is a zero of f(y), that is, a vector y*e 118" such that
f(y*) = 0. In the terminology of dynamical systems, a rest point is an
element p e M such that ir(p, t) = p for all t E R. Similarly, a periodic
orbit is one that satisfies ir(p, t + T) = -(p, t) for all t and for some fixed
number T. The corresponding solution of (3.1) will be a periodic function.
Dynamical Systems
an assumption that the theorem is false) and will not be given here; a
good reference is [ALGM].
THEOREM (Dulac criterion). Suppose that (3.1) is two-dimensional. Let
F be a simply connected region in R2 and let 13(x) be a continuously dif-
10
P2
Figure 3.1. Examples of limit sets for planar systems: a a rest point; b a periodic
orbit; c multiple rest points with connecting orbits.
in R". The solution 0(t, yo) is said to be stable if, for any e > 0, there
exists a S > 0 such that if IIYo -xoI< 8 then II0(t, yo) -O(t, xo)II < e for
all t > 0. The solution 4(t, yo) is said to be asymptotically stable if it is
stable and if there is a neighborhood N of yo such that if xo e N then
lim, -. II 0(t, xo) - 0(t, yo)II = 0. We shall be concerned with the case where
0(t, yo) is a constant solution or rest point, that is, where 0(t, yo) =Yo
for all t. We usually use y* to denote a rest point. Note that a rest point
y* is asymptotically stable if it is stable and an attractor.
The system
(3.2)
x'=fy(Y*)x
is said to be the linearization of (3.1) around the rest point y*, where
Dynamical Systems
[ax
Ifi I
11
Y=Y%
M+(P) =
=Pl;
M-(P) _ {x I lim,-
'Jr(x, t) = P.
The sets M+(P) and M-(P) are locally manifolds of dimension n-k
and k, respectively, and all trajectories with initial conditions on these
sets tend to the rest point as t tends to infinity (stable) or as t tends to
negative infinity (unstable). One should think of these manifolds as surfaces in the appropriate space. On these surfaces, trajectories tend to the
rest point as t tends either to positive or to negative infinity. (To assist
with the notation, the reader should associate the plus sign on M+ with
positive time and the minus sign on M- with negative time.) In particular,
a single eigenvalue with positive real part makes the rest point unstable.
The corresponding eigenvectors generate the tangent space to the respec-
12
tire omega limit set. Then w(x) has nontrivial (i.e., different from P)
intersection with the stable and the unstable manifolds of P.
Figure 3.1c, where P is any of the three equilibria, illustrates the theorem.
A short proof (due to McGehee) can be found in the appendix of [FW2],
are not empty. The intuition behind the result is that an orbit cannot
"sneak" into and out of a neighborhood of P infinitely often without
having accumulation points on the stable and unstable manifolds. The
proof simply makes this idea precise. (The proof may be skipped on first
reading.)
Proof of the Butler-McGehee Theorem. Since P is a hyperbolic equilibrium, there exists a bounded open set U containing P, but not x, with
the property that if 7r(y, t) e U for all t > 0 (t < 0), then y belongs to the
local stable (unstable) manifold M+(P) (M-(P)); see [H2]. (P is the
largest invariant set in U, or U isolates P from any other invariant sets.)
By taking a smaller open set V, P E V C PC U, we have that 7r(y, t) E V
for all t > 0 (t < 0) implies y e M+(P) (M-(P)).
Since PE w(x), there exists a sequence [tn), limn-,. t = 00, such that
limn-,o x = limn . 7r(x, tn) = P. It follows that x e V for all large n.
Since xeM+(P), else w(x) = P, from the property of the neighborhood
V one may conclude that there exist positive numbers r,,, Sn such that
rn < t,,, 7r(xn, t) E V for -rn < t < s,,, and 7r(xn, -rn), 7r(xn, sn) e aV By
the continuity of 7r(x, t), solutions that start near P must remain near P;
hence it follows that rn and sn tend to infinity as n tends to infinity. However, V is compact, so (passing to a subsequence if necessary) one may
conclude that limn -. 7r(xn, -rn) = q E V and limn 7r(xn, sn) = q e V
We continue the proof for q; the other case is similar.
It is claimed that 7r(q, t) e V for all t > 0. Recall that limn qn = q
where qn = 7r(xn, -r, ). Fix t > 0. By the continuity of 7r, limn -. 7r(gn, t) =
7r(q, t). Since -rn < t-rn < 0 for all large n, 7r(gn, t) = 7r(xn, t - rn) e V
-.
13
for all large n. It follows that 7r(q, t) e V. Since t > 0 was arbitrary, the
claim is established.
Since 7r(q, t) e V for all t > 0, we have q e M+(P) by the isolating property of V cited in the first sentence of the proof. However, q e y+(x) =
y+(x) U ,(x). Since q e M+(P), q e y+(x) and hence q e w(x), which establishes one case of the theorem.
For many of the systems of interest here, the dynamics restricted to the various boundaries of the positive cone in I&" will be dynamical systems in their
own right - the boundaries will be invariant sets. It may happen that a rest
point P will be asymptotically stable when regarded as a rest point of the
lower-dimensional dynamical system and yet have unstable components
when the full system is considered. If the entire stable manifold is contained
in the boundary, then the Butler-McGehee theorem can be used to conclude that no trajectory from the interior of the positive cone can have P as
an omega limit point. Indeed, the omega limit set cannot equal P because
the initial point does not belong to the stable manifold of P. If the limit
set contains P then it would also contain a point of the stable manifold distinct from P, by the Butler-McGehee theorem, and would therefore contain the closure of the entire orbit through this point since the omega limit
set is closed and invariant. However, this typically leads to a contradiction, since orbits in the stable manifold of P are either unbounded or their
limit sets contain equilibria that can be readily excluded from the original
limit set (e.g. are repellors). Section 5 will use the theorem in this way.
4. Analysis of the Growth Equations
For system (2.3) the positive cone is positively invariant (see Appendix B,
Proposition B.7). In simpler terms, if the system is given positive initial
conditions then the two components of the solution remain positive for
all finite time. Moreover, if one adds the two equations and defines E _
1-S-x, then one obtains a single equation
E'= -E
with E(0) > 0. It follows at once that lim,- E(t) = 0 and that the convergence is exponential. This not only gives the required dissipativeness
but also leads to the simplification of the system by the elimination of
14
x'=x m(1-x)
l+a-x -1 ,
0:5x:51.
It might seem at first that it was extremely fortuitous that the aforementioned limit should exist. However, there is a simple, intuitive explanation. If there were no organisms in the model - that is, if only nutrient
were present in the equation - then the nutrient would satisfy
A=
m-1'
(4.2)
[1+a-x
Clearly, if m < 1 or m > 1 and A > 1, then lim, -. x(t) = 0 (x'(t) is negative and x(t) is bounded below by zero). On the other hand, if A < 1 and
m > 1, then lim,_. x(t) = I -A (and hence lim,_. S(t) = A). If m < 1,
the organism is washing out faster than its maximal growth rate, whereas
if A >- I there is insufficient nutrient available for the organism to survive.
In either case, extinction is not a surprising outcome. The case m = 1 is
handled by using (4.1) directly.
5. Competition
To study competition in the chemostat, introduce two different microorganisms into the system, labeled x1 and x2, with corresponding parameters
Competition
15
-5- m1Sxl
- m2Sx2
a1+S az+S
_
mS -l
xl-xl(al+S
(5.1)
x2_xz
mzS
az+S
-l
E, = -E,
=x1(ml(1-E-xl-x2)
xl
al+l-E-xl-x2
xz
x2(m2(1-E-xl-x2)
a2+1--x1-x2
lime-, E(t) = 0,
where the convergence is exponential. Again this shows that the system is
dissipative and that, on the set E = 0, trajectories satisfy
=(ml(1-xl-xz)
xl
x1 al+l-xl-x2
x2(m2(1-xl-xz)
x2 -
a2+l-xl-x2
-l
/'
-l
(5.2)
'
xl =x1
m
1+a1
I x1-x2
M2
xz -x2 l+az x1-x2 [1-A2-x1-x21,
1
16
E, = (1-A,, 0),
E2 = (0,1-A2).
lim, x,(t)=1-A,,
lim,-- x2(t) = 0.
Proof. We begin by analyzing (5.2). The first step is to compute the stability of the rest points of system (5.2) by finding the eigenvalues of the
Jacobian matrix evaluated at each of these rest points. At (0, 0) this matrix takes the form
(m,-1)(1-A,)
l+a,
0
(m, -1)(1-A2)
1+a2
(A,-1)(a,ml)
(A,-1)(a,ml)
(A,+a,)2
(A, +a,)'
(m2-1)(A1-A2)
A,+a2
Competition
17
Since 0 < A, < A2 and m2 > 1, both eigenvalues are negative. Thus E, is
(locally) asymptotically stable. At (0,1-A2), the variational matrix takes
the form
(m1-1)(A2-A,)
A2+a1
(A2-1)(a2m2)
(A2+a2)2
(A2-1)(a2m2)
(A2+a2)2
One eigenvalue is negative since A2 < 1 and one is positive since A, < A2.
Thus the stable manifold is one-dimensional and, since E2 attracts along
the x, = 0 axis, the stable manifold lies there. In particular, the ButlerMcGehee theorem (stated in Section 3) allows one to conclude that no
trajectory with positive initial conditions can have E2 as an omega limit
point. Since the initial data are positive, the omega limit set cannot equal
E2. If it contained E2, then it must also contain an entire orbit different
from E2 belonging to the stable manifold of E2. There are only two possible orbits; one is unbounded, and the other has alpha limit set E0. But
the omega limit set cannot contain an unbounded orbit and it cannot
contain E0 since it is a repeller. Therefore, E2 is not a limit point.
Since E, is a local attractor, to prove the theorem it remains only to
show that it is a global attractor. This is taken care of by the PoincareBendixson theorem. As noted previously, stability conditions preclude a
trajectory with positive initial conditions from having E0 or E2 in its omega
limit set. The system is dissipative and the omega limit set is not empty.
Thus, by the Poincare-Bendixson theorem, the omega limit set of any
such trajectory must be an interior periodic orbit or a rest point. However,
if there were a periodic orbit then it would have to have a rest point in its
interior, and there are no such rest points. Hence every orbit with positive
initial conditions must tend to E,. (Actually, two-dimensional competitive systems cannot have periodic orbits.) Figure 5.1 shows the x1-x2 plane.
18
E,
Figure 5.1. The phase plane diagram for the system (5.2).
M+(E2)=1(E,0,x2)Ix2>O,x2+E<_1).
Let p = x(0) be an arbitrary initial point with x;(0) > 0. Then the initial
data do not belong to either stable manifold. Hence w(p) is not equal
to either Eo or E2, but it does lie on E = 0. Since it is invariant and since
every solution of (5.2) on E = 0 converges to an equilibrium, w(p) contains an equilibrium. By the Butler-McGehee theorem, Eo e w(p) since
M+(Eo) is unbounded. If w(p) contains E2, then w(p) also contains
either Eo or an unbounded orbit, again by the Butler-McGehee theorem
(see Figure 5.2). Since this is impossible, El must be in co(p). However,
El is a local attractor, so w(p) = El. This completes the proof.
If A = A2 then it is not difficult to show that coexistence is possible. This
The Experiments
19
6. The Experiments
The experiments of Hansen and Hubbell [HH] confirm the mathematical
result. By working with various microorganisms, Hansen and Hubbell
showed that it is the lambda value which determines the outcome of the
competition. It is worth noting that this was an example of the mathematics
preceding the definitive biological experiment on this type of competition.
Although the experiments may be described by a couple of paragraphs,
one should not conclude that such experiments were easy. Microbial experiments are always fraught with difficulties. The "proper" organisms
and limiting nutrient must be selected so that they can grow under the
circumstances where the chemostat is operated. The chemostat must be
operated at a turnover rate (washout rate) that allows no growth on the
cell walls and no buildup of metabolic products. The feed bottle must
not contain substances from whose molecules the organisms can synthesize the limiting nutrient. If the organism mutates during the experiment,
20
the experiment is invalid, so careful checks are run at the end to determine
that what is grown is actually what was introduced into the culture vessel.
The list could continue.
To motivate the form of the experiments, note first that two parameters
are properties of the organism: the m and the a of the chemostat equations. One might postulate that the competitor with the largest m or the
one with the smallest a should win the competition. Recall that m is the
maximal growth rate and that a (the Michaelis-Menten constant) represents the half-saturation concentration (and so is an indicator of how well
an organism thrives at low concentrations). Both of these quantities are
obtainable in the laboratory by growing the organism (without a competitor) on the nutrient. (Hansen and Hubbell used a Lineweaver-Burk plot.)
Three experiments were carried out (each reproduced three times) with
these postulates in mind. In the first experiment, the organism with the
larger m had the smaller lambda value. In the second, the organism with
the smaller Michaelis-Menten constant had the larger lambda value. In
both cases, the organism with the smaller lambda value won the competition as predicted by the theorem. Finally, organisms with differing m and
a values but with (approximately) the same lambda value were shown to
coexist (for a reasonably long time).
We reproduce here one table (as Table 6.1) and four graphs (as Figure
6.1) from [HH]. The table shows the organisms used, their parameters,
and the run parameters for the chemostat. The limiting nutrient was tryptophan. Hansen and Hubbell used different notation than that presented
here, so one should translate as follows: r = m - D, J = A, K = a, =
m, and So = 5(0). The graphs show the predicted time course (with the
unscaled variables) in dashed lines and the experimental values with dots
connected by solid lines.
An interesting part of the technique was the way in which the equal
lambdas were obtained. One strain of E. coli had a growth rate that was
inhibited by a chemical (Nalidixic acid) while the other strain was essentially immune to the compound (see the third graph in Figure 6.1). By
adding the proper amount of the chemical, it was possible to alter the
growth rate so as to make the resulting lambda values equal. The chemostat with an inhibitor will be studied in Chapter 4.
7. Discussion
Theorem 5.1 is an example of the principle of competitive exclusion: only
one competitor can survive on a single resource. Many of the well-known
Table 6.1
Other run parameters
It
(g/liter)
(per
hour)
(per
hour)
2.5x1010
3.8x1010
3.0x10-6
3.1x10-4
0.81
0.91
C-8 nal'specs
6.3 x 1010
1.6 x 10-6
C-8 nalsspec'
C-8 nal'specs
6.2 x 1010
6.3 x 1010
1.6 x 10-6
1.6 x 10-6
C-8 nalsspec'
6.2 x 1010
0.9 x 10-6
0.41
Bacterial
strain
Yield
Ks
(cell/g)
C-8'
PAO283b
31'
Vol-
(g/liter)
D
(per
hour)
1x10-4
6.0x10-2
200
5 X 10-6
7.5 x 10-2
200
5 x 10-6
7.5 x 10-2
200
(g/liter)
0.75
0.85
2.40x10'
0.68
0.61
1.98 X 10-'
0.96
0.68
0.89
1.35 x
0.61
0.34
1.98 x 10-'
ume
(ml)
2.19x10-'
10-7
1.99 x 10-'
a Escherichia coli.
b Pseudomonas aeruginosa.
INalidixic acid added (0.5 g/ml).
Source: [HH, p. 1491], Copyright 1980, American Association for the Advancement of Science. Reproduced by permission.
Observed
--- Predicted
40
20
60
Time (hours)
.4
.8
Time (hours)
20
40
60
Time (hours)
80
100
120
Discussion
23
models of competitive systems seem to satisfy this conclusion - for example, the two-dimensional Lotka-Volterra competition model. In the dis-
S'= (S(O)-S)Dx1=x1
m1S
a1+S
X2=x2(a
z
+S
m2Sx2
a2+S
a1+S
m1Sx1
-D '
-D),
mS
a;+S
Theorem 5.1 asserts that if the equations f (S) = D have unique positive
solutions S = A, (i = 1, 2) and if 0 < Al < A2 < 5101, then the population
x1 is the winner of the competition; it eliminates population x2 from the
chemostat. It follows that if, as in Figure 7.1a, one function dominates
the other for all positive values of S (or at least for 0 < S < 5(0)), then the
corresponding population should win the competition. For there to be
any hope of coexistence, the two functions must cross before S = 510).
Denote by S* the point where the functions cross. If S* < 5(O), let D* =
f (S*). If the dilution rate D is set equal to D*, then Al = A2 = S* (see
Figure 7.1b) and coexistence is possible.
Caption for Figure 6.1
a Experiment 1: Strains differ principally in their half-saturation constants for
tryptophan, and PA0283 loses to C-8 as predicted. b Experiment 2: Strains differ
in their intrinsic rate of increase but not in their half-saturation constants, and
C-8 nalrspecs loses to C-8 nalsspecr as predicted. c Effect of naladixic acid on the
intrinsic rate of increase of strains C-8 nal'specs and C-8 nalsspecr. d Experiment 3: Strains differ in the half-saturation constants and in their intrinsic rates
of increase, but nevertheless have identical J parameters; the strains coexisted for
the duration of the experiment, as predicted. In each experiment, the predicted
curves were obtained by numerical integration. Bars around points in experiments
2 and 3 indicate ranges of three replicate values. (From [HH, p. 1492], Copyright
1980, American Association for the Advancement of Science. Reproduced by
permission.)
24
a
S
f, (S)
f2(S)
b
S'=A1=A2
f, (S)
f2(S)
Figure 7.1. Example of Monod-type uptake (or growth) functions: a fl(S) dominates f2(S) for all values of S, and competitive exclusion holds; b f,(S) and f2(S)
cross exactly at the value of D, producing coexistence, the knife-edge phenomenon; c fl(S) and f2(S) cross, and competitive exclusion holds.
Discussion
25
attaching itself to the wall of the chemostat and so escapes the model's
implications (the dilution process does not affect the cells attached to the
wall). The wall is a refuge. This is a real, practical problem in the operation of the chemostat, one that frequently affects the allowable dilution
rates. (When the turnover is too slow, wall growth is a problem.) Another
possibility is that there is a small perturbation due to mutual interference
between the two populations. A mass action term -ex1x2 in one of the
equations would remove the instability. Neither of these situations meets
the basic "conservation" principle of the chemostat. G. E. Powell [Po]
studies the problem quite generally and considers the case where there is
an additional substitutable nutrient, which he calls P. Powell includes a
term eg1(P) in the equations of growth of the competitors, along with the
26
Discussion
27
1. Introduction
In the previous chapter it was shown that the simple chemostat produces
competitive exclusion. It could be argued that the result was due to the
two-dimensional nature of the limiting problem (and the applicability of
the Poincare-Bendixson theorem) or that this was a result of the particular type of dynamics produced by the Michaelis-Menten hypothesis on
the functional response. This last point was the focus of some controversy
at one time, inducing the proposal of alternative responses. In this chapter it will be shown that neither additional populations nor the replacement
of the Michaelis-Menten hypothesis by a monotone (or even nonmonotone) uptake function is sufficient to produce coexistence of the competitors in a chemostat. This illustrates the robustness of the results of Chapter 1. It will also be shown that the introduction of differing "death rates"
Liapunov Theory
29
x' = f(x),
(2.1)
XeG
dt V(x(t)) =
(2.2)
by the chain rule and (iii), so that V(x(t)) is nonincreasing along solutions of (2.1).
The following theorem is a minor variation of the LaSalle corollary of
Liapunov stability theory, taken from [WLu] (see also [H2], where V is
30
ant subset of E.
For the reader unfamiliar with Liapunov theory, the intuition goes like
this: V is some sort of measure of "height." The magnitude of dV/dt =
VV f <- 0 is a measure of how fast solutions run "downhill." The downhill slide stops at E. The theory of dynamical systems says that the omega
limit set is invariant and therefore contained in M.
3. General Monotone Response and Many Competitors
The equations describing competition between n competitors having concentrations x1(t), x2(t), ..., xn(t) for growth-limiting nutrient S(t) in a
chemostat are a straightforward generalization of equations (2.1) and
(2.2) of Chapter 1. Replacing the Michaelis-Menten response function by
general monotone response functions f (S) for the ith competitor, 1 <i <- n (precise hypotheses on these functions will be described shortly),
one obtains the system
1'r
x, = xi(f(S) -D),
i=1,2,...,n,
with S(0) >- 0 and x,(0) > 0.
By measuring S and x;/-y1 in units of S(0) and time in units of D-', one
obtains the nondimensional system
n
x; = xi(f(S)-1),
(3.1)
i=1,2,...,n.
(Actually, f (S) in (3.1) should really be D-'f (S (0'S), but we rename the
latter to be the former.) Assume that the f satisfy the following:
31
These assumptions are really quite mild; the f need only be increasing and sufficiently smooth. It is not even required that f be bounded
on 118+. From a biological perspective, of course, the model loses relevance for really large values of S. In addition to the Monod functions,
other functions that have been suggested include the exponential kinetics
m(1-exp(-S log 2/a)), hyperbolic kinetics m tanh(S log 3/2a), and piecewise linear kinetics given by mS/2a for S < 2a and by m for S > 2a. The
piecewise linear kinetics fails to satisfy the strict monotonocity of (iii)
and fails to satisfy (iv) at one point, but these assumptions could be weakened so as to include this case.
0<A1<A2<...<An<co.
This assures that competitor x1 has the least requirements for growth and
therefore is favored to win the competition.
As in Chapter 1, set
n
E=S+EXjj=1
and observe that in the variables E, x1, x2, ..., xn, (3.1) takes the form
E'= -E,
X1=XiI J'
1+E-JE1XjI-1I,
(3.2)
i=1,2,...,n.
Obviously,
lim,_0E(t)=0,
and so it follows that solutions of (3.1) and (3.2) exist and are bounded
for t ? 0. Both S(t) and xi(t) remain nonnegative from the form of (3.1).
Again, as in Chapter 1, we are led to consider the system (3.2) restricted
to the invariant hyperplane E = 0, to which all solutions are attracted at
an exponential rate. This system is given by
n
l
xjl-1
xXi(f(I-j=E1//
\\
\\
i=1,2,...,n.
The relevant domain for (3.3) is the set
(3.3)
32
XE R' ,: EXj<1
j=1
It is easily seen that Sl is positively invariant for (3.3); indeed, the vector
field (3.3) points into the interior of 12 on that part of its boundary where
Ej=1 xj = 1. To see this, observe that if x(0) lies on this component of the
boundary of S2 then
d
(x(t)=_1<0,
Y'
1=0 j=1
so Eij=lxj(t)<1 fort>0.
A competitor x; for which A; >- 1 is an inadequate competitor, since its
break-even concentration equals or exceeds the reservoir concentration
of nutrient. Our first result makes this mathematically precise.
XjI-11 <_Xi(f(1)-1)<-0
in G = S2. If A; > 1 then E= (xe S2: 1''(x) =01 _ [xe S2: x; = 0}, whereas
if A; = 1 then E = ( x e 9: x; = 0 or Ej=1 xj = 1 1. As the vector field (3.3)
points into the interior of S2 on Ej=1 xj = 1, it follows that the largest
invariant set M in E is equal to I x E 0: x; = 0 ). By the LaSalle corollary,
x(t) -*M as t oo, so the assertion of the proposition holds.
Now consider the case that the favored competitor x1 is an adequate one;
that is, suppose 0 < Al < 1. Let
E1=(1-A1,0,0,...,0)
be the rest point corresponding to the survival of only species x1. System (3.3) may have other rest points besides E0 = 0 and E1 in S2 if 0 <
Aj < 1 for some j >: 2, but the following analysis does not require their
consideration.
Our main result here states that competitive exclusion holds for n competitors in a chemostat provided each competitor possesses a monotone
uptake function. The proof follows [AM].
THEOREM 3.2. Let x(t) be a solution of (3.3) in J for which x1(0) > 0.
Then
lim,--x(t) =E1.
33
Proof. DefineO=(xeS2:Ejxj=1-Ai1,63=(xeS2:Ejxj<1-A11,and
C = (x e S2 : E j xj > 1- A11. It will be shown that a solution starting in e
either remains in C and converges to E1 or enters 63 and remains there.
Once in (B, we show that the solution converges to E1i completing the
proof.
First, observe that
n
CJEIxJ)
=JElxjl f (I-EXk)_I)<0
I- F, Xk<A1.
k=1
It follows immediately that if x(t) e L\EI for some t then x(s) e 63 for
s > t. In particular, once a solution enters 63, it can never get out.
Suppose the solution x(t) remains in C for all t > 0. The previous calculation shows that if V(x) = E,=1 xj in C then V(x) < 0. It is easy to see
that V(x) = 0 for x e G U A if and only if x = E1. By the LaSalle corollary
with G* = 9 and G = C, any solution remaining in C for t >- 0 converges
to E1.
verges to the largest invariant set in (xe 63U0: V(x) = 01. For our solution, obviously w(x(0)) C 0 since x1 increases along x(t ). The only invariant subset of 0 is E1, so necessarily x(t) -> E1 as t co. This completes
the proof.
We hope the reader will appreciate the elegance and simplicity of the
arguments supporting Theorem 3.2, which are based on the LaSalle corollary. In particular, a linearized stability analysis about each of the rest
points of (3.3), required in Chapter 1, was completely avoided. A careful
reading of the proof of Theorem 3.2 reveals that assumption (iii) on f,, is
not crucial to the proof; we will have more to say about this later. Finally,
it should be noted that the assumption (iv) on f can be relaxed somewhat.
It can be weakened to requiring only that f be locally Lipschitz continuous
34
tion that all of the removal is accounted for by the washout term. If,
for example, a competitor's mortality rate is a significant fraction of the
washout rate D, then the assumption is not valid. In this case, the removal
rate for that competitor should be the sum of D and the mortality rate.
Another possibility is that a filter on the output might slow the washout
of an organism but not the nutrient. This could result in that organism's
removal rate being less than D. A natural question, then, is whether a
species-specific removal rate changes the outcome.
This question was studied by Hsu [Hsul] in the chemostat with Michaelis-Menten dynamics, and his work is presented here. The equations
take the form (ignoring the yield constants)
n m; Sx;
xi=x;
m;S
(ai+S
-D;
i= 1, 2,...,n,
with S(O) > 0 and x,(0) > 0. Scaling by D and 5(0) as before yields the
normalized equations
S' 15
x; = x;
m Sx
`
;_1 a;+S
mS
a;+S
- D;
i=1,2,...,n,
(4.1)
35
_ a,D,
Al
m; - Di
'
where it is assumed that mi > Di if Ai is to be defined. Exactly as in Chapter 1, if mi < Di or if Ai ? 1 then 1im,- . xi(t) = 0; in this case, the entire
system merely tends to a lower-order dynamical system, that is, one with
fewer competitors. This statement provides necessary conditions for sur-
vivability, and one need only consider competitors that satisfy m; > Di
andAi<1.
The mathematical difficulties in treating (4.1) are immediately apparent the conservation principle is lost, and the equations cannot be combined
to eliminate one of the variables. Enough of the analysis survives, however, to at least show that (4.1) is dissipative. Adding the equations and
replacing Di by d = min(D,, D2,..., D,,,11 yields a differential inequality
for >G = S+ Z,%, xi of the form
' <- 1- d>G.
then it follows
limsup[S(t)+
xi(t)] <-d
'.
36
109I1
X,]]+
E Cixi,
X1
i=2
dt
V(S(t),
n mixiS
ai+S
1-5-E
1
ml -Dl (S-A1)x1
a1+S
m2-D2
C2
a2+S
an+
Cn
(S-A2)x2
Dn
(S-A,)xn
1-S
=(S-A1)
m1x*1
mixi
(A1-Ai)ai+S.
xl =
1-A1
(1-Al)(a1+A1)
m1A1
D1
1-S
mix]*.
a1+S
may be simplified to
al(S-A1)
A1S(a1+S)
This in turn may be substituted into the expression for dV/dt to obtain
dV
dt
(S-Al)2a1
(a1+S)SA1
n
2
m (A1 -A`)
X'
ai+S
'
37
is given by
S =A,,
x, _ (1-A,)/D1,
x,=0, i=2,...,n.
An application of the LaSalle corollary yields the desired result.
The biological conclusion is, of course, that differing removal rates do not
alter competitive exclusion in the chemostat. One anticipates that a similar conclusion is true if the Michaelis-Menten dynamics is replaced by the
general monotone term f (S) used in Section 3. However, the Liapunov calculations depend on this form and the general question is still unresolved.
The reader will have noticed that the Liapunov function used in the
proof of the theorem was not obvious on either biological or mathematical grounds. Its discovery by Hsu greatly simplified and extended earlier
arguments given in [HHW ]. This is typical of applications of the LaSalle
corollary. Considerable ingenuity, intuition, and perhaps luck are required
to find a Liapunov function.
5. Nonmonotone Uptake Functions
It has been shown that competitive exclusion - that is, the extinction of
all but one competitor - holds regardless of the number of competitors or
the specific monotone functional response. If one restricts attention to
the Michaelis-Menten functional response, then competitive exclusion
has been shown even in the case of population-specific removal rates.
There is evidence, however, that a monotone functional response may
be inappropriate in some cases. A nutrient which is essential at low concentrations may be inhibiting (or, indeed, even toxic) at higher concentrations. Butler and Wolkowicz [BWo1] consider this possibility; their
work has been recently extended in [WuL]. We will describe some special
cases of their work in terms of the unscaled system immediately preceding (3.1). Assume that the functional response f satisfies (i), (ii), and (iv)
of Section 3, but replace the strict monotonicity assumption (iii) by
(iii') there exist unique, positive, extended real numbers A, and i with
A, < , such that
38
f(S)<D
if S e [A,, Nil
and
f(S)>D if Se(a,,,).
Furthermore, f'(A,) > 0 if A, < oo and f'(,) < 0 if , < oo.
0<A,<A2<
<A,<S(O)<Aj, j=v+l,...,n.
Q = U(A1, pi),
i=1
39
0.5
0.4
0.3
a
0.2
0.1
0.30
0.25
D
0.20
0.15
0.10
0.05
Figure 5.1. a A monotone uptake function, where 0 < A < = oo. b A nonmonotone uptake function, where 0 < A < It < oo.
(i) If Al < 5(0) < j, then El attracts all solutions with xk(0) > 0.
(ii) If 5(0) > A j, then E0 and E1 are local attractors, their basins of attrac-
tion are non-empty open sets, and the complement of the union of
the two basins of attraction has zero Lebesgue measure.
40
We remark that Q will always be connected in the case where it, = oo. In
particular, if f, is monotone increasing then Q is connected.
Case (i) of the theorem corresponds to competitive exclusion: x, is the
winner. The second case represents a new phenomenon - namely, the possibility that too much nutrient can lead to the washout of all populations.
Both the competitive exclusion outcome (represented by E,) and washout
(represented by E) have significant likelihood; which one occurs depends
on the initial conditions. That washout of all populations (including x1)
is possible in case (ii), for some set of initial conditions, is not difficult to
anticipate. If S(0) > 5(0) then f,,(S)-D < 0 for all i, so all populations
decrease initially. If the populations are initially small then they may be
washed out of the chemostat before they are able to lower the nutrient
concentration to favorable levels.
Interesting possibilities can occur if Q is not connected. Rather than
formally stating a theorem, we apply results in [BWol] to a specific example. Consider two populations with growth functions as follows: fl(S)
is monotonically increasing to a maximum value M, = f1(S1), whereupon
it monotonically decreases. Similarly, f2(S) monotonically increases to
its maximum M2 = f2(S2), whereupon it also decreases monotonically.
Suppose that f'(S) = 0 only at S = Si and S, < S2, and that f1(S) = f2(S)
only at S = 0 and S = S*, where S, < S* < S2. Let D* = f (S*) and assume
that f (oo) < D* for i = 1, 2. Figure 5.2 depicts the situation.
If the dilution rate D is slightly smaller than D*, then Q is connected
for all values of S(0) since Al < A2 < , < 2. In this case, the theorem
applies. If 5(0) < A, then Q is empty and washout occurs. If A, < S(0) < A2
then Q = (A,, 1) and competitive exclusion holds with x, the winner; if
A2 < S(0) then Q = (A1, 2) and, again, competitive exclusion holds with
x, the winner. If 2 < S(O) then 5(0 Q = (A,, 2); both competitive exclusion and washout from too much nutrient are possible outcomes.
However, if D is slightly larger than D* then A, < , < A2 < 92 and Q is
disconnected if either (a) A2 < S(0) < 142 or (b) 2 < S(O) hold. If (a) holds,
then E, and E2 are local attractors and the complement of the union of
their basins of attraction has zero Lebesgue measure. In this case, the
winner depends on how the chemostat is charged at t = 0 - that is, on the
initial conditions. If (b) holds, then E0, E1, E2 are all local attractors and
the complement of the union of their basins has zero Lebesgue measure.
In this case, washout of all populations, competitive exclusion of x2 by
x1, and competitive exclusion of x, by x2 are all possible outcomes, depending on the initial conditions.
41
Figure 5.2. Two uptake functions are shown: f,(S) = 3S/(S2+S+1) and f2(S) _
2.5S/(0.2S2+S+3); f, peaks first, and f,(S*) = f2(S*) = D*.
These results follow directly from general results in [BWol]. The following intriguing (and fictional) scenario is a slight embellishment of one
discussed there. Suppose we would like to use one or both of two populations of microorganisms to remove a contaminant (to humans, a nutrient
to the organisms) entering our wastewater treatment plant (chemostat).
The contaminant serves as a nutrient for both populations; their uptake
rates are depicted in Figure 5.2. Suppose further that the dilution rate is
slightly larger than D* and that it is beyond our control. The Environmental Protection Agency (EPA) sets the maximum acceptable concentration of the contaminant in the water supply at S = A. The contaminant
concentration entering the plant is S = S(O), and
1Al<A<A2<S(O)<2.
We are then in an unfortunate situation. Population x, alone cannot control the contaminant since , < S(O), and therefore it may wash out owing
to excess nutrient. Population x2 alone cannot control the contaminant
because A2 exceeds the EPA acceptable level A. Both populations together
cannot be guaranteed to control the contaminant level either, since population x2 could competitively exclude x, and thereby leave the unacceptable
level A2 of contaminant in the water. What is needed is another organism,
42
x3, for which A3 < Al < A2 < 3. In that case, Q is connected (Q = (A1, 2)
or Q = (A1, 3)) and S(O) e Q, so xi competitively excludes x2 and x3 and
S(t) -> A, < A as t -> oo. In practice, x2 and x3 would need to be continually added to maintain this situation.
Recent work of Wolkowicz and Lu [WLu] extends the results of [BWo1]
described here to include, in some cases, the possibility of populationdependent removal rates. However, at the time of this writing it remains
an open problem to describe the global behavior of solutions of the equations modeling n competitors in the chemostat, allowing both for speciesspecific removal rates and for not necessarily monotone functional responses (e.g., assuming only (iii')).
6. Discussion
In the previous chapter it was shown that exploitative competition for a
single nutrient by two competitors, each with Michaelis-Menten uptake
functions, results in the elimination of one competitor; that is, competitive
exclusion holds. In the present chapter we have seen that the basic prediction - the elimination of all but one competitor - is unaltered by allowing
any number of competitors and allowing quite general, not necessarily
monotone, uptake functions. (It can be argued that any biologically realistic uptake function should satisfy the assumptions of Section 5.) The basic
prediction is unaltered even if population mortality cannot be neglected that is, in the case of differing removal rates - provided that MichaelisMenten uptake functions are used (but see [WLu] for more general uptake
functions). More work is needed in the case of differing removal rates to
show that they do not affect the prediction for general uptake functions.
We can reasonably conclude that coexistence of competitors can only
occur under different circumstances than have been considered so far.
The remaining chapters address various such possible circumstances.
1. Introduction
The previous two chapters showed that competitive exclusion holds under
44
case of parameters that yield a nonhyperbolic rest point or cycle. However, since the parameters in the system studied are measured quantities,
this is a reasonable assumption; it simply says that certain measured quantities are "unlikely" enough that they may safely be neglected. In these
cases stability fails to be determined from the linearization.
The new mathematics that is introduced here is elementary bifurcation
theory, in particular, bifurcation from a simple eigenvalue. Although the
necessary theorems will not be proved, the material will be discussed in
some detail.
2. The Model
We take as the model that of the simple chemostat of Chapter 1, with
input nutrient S(t) and organism x(t) growing on that nutrient, and add
two predators on x which we label y and z. It is assumed that both the
nutrient uptake from the lowest level and the predation from the highest
level follow Michaelis-Menten or Monod kinetics. The use of the Monod
formulation has already been discussed for the consumption of the nutrient. That the same format should apply in the case of a predator feeding on prey is not immediately clear. This formulation is one of a general
class known as a Holling's type-II functional responses [Hol]. A nice discussion can be found in [MD, p. 5], which we repeat here.
The object is to partition the time of an individual predator. Let N
denote the number of prey caught during a food-search time period of
length T. Let x denote the prey density (units are cells/unit volume), s
the search rate (units are volume/time), and b the handling time (units
are time/prey). One then has N = sx(T - bN). The functional response
(the consumption term) for one predator is NIT or, extending over a unit
volume,
sx
l+sbxThis is of the same form used for the consumption of nutrient where m is
1/b and a is 1/sb. Modifications and other models are discussed in [CN2,
sec. 3.2].
The basic equations with all of the parameters are
m1Sx
(o1-
y1(a,+S)'
x'= x
m1S
ai+S
-D-
m2Y
y2(a2+x)
m3z
y3(a3+x)
The Model
_
m2x
-D
M3 x
-D ,
Y -Y a2+x
Z=Z
a3+x
5(0)
a, =
y1S( '
a,
a2
d2 =
,
-
Y,YzS()'
IS(),
m,= -, i=1,2,3,
Z=
a3
a3 =
S(0),
F=Dt.
S'= 1-5
m,xS
a,+S
m,S
a,+S
Y
m2Y _ m3z
-1- a2+x
a3+x
m2x
-1
a2+x
(2.2)
M3 x
z=z a3+x-1'
S(0) = S ? 0, x(0) = x >- 0, Y(0) = Yo >- 0, z(0) = z >_ 0.
set (see Appendix F). Let E = 1-S-x-y-z. Then system (2.2) may be
written as
46
E'= -E,
mzY _
x'=x ml[1-E-x-y-zl -1- az+x
l+ai-E-x-y-z
Y' =
Z'=z
Ymzx
a2+x
m3x
a3 +X
-1
m3z
'
-1 ,
Clearly, lim, . E(t) = 0 and hence the omega limit set of any trajectory
lies in the set E = 0. (Alternatively, one could appeal to the theory discussed in Appendix F.) Trajectories in the omega limit set are solutions
of the following system:
mzx _ 1
az+x
'
(2.3)
/mix
The first step in the analysis is to eliminate the cases of inadequate competitors and inadequate prey - "inadequate" in the sense that they could
not survive in the chemostat without predators, or survive on this prey
even at the prey's maximum possible level. Define
a;
A;= m,-1'
i=1,2,3,
form1>1.
LEMMA 2.1. If m, <- 1 or if m, > 1 and A, >_ 1, then lim,-. x(t) =
lim,- z(t) = 0.
47
x'<xWa -1.
1
a,
or
lim,. x(t) = 0.
For t large, y' < -dy and z'< -dz for some d > 0, and hence
limy(t)=0 and
lim,_.z(t) = 0.
For mz <- 1 or m3 <- 1, the argument is similar except that only the relevant predator population tends to zero. The idea of the proof is the same
for the As. Suppose that m2 > 1 and A2 >- 1. Since x:5 1, it follows that
Y,= Y
(mz-1)
(2.4)
is the desired contradiction. Similar arguments work for the other dependent variables.
A2<A3,
(H1)
48
_ m1xS
-x
al+S
m2xy
a2+x
(These equations appear in [Cal; Ca2; CN2; JDFT; Se; TDJF] and no
doubt in many other papers.) The same simplification effected previously
may be applied to this system, letting E = 1-S-x-y. Then lim1-,, E(t) _
0. The resulting system of interest is
2
Y'=Y a2+x
1),
and
E2 = (1-A1, 0).
M=
m11
mil'
m21
m22J
where
mil
m1(1-x-y)
=
m1al
m2y -1+x(+ m2y
1+a,-x-y a2+x
(1+a1-x-y)2 (a2 +X)2
'
_
m12
-m1a1x
m2x
(1+a1-x-y)2
a2+x'
m2a2 y
m21
(a2+x)2'
m22
49
m2x
a2+x -1.
(m, -1)(1-A,)
M =
1+a1
0
-1
The eigenvalues are on the diagonal; one is negative and the other, under
our hypothesis (HI), is positive, so E1 is a saddle point. This is expected
because (3.1) is a predator-prey system; without the prey, the predator
cannot be expected to survive. The eigenvector corresponding to the negative eigenvalue lies along the y axis, reflecting this. The eigenvector corresponding to the positive eigenvalue lies along the x axis, reflecting the
prey's ability to grow in the absence of a predator.
Evaluated at E2, M takes the form
aim,
M=
(1-a1) (a1+A,)2
0
m2
m1a1
+ 1+a2-J11
(m2-1)(1-A,-A2)
1 +a2-Al
Again the eigenvalues are on the diagonal, and one is clearly negative; its eigenvector lies along the x axis. This corresponds to an attracting steady state when the predator is absent. The rest point will be a
local attractor for the full system if the remaining eigenvalue is negative.
Clearly, this can happen only if A,+A2 > 1, since we are assuming (Hl).
This corresponds to the (local) extinction of the predator and the survival
of the prey. If A, +A2 < 1 then E2 is a saddle point. In this case the eigenvector points into the positive quadrant, since both terms in the first row
have the same sign and the eigenvalue is positive. The Butler-McGehee
theorem, discussed in Section 3 of Chapter 1, may be applied at both E,
and E2 to conclude that no trajectory starting in the interior of the positive quadrant may have E, or E2 in its omega limit set.
50
m,(1-xx-Yc) _
1+a,-xx-yc
m2Yc
a2+xc =1'
m 2 xc
= 1.
a2+x,
Clearly, xx is given by
xc =A2=
a2
m2-1
l+a,-A2-yc
a2+A2
(3.2)
m,(1-A2-Yc) -1 =
l+a,-A2-Yc
YC
A2
or
(m,-1)(1-A2-A,-yy) = Y` (1+a,-A2-Y,).
(3.3)
Since A2+yy must be less than 1, by (3.2) one can see from this form of
the equation that if A,+A2 > 1 then there is no positive solution y, since
one side of the equation would be positive while the other side is negative.
On the other hand, if A,+A2 < 1 then the left-hand side of (3.3) is a line
with positive y intercept and a zero at 1-A, -A2. The right-hand side of
(3.3) is a parabola with zeros at 0 and at 1 +a, -A2. Since 1-A, -A2 <
1 +a, -A2 there is a unique value of yy, 0 < yv < 1-A, - A2, that satisfies
(3.3). The value of x, is already unique. We summarize this in the following remark.
al asymptotic stability of E2 in the first case. Since the system is twodimensional, the Poincare-Bendixson theorem provides a proof of the
global claim.
51
The next step is to analyze the stability of the interior rest point. To do
this one considers the variational matrix at EE
m,a2a,
M=
(1+a,-A2-yc)2
m2YcA2
-m,a,a2
(a2+A2)2
(1 + a, -A2 -yc)2
(m2-1)Yc
A2+a2
-1
<
m2Yc
miai
(l+a, -A2-yc)2
(a2+A2)2
This simplifies to
Yc
M2 A2
<
m, a,
(1+a,-A2-Yc)2
(3.4)
A standard reference for the material in this section is [CL]. Here, the
basic definitions and theorems are given but no proofs are presented.
Floquet theory deals with the structure of linear systems of the form
x'=A(t)x,
(4.1)
52
Y'= A(t)Y,
(4.2)
THEOREM 4.1. LetA(t) be periodic of period T. Then if F(t) is afundamental matrix, so is fi(t) = F(t+T). Corresponding to any fundamental
matrix (D(t) there exists a periodic nonsingular matrix P(t) of period T
and a constant matrix B such that
4)(t) =
P(t)eBt.
As customary, the exponential of a matrix means the sum of the matrix series corresponding to the exponential function. The eigenvalues of
cF(T) = eBT are called the Floquet multipliers. The eigenvalues of B are
called the Floquet exponents. (There is some delicacy about the uniqueness of B which we will ignore because it is not relevant to our use.) Usually it is not possible to compute the Floquet exponents or multipliers.
However, for low-dimensional systems of the kind we will investigate,
there is a general theorem about the determinant of a fundamental matrix
which is helpful. Let fl t) be a fundamental matrix for (4.1) with 4)(0) _
I. Then
det4) (t)=exp[ fo
ttrA(s)ds]
(4.3)
J
(4.4)
L
fo
x'= AX),
(4.5)
53
y'=f'(y(t))y,
(4.6)
1 dx
S(t)-S,= x dt
M2
m2-1
y`
)ldY
a2+A2 y dt
m1 a,
+ m2
a2+x
54
Proof.
x' _
x
m1S
m2y
a1+S
a2+x
_m1SC+m2Yc-m2Y
m1S
a1+S
a1+Sc
a2+x
a2+A2
mlal(S-Sc)
m2
(a1+S)(a1+SS)
a2+x
m2Yc
(Y-Y,)+ (a2+A2)(a2+x) (x-xJ
)(S_S,)
a2+xm2
(a1+S)(a1+Sc) +
m2Yc
+x + (a2+x)(a2+A2)
(x-A2)
_ (___m1a1
(a 1+S)(a1+Sc) + a2+x
Y'
m2
YC
1+
y m2-1 ( a2+A2
'
from which the remark follows by solving for S-Se. (Note that use has
been made of (3.2) and various arrangements of the fact that a2+A2 =
m2A2.)
x'=f(x,y),
Y'=g(x,y),
(5.1)
where we have in mind that the functions f and g are given by the righthand side of (3.1).
- f (a (x(t),Y(t))+ ay (x(t),Y(t)))dt.
(5.2)
(m2xc
(al +S )2
r
T +11 Q(x, y) dxdy,
(5.3)
55
Since S, = 1-xx-y' and x, = A2, if the quantity in brackets in (5.3) is negative then (3.4) is satisfied (and conversely). The use of this complicated
lemma has the following consequence for the stability of a periodic orbit.
COROLLARY 5.2. If (3.4) holds then r is asymptotically (orbitally) stable.
Proof. The quantity under the integral sign in the definition of A in (5.2)
is the trace of the Jacobian matrix for the system (5.1) evaluated along the
periodic orbit. Theorem 4.2 then applies. A periodic orbit for an autonomous system has one Floquet multiplier equal to 1. Since there are only
two multipliers and one of them is 1, e is the remaining one. The periodic
orbit is asymptotically orbitally stable because, in view of Lemma 5.1,
0<0.
Proof of Lemma 5.1. Differentiation and substitution yield
1T1
-J
m1(1-x-y)
1+a1-x-y
m2y -11
a2+x J
-mial
+xI
(a2+x)2ag+x
LL (1+a2-x-y)2
(5.4)
))
The quantity in the first square bracket is just x'(t)/x(t) and hence integrates to 0 since x(t) is periodic of period T. Similarly, the third square
bracket is just y'(t)/y(t) and integrates to 0 for similar reasons. Thus
one obtains that
Jo
m2y(t)
a1m1
(l+a, -x(t)-y(t))2)
((a2+x(t))2
x(t) dt.
(5.5)
that
('T m2x(t)Y(t) dt = (' T( x(t)
o
(a2+x(t))2
o
a2+x(t)
x(t)
m2Y(t)
a2+x(t)
m1S(t)
a2+x(t) \\a1+S(t)
fT x(t)
o
m1SC
a2+x(t) a1+S,
+JT
x(t)
dt
-I- x'(t)
x(t)
1
\m1S(t)
0 a2+x(t)a1+S(t)
dt
dt
m1SSc
a1+
dt,
56
where we have used Equation (3.1) and the fact that x%(a2+x) integrates
to 0. Thus one has that 0 = I,+I2. We investigate each integral separately, beginning with I2. First the terms are combined to give
I
2
x(t)
(mlS(t)(aI+Se)_mlSc(aI+S(t)))dt
- fa2+x(t)
a,+S(t))dt.
m, a,
IZ
m, a,
m2
a2+x)-1
I
x
M2
m2-1
yy
\l+ a2+A2
P, dx+Q, dy.
Green's theorem may now be applied to deduce that
IZ
- ay')dxdY,
a+ScJJR\ax
where
y` )xP(x,A
Qi(x,Y)=- M2 (1+ m2A2
Y
M2_1 \\
m2(a,+l-x-y))-
P(x,y) = I
\a,+Sc
a2+x
Q'
< 0,
aP' > 0
Y
I'
=(
M'S` -1
a,+S,
fT
x(t)
ho a2+x(t)
dt.
dy
57
(y
-1)r
I,-
m1Sc
al+Sc
m2
(miSc
-1)=T
T
m, a,+Sc
a2+A2
yeT
Yc
m2A2
The same technique will be applied to the second integral in (5.5), although it is much more complicated. First of all, write
_f T r alm,x(t)
(a1+S(t))2
dt
r x(t)
mla1(a2+x(t)) _ m1a1
Ja2+x(t) { [ (a1+S(t))2
(a, +Sc)2 (a2+x(t))]
=-f,)
+[mla1(a2+x(t))
mla1(a2+A2)I
(a1 + Sc) 2
(a1+Sc)2
+[mla1(a2+A2)1) dt
(a1+Sc)2
= I3 + I4 + I5.
We begin with 13. As before (suppressing the notation for the dependence
=a m1
1
dt.
(aj+Sc)2(al+S)2
('r x(S-Sc)
a,ml
(a,+Sc)2 Jo
a1+S
dt+
a, m,
('r x(S-SC)
a,+Sc Jo
(a,+S
dt.
aQ2
as
2l dxd ,
Y/
y
where
P2=P(x,Y),
m2 (I+
M2_1
Yc
xP
m2A2 y
58
and
rT X(S-Sc) dt
(a, +S)2
Jo
P =
3
f f( aQ3
JR( ax
- aP3l
dxdy,
ay)
P(x, Y)
1+a,-x-y'
m2
Q3= m2-1
1+
xP(x, Y)
YC
m2A2 Y(1+a,-x-y)
Note that
01
aQ3
< 0,
as22 > 0,
as 3 > o.
Y
Thus 13 can be written as an integral over R which has a negative integrand. The remaining two integrals are easy. By (2.4), one has
m,a, fTX(XA2)
dt
(a,+Sc)2 o a2+x
m1a,
(m2-1)(a,+Sc)2
T Xy dt
y
m,a,
(m2-1)(a1+S,)2 r y
m1 a1
dY
1 dxdy
(m2-1)(a,+S,)2
J JR Y
m1a1(a2+A2) ('T
and
Is
(a,+S,)2
o a2+x
m,a,(a2+A2) (T(Y
m2(a,+SC)2
dt
+1 I dt
(a,+S,)2
T.
(a1+Sc)2
Y`
m2A2
T+ f f Q(x,y)dxdy.
R
59
Before giving a proof, note that the components of Ec are readily obtainable: x, = A2, and y, is the root of a quadratic. The condition is stated in
the form (3.4) to avoid the complicated expression that would result from
using the quadratic formula.
mial
M2 A2
(5.6)
then there exists an asymptotically orbitally stable periodic orbit for (3.1).
Kuang [Kl] has shown that if the parameters are such that
Yc
m1ai
m2A2
(1+a,-A2-yc)2
is small and positive, then the limit cycle is unique and asymptotically
stable.
60
(2.3), this food chain corresponds to the absence of a predator z competing with y. After the simplification, one can view (3.1) as (2.3) with the
initial condition z(0) = 0, since the x-y subsystem represents an invariant
set. A natural question, then, is: If one chooses the parameters so that
the food chain has a periodic limit cycle, can the full system have a limit
cycle in the positive cone in ff83? The tools to answer questions of this type
come from bifurcation theory. The kind of theorem needed is of the simplest sort, bifurcation from a simple eigenvalue. Although no theorems
will be proved, this section attempts to sketch the basic ideas needed.
Some familiarity is required with the fundamental theory of ordinary differential equations, particularly the Floquet theory, so the reader who is
interested in only the results should skip ahead to Section 7 for application to the system under study.
A good explanation of the theory can be found in Smoller [Smo] (see
also [MM]). All forms of the basic result are essentially equivalent. Bifurcation theory is not restricted to differential equations but is actually concerned with mappings or functions. A principal tool in developing the
theory is the implicit function theorem. When the theory is used in infinitedimensional spaces, quite sophisticated mathematics is required. However, the problem here can be dealt with in a finite-dimensional setting.
Suppose one has an equation
f(x,A)=0,
(6.1)
where f is a smooth function from 118" x R into ff8" and where we think of
A as a parameter. One seeks the set of solutions - that is, the set
61
P=(0,A)=trivial solutions
(0, Ao)
J=IfI,
8x;
is nonsingular at (xo, A0), then the implicit function theorem implies that
there exists an interval J containing A0 as an interior point and a smooth
for A e Jfl I. Consequently, we see that a necessary condition for (xo, A0)
to be a bifurcation point is that J be singular at (x0, A0).
Fortunately, although the implicit function theorem would appear to
be inapplicable as a tool to discover the structure of solutions of (6.1) in a
neighborhood of a bifurcation point, it can be successfully applied once
62
P={(O,A)IAeI).
In stating the result used in this chapter, we assume this to be the case.
The ideas of dynamical systems were introduced in Chapter 1 (Section
3). Two mappings appear in the discussion here, both based on ideas best
expressed in terms of dynamical systems. The discussion is limited to dimension 3 because that is what we need, but it is valid for R". Suppose
one has a differential equation
x'= AX),
(6.2)
where xe 083. Given an initial point x0, there is a trajectory through it,
lr(xo, t). If the time t is fixed, say at t = T, then 7r(x, T) is a function from
083 to 083. This is called the solution mapping. Now suppose that xo is a
point on a periodic orbit y of period T. Let P be a plane through xo and
orthogonal to the tangent vector f(xo). Let N be a sufficiently (in a sense
to be made clear in what follows) small neighborhood of xo in P. By continuity, solutions corresponding to initial conditions near xo will remain
"close" to the periodic orbit and return to a point "near" xo at time T. Indeed, if we are close enough, the orbit will reach the plane P although
not necessarily at time T. By using the implicit function theorem, one
can show that given x close enough to xo there is a time r(x) such that
7r(x, r(x)) e P. (There is a good discussion of this in the textbook of Hartman [Har].) Hence, for xe N, lr(x, r(x)) defines a mapping from R2 into
R2. This mapping is called the Poincare map associated with y. The two
mappings are related and we shall make use of this relation. The Poincare map is what we need, but the solution map makes the computations
tractable. The principal approach may be summarized in the following
statement.
REMARK 6.1. Let y and xo be as before. A fixed point of the Poincare
mapping (different from x0) gives the initial conditions for a periodic
orbit of equation (6.2) in a neighborhood of x0.
Since each of these mappings is from 08" to 08", where n = 3 for the solu-
tion map and n = 2 for the Poincare map, their linearizations are given
by a matrix. The following is a basic result connecting the linearization
of the two maps.
63
LEMMA 6.2. Let -y be as before, and consider the solution map and the
Poincare map in a neighborhood of a point of y. Then the eigenvalues
of the linearization of the Poincare map together with [1) are the eigenvalues of the linearization of the solution map.
A proof of this lemma is given in Appendix E.
Now replace the differential equation by one with a parameter,
(6.3)
Suppose now that -y is a periodic trajectory for every value of the parameter. This corresponds to a curve of fixed points - a trivial curve - of
the Poincare map in the terms discussed for equation (6.1). We can take
coordinates in the plane P such that xo corresponds to the origin, and
then look for conditions that will yield a bifurcation point of the Poincare mapping. The following theorem provides such conditions in very
general terms.
THEOREM 6.3. Let W be an open neighborhood of 0 e R" and let I be an
open interval containing AO in R. Let P(x, A) be a twice continuously differentiable mapping of W X I into 1f8" satisfying P(0, A) = 0 for all A C- I.
Let L (A) be the Jacobian of P with respect to x evaluated at (x, A) = (0, A).
Suppose that 1(A) is a real, simple eigenvalue of L(A) satisfying 1(Ao) = 1
with dl(A0)/dA * 0 and that vo spans the null space of L(A0) - Id. Then
there exist S > 0 and a continuously differentiable map (0,A): (-S, S)
VOL x (f8 such that
64
must be inside the unit circle because of the stability. For the Poincare
map corresponding to (2.3), then, one eigenvalue will be inside the unit
circle; the object of applying this theorem is to have the remaining one
move across the unit circle as prescribed in the theorem's hypotheses. The
linearization of (2.3) about the periodic orbit in the z = 0 plane takes a
special form; the following useful lemma is used to compute the needed
eigenvalues.
A(t)
B(t) =
b1(t)
b2(t)
b3(t)
where
z(t) = f Y(t)Y-'(s)b(s)z3(s)ds
0
with b(s) = col(bl(s), b2(s)). In particular, if P1, P2 are the Floquet multipliers associated with the 2 x 2 system (eigen values of Y(T)) then the
Floquet multipliers of the 3 x 3 system are p1i P2, and p3 = z3 (T ).
Proof. The proof is a straightforward computation. The z3(t) term decouples from the system and can be obtained by a quadrature. The formulas for zi(t), i = 1, 2, can be obtained by substituting z3(t) into the
two-dimensional system
zi(t) = anz,+a12z2+b1(t)z3(t),
zi(t) = a21z1 + a22z2 + b2(t)z3(t),
Competing Predators
65
7. Competing Predators
Armed with the preceding discussion, we can now state the main theorem
of this chapter. It shows that, in contrast to the basic chemostat, coexistence can occur if the competition is at a higher trophic level. (We remind
the reader of the general assumption of hyperbolicity of limit cycles.)
THEOREM 7.1. Let ai and m; (i = 1, 2) be fixed so that m; > 1, let Ai < 1,
and assume that the parameters are fixed so that (5.6) holds. Then there
exists a number a3 such that for some values of a3 with I a3 - a3 I sufficiently small, (2.3) has a periodic orbit in the positive octant near the x-y
plane.
Proof. Let y = (x(t), y(t), 0) be the orbitally asymptotically stable periodic orbit of period T given by Theorem 5.4. (We have already noted
that if there are several orbits then one must be asymptotically stable, by
our assumption of hyperbolicity.) Let the Floquet multipliers of y, viewed
as a solution of (3.1), be 1 and p,, where 0 < p, < 1. This trajectory remains a solution of the system (2.3) for all values of m3 and a3. Fix m3.
For this periodic orbit, define (a3) by
fT
(a3) = T3
a3+x()
Note that (0) = m3 > I and (a3) is decreasing in a3. In fact, it follows
that
a
aa3
-m3 rT
T
xO
(7.1)
hl3
fT
a3T Jo
r
r
x(S) d5.
(7.2)
66
P2(Y,0;a3)=0
Competing Predators
67
aP,
J=
aY
aP,
aZ
aP2
az
aP,
pi = ay (0, 0; a3)
is independent of a3.
To compute the remaining eigenvalue of P, one computes the eigen-
values of the solution map. The linearization of (2.3) about y takes the
form
A(t)
fZ
0
m3x(t)
a3+x(t)
-1
where fZ denotes the partial derivative of f, the right-hand side of the first
equation in (2.3), with respect to z. The matrix A(t) is just the linearization of (3.1) about (x(t), y(t)). Its fundamental matrix possesses Floquet
multipliers 1 and p, e (0, 1) as noted previously. By Lemma 6.4, the other
multiplier is P2 =
Consequently, from the upper triangular
form of J and Lemma 6.2,
eT(A(a3)_,).
P2 =
a3)-1).
aZ2(0,0;a3)=eT(
This eigenvalue, which we denote by 1(a3) (in anticipation of verifying Theorem 6.3), clearly satisfies 1(a3) = 1 and d1(a3)/da3 < 0, since d1A/da3 < 0.
68
8. Numerical Example
The limit cycle found in the previous section holds only for I a3 - a3 I small.
Obviously, once the limit cycle exists, it can be continued, either globally
or until certain "bad" things happen such as the period tending to infinity
69
S
a3
a3
Figure 7.2. a A possible bifurcation diagram with positive solutions for a3 > a3,
b A possible bifurcation diagram with positive solutions for a3 < a3.
70
Figure 8.1. Numerical solutions showing bifurcation from a limit cycle in a plane
into a limit cycle in the interior. The fixed parameters are a, = 0.3, a2 = 0.4, m, =
10.0, m2 = 4.5, and m3 = 5.0. The parameter a3 was varied between 0.46 and 0.48.
The limit cycle moves from one plane (a face of 118+) through the interior of 118+
(showing coexistence) and collapses into the opposite face of IfB+.
Discussion
71
another trophic level to the model of Section 3 can produce complex dynamics. This was suggested by recent (unpublished) work of Wolkowicz
on a food chain with a reproducing prey at the lowest level, and by the
results of Hastings and Powell [HP].
If one adds another predator to the model of Section 3, then the system
is governed by the equations (in scaled form)
S'=1-S-xf (S),
X'= x(.f1(S) -1) -Yf2(x),
Y'=WAX) -1)-zf3(y),
Z'= Z(f3(Y) - 1),
where
m;u
f(u) = a,+u
As in the earlier work, one has immediately that
lim1_ S(t)+x(t)+y(t)+z(t) = 1.
On the omega limit set, then, one may eliminate the S variable by using
z'=z(f3(y)-1).
The parameters of the first two levels were fixed: ml = 10.0, m2 = 4.0,
a, = 0.08, and a2 = 0.23. The values are consistent with those suggested
in [CN2]. The growth rate of the top predator was fixed at m3 = 3.5, and
the value of a3 varied to obtain Figures 9.1-4. Figure 9.1 shows the attractor as a limit cycle; Figure 9.2 indicates the appearance of a perioddoubling bifurcation. Figure 9.3 shows the attractor after two such bifurcations, while Figure 9.4 suggests a chaotic attractor.
10. Discussion
The final conclusion of Section 7 was that two predators could survive
on a common prey. It is significant that this can happen only as an oscillatory phenomenon. This is easy to understand intuitively. If the prey
72
Figure 9.1. A periodic orbit in the long food chain. Parameters are a, = 0.08,
a2 = 0.23, a3 = 0.40, m,=10.0, m2 =4.0,andm3=3.5.
level came to a steady state, it would favor one predator or the other.
While it is oscillating, sometimes the concentration favors one competitor and sometimes the other. Each manages to grow enough while it has
the advantage to survive. One does well at higher concentrations and
then crashes rapidly; the other is steady at lower concentrations and improves while the first is "crashing." This is a manifestation of the common
r-K strategist discussion in the ecology literature.
Discussion
73
Figure 9.2. Period doubling in the long food chain. Parameters are as in Figure
9.1 except that a3 = 0.3.
the literature. Figure 10.1 shows the results of the experiments of Jost
et al. [JDFT]. A food chain was constructed in a chemostat consisting
of a nutrient (glucose), a bacteria living on that nutrient (Azotobacter
vinelandii), and a ciliate (Tetrahymena pyriformis) feeding on the bacteria. Both sustained oscillations (Figure l0.la) and convergence to a
steady state (Figure 10.1b) were observed, depending on the washout rate.
74
Figure 9.3. Two period doublings in the long food chain. Parameters are as in
Figure 9.1 except that a3 = 0.24.
Delays can also introduce oscillations. These problems are infinitedimensional ones and much more difficult to analyze rigorously. Comments on the delayed chemostat can be found in Chapter 10, along with
Discussion
75
Figure 9.4. A complicated attractor in the long food chain, possibly chaotic.
Parameters are as in Figure 9.1 except that a3 = 0.20.
76
Sf=.48 mg /ml
10
20
15
25
Time, days
12
A. rinelon6i
a
12
16
20
Time, days
Figure 10.1. A food chain constructed of a nutrient (glucose), a bacteria (A. vinelandii) living on that nutrient, and a ciliate (T. pyriformis) feeding on the bacteria:
a sustained oscillations; b convergence to a steady state. (From [JDFT], Copyright 1973, American Society for Microbiology. Reproduced by permission.)
Discussion
77
1. Introduction
In the first two chapters the general theory of the chemostat was developed,
and it was shown that competitive exclusion is the expected outcome. In
Chapter 3, coexistence was shown to occur when the competition was at
a higher trophic level; the mechanism was simply the oscillation of the
object of the competition - the prey in the case being considered. In this
chapter, we return to the basic chemostat model but add another factor,
the presence of an inhibitor. The inhibitor affects the nutrient uptake rate
of one of the competitors but is taken up by the other without ill effect.
The use of Nalidixic acid in the experiments of Hansen and Hubbell
[HH], discussed in Chapter 1, is an example. Its effect on one strain of
E. coli was essentially nil while the growth rate of the other was severely
diminished.
The interest in this subject goes far beyond laboratory examples. It is
The Model
79
inhibited strain is the weaker competitor without the presence of the inhibiting agent, then one expects it to be eliminated. The focus is thus
on the case where the superior competitor is the one sensitive to the inhibitor. The question remains the same: Does competitive exclusion still
hold?
Taking the chemostat as the basic model, the inquiry focuses on determining the outcome as a function of the basic parameters of the organisms and the operating parameters of the chemostat. In the MichaelisMenten formulation, the parameters of the organisms are the a,s and m,s
of Chapter 1; the operating parameters of the chemostat are S(0) and D,
the input concentration of nutrient and the washout rate. If the inhibitor
is assumed to be input into the chemostat at a constant rate, then new
parameters are introduced. One new operating parameter would be this
constant input concentration. The uptake characteristic of the uninhibited organism and the effect of the inhibitor on the growth of the inhibited
organism are the other new elements that can be expected to play key
roles. The former will be treated in Michaelis-Menten form whereas the
latter will be treated in a quite general fashion.
The basic model is that of two competitors in the chemostat, and the
new element is the effect of the inhibitor. The model was first proposed
by Lenski and Hattingh [LH], who investigated the various outcomes
with a computer simulation. The mathematical analysis here follows that
contained in Hsu and Waltman [HWI].
2. The Model
As noted in the introduction, the model is that of a standard chemostat
with two competitors, but with the added feature that an inhibitor is also
input from an external source. The nutrient (and inhibitor) uptake and
conversion (in the case of nutrient) are assumed to follow MichaelisMenten dynamics. The results are probably valid for general monotone
dynamics, although this has not been established.
Let S(t) denote the nutrient concentration at time tin the culture vessel,
x1(t), xz(t) the concentration of the competitors, and p(t) the concentration of the inhibitor. The equations of the model take the form [LH]:
S'= (S (0)_ S)D -
mix IS
al+S f(p)
m1S
xi= xi(a +S
f(p)-D
m2 x2S
az+S
80
Cm2Sa2+S D/
p'=(p(0l-p)D-6X2 P
K
+p
where S(0) is the input concentration of the nutrient and p(0) that of
the inhibitor, both of which are assumed constant; D is the dilution
rate of the chemostat. The terms mi, a; (i = 1, 2) are the usual parameters, the maximal growth rates of the competitors (without an inhibitor)
and the Michaelis-Menten (or half-saturation) constants, respectively.
The parameters S and K play similar roles for the inhibitor, with S the
maximal uptake rate by x2 and K a half-saturation parameter. The func-
To reduce the number of parameters - and to provide a standard environment so that comparisons can be made in terms of the parameters
of the competing populations - the equations will be scaled, much as was
done in the first three chapters. First, scale the units of concentration
of S, x1, x2, by the input concentration S(0). This includes the parameters a,, i = 1, 2. (We have already tacitly scaled out the yield parameters,
which scale the conversion of nutrient to organism.) Then scale time by
the dilution rate (with units 1/time). This reduces D to unity and replaces
m; by m,/D (i =1, 2) and S by S/D. Finally, scale p by p(0), which has the
effect of scaling p(0) to unity. In [LH],
f(p) = e-Ap,
so this would now be written
.f(p)=e
Ap"(plp(0))
The new variable is p/p(0) and the new parameter is Ap". If one makes
these changes and then returns to the "old" names (e.g., using m, as the
new "maximal growth rate," the "old" m1/D), the system (2.1) takes the
form
mixIS f(p)- m2x2S
S'=
xi_
-x,
1-S-
aj+S
a2+S
7mS
a
f(p)-1
I
- x2
( am2S
2+S
81
(2.2)
'
Sx2 P
K+p,
E=1-S-xl-X2.
Then
82
E'= -E,
/m1(1-E-x1-x2)
a,+1-E-x,-x2 AP)-1
x, = x,
xz-x2
/m2(1-E-x1-x2)
a2+1-E-x1-x2
X2 P
K+p
Clearly, lim1_. E(t) = 0. Hence, the solutions in the omega limit set of
(3.1) must satisfy
xI-x1/m1(1-x1-x2).f(P)-l
(\
X2'
l + a, - x, - x2
_ X2( m2(1-x1-x2)
I,
1+a2-x,-x2
(3.2)
pP,
K
xi(0) > 0 (i = 1, 2), x1(0) +x2(0) < 1, p(0) > 0.
Let
a1
1 m1-1'
a2
m2-1
(3.3)
These are the determining parameters for the simple chemostat, as discussed in Chapter 1, and would determine the outcome if the inhibitor
p were not present. The form of the equations (3.2) guarantees that if
xi(0) > 0 (i = 1, 2) then xi(t) > 0 for t > 0. Moreover, p' In=o = 1 > 0,
so if p(O) >- 0 then p(t) > 0 fort > 0. The terms x1(t) and x2(t) satisfy
x'<X (m1(1-x1-x2) -1
l+a,-x1-x2
1
,_
X2'
m2(1-x1-x2)
x2( 1+a2-x1-x2
(3.4)
Taking advantage of the monotonicity (in the variable 1-x, -x2) in the
right-hand side of (3.4) yields a set of two scalar differential inequalities
of the form
X2' <x 2
83
f\
Cm2(1-x2)
1+a2-x2
The comparison with the growth equations for the chemostat (equation
(4.1) or (4.2) of Chapter 1) establishes the following proposition.
PROPOSITION 3.1. If m; s 1 or if m; > 1 and A, >- 1, lim,_. x,(t) = 0. If
m, > 1 and 0 < A, < 1 then lim sups _ xi (t) <- 1- A, with i = 1 or 2.
This simply states the biologically intuitive fact that if one of the competitors could not survive in the simple chemostat, that competitor will not
survive in the chemostat with an inhibitor. Thus we may assume m, > 1
and 0 < Ai < 1, i = 1, 2. There are some other simple cases that can also
be easily dispatched.
PROPOSITION 3.2. If 0 < A2 < A, < 1, then
lim,- 0 X, (t) = 0,
(1-p)(K+p)-S(1-A2)p = 0.
(3.5)
(The reason for labeling the root p2 will become clear in the next section.)
Biologically, the proposition states that if x2 eliminates x1 in a chemostat
without x, being inhibited, x2 eliminates x, when x, is inhibited.
Proof of Proposition 3.2. That p2 < 1 follows from the fact that p(t)
satisfies
p'<1-p
and the basic comparison theorem. The boundedness of p and the fact
that E tends to zero also show that all solutions of (3.1) are bounded. The
other assertions in the proof will follow from the more general comparison theorem (see Appendix B, Theorem B.4) and a knowledge of the
behavior of solutions of the basic chemostat equations, system (5.2) of
Chapter 1. Since "=" is also "?," (3.4) may be written
84
(m1(1-x1-x2)
1+a1-x1-x2
/m2(1-x1-x2)
and
x,(t) ? u2(t).
If A2 < A1i then limf_. u1(t) = 0 and limn-, u2(t) = 1-A2 (by Theorem 5.1 of Chapter 1). Thus one can conclude that lim1 . x1(t) = 0 and
lim inf1- x2(t) >_ 1-A2 (Theorem 5.1 of Chapter 1). Since we already have
(by Proposition 3.1) that lira SUP, - x2(t) <- 1-A2i the proof is complete.
The hypothesis of Proposition 3.2 excludes the case Al = A2. This is ordinarily not biologically important because the A, are computed from measured quantities; it is unlikely that they would be exactly the same (or the
same with respect to this environment). However, an interesting potential application is the case where the organisms are indeed the same (mutants of the same organism) except for their sensitivity to the antibiotic.
Intuitively, if the organisms are the same except for sensitivity to the inhibitor, one expects the x1 population to lose the competition when the
inhibitor is present. However, establishing this mathematically cannot be
done directly from the comparison theorem as used before, since if Al =
A2 then coexistence occurs with the chemostat equations used for comparison purposes. In order to use the comparison principle, one needs a
better estimate than f (p) <- 1, p >- 0. This is the purpose of the following
lemma.
LEMMA 3.3. There exists a number y > 0 such that p(t) > -y for t sufficiently large.
Proof. Suppose liminf1-. p(t) = 0. If p(t) decreased to zero monotonically then there would be a point to such that, for t> to, p(t)+Sp(t)/
(K+p(t)) < 1. For such values, p'(t) > 0, which contradicts p(t) decreasing. Hence there exists a set of points tn, t, -> oo, such that p'(tn) = 0 and
p(tn) -> 0 as to -> oo. For such values of tn,
0 = 1-P(tn) - bP(tn)x2(tn)
K+p(tn)
> 1-P(tn)-
85
SP(t1)
K+p(tn)
>0
for n large. This establishes the lemma.
THEOREM 3.4. If 0 < A2 <- Al < 1, then
limt- , xl(t) = 0,
limt-,,, x2(t) = x2 > 0,
I+al-x1-x2
m2(l-xl-x2)
-1 ,
I+a2-x1-x2
m,>1, i=1,2,
(3.6)
0<A1<A2<1.
The results in this section provide conditions for one or both of the competitors to wash out of the chemostat; these are the uninteresting cases.
To avoid "unlikely" cases in the analysis to follow, we shall tacitly assume
that all rest points are hyperbolic - that is, their stability is determined by
their linearization.
It is easy to anticipate that oscillations in the concentrations of x1, x2, and
p may be possible when 0 < Al < A2 < 1. If p(0) is small then the system
(3.2) behaves as if there were essentially no inhibitor. The superior competitor xl will begin to prevail over its rival x2, driving its concentration
86
E0=(0,0,1),
(xe,0,1), E2=(0,4Z,pz).
E1
parisons between the subscripted As. The local stability of each rest point
depends on the eigenvalues of the linearization around those points. The
Jacobian matrix for the linearization of (3.2) at E;, i = 1, 2, takes the form
J=
m11
m12
m13
m21
m22
m32
m33
At E0,
m1J(1)
1+a1
J=
-1
87
The eigenvalues are the diagonal elements. One eigenvalue is -1, and
the associated eigenvector lies along the p axis. This corresponds to the
growth of the inhibitor to its limiting value in the absence of a consumer.
The set ((0, 0, p) I p > 01 is positively invariant and is part of the stable
manifold of E0. Because A2 < 1, m22 = m2/(1+a2)-1 is positive. Similarly, the remaining diagonal term, m11, is positive if 0 < A0 < 1 and negative otherwise. When this eigenvalue is negative, the stable manifold of
E0 is the entire x, -p plane.
At El, m21 = 0, which means (since m23 = m31 = 0) that the eigenvalues
are just the diagonal elements of J. Thus
mla,(1-Ao)
/11 = - (a,+Ap)2
N2 =
f(1)
m2a2(Ao-A2)
(a2+Ao)(a2+A2)
la3 = -1.
If 0 < A0 < A2 < 1, then E, is asymptotically stable. This reflects the fact
that x1, in the presence of the maximal inhibitor concentration, is still a
better competitor than x2. If A0 > A2 then E1 is unstable and, of course,
if A0 > 1 then E, does not exist.
At E2, m12 = m13 = m23 = 0, so again the eigenvalues are just the diagonal elements
] =
m1A2.f(Pi)
a +1
1
-1'
m2a2(1-A2)
(a2+A2)2
3
= -1- SK(1-A2)
(K+pz)2
1-x1c-x2c = A2,
(4.2)
for this is the only nontrivial zero of the derivative of x2 in (3.2). Using
this, one sets the derivative of x1 equal to zero to find
88
Table 4.1
Point
Existence
Stability
Eo
Always
E,
0 < AO < I
1- or 2-dimensional
stable manifold
Asymptotically stable
E2
Always
if0<A0<A2
Asymptotically stable
if m,A2f(P2*) <a, +A2
MIA2
.f =1
al+AZ(p)
al+A2
m1A2
=f
-1 a1 +A2
m1A2
(4.3)
1-pC- SxZcp** =0
K+pc
or
x2c
(1-pC)(K+pC)
=
bpi
(4.4)
xic =I -xzc-A2.
(4.5)
Thus Ec exists if and only if f(1) <- (a1+A2)/m1A2 and x2c < 1-A2. Since
1-A2 =xz, it follows that if x2c exists then x2c < x2. This last inequality is the biologically expected statement that x2 will do less well in the
89
coexistent steady state than in the steady state where it is the sole survivor.
The inequality is true if and only if (see (3.5))
x2c=
(1-p*)(K+p*)
sp*
<
(1-p2)(K+p2) _
-1-A2=x2.
.f(p2) >
(4.6)
m1A2
Proof. To see the sufficiency of the conditions, note that if the Jacobian
of E2 has a positive eigenvalue then
.f(p2) >
a1+A2
m1A2
If E1 exists and its Jacobian has a positive eigenvalue, then A2 < A0. Consequently, one has f(l) < (a1+ A2)/m1A2. It is easily seen from the monotonicity of A/(a1+A) that this inequality also holds if E1 does not exist.
There are two cases. If A0 < 0 then
a1+A1
< a1+A2
m1A1
m1A2
f(1) <
a1+1
<1 -
MI
m1A1
m1A2
f(1) <
a1+A2
m1A2
<.f(p*)
one can conclude from the mean value theorem that there exists a unique
number p* with f(p*) = (a1+A2)/m1A2 and p2 < p* < 1.
90
Since
Sx2 P
2
P2+K+p2'
it follows from (4.4) that x2c < x2 = 1 +A2. Therefore Ec exists and is
positive.
Conversely, if Ec exists then p* < p7 < 1 and x2c < x2, so
f(P,*)
f(P*) >
a,+A2
= m,A2
(a,+A2)2f(pc*)xlc*
J=
m2a2
m, a,
- (a,+A2)2
f(p*) xis
m2a2
x 2c
(a2+A2)2
*
x2c
bKx*
-1- (K+pc
)2
bpC
m,a2
ic f'(p7 )
a, + .A2 x
K+p
3+2 +
6Kx2*c
(1
(K+p$ )2
+ I l +
a,xjc
(a,+A2)A2
a2 x 2c
(a2 +A2)A2/
SKxic
a2x2c
l aixic
(K+p7)2/C(ai+A2)A2 + (az+Az)Az/
f'(Pc)
a2
bpi
X]c x2c= 0.
*
(4.7)
91
Since f'(p) < 0, the constant term is positive, so the Routh-Hurwitz criterion (Appendix A) says that EE will be asymptotically stable if and only if
+
C
6Kx2c
aIxi,
(K+p*)z
(a,+A2)A2
6Kx2,
a2 X2,
(a2+A2)A2
a,.,!(,*,
az x2c
>-f(ps)
5p,
x*x*
(4.8)
x, (t) = x],
lim,x2(t) =0,
lim, p(t) = 1.
Proof. Since
0<
a,
m, f(1)-I
< A2,
a,
ins f(l+e)-1
< A2.
Since p(t) < 1 +e for all large t, for such t one has that
92
Cm 1-x x
f(1+E)- 1
xt ? xl
1+(al-x1-x2
m2(1-x1-x2)
X2' _ -x2(1+a2-xi-x2 -1
at
mi f(1+E)-1
limt_ p(t) = 1.
Then, for c small,
x1 _ x1
(ml(1 -XI)
l+a1-xi f(1-E)-1
holds for all large t (where t depends on E). By comparison with the
growth equations (4.1) of Chapter 1, it follows that
lim sup x1(t) <- 1t-W
al
m1f(1-E)-I
lim,xl(t) = 0,
lim,- x2(t) = xz,
p(t) = P2
93
K+p
=0.
Clearly,
Since limsup,-,,, x2(t) < 1-A2 (by Proposition 3.1), for each E > 0 there
exists TI(E) > 0 such that x2(t) <- 1-A2+E for t > TI(E). It follows that
P - 1-P-
S(I -A2+E)p
K+ p
and consequently there exists T2(E) >- TI(e) such that p(t) >- p*(E) -E for
t > T2(E). One therefore has that
m 1-x
f(P*(E)-E)-1
m,1-x,)
<x
and
m2(1-x, -x2)
-1
fort >- T2(E). If m, f(p*(E) - E) <- 1, then the second differential inequality
lim,- x,(t)=0.
If m, f(p2*) > 1, then m, f(p*(E) - E) > 1 for sufficiently small E. Since
x'=.f(x),
xe 1183,
(6.1)
94
ax;
-< 0,
i # j.
(6.2)
Such systems are said to be competitive (as noted in Chapter 1). When
(6.1) represents a population growth equation, (6.2) indicates that an increase in the size of one component inhibits the growth of the others.
Such a system is not necessarily order-preserving, so the theory of monotone systems does not apply. However, if solutions exist for all time and
if one runs time "backwards" (more correctly, if one makes the change
of variables t = --r and regards r as "time"), then the corresponding dynamical system is monotone. More formally, the system
x'= -f(x)
(6.3)
is monotone and the results of Appendix C apply. Thus the limit properties in the theory of monotone dynamical systems (discussed in Appendix
C) apply to the alpha limit sets, not the omega limit sets, of competitive
systems. Observe that periodic orbits are both alpha and omega limit
sets. The results on monotone systems may be reinterpreted for (6.1) satisfying (6.2).
In Remark 6.1 we have used < rather than <-K because K is the usual
positive cone. A consequence of these remarks is that planar cooperative
or competitive systems do not have periodic orbits. For example, consider a planar cooperative system and let P be an arbitrary point on a
periodic orbit. Impose the standard two-dimensional coordinate system
at P. The orbit cannot be tangent to both the x and they axes at P and so
must have points in both quadrants II and IV (the sets unordered with
respect to P), since points in quadrant I or III would be ordered. The
orbit cannot pass through P again and cannot have points in quadrants I
95
and III, and so is not a closed orbit. This argument fails in three dimensions, which can support periodic orbits for competitive systems.
A major difference between competitive and cooperative systems is that
cycles may occur as attractors in competitive systems. However, threedimensional systems behave like two-dimensional general autonomous
equations in that the possible omega limit sets are similarly restricted.
Two important results are given next. These allow the Poincare-Bendixson conclusions to be used in determining asymptotic behavior of threedimensional competitive systems in the same manner used previously for
two-dimensional autonomous systems. The following theorem of Hirsch
is our Theorem C.7 (see Appendix C, where it is stated for cooperative
systems).
THEOREM 6.3 [Hi4]. Let L be a compact omega limit set of a competitive system in 1183. If L contains no equilibria, then L is a closed orbit.
The system (6.1) is said to be competitive and irreducible if it is competitive and if the Jacobian of f(x) in (6.1) is an irreducible matrix (see Appendix A) for all x.
96
THEOREM 7.1. Let E, exist and let (x1(t), x2(t), p(t)) be a solution of
(3.2) with xi(0) > 0, i = 1, 2. Then
Proof. Note that M+(Eo), the stable manifold of E0, is either the p axis
if E1 exists or the x1-p plane if E1 does not exist. The manifold M+(E2) is
the x2-p plane less the p axis; if El exists, M+(E1) is the x, -p plane less
the p axis. Since (x1(0), x2(0), p(0)) does not belong to any of these stable manifolds, its omega limit set (denoted by w) cannot be any of the three
rest points. Moreover, w cannot contain any of these rest points by the
Butler-McGehee theorem (see Chapter 1). (By arguments that we have
used several times before, if co did then it would have to contain E0 or
an unbounded orbit.) If w contains a point of the boundary of R then,
by the invariance of w, it must contain one of the rest points E0, E1, E2 or
THEOREM 7.2. Suppose that system (3.2) has no limit cycles. Then E, is
globally asymptotically stable.
97
Proof. In view of Theorem 7.1, the omega limit set of any trajectory
cannot be on the boundary x1 = 0 or x2 = 0. Away from the boundary,
the system is irreducible. Since there are no limit cycles, all trajectories
must tend to E, by Theorem 6.3.
THEOREM 7.3. Suppose that EE is unstable and hyperbolic, and let (x1 (t),
Proof. Theorem 7.1 guarantees that none of the boundary rest points
belong to its limit set, w(q). It must be shown that EE does not belong to
w(q). The result will then follow from Theorem 6.3.
Suppose EE e w(q). Since q ff M+(E,), it follows that w(q) # E, The
Butler-McGehee theorem implies that w(q) contains a point r of M+(E,)
To show the instability of E, one must show that the inequality (4.8) is
violated. This, however, is a very formidable task without further information about the function f(p), which gives the effect of the inhibitor. In
the next remark, a specific function is chosen, the function used originally
by Lenski and Hattingh [LH]. The proof then reduces to the application
of the Routh-Hurwitz condition, still a computationally formidable task.
However, the point of interest is deriving any oscillation at all, given the
tendency of the chemostat to approach a steady state.
REMARK 7.4. Let f(p) = e-'"Pin (3.2). Then font sufficiently large there
exists a b0 > 0 and a K0 such that, for 8 > 60 and K < KO, (3.2) has a limit
cycle.
a1+A2
98
1-p*
(ri - c)
77
77
C=1--=
Note that when q is fixed, pc is fixed. From the definition of x2c (see (4.4)),
it follows that
SKx2c
1- p*
(K+p*)2
c -1
K+p
K+p* p*
(7.1)
> c+2(3+
ai(1-A2) +
a1(1-A2)
(ai+A2)A2/ \(ai+A2)A2
)((I
2(a2+A2)A2
-A2)a2
()
7.2
and
m,
ae-<1.
i+l
1+A2-x2c, limb_, x2c=0 and limb _,xic=1+A2. Hence, for S sufficiently large,
(a2+A2)A2
2)A2
and
2A2
1
2(a2+A2)A2
(1-A2)(f1-c).
The left-hand side of (4.8) has three factors, which we denote by F1, F2, F3
in the order given. By the discussion so far, for S sufficiently large we have
Fi < (3+
\
F2<2,
F3 <
a,(1-AZ)
(a,+A2)A2
a1(1-A2)
+ 1.
(a, +A2)A2
which contradicts (4.8). Hence, Ec is unstable and the conclusion of Remark 7.4 follows from Theorem 7.3, completing the proof.
Discussion
99
If, in addition, the vector field is analytic, then it can be shown (see [ZS])
that there must be an attracting periodic orbit.
8. Discussion
The material in this chapter makes several interesting points. First of all,
the need for the more sophisticated tools is clear; without them, there can
be no global results. It also illustrates that three- and four-dimensional
systems can be analyzed. Better mathematical tools might make even more
complex systems tractable. Although coexistence can occur either as an
attracting rest point or as a stable periodic orbit, the interesting case is the
oscillatory one. The time course is shown in Figure 8.1 for a sample problem; Figure 8.2 shows a three-dimensional phase plot of the limit cycle.
Experiments involving the chemostat with an inhibitor should be no more
difficult than those of ordinary chemostat experiments (which, however,
are not easy). The oscillatory case suggests an interesting experiment,
which is hinted at in the work of Hansen and Hubbell [HH] discussed
in Chapter 1: they used an inhibitor (in a very limited way) as an alternate
explanation of the coexistence case. A definitive experiment remains to be
done to see if the oscillations can occur for parameters in a realistic region.
A
0.7
0.6
0.5
0.4
0.3
0.2
0.1
20
40
60
80
100
Time
Figure 8.1. Plot of 100 time steps in the case of oscillatory coexistence. Parameters are a, = 0.5, a2 = 3.4, m, = 5.0, m2 = 6.0, K = 0.1, n = 5.0. (From [HWI],
reprinted with permission from the SIAM Journal on Applied Mathematics, volume 52, number 2, pp. 528-40. Copyright 1992 by the Society for Industrial and
Applied Mathematics, Philadelphia, Pennsylvania. All rights reserved.)
100
X,
Figure 8.2. Plot in E3 of the limit cycle given in Figure 8.1. (From [HW1], reprinted with permission from the SIAM Journal on Applied Mathematics, volume 52, number 2, pp. 528-40. Copyright 1992 by the Society for Industrial and
Applied Mathematics, Philadelphia, Pennsylvania. All rights reserved.)
1. Introduction
In the preceding chapter we saw how the chemostat could be modified
to account for a new phenomenon - the presence of an inhibitor. In this
chapter we extend the idea behind the simple chemostat to a new apparatus in order to model a property of ecological systems that it is not
possible to model in the simple chemostat. The idea is to capture the
essentials of the new phenomenon without destroying the tractability of
the chemostat either as a mathematical model or as an experimental one.
A very simple situation will be described here; a more complicated one with a less explicit (in the sense of less computable) analysis - will be dis-
cussed in the next chapter. Just as the chemostat is a basic model for
competition in the simplest situation, the apparatus here shows promise
of being the model for competition along a nutrient gradient.
The "well mixed" hypothesis for the chemostat does not allow a nutrient
gradient to be generated. A basic tenet is that the nutrient concentration
is the same everywhere; hence any advantage in nutrient consumption is
present everywhere. The model that incorporates a true gradient would be
one involving partial differential equations; a new variable, space, must
be accommodated. Systems of nonlinear partial differential equations are
difficult mathematical objects to understand and analyze. Even numerical
solutions pose added and significant difficulties. Moreover, even if an experimental gradient is devised, measurements that do not disturb the local
environment take on new difficulties.
A piece of laboratory apparatus was devised by Lovitt and Wimpenny
[LWI; LW2; WL] for experiments along a nutrient gradient. It is a concatenation of chemostats in which the adjacent vessels are connected in
both directions. Output occurs at the first and last vessels, and those in
101
102
between exchange their contents - nutrient and organisms. The flow rates
in, out, and between vessels are constant and equal. The apparatus was
named a gradostat. It does not occur in nature, at least in this form. Indeed, although we shall think of the apparatus as connected horizontally,
the closest approximation in nature may be vertically, as in a water column. In Chapter 6, we shall see that much more imaginative connection
patterns are possible. Growth along nutrient gradients does occur in abundance in nature. For example, the surface films in dental plaque represent
growth along such a gradient, as does growth along the banks of a stream
or along a seacoast. In a water column, sunlight replaces the nutrient as
an essential source needed for growth.
In a loose sense the apparatus generates a "discrete" gradient; the nutrient concentrations will vary from vessel to vessel, so the "parameters" of
competition change from vessel to vessel. If there is no consumption, the
nutrient concentrations arrange themselves as discrete points along a linear gradient. The effect of a nutrient gradient on growth and competition can be studied with such a device.
Figure 1.1 shows the device used by Lovitt and Wimpenny, and Figure
1.2 is a schematic of the device to be analyzed in this chapter. The mathe-
rn
-_n
R2
RI
V5
V4
V3
V2
Vl
B2
BI
Figure 1.1. The device of Lovitt and Wimpenny. (From [LW2], Copyright 1981,
Society for General Microbiology. Reproduced by permission.)
The Model
sl,ul,vi
103
2'U2,V2
nutrient. The analysis in this special case takes a particularly elegant form,
the relevant conditions being expressed as a set of basic inequalities. As
we shall see in the next chapter, much more general formulations can
cover many more interesting situations (more interesting gradients), but
at the expense of a more complex analysis. The analysis will make extensive use of the theory of monotone systems discussed in Appendix C. The
presentation closely follows that of [JSTW]. Since there are only two
vessels, the model can be completely analyzed.
To keep the volume in each vessel constant, it is necessary to have medium (without nutrient) input at the right-hand end (as shown in Figure
1.2).
2. The Model
The competitors will be labeled as u and v with a subscript denoting the
vessel: ul is the first competitor in the first vessel, v, the other competitor
in the first vessel, and so on. The nutrient concentration in each vessel is
also labeled this way. The equations will differ from that of the basic
chemostat only in the flow of nutrient between vessels; for example, there
are two outputs from each vessel, one out of the system and one into the
adjoining vessel. If there were more vessels in the gradostat, then the
middle ones would have two connections as well - that is, connections to
the two adjoining vessels.
The equations take the form
S'= (S(1-2S,+S2)D-.fu(S1)ui/'Yu-,ft,(Si)vI/y,,,
Sz = (Si-2S2)D-.fu(S2)u2/'Yu-.f,,(S2)v2/'Y,,
104
of = (-2u,+u2)D+fu(Si)ul,
ui = (u,-2u2)D+fu(S2)u2,
(2.1)
v'=
vi = (v1- 2v2)D+fr,(S2) v2,
mRS.
fu(Si) = au
Sir
+fv(Si) =
a++Si
i=1,2,
,
rate. These terms are used exactly as in Chapter 1. As noted there, the
last two quantities are under the control of the experimenter, while m, a,
and y depend on the population being cultured. Since the focus is on
competition, it is reasonable to assume that the functions fu and fv are
different. Specifically, it is assumed that either mu # m, or au # av.
The quantities au and av have units of concentration, so if all concentrations (nutrients, organisms, and Michaelis-Menten constants) are measured in units of 5(0) then S(0) may be scaled out of system (2.1). Similarly,
= S1-2S2-fu(S2)u2-fv(S2)v2,
of = -2u,+u2+fu(Si)u1,
u2 = u,-2u2+fu(S2)u2,
of = -2v,+v2+fv(S1)v,,
v2 = v1-2v2+fv(S2)v2,
(2.2)
The Model
105
The as and ms have changed their biological meaning, although the new
fs are formally the same as those in (2.1).
Now define
E1(t) _ ; - S1(t) - u1(t) - v1(t)
and
E2(t) _ ; -S2(t)-u2(t)-v2(t).
Adding the equations that correspond to a dependent variable with a
subscript 1 and those with a subscript 2 yields
Ej=-2E1+Ez,
EZ = E1-2Ez,
z
u1 = -2u1+Ll2+fu(3
-E1-ul-Ul)u1,
(2.3)
uz = ul-2uz+.fu(3 - Ez - uz - vz)uz,
z
v1= -2U1+U2+fv(3
-E1-ul-Ul)Ul,
vz = vl-2vz+fU(, -Ez-uz-vz)vz.
The form of (2.2) and the uniqueness of initial value problems guarantee
that the nonnegative cone (in 1186) is positively invariant (see Appendix B,
Proposition B.7). The following "conservation" result parallels that for
the simple chemostat.
LEMMA 2.1. The system (2.2) is dissipative. Moreover,
is positively invariant.
z'= -2z1+z2,
z2 = z1 - 2z2,
and the assertions about limits follow by solving this constant coefficient
system.
106
The difference between this and the previous conservation results is that
it holds in each vessel rather than for the entire "biomass." As a consequence of this lemma, the omega limit set of any trajectory is nonempty, compact, connected, and contained in F. Since every trajectory is
asymptotic to its omega limit set, it is important to analyze the system on
this set. (See Appendix F for a rigorous justification.) Trajectories in the
omega limit set satisfy
(2.4)
-u, -v,)v,,
vz = v,-2vz+fll(; -u2-v2)v2
on the region that we again call r to conserve notation. This region r is
defined by
3).
System (2.4) is the one that will receive most of the analysis. Several of
the results in the appendices will be used; the theory of monotone systems
and the persistence results will be particularly useful. It is generally not
possible to analyze a four-dimensional system such as (2.4) because the
dynamics can be very complicated; indeed, they can be chaotic. One must
work very hard, using the theory developed, to show rigorously that the
LEMMA 3.1. Let (Si , Sz, u*, u2, v*, v2) be an equilibrium point of (2.2).
Then:
107
fv(Sl) >fv(S2);
(c) if ui > 0 then
<2-fu(S*)<1,
1 < 2 - fu (SZ) < 2, and
(2 - f. (S*)) (2 -fu (S*))
(d) if v* > 0 then
1;
(2-fv(Si))(2-fv(S2)) = 1.
Before proving this technical lemma, we make some observations. One
might think that coexistence in the gradostat is possible by having one
competitor win in one vessel and the other in the second vessel. Statement
(a) shows this is not possible (at least, not as a steady state). The last state-
ments in (c) and (d) provide a crucial key to actual computation of the
coordinates of the rest points. The other inequalities provide estimates
that will be useful in stability considerations.
Proof. The proofs of these assertions are all relatively straightforward.
The results will be established for u7; the others follow similarly. The
equations for the rest point are just the equations with the left-hand side
of (2.2) set equal to zero. The equations for u, yield the first statement in
(a) directly. The second equation in (2.2) at equilibrium becomes
108
(2-.fu(Si))(2-.fu(Sz)) = 1.
The inequalities then follow from (b). This completes the proof of the
lemma.
There is an equilibrium point of the form (; , i , 0, 0, 0, 0). This corresponds
to both organisms washing out of the gradostat. In view of Lemma 3.1(a),
there are potential equilibrium points of the form (Sb S2, 0 1, u2, 0, 0) and
(Sl, S2, 0, 0, v"1, v2), with all nonzero entries positive. These correspond
to one of the competitors washing out of the gradostat. For coexistence
to occur as a steady state, it must be shown that there also is an equilib-
rium point with all components strictly positive. The existence and the
stability of these rest points are closely related, as the following sections
show. To determine the stability of rest points it turns out that certain
functions of the coordinates of the rest points must be evaluated, so it is
important to be able to compute them explicitly. Our inability to make
these computations when the number of vessels is more than two restricts
the analysis, as the next chapter will show.
Both ul and u2, if they exist, are solutions of
-2u1+u2+.fu(z3-u0ui =0,
uI -2u2+.fu(3 -u2)u2 = 0.
A straightforward analysis reduces the question of solutions to finding
the roots of a cubic equation. Of course, these quantities must be positive
for the rest points to be meaningful. Conditions for this, based on stability considerations, will be given in the next section. Numerically solving a cubic is simple; in fact, there is an unlovely formula that does it
explicitly. The important point is that since solving the cubic is possible, conditions based on the numerical coordinates are testable. A similar
computation gives the coordinates of v, and v2.
The more difficult question is the existence of an interior rest point that is, where Si, ui, and vi are all positive. (Stability considerations, or
the arguments that follow here, show that there is at most one interior
rest point.) From Lemma 3.1 one has
(2-.fu(Si))(2-.fu(Sz)) = 1,
(3.2)
1.
109
3a, -Allaurl+A12r2 = 0,
(3.3)
3a, -A21avr1+A22r2 = 0,
where
A11 = (2mu-3),
A12 = (mu-1)(mu-3),
I-All au
A12
-A21av A22 11
not vanish is
and
r2 = C3/CI,
where
cl = A12 A2l av -All A22 au,
c2 = 3A12av -3A22a2u,
2
Sl +Sz = C21CI,
c2 t c2 -4C1C3
2 c1
S*
2
C2
(3.4)
C2-4C1C3
2c1
Since S* > Sz, there is only one choice of signs. Hence S* and S2, if they
exist, are uniquely determined by (3.4). If Si and SZ are determined from
(3.4), then the components of u* and v* can be determined from the basic
equations (2.2).
110
Table 3.1
Point
Coordinates
Eo
(0,0,0,0)
El
(z 1, u2, 0, 0),
ui>0, i=1,2
(0, 0,
E2
V1, v2),
t;>0, i=1,2
E.
U1
Similarly,
U2
2S2+fv(S2)(3-Sz)-Sj
fv(S2* )-fu(S2*)
.
V1 = 23 -S1. -u1,
*
*
V2= 3-S2-t[2.
I
the solution of (3.2) represents two parallel lines. If they are coincident,
111
S2 = Sl-2S2-fu(S2)u2-fv(S2)v2,
uf=-2u1+u2+fu(Si)ui,
(4.1)
u2 = ui -2u2+fu(S2)u2,
Si(0) ? 0, ui(0) >- 0, i = 1, 2.
ui = -2ul+u2+fu(; -ul)uJ,
u2 = uI-2u2+.fu(3 -u2)u2,
(4.2)
ui(0)>-0, i=1,2.
Of course, the equations are restricted to the region defined by 0 < ul(t) <
112
a
au2
(-2u,+u2+f,,(; -ui)ui) = 1,
au,
(ul-2u2+J (3 -u2)u2) = 1.
and
Q2(Z)=2-fu(!-z).
J_
a,(0)
-a2(0)
with eigenvalues
-(a,(0)+a2(0))f (a,(0)-a2(0))2+4
A
The eigenvalues are real, and if a, (0)a2(0) < 1 then the origin is a saddle
point.
Proof. If a,(0)a2(0) > 1 then both eigenvalues have the same sign, so
the origin is an attractor or a repeller according to whether a,(0)+a2(0)
is positive or negative.
Note that for (4.2) the nonnegative quadrant is positively invariant.
When a,(0)a2(0) < 1 (i.e., when the origin is a saddle point), we will
show that the stable manifold of (0, 0) does not intersect the interior of
the positive quadrant. Let A+ and A- denote the positive and negative
eigenvalues of J,,, respectively. One has that
a,(0) +A+ =
Similarly, one has that a2(0)+A- < 0. An eigenvector (z,, z2) corresponding to A- satisfies
z, - (a2(0) +A-)z2 = 0,
113
(a) there is no rest point of the form (I , u2) with u; > 0, i = 1, 2, and
(b) the origin attracts all trajectories with initial conditions in A+ n F,
where (f8+ denotes the interior of R
(2-fu(3-ul))ul-u, =0,
-ul+(2-fu(3 -u2))u2 =0.
(4.3)
A nontrivial rest point can exist if and only if the determinant is zero;
hence, a1(u1)a2(u2) = 1. However, ai(z) is monotone increasing, so
a1010a2012) > a1(0)a2(0) ? 1.
Proof. The existence of a rest point with positive coordinates has been
established in the paragraph preceding the statement of the lemma. Suppose there were two distinct rest points (i1, u2) and (u1, u2) with
114
u2=u,a,(u,)<-u,a,(u,)=u2,
with equality only if ii, = u,. Hence, if (u,, u2) * (u,, u2) it follows that
with a3(z) =
a2(z), respectively.
The conclusion of the preceding arguments is very simple. There are only
two possibilities: The origin is an attractor and all trajectories tend to it;
or the origin is not the omega limit point of any trajectory with positive
initial conditions, and there exists a unique rest point with positive co-
5. Local Stability
We turn now to computing the local stability of the rest points of the full
system. The arguments are based on standard linearization techniques,
but the size of the variational matrix makes some of the computations
difficult. The variational matrix for (2.4) takes the form
-a,-,6,
-a,
-a2-a2
-a2
-a3
-a3 -03
-(34
-a4-a4
Local Stability
115
u,
i -
a=
(au+3-ul-vl)2'
mu au u2
(au+j-u2-v2)2'
- (av+3-ul-v1) 2'
z
mvavv2
-u2-v2)2.
When evaluated at a hyperbolic rest point, J will determine the (local) asymptotic stability of that point. The computational problem is increased
by the size of the matrix and the complexity of the entries. Fortunately,
for some of the rest points there will be a large number of zero entries in J.
For a fixed set of parameters, let SZ denote the rest point set of the sys-
tem (2.4) in F. There are four possible types of rest points, which we
denote as follows:
E0=(0,0,0,0);
E1 = (u1, u2, 0, 0)
with ui > 0, i = 1, 2;
0J'
J= [Ju
where J and JJ are the variational matrices for the two-dimensional systems considered in Section 4. Hence the origin is asymptotically stable if
and only if both of the two-dimensional systems (the systems without
competition) are stable. From Section 4, this is the case if and only if
116
al(0)a2(0) > 1,
and
3(0)4(0) > 1,
a1(0) > 0
(5.2a)
a3(0)>O-
(5.2b)
a1(0) < 0, and v1-v2 if a3(0)a4(0) < 1 or a3(0) < 0), and hence a nontrivial equilibrium point exists in the corresponding two-dimensional subset of the boundary.
The rest point El corresponds to a rest state without the v competitor.
When it is asymptotically stable, the v competitor will become extinct
(will wash out of the system) for nearby initial conditions. If the stability
is global (which will turn out to be the case when stability is local), then v
becomes extinct for all positive initial conditions. At E1, J takes the form
r-al-01
-a1
-a2-/32
-I2
-a3
-a4
where the a;s and (3,s are evaluated at (u1, u2, 0, 0). The zero block in the
lower left corner makes the computation of eigenvalues easy, for (using
a1a2 = 1) they satisfy
U2+(a1+a2+/31+(32)1+x2(31+132+,61(32]
x [A2+(a3+a4)A+a3a4-1] = 0.
(5.3)
from the first square bracket are real (the radical simplifies) and always
have negative real parts. These eigenvalues correspond to eigenvectors in
the subspace (u1i u2, 0, 0), which is the stable manifold of (ul, u2) viewed as
a rest point of the two-dimensional system (4.2). This is expected, in view
-(a3+a4)t (a3-a4)2+4
2
The eigenvalues are real, and the sign depends on the values of a3(u1)
and a4(u2). Clearly, if a301)a402) > I then the two eigenvalues are of
Local Stability
117
the same sign. Then E1 will be an attractor if a3(u1) > 0 and have a twodimensional unstable manifold if a3(u1) < 0.
If CO (u1)a4(u2) < 1, there is one negative and one positive eigenvalue.
Let A- denote the negative eigenvalue and let z = (z1, z2, z3, z4) be the
corresponding eigenvector. Since
+U3
a3-a4- (03-a4)2+4
2
<0
and (A-+a3(u1))z3 = z4, z3 and z4 are of opposite signs and hence z must
point out of the positive cone at (u1, u2i 0, 0). In particular, the stable
manifold of El does not intersect the interior of the positive cone. This
statement is trivially true in the other case, since the unstable manifold
is two-dimensional and the stable manifold lies in the boundary. In this
case no trajectory with initial conditions in the interior of r can tend to
El. This argument is summarized in the following lemma.
LEMMA 5.1. If a3(U1)a4(L 2) > 1, then E1 is an attractor if a3(u1) > 0 and
has a two-dimensional unstable manifold if a3(u1) < 0. If a3(u1) a4(u2) <
1, or if a3(u1)a4(u2) > 1 and a3(u1) < 0, then E1 is not the limit of any
trajectory with initial conditions in the interior of P. Similar statements
apply at E2 using a1(01) and a2(02). When either E1 or E2 is unstable, the
stable manifold of that rest point lies in the corresponding two-dimensional face containing E1 or E2.
The difficult part remains - the stability of an interior rest point. Conditions for the existence of such a rest point will be established, on geometric grounds, later in the chapter.
118
03 =
mu(Si )2 ( a1 -2)2
( a3
0z =
u,
-2)2 v i ,
au
mu(SZ)2
04 =
( C12
-2)' u *
-2)2 v2,
by multiplying and dividing by the square of the appropriate S. Then
replace a2, a3, u2, and v* by the substitutions (obtained from Lemma
mU(Si )2
( a4
a3 = 1/a4;
u2=a1U
vj =a4v2.
-2)2(
a, av
1
(Si SZ)2
al
a4
-2/ al a4u1
v2,
Order Properties
119
(a2a3-1)0i$4 =
CaJa4
(al-2)2(a4-2)2u1 vz.
d4=
where
/1
R(a1,a4) _ (al4I
G
-2)2(
1
4 -2I aia4
-1 I(ai-2)2(a4-2)2.
+I
041 C(4
3(a,a4-1)2
ala4
Q(a1, a4),
where Q(a1, a4) = 5+5a1a4-4a, -4a4. Because Q is an increasing function of a, in the allowable range of a4, Q > 3 - 3a4/2 which must then be
positive.
In the preceding sections, the possible rest points for the gradostat equations were determined and their stability analyzed. The problem that remains is to determine the global behavior of trajectories. In this regard,
the theory of dynamical systems plays an important role. First of all,
some information can be obtained from the general theorem on inequalities discussed in Appendix B. We illustrate this with an application to the
gradostat equations.
Let u(t) = (u1(t), u2(t), v,(t), v2(t)) be the solution of (2.4) with omega limit point p, and let x(t) = (x1(t), x2(t)) be a solution of (4.2) with
120
x](t)=x2(t)-(2-fu(3-xl(t)))xl(t),
x2(t) =x1(t)-(2-fu(3 -x2(t)))x2(t)
Hence, for all t > 0, Theorem B.1 (the basic comparison theorem) states
that
ul(t) < x1(t) and u2(t) < X2(0-
121
in Appendix B). Let Df(x) be the variational matrix evaluated at an arbitrary point. A sufficient condition for strong monotonicity of a monotone
system is given in Theorem C.1 - namely, that the matrix J be irreducible.
From (5.1) this is easily seen to be the case in the interior of P and some of
its boundary. The dynamical system generated by (2.4) is strongly monotone in the interior of U.
PROPOSITION 6.2. The system (2.4) is uniformly persistent if and only if
the rest points El and E2 are unstable.
In the previous section it was shown that the rest points El and E2, when
unstable, had no part of their stable manifolds in the positive cone (Lemma 5.1). This shows that (H) of Theorem D.2 holds. If the covering is
taken to be the set of rest points, the flow on the boundary is acyclic (in
terms of Appendix D), for the rest points in the faces ul = u2 = 0 and the
VI = v2 = 0 attract all points in that face. An application of Theorem D.2
completes the proof: take X = lRl and E = ((u1i u2, v1, v2) E l j ui > 0
for some i and vj > 0 for some j).
Recall that hyperbolicity is a generic assumption in this chapter and that
dissipativeness has been established. This has the following consequence.
PROPOSITION 6.3. The rest point E* exists if and only if El and E2 are
unstable.
Proof. If El and E2 are unstable, then dissipativeness and uniform persistence (previous proposition) yield the existence of an interior rest point
E. for 7r(x, t) (Theorem D.3). If E. exists then it is unique and has all
eigenvalues negative (Lemma 5.2). Suppose that E* exists and that El is
asymptotically stable. Then, since E. <K El and both are asymptotically
stable, Theorem E.1 contradicts the uniqueness of E. A similar argument
applies if E2 is asymptotically stable. Note that the computations leading
up to Lemma 5.1 explicitly determine the signs of the eigenvalues for
linearization about E1 and E2.
7. Global Behavior of Solutions
Enough information has now been collected to classify the behavior of
all solutions of the gradostat equations. This classification will be given as
a function of the set SI of rest points. The existence and stability of the rest
points has already been established. Indeed, the coordinates of the rest
122
points can be given in terms of the system parameters - the au, a,,, mu, m
that are functions of the organism being cultured. The (local) stability of
equilibria is determined by a and 0 evaluated at these points. The set Sl
belongs to one of four categories, and for each category the global behavior is given.
THEOREM 7.1. Suppose all of the rest points of (2.4) are nondegenerate.
Proof. Case (a) is covered by Lemmas 4.2 and 6.1. This case occurs if
and only if the inequalities (5.2) are satisfied.
Case (b) occurs if and only if one of the inequalities in (5.2a) or (5.2b)
is reversed and the other set of inequalities holds. Lemma 6.1 again yields
that E, belongs to the omega limit set of any trajectory with positive
initial conditions.
For case (c) to hold, one inequality in each part of (5.2) must be reversed and (by Proposition E.2) both E, and E2 cannot be stable. Thus,
one of E, and E2 is a local attractor, say E,. (Lemma 5.1 provides the explicit conditions for determining which is stable.) Similarly, if case (d)
holds, both E, and E2 must be unstable.
The next lemma will be useful in our proof. It will be convenient to let
[P,, P2]K denote the order interval, that is
[P1,P2]K={PIP1
KP-5KP21-
LEMMA 7.2. Suppose that E, and E2 belong to fl and that E, is unstable. If hypothesis (c) holds, then every solution starting at a point xo =
(u,(0), u2(0), v,(0), v2(0)) belonging to I' fl [E2, E,]K satisfies
. lr(xo, t) = E.
123
Proof. Let x denote a point (ul, ul, v1, v2), and let F(x) denote the vector
field on the right-hand side of (2.4). Let A denote the largest eigenvalue of
J, the Jacobian of F at x = El. (J is displayed just prior to (5.3).) Since
EI is unstable, A > 0. An eigenvector z = (u, v), u = (ul, u2), v = (vl, v2)
corresponding to A must satisfy
Au + By = Au,
Cv=Av,
where A, B, and C are (respectively) the three nonzero matrices in the
upper left, upper right, and lower right corners of J, A is an eigenvalue of
C, and the eigenvalues of A are both negative. By direct calculation (or
by an appeal to Theorem A.5), one can choose v > 0. Then u satisfies
u = -(A-AI)-IBv.
Since By < 0 and -(A-AI)-I > 0, by Theorem A.12(i) (or by direct calculation) one concludes that u < 0 and x <K 0.
Consider a point xr = E, + rx for 0 < r and r sufficiently small. A calculation yields that
= rJx+o(r),
where o(r) represents a term satisfying
1imr-o(o(r)/r) = 0.
Thus
and 7r(xr, t) converges monotonically to a rest point for all small positive
r. In Theorem 7.1(c) it is clear that
lim, - 7r(xr, t) = E2,
since E2 is the only rest point contained in [E2, xr]K. In Theorem 7.1(d),
the inequalities may be strengthened to read
E. <K 7r(xr, t) :5K Xr <K El,
lim,-.7r(xr, t) = E,,.
124
such that
E2<KXs<KXO<KXr<KEl.
Since
lim,.. lr(x, t) = E2
for xeP\[E2,El]K. Assume this is false; that is, assume there is an xe
P\[E2,El]K such that the y+(x) does not enter [E2iE1]K.
By Lemma 6.1, every limit point p of lr(x, t) must satisfy p e [E2, EI]K.
By Lemma 5.1, lr(x, t) cannot converge to El. Hence, one can find a limit
point P = (p], p2, P3, p4) with p3 > 0 or p4 > 0 or both. By the invariance
of limit sets, 7r(p, t) must have third and fourth entries positive for t
positive and hence belong to [E2, El]K. By Lemma 7.2,
since its limit set contains the asymptotically stable rest point E2. This
completes the proof of case (c).
In case (d), both El and E2 are unstable. An argument similar to Lemma 7.2 shows that
lim,-. 3r(x0, t) = E.
for xo e f fl [E2 i E* ] K. If x0 e P fl [E2, El ] K, choose x1 E P fl [E2, E* ] K and
X2 E f fl [E*, El ]K such that
Numerical Example
125
X1 SK x0 SK x2.
enough to E1.) Strong monotonicity and the fact that lim,-,, lr(xi, t) =
E, i = 1, 2, implies that 1im,
lr(xo, t) = E. Therefore, the positively
invariant set f' fl [E2, El1 K belongs to the basin of attraction of E.
Suppose that x0 is a point such that
lr(xo,t) Ffl[E2,EllK
for any t > 0. As before, to prove the theorem it is sufficient to show
that a limit point p of ry+(xo) belongs to f fl [E2, EI]K, since all such
points are attracted to E. Since 7r(xo, t) cannot converge to either El or
E2 and remains outside of f' fl [E2, El ]K, there must be a limit point of
-y+(xo) distinct from E] and E2 in f fl [E2, El 1K, with p, >_ 0, PI +p2 > 0,
and p3+p4 > 0. By the invariance of limit sets and strong monotonicity, ir(p, t) E I' fl [E2, El 1K. This completes the proof of case (d), since
lim,-.. ir(p, t) = E*.
8. Numerical Example
The theoretical results in section 6 show the global asymptotic behavior
of the gradostat. The outcome of the competition was always a steady
state, but there were several different possibilities. In each case, however,
computations were provided not only to locate the rest points but also to
determine their stability. It remains to show that these computations can
be carried out. In doing so, a single parameter will be changed to illustrate all three of the nontrivial cases. (The total washout case is not illustrated because it is uninteresting and clearly achievable.)
The outcomes are presented in Table 8.1. The parameters a and m are
fixed, which in turn fixes the value of El as shown. The a, term is fixed
but m is assigned three different values (as shown), which in turn yields
three different sets of coordinates for E2. Given the coordinates of these
points, a can be computed. The terms a, and a2 determine the stability
of E2, while CO and a4 determine the stability of El. All three permutations are illustrated: El stable and E2 unstable; El unstable and E2 stable;
both E, and E2 unstable, in which case E,, exists and is stable. As a consequence of Theorem 6.1, stability means global stability (with respect to
the interior of the cone).
The interesting case is coexistence. Figure 8.1 illustrates the time course
for this choice of parameters, showing the asymptotic approach to a steady
state with all limits clearly above zero.
126
20.18
25
E,
(0,0,0.3092,0.1348)
(0,0,0.3123,0.2076)
(0,0,0.3794,0.2434)
a,
0.3917
0.6918
0.8841
a,a,
0.4590
0.9972
1.4036
E,
Unstable
Unstable
Stable
a3
1.0006
0.6557
0.3344
a3 a4
1.6463
0.9984
0.4711
E,
Stable
Unstable
Unstable
E.
(0.2140,0.1471,0.0968,0.0638)
u wins
u and v coexist
v wins
Outcome
0.4
Si
0.3
0.2
sZ
vl
0.1
00
0
Time
10
15
Figure 8.1. The coexistence case with the parameters from Table 8.1. (From
[SW2], Copyright 1991, Microbial Ecology. Reproduced by permission.)
Discussion
127
9. Discussion
The important conclusion from this chapter is that coexistence for two
populations can occur in the gradostat for an open set in the parameter
space. Since the purpose of the gradostat was to mirror behavior along
a nutrient gradient, one can speculate that some of the coexistence observed in nature can be attributed to the existence of such gradients. The
gradostat analyzed had only two vessels, but we anticipate that coexistence of two populations could occur in more complicated gradostats.
A complete analysis of the asymptotic behavior was obtained in terms of
the parameters of the system, and the computability of the conditions
was also demonstrated. In the next chapter, the general gradostat will be
studied and, unfortunately, some of this computability will be lost.
A surprising consequence of the analysis was that, when the interior
rest point existed, it was unique and globally asymptotically stable.
Competition in a modified gradostat was considered in Smith and Tang
[STa]. There the rate E between the vessels (called the communication
rate) was allowed to differ from the rate D (the dilution rate) from the
feed bottle and to the overflow vessel. This, of course, still maintains the
assumption that the volume in each vessel is constant. It was shown that
the outcome of competition can be sensitive to the ratio E/D in the following sense: As E/D is increased, first one competitor wins the competition, then coexistence occurs, and finally the second competitor wins. The
analysis in the case E # D is entirely similar to the case E = D discussed in
this chapter. In [STa], a number of operating diagrams were determined
numerically. For fixed population parameters au, a,,, mug m,,, these operating diagrams depict regions in the E-D plane in which the various outcomes occur. One such diagram is shown as Figure 9.1 (see page 128).
128
V1
0.25
0.50
0.75
1.00
1.25
1.50
Communication rate E
Figure 9.1. Operating diagram for two-species competition with a varying communication rate E different from the washout rate D. In region VI, SI = (E01;
in region V, SI = [Eo, E21; in region II, f = [Eo, Ell; in regions I and IV, fI =
(Eo, E,, E21; in region III, SI = (EO, E, E2, E. 1. (From [STa], Copyright 1989,
Journal of Mathematical Biology. Reproduced by permission.)
1. Introduction
In the previous chapter the gradostat was introduced as a model of competition along a nutrient gradient. The case of two competitors and two
vessels with Michaelis-Menten uptake functions was explored in considerable detail. In this chapter the restriction to two vessels and to MichaelisMenten uptake will be removed, and a much more general version of the
gradostat will be introduced. The results in the previous chapter were obtained by a mixture of dynamical systems techniques and specific computations that established the uniqueness and stability of the coexistence
rest point. When the number of vessels is increased and the restriction to
we are considering only one nutrient, the input at the right-hand end is
only medium without nutrient. Let S denote the nutrient concentration
129
130
D
2
),D
n
Thj
Figure 1.1. The standard n-vessel gradostat. The left vessel labeled R is a reservoir
containing nutrient at concentration S(O), C is an overflow vessel, and D denotes
the dilution rate. All vessels have the same volume.
and u and v the concentration of the two competitors. Then, using the
subscript i to denote concentrations of S, u, and v in vessel i, the equations take the form
S,=(Si_1-2Si+Si+1)D-
Ui
fu(Si)-
'YU
Ui
fv(Si),
'Yv
u, = (ui_1 -2ui+ui+1)D+uifu(Si),
vi '= (vi_1-2vi+vi+1)D+vifv(S),
i=1,...,n,
where
SO=S(O),
uO=vO=0,
is continuously differentiable;
f(S) =
Introduction
131
The 5(0) is the input concentration of nutrient (to the leftmost vessel),
and D is the washout rate. These two parameters are under the control of
the experimenter. The terms yu and yu are the yield coefficients. For convenience, one can scale substrate concentrations Si by S(), time by 1/D
(making m, nondimensional and D = 1), and microorganism concentrations by yu5(O) and yU5() to obtain the less cluttered system
S, =
51-1-2S; + Sr+i
- uifu(Si) - vr.fu(Si),
u, = u,_j-2u;+u;+i'+u;.fu('Sl),
v, =
i=1,...,n,
where we use the same conventions as in the unscaled equations except
that SO = 1. Hereafter, we refer to (1.1) as the "standard" n-vessel gradostat model.
Mathematically and experimentally there is no reason to connect the
vessels linearly, to restrict the source to the left-hand vessel, or to keep
the washout rates D equal so long as the volume of the fluid in each vessel
is kept constant (see [S7]). We next describe a class of gradostat models
which is sufficiently general to include all cases of biological interest and
yet remain mathematically tractable.
Suppose that our gradostat consists of n vessels. Let E,1 be the constant
(volumetric) flow rate from vessel j to i (i # j), with the convention that
Ei, = 0 for i = 1, ..., n. Let V be the volume of fluid in the ith vessel,
Di the flow rate from a reservoir to vessel i (D, = 0 if no such reservoir
exists), SP) the concentration of substrate in the reservoir feeding vessel i
(5;(0) = 0 if D; = 0), and C, the flow rate from vessel i to an overflow vessel
(Ci = 0 if no such vessel exists). The notation diag(f3) is used to denote a
diagonal matrix whose diagonal elements are given by 0,; E is the matrix
of flow rates E,1.
The rate of change of the vector S(t) = (S1(t), ...,
at time tin a
general gradostat, in the absence of any consumers, is given by
[diag(V,)]S'=AS+g,
where
A =E-diag[C;]-diagl E Eii
L 1=1
11
g = (DIS(), D2Sz),
132
EE,j+Di=EEli+Ci
1
(1.2)
or that the volumetric flow rates in and out of any fixed vessel are the
same.
It is convenient to multiply through by [diag V, ] ' and obtain
S'= AS+e,
(1.3)
i#j,
A,j>-0,
(1.4)
and
n
EA,j _-V-'D,<_0
(1.5)
i=1
by virtue of (1.2). Our assumption that S; > > 0 for some i implies D, > 0,
and hence strict inequality holds in (1.5) for some i.
Our principal hypothesis is that the matrix A (or, equivalently, the ma-
trix E) will be assumed to be irreducible (see Appendix A for a mathematical definition). This means that the set of vessels comprising the
gradostat may not be partitioned into two disjoint non-empty subsets,
I and J, such that no vessel in subset J receives input from any vessel in
subset I. Note that the standard gradostat has this property but that the
gradostat in Figure 1.2 does not. In that figure, the subset J consisting of
the two vessels on the left receives no input from the vessel on the right,
which comprises subset I. If that gradostat were of interest, one could
D
2
Figure 1.2. A three-vessel gradostat that does not satisfy the irreducibility hypothesis. The two vessels on the left receive no input from the vessel on the right.
Arrows pointing down from the first and last vessel represent flow to overflow
vessels not depicted. Notation is the same as in Figure 1.1.
133
simply treat the third vessel as an overflow vessel and effectively ignore
it. The subset J could be viewed as the gradostat and the results to be
described in this chapter could be applied to it. Once this subgradostat is
understood, the input to the third vessel is known and it is then a simple
matter to describe what happens in this last vessel. Thereby, the behavior
of the entire gradostat can be worked out. More generally, if the matrix
A does not have the property of being irreducible, then one can always
partition the gradostat into irreducible subsets (subgradostats) that can
be studied sequentially (see [BP]). In this sense, there is really no loss
in generality in assuming irreducibility from the start. We also mention
another way to view the hypothesis of irreducibility: for any pair of distinct vessels i and j, material from vessel i can travel to vessel j, though
perhaps indirectly by first passing through intermediate vessels before
entering vessel j.
While we focus on irreducible gradostats, reducible gradostats may be
of biological interest as well. They could be used to model a system of
mountain lakes situated at different elevations, where a lake at higher
elevation feeds a lake at lower elevation.
Let F. = diag[ f (S1), fu(S2), ...,
and let FU be defined analogously with subscript v replacing subscript u. Then, introducing consumption, the general model takes the form
S'=
-FF(S)v,
u'= Au+FF(S)u,
(1.6)
v'=
The standard model (1.1) is a special case of (1.6), where eo is the vector
with first component equal to 1 and all others equal to 0 and where A is the
134
D
2
D
2
Figure 1.3. Two irreducible gradostats: a cyclic gradostat; b "dead-end" gradostat. Note that the inflow to each vessel balances the outflow.
of M. The next result is crucial for establishing the conservation principle. The properties of the matrix A in the hypotheses of this result have
already been noted in the previous section.
LEMMA 2.1. Let A = (aij) be an irreducible matrix with nonnegative offdiagonal entries. Suppose that
n
Z aij50
j=1
for each i and that strict inequality holds for some i. Then s(A) < 0 and
-A-'>0.
135
IEJ
aii(xi-1)
Zaij +
IEJ
Z au(xl-1).
IE
As xi < 1 for 1 E J and ail > 0, it must be that ail = 0 for all 1 E J; otherwise, the sum is negative. Thus aid = 0 for all i c- I and j c- J. This contradicts the irreducibility of A and so proves the lemma.
The conservation principle is stated next. Like its analog, Lemma 2.2 of
Chapter 5, it states that the total nutrient in each vessel, consisting of
both pure nutrient and that making up the biomass of microorganisms,
approaches a constant value (which depends on how the vessels making
up the gradostat are configured) exponentially fast. As in the previous
chapter, the conservation principle is crucial for our analysis because it
allows the reduction of (1.6) to a lower-dimensional dynamical system.
LEMMA 2.2. Solutions of (1.6) with nonnegative initial data exist and
are nonnegative and bounded for t ? 0; moreover,
limf- S(t)+u(t)+v(t) = z,
where z > 0 is the unique solution of
Az + eo = 0.
w(0) = S(0)+u(0)+v(0) ? 0.
136
one can easily calculate that z; = S 0)[1-i/(n+1)], 1 s i <- n. The concentration of nutrient declines linearly from the value 5(0) in the leftmost
vessel to the value S (0)1(n + 1) in the rightmost vessel, just as one might
expect by analogy with a diffusion process. Thus, a nutrient gradient is established. This observation explains the term "gradostat" coined by Wimpenny and Lovitt to describe this continuous culture device.
Lemma 2.2 says that on the omega limit set, solutions of the system
(1.6) satisfy
u'=[A+Fu(z-u-v)]u, u(0)=uO?0,
v'= [A+FF(z-u-v)]v,
v(0) = vO >:0
on
r=1(u,v)El+u:u+v<z].
Note that P is positively invariant for (2.1). In fact, solutions remain
nonnegative (by Proposition B.7) and, if u, + v, = zi,
(u;+v;)'= [A(u+v)],
From this inequality we conclude that the vector field points into r on
the hyperplane u; + v; = z,.
We will use several notations for a solution of (2.1). The notation
(u(t, u0, vo), v(t, u0, v0))
The initial condition (uO, vO) may be dropped from the notation when no
confusion can occur over which initial condition is being considered. We
will also find it convenient to let x = (uO, vO) and write lr(x, t) for the
solution of (2.1), that is,
lr(x, t) = (u(t, u0, vO), v(t, u0, v0)) = 7r(u, vO, t).
137
vifu0=0.
The next result states that a solution of (2.1) - starting at t = 0 from an
initial distribution of each microbial population among the vessels of the
LEMMA 2.3. If (u(t), v(t)) satisfies (2.1) and u;(0) > 0 (resp. v;(0) > 0)
for some i, then u(t) > 0 (resp. v(t) > 0) for all t > 0.
(3.1)
The initial data u0 must belong to the order interval [0, z] = [ u : 0s u:5 z j.
It is easy to see that [0, z] is positively invariant. Solutions of (3.1) exist for
all t >- 0 and are nonnegative and bounded, by Lemma 2.2 and the fact that
(u(t), 0) is a solution of (2.1). The next result is the analog of Lemmas 4.2
138
so u(t) <-y(t) where y(O) = u(0) and y satisfies y'= [A+Fu(z)]y (by
Theorem B.1). Since y(t) -- 0 as t --> oo, the same holds for u(t). The case
where s = 0 is more difficult; we refer the reader to [S7]. Now suppose that
s > 0 and let v be a corresponding eigenvector for A +Fu (z) such that v > 0.
Such an eigenvector exists by Theorem A.5. Put G(u) = [A+F,, (z-u)]u,
Tu = [-(A-p)-'] [Fu(z-u)+pjlu,
139
where I is the identity matrix. Note that the matrix in the first bracket on
the right side is positive for the same reasons that [-A]-1 is. The map T
140
4. Competition
We turn now to the question of competition. In the previous chapter,
we established a classification of the dynamic behavior based on the set
of rest points. Unfortunately, our computations - which established the
stability of any interior rest point and thereby led to the conclusion that
such an equilibrium is unique in the case of two vessels and MichaelisMenten uptake functions - are extremely difficult for n vessels and general uptake functions [HSo], so the results in this case are not as simple
as in that chapter. In the present context we attempt to classify the dynamics in terms of both the set 0 of rest points and the sign of the stability
El = (u, 0),
E2 = (0, 6),
Proof. Part (a) is clear, and (b) and (c) follow from Theorem 3.1. Part
(d) follows from Lemma 2.3.
J= [A+Fu(z-u_v)_Du
-Du
-D1
J'
where
-vj),...,vnfu(zn-un-vu))-
(4.1)
Competition
141
At E0, both Du and D vanish so that J has only the two diagonal blocks
A+FF(z) and A+FF(z). Clearly, the eigenvalues of J consist of the union
We now turn to the question of the stability of El. The main results
are stated formally, since they are crucial for our later analysis. The basic
idea is simple. Imagine that a microbial population is allowed to grow
in the gradostat in the absence of competition. This population will approach the equilibrium concentration u; in vessel i corresponding to the
rest point El. Now add an infinitesimally small concentration of the competing population and ask whether or not it can survive. If the vector of
concentrations of the competing population is v(t), then v(t) approximately satisfies the linear system v'= [A+FU(z-a)]v, since v is assumed
to be negligibly small. Consequently, the competing population v survives or decays as s(A+Ft,(z-a)) is positive or negative. This is all made
precise in what follows.
LEMMA 4.2. If El exists then s =
(i)) is an eigenvalue of the
variational matrix corresponding to E,, which is asymptotically stable in
the linear approximation if and only ifs < 0. Ifs > 0, then the variational
matrix at E, has a corresponding eigenvector w satisfying w <K 0.
J- [A+F,(z-fi)-Du
0
-Du
A+FF(z-a),'
[A+Fu(z-a)]x = 0
has the solution x = u > 0. This fact and Theorem A.5 imply that
s(A+FF(z-a)) = 0
and that a is the corresponding eigenvector. Since Du >- 0 is not the zero
142
[A+Fu(z-u)-Du-sI]-1Duw2 <
0,
A similar result holds concerning the stability of E2, which is asymptotically stable if s = s(A+Fu (z - v)) < 0 and unstable if s > 0.
An important observation concerning (2.1) is that it generates a strongly
monotone dynamical system in the interior of P. This observation is immediate from (4.1) and Theorem C.1. Let x = (u, v) and y = (u, v) be two
points of R2n. We write x <K y in case u < u and U <- v; we write x <K y
(a) if there exists an equilibrium E. e 0 satisfying E, > 0, then bl contains El and E2;
(b) if 0 = [E0], then E0 is a global attractor for (2.1);
Competition
143
(c) if 0 = [E0, EI) (resp. (ll = (E0, E2}), then the nontrivial rest point attracts all orbits of (2.1) with initial conditions (u0, v0) E F satisfying
u0 # 0 (resp. v0 * 0).
We can now state the main result of this chapter. It provides sufficient
conditions for the coexistence of the two populations in the gradostat
and ensures that SI contains a positive rest point E. In fact, it guarantees
that the two populations are uniformly persistent in the sense of Appendix D. Briefly, Theorem 4.4 states that coexistence holds if each population can successfully invade its competitor's rest point.
THEOREM 4.4. Suppose that El and E2 exist and
s(A+Fu(z-v)) > 0,
0.
144
(4.2)
0=[E**,E*]Knr
attracts all orbits corresponding to initial data (u0, vo) E r satisfying uo
0 and vo # 0. If E* = E** then E* attracts all orbits as before. If Sl has no
accumulation points in r, then there exists a subset of r whose complement has zero Lebesgue measure consisting of points (u0, vo) for which
lr(uo, vo, t) approaches a rest point in 0 as t-> oo. Both E* and E** have
the property that the stability modulus of the Jacobian of (2.1) at these
points is not positive.
Figure 4.1 describes the theorem schematically.
Competition
145
...........
Eo
Et
Figure 4.1. The attracting hypercube 0 is shaded. Each axis represents a copy
of R". The monotone trajectory emanating from near E, and converging to E*,
and a similar one emanating from near E2 and converging to E**, are described
in the proof of Theorem 4.4.
If (uo, vo) e [E*, EI]K and v0#0 then E * <K lr(uo, vo, t) <K E1 for
t > 0, by strong monotonicity and Lemma 2.3. For all sufficiently small
positive r, 7r(uo, vo,1) <K Xr and monotonicity implies that
limi
146
E. in the direction w. could be constructed exactly as in the first paragraph of the proof. This would clearly contradict the conclusions of the
second paragraph of the proof. Thus s(J.) <_ 0 and a similar argument
shows that s(J..) <_ 0, where J.,, is the Jacobian of (2.1) at E*..
If (uo, v0) E F satisfies E2 <K (uo, vo) <K El, then choose a point Xr on
the ray through El in the direction w such that lr(uu, vo,1) <K xr. Since
ir(xr, t) -> E. as t oo, monotonicity implies that (u, v) <_K E. for every
point (u, v) of the positive limit set of the orbit of (2.1) starting at (uo, vo).
A similar argument establishes E.. <_K (u, v). Therefore, the limit set corresponding to the orbit through (uo, vo) is contained in O.
We complete the proof by showing that the omega limit set A of a point
(u(0), v(0)) E F satisfying u(0) # 0 and v(0) # 0 is contained in O. We first
show that E2:5 K x <_K El for each xE A. By Lemma 2.3, (u(t), v(t)) > 0
for t > 0. The solution (u(t), v(t)) of (2.1) with (u(0), f)(0)) = (u(0), 0)
must satisfy (u(t), v(t)) <K (u(t), 0(t)) = (11(t), 0) for t > 0 by strong
monotonicity, since (u(0), v(0)) <_K (u(0), 0). Similarly we have
for t > 0,
where (u(t), v(t)) is the solution of (2.1) satisfying (u(0), v(0)) _ (0, v(0)).
Letting t - oo in the inequalities (u(t), 0(t)) = (0, 0(t)) <K (u(t), v(t)) <K
(u(t), 0) and observing that (0, v(t))-4E2 and (u(t), 0) -+ El as t --goo, we
see that every limit point of the orbit starting from (u(0), v(0)) must be-
long to Ffl[E2iEl]K.
If x c- A then x = ir(y, l) for some y E A, since A is invariant. This and
Lemma 2.3 imply that if x = (u, v) e A then either u = 0 (v = 0) or u > 0
(v > 0). Furthermore, if u > 0 and v > 0 then E2 <K X <K El. In this case,
as x is a limit point of (u(t), v(t)), it follows that E2 <K (u(to), v(to)) <K El
for some to > 0. By arguments in a preceding paragraph we may then
conclude that A C O. Therefore, this containment holds whenever A contains a point x = (u, v) with u > 0 and v > 0. Suppose that A contains no
such point. Then every point x = (u, v) E A satisfies either u = 0 or v = 0
but not both, since Eo does not belong to A. Furthermore, A contains the
entire orbit through x together with the alpha and omega limit sets of this
orbit. Consider the case where x = (u, 0). The omega limit set of the orbit
through x is clearly El, and the alpha limit set of any entire orbit that
belongs to [0, z] is Eo unless x = El. Since Eo e A, we conclude that El is
the only point of the form (u, 0) in A. A symmetric argument shows that
E2 is the only point of A of the type (0, v). Since A is connected, it follows that A C 0, A = El, or A = E2.
Suppose that A = El. Then u(t) + v(t) --> a as t -> oo. By continuity of
s(-), we can find e > 0 such that
Competition
147
-u)) > 0.
s=
v(t) ?
8ves(`-ra),
t >- to.
This contradiction to the boundedness of v(t) shows that A = El cannot hold. A similar argument shows that A = E2 cannot hold. Therefore,
A C 0 as asserted.
Our assertion - that almost all initial conditions (uo, vo), in the sense
of Lebesgue measure, belong to orbits converging to a rest point in 0 if
the set Sl has no accumulation points - follows from Theorem C.B.
case, when E* * E**, by Theorem E.1 there must exist another positive
rest point belonging to [E**, E*]K for which the corresponding Jacobian
matrix is not stable.
In [STW ], a global bifurcation theorem is used to show the existence of
a connected global branch of ordered pairs (m,,, E*) connecting (m*, E1) to
(m *, E2), where the bifurcation parameter is the maximum growth rate m
of population u and where in;, mu* are critical values of this parameter.
148
The idea in the proof of Theorem 4.4 can be used to prove the following result, which describes some of the possible behavior of solutions of
(2.1) when the two key stability moduli in Theorem 4.4 have different
signs.
sl =
0,
s2 =
v)) # 0.
(c) If sl < 0 and s2 < 0 then there exists a positive rest point E+ such
that s(J+) > 0, where J+ is the Jacobian (4.1) at E+, E2 <K E+ <K El. If
s(J+) > 0 then there exist rest points Ep and Eq satisfying
E2 <_K Eq <K E+ <K E# :f:-:K El,
where equality may hold in either the first or last inequalities. The rest
point Ea attracts all solutions satisfying Eq <K (u(0), v(0)) <K E+ except
E+; Ep attracts all solutions satisfying E+ _<K (u(0), v(0)) <_K Eq except
E+. Both s(Jq) <- 0 and s(JJ) <_ 0, where Jq and Jq are the Jacobians (4.1)
respectively.
at Eq and
Figure 4.2 describes some of the possibilities schematically.
Proof. If sl > 0 then the rest point E*, obtained in the proof of Theorem
4.4, exists and is either positive or coincides with E2. The rest point E.
E0
E1
E,
E0
E1
L
E0
E1
Figure 4.2. a Case (a)(i) of Theorem 4.5, where E2 is the global attractor. b One
of several possible scenarios in case (a)(ii); there could be more rest points. c A
possible scenario in case (c); this case cannot be eliminated, but it is believed to be
unlikely for biologically reasonable uptake functions.
150
attracts all solutions of (2.1) satisfying E. <-K (u(0), v(0)) <-K El and
v(0) * 0, by the argument given in the proof of Theorem 4.4. If E. = E2
then E2 attracts all solutions with v(0) * 0 because every such solution
either converges to E2 or eventually enters and remains in [E2, El I K, in
which case it converges to E2. The argument is similar to the one in the
proof of Theorem 4.4 showing that 0 attracts all solutions. The assertion
s(J*) <- 0 follows as in the proof of Theorem 4.4. This establishes all the
assertions of case (a) except for those concerning the rest point E,, which
follow from Theorem E.I. The proof of case (b) is analogous to that of
case (a).
Consider case (c). The existence of the rest point E+ with s(J+) ? 0 is
proved in Proposition E.2. If s(J+) is positive then the main construction
of Theorem 4.4 can be applied to obtain a monotonically converging solution starting at E++rw, where r > 0 and 0 <K w is the principal eigenvector of J+. The limit of this solution is Ep, which may coincide with El. Another monotonically converging solution starts at E+-rw, where r > 0 and
this solution converges to Eq, which may coincide with E2. The remaining
assertions in this case follow from now standard arguments.
It is important to stress that, in the case (considered in the previous chap-
S2 < S,
=.fvand
151
e3 = 510,249/160. The rest point E. = (u, v) (where u = (u1, u2, u3) and
similarly for v) given by ul = 10, u2 = 11, u3 = 9, vl = v2 = v3 = 22,000 is
unstable. One concludes that unstable positive rest points are possible for
the class of gradostat models considered in this chapter.
set u'= 0 and v'= 0 in (2.1) where zi = 1- i /(n + 1) (see the paragraph
following Lemma 2.2). We obtain
0=(ui_1-2ui+ui+1)+uifu(1-i/(n+1)-ui-vi),
0 = (vi_1-2vi+vi+l)+vifu(1-i/(n+l)-ui-vi),
(5.1)
u0=V0=un+1 =vn+l=0,
where the index i runs from 1 to n. If the vessel i is imagined to be located
152
0=E2 Uxx+ufu(l-x-u-v),
(5.2)
0=
The boundary conditions follow naturally from the conventions uo u, 1 = 0, and similarly for v, which say that there are no microorganisms
in the two reservoirs. They are justified by the agreement between the
numerically computed rest points of (5.1) and the solutions of (5.2) obtained by using singular perturbation theory (as in [S9]).
Consider first the single-population equilibrium for u. Setting v = 0 in
(5.2) leads to the boundary value problem
0 = EZUxx+ufu(1 -X-u)+
(5.3)
U(O) = u(1) = 0.
A positive solution u must be concave on 0 < x < 1, since uxx < 0. The
function
u = 1 - x
(5.4)
u = U(x/e)-x,
(5.5)
0= U"+Ufu(1-U),
(5.6)
U(0) = 0, U(+oo) = 1.
The condition U(O) = 0 ensures that the right side of (5.5) vanishes at
x = 0. The condition U(+oo) = 1 implies that U(x/E) = 1, so that (5.4)
is approximated outside a neighborhood of x = 0 of size order E. The
system (5.6) is derived by passing to the variable X = x/E in (5.3) and
dropping all but the lowest-order terms in E. Figure 5.1 shows the phase
plane associated with (5.6), and Figure 5.2 gives a qualitative sketch of
the approximation (5.5).
153
Figure 5.1. The phase portrait of (5.6). The orbit corresponding to the solution
satisfying U(O) = 0, U(oo) = 1, is the upper orbit asymptotic to the saddle point
(1, 0). (From [S9], Copyright 1991, Journal of Mathematical Biology. Reproduced
by permission.)
Figure 5.2. A qualitative sketch of the approximation (5.5). (From [S9], Copyright 1991, Journal of Mathematical Biology. Reprinted by permission.)
S = 1- U(X/e).
154
with many vessels is that all growth and consumption occur in the first
few vessels.
A coexistence equilibrium corresponds to a solution (u, v) of (5.2), where
u and v are positive in some region and satisfy the boundary conditions.
By the results of the previous section, we expect that such a solution exists
provided each single-population equilibrium is unstable to invasion by its
rival. The stability of these equilibria can be considered (see [S9]), but we
do not require the formalities for our brief treatment here. Both u and v
must be positive and concave on the interval 0 < x < 1. There is a family
of solutions of (5.2) given by
u = r(1-x),
v = (1-r)(1-x)
(5.7)
(where r is a parameter satisfying 0 < r < 1), which satisfies the boundary
conditions at x = 1. Of course, the boundary conditions at x = 0 are not
satisfied, and this suggests that (5.7) is valid outside a small neighborhood of x = 0 for some value of r which must be determined. In order to
determine the solution in the boundary layer near x = 0, we set X = x/e,
u(x) = U(X), and v(x) = V(X) in (5.2), keeping only the lowest-order
terms in E. This gives
0 = U"+Ufu(1-U-V),
(5.8)
0=V"+Vfz,(1-U- V),
where U and V must satisfy
U(0) = 0,
V(0) = 0,
U(+oo) = r,
V(+co) = 1-r.
(5.9)
The first two boundary conditions are necessary in order for u and v to
vanish at x = 0. The second two conditions are required in order for the
solution to match up with (5.7) near x = 0. Of course, we also expect that
(5.10)
155
is negative and there are two zero eigenvalues). The corresponding eigenvector can be chosen to point into the region (5.10). This implies that the
stable manifold of P5 consists of two orbits, one of which lies, at least in
part, in the region (5.10). In [S9], sufficient conditions are given for the
existence of a value of r satisfying 0 < r < 1 for which there is a solution
of (5.8) satisfying (5.9) and (5.10). These sufficient conditions, which will
not be given here because they are of a technical nature and shed no light
on the biology, are at least consistent with the hypothesis that both single-
u = U(x/e) - rx,
v = V(x/e) -(I - r) x,
(5.11)
where (U, V) is the solution described previously. Note that the form of
(5.11) guarantees that (5.7) holds away from an order-e neighborhood of
x = 0, since U(x/e) - r and V(x/e) = 1-r when e is small.
The biological content of this analysis is revealed by considering also
the nutrient, which is given approximately by
S = 1-U(x/e)-V(x/e).
Away from the boundary layer near x = 0 there is a negligible amount
of nutrient, and population densities are dominated by simple diffusion
from the source which, in effect, is contained in the boundary layer. This
is where most growth and consumption of nutrient occurs. The implications for the n-vessel standard gradostat are that if n is large then all of
the interesting biology occurs in the first few vessels. It might reasonably
be concluded that, at least for the standard gradostat, there is little point
in considering gradostats with a large number of vessels.
It is particularly unfortunate that our analysis sheds no light on the
value of r, since (5.11) implies that r determines which population is dominant in the gradostat.
In Figure 5.3 and Figure 5.4 (taken from [S9] ), the numerically computed steady states of (5.1) for a five-vessel gradostat are displayed and
156
1.0
0.6
0.6
0.4
0.2
01
0
Figure 5.3. The five points represent computed values of the steady-state concentrations of a single species with m = 5 and a = 1 in a five-vessel gradostat. The last
three points are very close to the solid line. (From [S9], Copyright 1991, Journal
of Mathematical Biology. Reprinted by permission.)
1.0
0.8
0.6
0.4
0.2
Figure 5.4. a The five points depicted represent computed values of the concentrations of population U in each of the five vessels for the coexistence rest point.
For U, m = 5 and a= 1. The points are close to the line a(l-x), where a =
U5/V5, the ratio of computed values of U and V in vessel 5. b The corresponding concentrations for the competitor V, with m = 22.2 and a = 5.08; compare
with the line (1-a)(1-x). (From [S9], Copyright 1991, Journal of Mathematical
Biology. Reprinted by permission.)
(r, 0, 1-r, 0), with r as before, was computed using PHASE PLANE [Er].
Its projection in the U-V plane passed through the origin to within screen
resolution. The estimated value of r was used in Figure 5.4 to give a qualitative (not computed) picture of the approximation (5.11).
Discussion
157
6. Discussion
In the previous chapter, a fairly complete analysis was given of the twovessel gradostat with Michaelis-Menten uptake functions. In the present
chapter we have seen that much, but not all, of this analysis carries over
to very general gradostats consisting of an arbitrary number of vessels
connected together in a wide variety of ways and using arbitrary monotone uptake functions. The main result of this chapter, as in the previous
one, is the prediction that coexistence of two microbial populations in
the gradostat is possible provided that each population can successfully
invade its rival's single-population equilibrium. Coexistence occurs in the
form of convergence to a positive equilibrium - which may not be unique,
although no example of non-uniqueness is known. By extrapolation, the
model suggests that coexistence is possible in a natural environment supporting a nutrient gradient.
Two key results of the previous chapter - the uniqueness of a positive
rest point and the local asymptotic stability of a positive rest point - are
left unsettled in the present chapter. As already mentioned, the second
result no longer holds in the generality considered in the present chapter.
In [HSo], examples are given in which an unstable positive rest point
exists. While we know of no case where non-uniqueness of the positive
rest point has been shown, the fact that positive rest points may be unstable suggests the possibility that uniqueness too may fail in the generality
considered in this chapter. It is not clear at the time of this writing whether
unstable positive rest points can occur in biologically interesting cases.
The examples of Hofbauer and So are not conclusive in this sense, since
apparently there are no counterexamples to the conjecture that the results
of the previous chapter carry over to the standard n-vessel gradostat with
monotone uptake functions whose graphs intersect in at most one positive value of S.
We know that coexistence of two microbial populations in the gradostat is possible. It is natural to ask how many microbial populations can
coexist in an n-vessel gradostat. In [JST] it is shown that the number is
not greater than n. This is done by first showing that, generically, there
does not exist a coexistence rest point, and second by showing that there
are solutions corresponding to the presence of all populations which converge to rest points on the boundary - that is, to rest points with certain
populations absent. Numerical simulations in [BWu; CB] suggest that
three competitors cannot coexist in a three-vessel gradostat but can in a
158
four-vessel gradostat. Very little has been rigorously established concerning competition between more than two competitors in a gradostat. For
more than two competitors, the equations no longer give rise to a monotone dynamical system.
1. Introduction
The results of Chapters 1 and 2 demonstrate that coexistence cannot occur
160
X2X' S.
?1
X2
Figure I.I. If D = D2, then A, < A2 and population 1 has a competitive advantage
over population 2. If D = D,, then A2 < A, and the reverse holds. If D = D(t)
oscillates periodically between D, and D2, then each population enjoys the competitive advantage during a part of the cycle.
the washout rate. We will vary the washout rate in this chapter. The mathematics, however, is the same regardless of which operating parameter is
varied, and the end results are similar - coexistence is possible.
larger than f2 for S larger than S*; see Figure I.I. Then, in a chemostat with constant values of the operating parameters, competitor x, is
favored when D is larger than D* since A, is smaller than A2, and competitor x2 is favored when D is less than D* since in that case A2 is the
smaller. It seems plausible to expect that if D is allowed to vary with time
in such a way that alternately D is less than D* and D is larger than D*,
then coexistence of the two competitors is possible. As simple as this
reasoning sounds, it turns out to be surprisingly difficult to prove rigorously. The reason is that the corresponding equations are non-autonomous
(i.e. time-dependent), and one can no longer explicitly compute the analogs of the steady states as was done for the autonomous equations of
Chapter 1.
Introduction
161
x,=x,(f(S)-D(t)), i=1,2,
x;(0) > 0, S(0) > 0.
D(t+w) = D(t)
for some positive co. A typical example is
'g(s)
<g> = w-'
ds.
S'=(1-S)D(t)-x1f1(S)-x2f2(S),
x;=x,(f(S)-D(t)), i=1,2.
(1.1)
We have relabeled <D> as 1 in the equations (1.1). The f (S) in (1.1) are
actually (D>-'f (SS), but again we have relabeled. Furthermore, since
162
Equations (1.1) are periodic in the time variable. As this is the first encounter with such equations, it is appropriate to review the basic strategy
for dealing with them. For the purposes of this review, consider the general periodic system
(2.1)
f(t+w, x) = f(t, x)
holds for all (t, x). The w-periodic (hereafter simply "periodic") solutions x(t+w) = x(t) play the same role for (2.1) as rest points do for autonomous systems.
Just as the stability of rest points can be determined by linearization,
the stability of a periodic solution x(t) can often be determined by linearizing (2.1) about x(t). The linearization, or variational equation, corresponding to x(t) is
z
ax(t,x(t))z.
(2.2)
This system is periodic and therefore the Floquet theory described in Section 4, Chapter 3, applies. Let 4i(t) be the fundamental matrix solution
of (2.2). The Floquet multipliers of (2.2) are the eigenvalues of 4'(w); if it
is a Floquet multiplier and p. = e"A then A is called a Floquet exponent.
Only the real part of a Floquet exponent is uniquely defined.
A fundamental difference between (2.2) and (4.6) of Chapter 3 is that 1
163
such that, if I x(to)-y(to)I < S for some to ? 0 and some solution y(t),
then Ix(t) -y(t) I < e for all t >- to; and (ii) there exists b > 0 such that, if
I x(to) -y(to)l < b for some to 0, then lim,.,Ix(t)-y(t)l=0 and this
convergence is uniform in to.
In terms of Floquet exponents, the condition for stability is 92(.x) < 0
for all exponents and the condition for instability is that 92(A) > 0 for
some exponent A. Here, 92(A) denotes the real part of A.
A powerful conceptual tool for the analysis of (2.1) is the Poincare
map. Let x(t, xo) denote the solution of (2.1) satisfying x(0) = xo. The
Poincare map is defined by
P simply advances a point one period along the trajectory through the
point. The basic properties of P follow from standard theorems for ordinary differential equations: P is continuous and one-to-one, and x(t, xo)
is a periodic solution of (2.1) with period w if and only if Pxo = xo. The
orbit of a point xo under P is the set O+(xo) _ [xo, Pxo, P2xo, ...1, where
P" denotes the n-fold composition of P with itself. Note that P'xo =
x(nw, xo), so that the orbit of xo under P is just a sampling of the solution of (2.1) through xo at integral multiples of W. The map P captures all
of the dynamical features of (2.1). For example, if limo _. P"xo =.k then,
P'(xi) =
ax
where 1(t) is the fundamental matrix for (2.2) with x(t) = x(t, x1) (see
[H2]). If Px1 =x, so that x(t,x,) is a periodic solution, then the eigenvalues of the derivative of P at x, are precisely the Floquet multipliers.
The behavior of orbits of P near a fixed point z can be described in the
case where z is a hyperbolic fixed point, that is, when no eigenvalue (multiplier) of the Jacobian of P at z has modulus equal to 1. In this case there
exist (local) stable and unstable manifolds M+(z) and M-(x) (respectively) containing the point X which are tangent to the stable (resp. unstable) subspace of the Jacobian of P at z. (The stable (unstable) subspace
164
is that space spanned by the real parts of the generalized eigenvectors corresponding to eigenvalues with modulus less (greater) than 1.) The stable
and unstable manifolds have the dimensions of the stable and unstable
subspaces, respectively. The manifold M+(x) contains all those points
near z whose orbit under P converges to x, and M-(x) contains all those
points near x whose orbit under P-' converges to k. Points near z not
belonging to the stable manifold have the property that their orbit cannot
remain in a neighborhood of,?.
The (local) unstable manifold M-(x) can be extended to a (global) object, which we will also call the unstable manifold (although its geometry
can be quite complicated; in particular, it need not be a submanifold),
simply by taking the union of the images of M-(x) under P", n = 0, 1,
2, .... Then the (global) unstable manifold can be characterized as (x I
P-"x x as n -> ol. A similar construction gives the (global) stable manifold. Hereafter, we will not distinguish between the local and global
objects.
In concluding this brief review, we remark that the principal motivation for working with the Poincare map as opposed to the solutions of
(2.1), neither of which is explicitly computable in general, is that the Poincare map has its domain in Euclidean n-space whereas solutions of (2.1)
must be viewed in Euclidean (n+1)-space - that is, as (t, x(t)). The advantage of working in a lower-dimensional space is the key. We hope the
utility of the Poincare map will be evident in the remaining sections of
this chapter.
3. The Conservation Principle
E'= -D(t)E.
Therefore, as (D> = 1, we have
E(t) = q(t)e-',
where
165
E(t) = 0.
In other words, independently of initial conditions, solutions of (1.1) asymptotically approach the plane, S+x1+x2 = 1, at an exponential rate.
The asymptotic behavior of (1.1) is therefore determined by the twodimensional system obtained from (1.1) by deleting the equation for S
and replacing S by I-x1-x2i just as in the previous chapters. This yields
xj =x,[f (1-x1-x2)-D(t)],
i=1,2.
(3.2)
=[(x1,x2)el+: x1+x2<-1},
which is positively invariant for (3.2).
In the remainder of this chapter, our attention will be restricted to the
system (3.2). The results of Appendix F can be shown to hold for discrete
dynamical systems (e.g., generated by the Poincare map for the system
of equations for (E, x1, x2)); this makes it possible to deduce the asymptotic behavior of (1.1) from that of (3.2).
We begin by obtaining a sufficient condition for the washout of a competitor from the chemostat which is independent of the presence or absence of an adversary. The following result is "sharp" in the sense that if
it fails to hold then the competitor can survive in the absence of competition in the chemostat.
liminf,_.xi(t)=e>0.
Then, for all large t, one has 1-x1(t) -x2(t) <- 1- (E/2), and for such t it
follows that
166
xi(t+s)<-xi(s)exp -J
[f(1-(e/2))-D(r)]dr
It is easy to see that the numbers (f (1) -1), i = 1, 2, are the Floquet exponents corresponding to the identically zero periodic solution of (3.2).
Consequently, the solution is asymptotically stable when both exponents
are negative and unstable when one of the exponents is positive. Proposition 3.1 says more than this; it states that competitor xi is washed out of
the chemostat if f (1) <- 1, but this outcome has nothing to do with competition since it occurs even in the absence of the other competitor. As
our main interest is in the effects of competition, we assume hereafter that
f(l) > 1, i = 1, 2.
(3.3)
When (3.3) holds for the ith competitor, that competitor can survive in
the chemostat in the absence of competition and with its concentration
oscillating in response to the periodically varying washout rate. This is
the content of the next result.
PROPOSITION 3.2. There exist unique, positive, periodic functions E(t)
and ii(t) such that (E (t), 0) and (0,,1(t)) are solutions of (3.2). If (x1 (t), 0)
is a solution of (3.2) satisfying xr(0) > 0, then
limr--Ix2(t)-)1(t)I = 0.
Proof. The assertion essentially concerns the scalar equation
x'=x[f(1-x)-D(t)],
where we have omitted the subscripts. It is assumed that f (l) > 1. Let
P denote the Poincare map P: [0, 1] -> [0, 1] given by Px0 = x(w, x0).
Then P is continuously differentiable, one-to-one, and satisfies PO = 0
167
and P1 < 1. The derivative P'(xo) = (ax/axo)(w, xo) = v(w), where v(t)
is the solution of
E1(t) = (fi(t), 0)
and E2(t)=(O,-q(t))-
168
xz
as
f2(1-E) -D
(3.4)
where we have suppressed the t dependence of the coefficients. A computation yields the fundamental matrix 1(t):
(t)exp[foau(s)ds]
0
u(t)
exp[ foa22(s)ds]/'
where (a1) denotes the coefficients in (3.4) and where u(t) is given by
fot
u(t) =
Note for future reference that u(t) < 0 for t > 0 since a12(r) < 0. Evaluating F at t = w, we obtain the multipliers exp[ f o ail(s) ds], i = 1, 2. It follows immediately that all and A12 are Floquet exponents. The remaining
assertions follow from the discussion of Section 2.
169
Denote by A21 and A22 the Floquet exponents of the periodic solution E2:
A22 = -<n.f2(1-fi)> and A21 = <.f1(1-_7))-l.
The solution E2 is asymptotically stable if A21 < 0 and unstable if A21 > 0.
Further analysis of the system (3.2) requires understanding the dynam-
ics generated by the Poincare map in the interior of Q. For general periodic systems, such an understanding is beyond current knowledge. Fortunately, system (3.2) - in addition to being two-dimensional - has the
property that it is competitive. A beautiful theory for such systems has
recently been constructed. The next section is devoted to the principal
result of this theory.
4. Periodic Competitive Planar Systems
The study of mathematical models of competition has led to the discovery
of some very beautiful mathematics. This mathematics, often referred to
as monotone dynamical systems theory, was largely developed by M. W.
(4.1)
where f(t+w,x1ix2)=f(t,x1ix2)and
aj;
(t, x1, x2) < 0,
ax
i * j.
(4.2)
The requirement (4.2) means that (4.1) is a competitive system - an increase in x2 has a negative effect on the growth rate of x1 and vice versa.
The system is said to be a cooperative system if the reverse inequalities
hold in (4.2). Our interest here will be in the competitive case since (3.2)
satisfies (4.2) in 12, as is easily checked. However, the cooperative case is
170
The key facts about competitive and cooperative planar systems are
summarized next in terms of the Poincare map P for (4.1).
LEMMA 4.1. The following inequalities hold for (4.1) satisfying (4.2):
xf = -fl(t,x1,x2),
x2 = -f2(t,x1,x2).
This system is cooperative and so, as u(0) <- v(0), we conclude from Corollary B.2 that u(w) <- v(w). This proves (ii).
lim1.,0Ix(t)-p(t)I = 0.
Proof. Given two points x, y e R2, one or more of the four relations
X:5 y, Y:5 x, X :5K y, or Y <K X must hold. Now, if P"OXO <_K
P"0+lxo
(or the reverse inequality) holds for some no >: 0, then Lemma 4.1 implies that P"xo <-K P"+l xo (or the reverse inequality) holds for all n > no.
Therefore, (P"xo)" converges to some z, since the sequence is monotone
and bounded. The proof is complete in this case, so we assume that there
Coexistence
171
is not a fixed point of P. Then it follows that for each n we must have
either that P"+lxo <- P"xo or that the reverse inequality holds. Suppose
for definiteness that xo -< Pxo, the other case being similar. We claim that
P"xo <- P"+Ixo for all n. If not, there exists no such that
p"oxo
xo<pxo<p2xo<...
<p"o-Ixo<-
Pno+ixo.
We remark that if (4.1) is autonomous and is either competitive or cooperative then we are free to choose w and a corresponding Poincare map
P, Theorem 4.2 implies that every bounded solution of (4.1) is asymptotic to an w-periodic solution. Since w is arbitrary, it follows that every
bounded solution converges to a rest point.
As an immediate corollary of Theorem 4.2, we have the following result for (3.2).
COROLLARY 4.3. Let x(t, xo) be a solution of (3.2) corresponding to xo E
SI. Then there exists a periodic solution p(t) of (3.2) such that
172
above some fixed positive concentration for all future time, then the main
result of the previous section shows that the corresponding solution of
(3.2) must approach a positive periodic solution.
This fact can be exploited to give simple necessary conditions for coexistence (or sufficient conditions for competitive exclusion). Suppose that
x(t) is a positive periodic solution of (3.2) and let S(t) = 1-x1 (t) _X2(0It is apparent that 0 < S(t) < 1. From (3.2) we have
E0=(0,0),
El = E1(0) = (E(0), 0),
Coexistence
173
our main result here, the following definitions are used. For vectors x
and y in (182, X <K y means that xl < yl and x2 > y2. Recall from Section 3
that x <K y means that the corresponding weak inequalities hold - for
example, the inequalities, E2 <K El and E2 <K E0 <K El hold.
One of the main results of this section is the following.
THEOREM 5.1. If A12 > 0 then there exists a fixed point E* of P, possibly
coinciding with E2, with the following properties:
(i) E2<_KE*<KEl.
(ii) For all x0 e 12 satisfying E. <K x0 <K El,
(5.1)
(iv) The Floquet multipliers of E** have property (iv) of Theorem 5.1.
174
E2 4
E0
E1
E0
E1
E.=E2
b
Figure 5.1. The shaded areas denote W+(E), the region of attraction of E..
a E. is positive and represents a positive periodic solution (coexistence). b E. _
E2 which therefore attracts all positive initial data.
Obviously, the cases where A12 > 0 and E. = E2, or where A21 > 0 and
5.1(iii) implies that both E, and E.. are positive. In fact, in this case
every solution starting with positive initial values converges to a positive
periodic solution by the following result, which is the main result of this
chapter. Its proof is given following the proof of Theorem 5.1.
COROLLARY 5.2. Suppose that both A12 and A21 are positive. Then there
Coexistence
175
(v) the multipliers of both E** and E. satisfy Theorem 5.1(iv); and
(vi) if E** = E* then limn
P"x0 = E. for all positive x0 e U.
Corollary 5.2 can be paraphrased as follows. Provided both A12 and A21
are positive, so that both E1 and E2 are unstable, the orbit of every positive point of ) is attracted to a positive fixed point of P belonging to the
"box"
B= Ix eQ:E** <Kx<_KE*).
Equivalently, every solution of (3.2) corresponding to positive initial data
is asymptotic to a positive periodic solution. The two populations coexist
in the chemostat. Figure 5.2 illustrates the result.
It is important to point out that the box B is a fixed positive distance
from the two coordinate axes. Because every orbit of P corresponding to
a positive starting point approaches B, it follows that there exists a positive number S, independent of the positive initial value, such that each
coordinate of the corresponding solution of (3.2) exceeds S for all sufficiently large t. This fixed distance 6 provides a cushion against the extinction of either population. Later, we will say that the system (3.2) is uniformly persistent when this situation holds. (See Appendix D.)
Another important observation concerning Corollary 5.2 is that if the
hypotheses are satisfied for a particular set of growth functions f,1 and
parameters, then the hypotheses continue to hold if these functions and
parameters are perturbed by a small amount. Such a property is clearly
important, because parameters and functions are never known precisely.
The stability of the conditions to perturbations follows from the wellknown continuity of simple eigenvalues of matrices to changes in their
entries.
It must be stressed that the hypotheses of Corollary 5.2 give sufficient,
but not necessary, conditions for the existence of a positive periodic solution possessing strong stability properties. Furthermore, since the singlepopulation periodic solutions E1(t) and E2(t) are not explicitly computable, as the corresponding rest points were in Chapter 1, it does not seem
possible to obtain explicit formulas for A12 and A21. However, these crucial
176
Eo
El
E2
E* =E**
Eo
E1
Figure 5.2. a The general case E. # E**; all positive starting points iterate to a
fixed point in the box B. Points in the uppermost box are attracted to E**, while
those in the lowermost box are attracted to E*. b The case E* = E**. Here, E*
attracts all positive starting points.
estimated.
In our numerical simulations of (3.2) using PHASE PLANE [Er], we
chose Monod-type functional responses f (S) = m,S/(a,+S) and a sinus-
Coexistence
177
.0.600
20.000
0.000
P'(xo) =
a b
c d
satisfies:
(i) a>0,d>0,c<0,b<0;
(ii) P'(xo) has eigenvalues , satisfying 0 < t < A2;
(iii) there exists an eigenvector u2 corresponding to 2 and satisfying
0 <K u2;
Proof. Assertion (i) follows from Theorem B.6 since, for (3.2), strict
inequality holds in (4.2). (ii) is a consequence of Theorem A.6, or can
easily be established by observing that ad - be > 0 by (4.3) of Chapter
3. (iii) follows from Theorem A.6, and (iv) is a consequence of Theorem B.6.
178
1' '(EI)
= (W) =
e'A12
u (W)
(see the discussion following (3.4)), where I(t) is the fundamental matrix of (3.4). Clearly, (1, 0)T is an eigenvector for P'(E1) corresponding to
the eigenvalue e' All < 1. An eigenvector corresponding to the eigenvalue
eW\12 > 1 is easily computed to be
El for 0 < E:5 Eo. by Lemma 5.3(iv) we conclude that E2 <K P"+'w(E) <K
P"w(E) <K E1 for n = 1, 2, .... Obviously, P"w(E) -* EE as n -> co, where
EE is a fixed point of P satisfying E2 5K EE <K El.
If 0 < E < E'then E2 <K WW) <K W(E) <K EI, so E2 <-K EE. 5K EE <K El.
does not hold. Applying P to the former inequality and using Lemma 4.1
gives EE, <-K Pw(E2) <K w(E2), contradicting that EEI <K w(E2) does not
hold and so proving the assertion. Therefore EE, <K w(E0) and, since E1 was
arbitrary, EE <K w(E0) for all E, 0 < E:5 E0. Consequently, E. <K w(EO).
Combining inequalities, for 0 < E :s EO we have EE0 <-K EE <-K E* 5K W(60);
hence, by Lemma 4.1, E,, !5K EE <--K E* <K P"w(E0). Letting n go to infin-
ity, we find that E,0 :5K EE <K E, :!5K EEo. This proves that EE = E. for
0 < E:5 EO, and (i) follows.
If x0 e SI and E. <K x0 <K El, then xO <K w(E) for all small E. Therefore
E. <K P"x0 <K P"w(E) and, as P"w(E) -> E*, (ii) follows immediately.
Coexistence
179
If E. = E2 then P"xo -> E2 for all xo with E2 <-K xo <K EI. This is obviously incompatible with a positive value of A21 since an argument parallel to that just given for El shows, when applied to E2, that (5.1) cannot
hold for an xo satisfying E2 <K xo <K E1. Indeed, the stable manifold of
E2 is contained in the x2 axis if A21 is positive. This proves that A21 <- 0 if
E. = E2.
Consider the first assertion of (iii). If xo a 1 is positive then we can find
w = (0, w2) and v = (v1, 0) in t such that w <K xo <K v. Applying Lemma
4.1, we have P"w <-K P"xo <-K P"v. As P"w -> E2 and P"v -> EI, and since
P"xo -p z and Px = z, by Theorem 4.2 we have E2:5 K x <-K E. Since A12
is positive, the stable manifold of E1 belongs to the x1 axis and so z # E1.
Therefore, by the fundamental theorem of calculus,
1
E1-z=PE1-P2=f P'(tE1+(1-t)z)(EI-z)dt.
0
exists a positive fixed point E*. of P satisfying E2 <K E** <K EI. By
(ii) of the theorem, we can find xo e f satisfying E** <K xo <K E1 sufficiently near E1 such that (5.1) holds. It follows by monotonicity of P that
<K E. Therefore, (i) of the corollary holds. Assertions (ii) and (iii)
180
are immediate from (ii) of the theorem. Assertion (v) of the corollary
follows from (iv) of the theorem.
It remains to show that (iv) holds, since (vi) follows from (iv). Let
xo e I be positive. Arguing as in the proof of the theorem, it can be
shown that lim" -_ P"xo = x, where X is a fixed point of P satisfying
E2 <_K
<_K El. Since the stable manifold of E, is the x, axis, X # E,. Sim-
ilarly, . # E2 and X * E0. Consequently, as in the proof of the theorem, E2 <K X <K E, and X is positive. Choose x,, i = 1, 2, such that x1 <_K
X 5K X2 and E2 <K X1 <K E** <K E* <K X2 <K E,. By monotonicity of P,
P"x1 <_K z <_K P"x2 and, letting n oo, we find that E** sK 9:5K E*. This
completes the proof.
6. Discussion
It was observed in Section 1 that coexistence of two competitors competing for a single growth-limiting nutrient in a chemostat with a periodically
varying washout rate should be possible, provided that the graphs of the
functional responses of the two competitors intersect for some positive
value of S and that the washout rate varies in such a way that alternately
one, then the other, competitor has the advantage. In Section 5 it was
shown that coexistence of the two competitors is only possible if there
exists a positive periodic solution of (3.2). This allowed us to show rigorously that the intuition described previously is, in fact, necessary for
coexistence to occur.
The very beautiful theory of the dynamics generated by planar, competitive maps, developed in [DS], was used to show that every solution of
(3.2) asymptotically approaches a periodic solution. Building on this result, the main result of this chapter, Corollary 5.2, gives a sufficient condition for the coexistence of two competitors (equivalently: for the existence of a positive periodic solution of (3.2)). This condition is that each
ponent, and one only needs to know the sign of each of these - if both
are positive, then coexistence is assured.
Discussion
181
fore, coexistence holds for these parameter values. Furthermore, as remarked in that section, coexistence continues to hold for all sufficiently
nearby values of the parameters.
Very little information exists on the robustness of the parameter region
for which coexistence occurs. Studying the case where the nutrient concen-
tration, rather than the washout rate, is varied periodically, Hsu [Hsu2J
obtains very interesting information about the parameter region corresponding to coexistence. Perturbation methods are used in [S1] to explore
this region in the case studied by Hsu. See also [SFA] for other numerical
work in both cases.
It is worth noting that positive periodic solutions of (3.2), though possibly unstable, can be shown to exist by using the bifurcation techniques
described in Chapter 3, Section 6. We have chosen not to pursue this route
here because we are unable to carry out the necessary calculations to determine the stability of the bifurcating periodic solutions. See [BHW2;
Cul], where this approach is used.
More is known about the dynamics generated by P than has been discussed in this chapter. In [DS] and [HaS] it is shown that there is a curve
C joining E1 and E2 which is the graph of a strictly decreasing continuous
function. This curve C forms the boundary of the unstable manifold of
E0 and every fixed point, except E0 must lie on C. Therefore, every orbit
of P except E0 is attracted to a fixed point on C. If each fixed point of P is
assumed to be hyperbolic, then there are finitely many fixed points. Moving along the curve C, the fixed points alternate between saddle points
and attractors. In particular, if the hypotheses of Corollary 5.2 hold then
there are an odd number of positive fixed points on C, at least one of
which is an attractor. See [S5] for more details.
Variable-Yield Models
1. Introduction
In the classical model of the chemostat, discussed in Chapter 1, it is assumed (following Monod [Mol; Mo2]) that the nutrient uptake rate is
proportional to the reproductive rate. The constant of proportionality,
which converts units of nutrient to units of organism, is called the yield
constant. As a consequence of the assumed constant value of the yield,
the classical model is sometimes referred to as the "constant-yield" model.
In phytoplankton ecology, it has long been known that the yield is not
constant and that it can vary depending on the growth rate [DI]. This led
to the formulation of the variable-yield model, also called the variableinternal-stores model [Gl] and the Caperon-Droop model [CNI]. This
model effectively decouples specific growth rate from external nutrient
concentration by introducing an intracellular store of nutrient. The specific growth rate is hypothesized to depend on a quantity, called the cell
quota, which may be viewed as the average amount of stored nutrient in
each cell of the particular organism in the chemostat. The cell quota increases with nutrient uptake and decreases with cell division, which acts
to spread the total stored nutrient over more cells. The uptake rate is
assumed to depend on the ambient nutrient concentration and, perhaps,
the cell quota. In fact, it is reasonable to assume that when the latter
is at a high value then uptake will be at a lower level, for a given nutrient concentration, than would be the case if the quota were at a low
level.
183
184
The principle simply states that if everything is expressed in nutrient equivalents then the sum of the variables should behave as a chemostat without
E'= (S(0)-E)D.
In previous chapters we have usually made use of the consequence of
this - namely, that
E(t) = S(t)+x(t)Q(t)
so
E'= S'+x'Q+xQ'
= S (O)D - SD - xp (S, Q)+x(Q)Q-xQD+xQ'
= (S(0)-S-xQ)D.
From the last equality it follows that
x(t)[Q'(t)-P(S(t), Q(t))+(Q(t))Q(t)l = 0.
Assuming that x(t) remains positive, one has the following equation for
the cell quota:
(2.1)
some comments are in order. First of all, this division means that for
x = 0 there is no biologically meaningful equation for Q. However, the
equation for Q makes mathematical sense even if x is zero. In the mathematical analysis, the case where x = 0 and S = 5(O) corresponds to a biological steady state of the system, with no organism and with free nutrient at the input level. To mathematically achieve a rest point for (2.1),
there must also be a value of Q that satisfies p(S 0), Q)-(Q)Q = 0. Although this may seem strange, and biologically unmeaningful, the result-
185
ing linear analysis is benign (and valid). (The reader may be more familiar
with this anomaly as it occurs in the analysis of linear oscillators in polar
coordinate form, when the equation for the polar angle makes mathematical sense even if the polar radius is zero.)
The functions (Q) and p(S, Q) are, respectively, the per-capita growth
rate and the per-capita uptake rate. To motivate appropriate hypotheses
for these functions, we consider some examples from the literature. The
following form for the growth rate is attributed to Droop [D1; D2; CNI;
CN2]:
P = Amax
(Q - Qmin)+
K+(Q-Qmin)+'
where Qmin is the minimum cell quota necessary to allow any cell division.
The term (Q-Qmin)+ is the positive part of Q-Qmin and therefore vanishes when the quantity is negative. Motivated by this example, we assume that is defined, continuous, and nondecreasing, and that there
exists P > 0 such that:
(Q) ? 0,
(2.2)
(P) = 0.
The growth rate increases with cell quota. The following form for the
uptake rate appears in [G2], where Q has the range Qmin :5 Q:5 Qmax:
P(S, Q) = Pmax(Q)
Pmax (Q)
K+S
Q - Qmin
- Pmax - (Pmax - Pmax) Qmax
-Qmin
high
low
In other words, p has the Monod form in S but the saturation value of
the Monod function, Pmax, decreases with cell quota Q. Cunningham and
Nisbet [CN1; CN2] take Pmax to be constant. Therefore, we assume that
p is continuously differentiable in (S, Q) for S ? 0 and Q >- P and satisfies
(2.3)
ap <0.
aQ
186
In particular, p(S, Q) > 0 when S > 0. Equation (2.2) requires that the
uptake rate vanish in the absence of nutrient, increase with increasing
nutrient, and decrease as the cell quota increases.
Observe that (2.2) and (2.3) imply that Q'? 0 if Q = P, and that therefore the interval of Q values, [P, oo), is positively invariant under the
dynamics of (2.1).
It will be convenient to scale the variables appearing in (2.1) as follows:
Dt,
S/5(0),
Q/Q*,
x = xQ*/S(0 .
(Q) = D-1(QQ),
P(S, Q) = (DQ*) 'p(S5, Q*Q),
then (2.1) becomes
x'=x((Q)-1),
Q' = p(S, Q) - (Q)Q,
(2.4)
S'=I-S-xp(S, Q),
where for convenience we have omitted the bars over the variables.
To determine the equilibrium points of (2.4), we use the following con-
Eo=(x,Q,S)=(0,Q,1)
and it always exists. Here Q0 is the unique solution of p(1, Q) -Q(Q) _
0. The other possible equilibrium, labeled El, corresponds to the presence
of the population:
El =(x,Q, S),
where
(Q) =
p(S, Q) = Q,
z=(1-S)/Q.
187
(2.5a)
(2.5b)
T=S+Qx,
where T consists of unbound free nutrient plus stored nutrient. An easy
computation shows that T satisfies
T'=1-T.
Therefore, all solutions of (2.4) asymptotically approach the surface
S+ Qx = 1
(2.6)
x'=x((Q)-1),
Q'=p(1-Qx,Q)-QA(Q)
(2.7)
in
L =[(x,Q)e R+
where L is positively invariant for (2.7).
The equilibria for (2.7) are obtained from those of (2.4) by deleting the
S coordinate and replacing S by 1- Qx. To conserve notation we use the
same letters, Eo and El, to denote the equilibria of (2.7). This should not
result in confusion so long as the equation is clear from the context.
A straightforward calculation shows that if E, exists as an equilibrium
of (2.7), then it is locally asymptotically stable. In this case, Eo is a saddle
point. The first result of this section describes the asymptotic behavior
of (2.7).
THEOREM 2.1. If El does not exist, then every solution of (2.7) satisfies
Variable-Yield Models
188
ax +
aQ =
(Q)-I+aQ
-x as
-(Q)-Q'
-Q <0,
there can be no periodic orbits in L.
Using the results from Appendix F, we obtain the following result for
(2.4).
THEOREM 2.2. If E _ (0, Q0, S) is the only steady state of (2.4) and
(Qo) # 1, then EO attracts all solutions of (2.4). If EO and E, = (X, Q, S)
exist as steady states, then E, attracts all solutions of (2.4) for which
x(0) > 0.
As usual, the proof of Theorem 2.2 involves consideration of the equations
x'=x((Q)-1),
Q'= P(l -Z-Qx, Q) -(Q)Q,
Z'= -Z,
where Z = I - T. The condition (Qo) # 1 ensures that EO is hyperbolic.
3. The Competition Model
Consider two populations, with densities x, and x2, competing for a single
nutrient of concentration S in the chemostat. Competition occurs in the
sense that each population consumes nutrient and so makes it unavailable for the competition. The average amount of stored nutrient per individual of population x, is denoted by Q,, and for population x2 by Q2.
Following the derivation of Section 2, we have the following equations:
189
x(=x1(1Ai(Qi)-D),
Qi = PI (S, Qj)-l(Q])QI,
xi = x2(P2(Q2) -D),
(3.1)
The functions i(Q,) and pi(S, Q,) are, respectively, the per-capita growth
rate and the per-capita uptake rate of population xi. We assume that gi
is defined and continuously differentiable for Qi ? Pi, where Pi >- 0, and
satisfies:
i(Qi) >_ 0,
r(Qi) > 0,
(3.2)
i(1')=0.
We assume that pi is continuously differentiable in (S, Qi) for S ? 0 and
Qi >- Pi, and that pi satisfies
Pi(0, Q) = 0,
api
as
>0
(3.3)
api
aQi
t = Dt,
S = S/S,
Qi = Qi/Q*,
Xi = XiQ*/S.
variable-Yield Models
190
xi = xl(l(Q1) -1),
(3.4)
where we have omitted the bars over the variables for convenience and
because, hereafter, we treat only (3.4). The hypotheses (3.2) and (3.3) are
carried over without a change in notation. In particular, the Q, range
over the interval Q, ? Pi.
Generically, (3.4) has at most three steady-state solutions. One of these,
which we label E0, corresponds to the absence of both competitors. It is
given by
Eo=(x1,Q1,x2,Q2'S)=(0,Q, 0,Qi,1)
and it always exists. Here, Q is the unique solution of
Pi(1, Q,) -QilAi(Q) = 0.
The two other possible steady states, labeled El and E2, correspond to the
presence of one population and the absence of the other. For example,
E1 = (Xl, QI, 0, 102, S),
where
A1(Q1) = 1,
xl = (I-S)/Q1,
P2(S, 02) -Q22(Q2) = 0
Examination of (3.5) reveals that El exists - in the sense that all components are nonnegative, Q, >_ Pi, and xl is positive - if and only if
191
(3.6a)
(3.6b)
where
2(Q2) = 1,
P2(9,02) = D02,
(3.7)
X2 = (I -.S)/Q2,
P1 (S' Qi)-Qi1A1 (Q1) = 0.
(3.8a)
(3.8b)
It is possible, but highly unlikely, that there exist steady states with
both xl and x2 present. This can happen if and only if both (3.6) and (3.8)
are satisfied and
9= S,
(3.9)
where S and S are as defined in (3.5) and (3.7), respectively. In this case,
there is a line segment of steady states of (3.4) joining El to E2. Since (3.9)
is highly unlikely, this case will be ignored. That is, we assume that S' * S
when both are defined.
It will be assumed hereafter that if both (3.6) and (3.8) hold then
S < S.
(3.10)
192
Variable-Yield Models
T=S+Q1x1+Q2x2,
where T consists of unbound free nutrient plus stored nutrient and satisfies
T'= 1-T.
(4.1)
S+Q1x1+Q2x2 = 1
(4.2)
as t - oo; that is, T(t) -+ 1 as t -> oo. Consequently, as a first step in the
analysis of (3.4), we consider its restriction to the exponentially attracting
invariant subset given by (4.2). Dropping S from (3.4), we obtain the
system
Q1=PI(1-Q1 x1-Q2x2,Q1)-l(QI)QI,
(4.3)
SI=f(x1,Q1,x2,Q2)el+:Q1x1+Q2x2:51,Q;?P1
It is immediate from the form of (4.3) that Sl is a positively invariant set.
We will refer to (4.3) as the "reduced system."
The equilibria of (4.3) are obtained from those of (3.4) by deleting the
S equation and using (4.2) to replace S. In order to conserve notation, we
retain the labels E0, El, E2 for the equilibria of (4.3). For convenience,
and since we require some of the relations below, we restate the equilibrium conditions here. The steady state Eo is given by
Eo = (0, Q, 0, Q2),
where the Q are uniquely determined by pi (I, Qo) = Qo (Qo) The steady
state El is given by
El = (xl, Ql, 0, Q2),
193
provided that 1(Q1) =1 has a solution Ql > 0 and p, (1, Q1) > Q. In this
case,
1(Q1) = 1,
provided that 2(Q2) = 1 has a solution 02 and pl (1, Q2) > Q. In this case,
2(Q2) = 1,
=S.
If (3.10) holds, then E0, El, and E2 are the only possible steady states of
(4.3).
The local stability of E0 is determined by Jo = [a,1], the Jacobian matrix
of (4.3) at E0. The nonzero entries of Jo are
a11=1(Q1)-1,
a21=-Qo
ap,
a22 = -Q101A1(Q10)-1A1(Q10)+
0
a33 = 2(Q2) - 1,
aPl
as
o aPl
a23 = -Q2
aQl
aS
aP2
oaP2
a41 = -Ql as , a43 = -Q2o as
and
aaP2
The arguments of the partial derivatives of pi are (1, Q). It follows that
the eigenvalues of Jo are its diagonal entries and that the two eigenvalues 20(Q9) -1 (i = 1, 2) determine the stability of E0, since the other two
eigenvalues are negative.
PROPOSITION 4.1. E0 is locally asymptotically stable if both ,(Q) < 1,
i = 1, 2, and unstable if ; (Q) > 1 for some i. Furthermore, i (Q) > 1 if
and only if E, exists.
variable-Yield Models
194
Proof. The first assertion has already been noted. If 1(Q0) > 1 then, by
our assumptions about ,, Q1 exists such that ,(Q,) = 1 and Q1 < Q.
Therefore p1(1, Qo) = Q01(Q0) > Qo > Ql This implies that E, exists.
Conversely, if E, exists then p1(1, Q,) > Q1 = Ql,(Q,), so
Q101(Q10)-P1(l, Q) = 0> 01A101) -P10, Q1)
C12 =x1Ii1(G1),
c22
ap,
c21 = -Q1
as -Q11(Q1)+ aQ1
aP2
C41=-(21 as, ,
aP2
C43 = -Q2 as
aPt
aP1
3P1
C44 =
as
aP2
as
an22 -2(Q2)-Q2/A2(Q2),
195
=0
= Q2N (Q2)-P2(S, 02)
[A'1 aPl
as
ap,
as
aP2
aP2
(4.4)
variable-Yield Models
196
of this mathematically technical section. Essentially, we claim that competitive exclusion holds as expected; the winner is the organism that can
grow at the lowest nutrient concentration.
THEOREM 5.1.
(i) If EO is the only steady state, then all solutions tend to EO as t - oo.
(ii) If EO and El are the only steady states, then all solutions with x1(0) >
0 approach El as t -> oo.
(iii) If EO and E2 are the only steady states, then all solutions with x2(0) >
0 approach E2 as t -> oo.
(iv) If Eo, E, and E2 exist and (3.10) holds, then all solutions with x2(0) >
0 approach E2 as t -+ oo.
Case (iv) is the interesting one, since both organisms can survive in the
absence of competition in the chemostat. Recall that (3.10) is simply our
convention of labeling as x2 the organism that can grow at the lowest
nutrient concentration.
The proof will be divided into various cases and presented as separate
propositions. The key to the proof is the use of new variables defined by
x, =x1,
U1 =x1 Q1,
x2 = x2,
U2 = x2 Q2.
(5.1)
In the new variables (x1, U1, X2, U2), system (4.3) takes the form
x1=x,(l(U1/x,)-1),
U(=P1(1-U1-U2, U1/x1)x1-U1;
(5.2)
xz = x2(2(U2/x2) -1),
UU
=P2(1-U1-U2, U2/x2)x2-U2.
={(x1,U1,x2,U2)cR 1xi>0,U1+U2-1),
which is positively invariant for (5.2). In fact, notice that (U1 + U2)' = -1
when U, + U2 = 1, so this hyperplane repells in A.
Although (5.2) appears to be singular at xi = 0, it is not hard to see that
the functions i(U,/xi)xi and pi(1 - U1- U2, Ui/xi)xi are locally Lipschitz,
vanishing at xi = Ui = 0, in a wedge shaped region 0 < c < Ui/xi < C of
197
the origin in the Ui-xi plane. If Pi > 0, so that pi and pi are defined for
Q, > Pi, then the lower bound c may be chosen as Pi. Therefore, we can
view
E0=(0,0,0,0),
El = (X1, (J1, 0, 0),
E2=(0,0,X2,02),
as steady states of (5.2), where U] = xl Q] and U2 = Q2 X2 - provided, of
course, that E0, E1i E2 exist for (4.3).
The principal reason that (5.2) is a useful way of viewing (4.3) is that
(5.2) generates a strongly monotone dynamical system in A. Observe that
for fixed U1 (U2), the (x1, U1) subsystem ((x2i U2) subsystem) is cooperative, while the two subsystems compete in the sense that an increase in
U2 (U1) has a negative effect on Ui (U2). By Theorem C.1 of Appendix C,
(5.2) preserves the partial ordering defined by
(XI, U1, X2, U2) :5K (xl, UI, X2, U2)
if and only if x1 <- Xl, U1 <- U1, X2 > X2, and U2 ? U2. By this we mean
that two solutions with initial data so related remain related in the future.
Furthermore, as the variational matrix of (5.2) is irreducible in 0, Theo-
rem C.I implies that if the initial data are distinct and ordered as shown
then the strong order relation, denoted by
(XI, U1, X2, U2) <K (-ti, Ul, X2, U2),
Xi=Xi(lji(U1/xi)-1),
(5.3)
(5.4)
Ui = Pi (I -U,, Ui/21)X1-U,
Variable-Yield Models
198
and
for t ? 0, i = 1, 2.
(5.5)
(0,0)
lim(zi(t), U; (t))
1-o
Ui)
(.X2, U2)
(5.6)
In any case, (5.5) and (5.6) imply the boundedness of solutions of (5.2)
and hence the boundedness of solutions of (4.3). Furthermore, (5.5) and
(5.6) essentially imply the first assertion of Theorem 5.1.
The second and third assertions of Theorem 5.1 are symmetric, so it is
sufficient to prove only the second one. It will often be necessary to use
our knowledge of the behavior of solutions of (5.2) in order to draw conclusions about the corresponding solution of (4.3). In particular, it will
be necessary to use knowledge of the behavior of xi (t) and U; (t) to determine the behavior of Qi(t) = Ui(t)/xi(t). From (5.2), we find that Qi(t)
satisfies
Qi =Pi(1-U1(t)-U2(t), Qi)-Y"i(Qi)Q,,
(5.7)
where we have selectively introduced the argument tin order to make the
point that we may view this equation as a non-autonomous equation for
Q;, particularly when we know the limiting behavior of the Ui. The next
lemma addresses this issue.
LEMMA 5.1. Let (XI (0, U1 (0, x2(t), U2(t)) be a solution of (5.2) satisfying xi(0) > 0 and Ui(0) > 0 for i = 1, 2. Then there exist constants c, C
such that 0 < c < C and
Proof. The upper bound for Qi is obvious from (5.7), which implies that
Qi < 0 whenever Q, is large, so it suffices to show that lim inft - ,. Qi (t) >
0. If not, then there exists t -> co such that Qi (tn) -> 0 and Q, (tn) <_ 0. Now
0 < U1(t)+U2(t) < 1 fort > 0, and we may assume that U,(tn)+U2(tn) ->
c e [ 0,1 ]. Furthermore, c < 1 since (as noted previously) the line U, + U2 = 1
199
=P1(1-c,0):5 0,
contradicting that p1(1- c, 0) > 0 since 1-c> 0.
The second assertion of the lemma follows easily because the right side
of (5.7) is strictly decreasing in the variable Q,. If the limit superior of
Q,(t) as t oo differed from the limit inferior, then we could find two
sequences of times t and s tending to infinity along which Q;(t) approaches distinct limits and along which (Q1)'(t) vanishes. Taking the
limit as n -> oo of (5.7) along each sequence produces a contradiction to
the monotonicity mentioned previously.
The importance of the first assertion of Lemma 5.1 lies in the local Lipschitz continuity of (5.2) in the domain ((x1, U1, X2, U2) e OR+: U1 + U2 <_ 1,
x1 <- z1, x2 <- x2, and either (x1, U,) = 0 or c < x;/ U; < C}. This domain
contains the positive limit set of any solution satisfying the hypotheses of
Lemma 5.1. Therefore, this limit set is invariant for (5.2).
The next result establishes part (ii) of the theorem.
PROPOSITION 5.2. If Eo and E1 are the only steady states, then all solutions of (4.3) with x1(0) > 0 approach E1 as t-* oo.
Proof. If x2(0) = 0 then the result follows from Theorem 2.1. Therefore,
assume that x2(0) > 0. By Proposition 4.2, E1 is locally asymptotically
stable. Conditions (5.5) and (5.6) together imply that (x2(t), U2(t)) -* 0
as t oo and
lim s u p ,
X,
<- X1,
Suppose that x1(t) does not converge to 0 as t -*oo. Then the omega
limit set of the solution x(t) = (x1( t), Q1 M, x2(t), Q2(t)) contains a point
(X1, Q1, 0, Q2) with X1 > 0. As this point belongs (by Theorem 2.1) to the
domain of attraction of E1, and since E1 is locally asymptotically stable
(by Proposition 4.2), it follows that x(t) -* E1 as t oo and we are done.
If x1(t) -> 0 as t oo then, by Lemma 5.1, U, (t) 0 and Q, (t) - Q as
t -> oo. Now Q > 01 (see proof of Proposition 4.1) and so, taking Q with
Q1 < Q < Q1, we conclude that Q1(t) >- Q for all t 2t to, for some to > 0.
Therefore
xf? x1(N-1(Q1)-1)
Variable-Yield Models
200
x(r) =El+rx+o(r)
as
We will follow the orbit of (4.3) through x(r), for small r > 0, by considering the orbit of (5.2) through y(r). Let F = (FI, F2, F3, F4) denote
the right side of (5.2). Then straightforward calculation shows that
F1(y(r)) = ri1
Qd
'1 (Ql)Q,+o(r),
F3(y(r)) = rA1+o(r),
F4(y(r)) = rQ2A1 +o(r).
Therefore,
F(y(r)) <K 0
201
for all sufficiently small r > 0. It follows from Theorem C.2 that the solution (xi (t), Ul*(t), xz (t), Uz (t)), starting at t = 0 and at such a point y(r),
satisfies assertions (c) and (d) of the proposition. Therefore, the x7(t)
and U,*(t) have limits as t ->oo. By (5.5) and (5.6), these limits are finite.
In fact, since (x2(0), U2 (0)) <- (z2, U2), it follows that (x2* (t), U2* (t)) <_
(12,U2)forallt>0.
Since x2(t) is monotone increasing to a positive limit, it follows from
(4.3) that (x2)'(t) -4 0 as t- oo and therefore that Q*(t) --> 02- If x* (t) has
a positive limit as t oo then, since (x*)'(t) --> 0 as t-> oo, it follows that
Q* (t) -*Ql. But then (x* (t), Q* (t), x2 (t), Q2 (t)) has the limit (x* (oo),
Ql, x2(00), Q2) as t - oo, which is not a rest point of (4.3). This is impos-
sible, so we conclude that x* (t)-*0 as t --goo. Now, using that QZ(t) 02 (and therefore (Q*)'(t) 0) as t - oo, we conclude that
P2(1-Q2x2(O0), Q2)-i 2(Q2)Q2 = 0.
(x*(t),Qi(t),x2(t),U2(t))-E2 as t,oo.
Since (x, (0), Qi (0), x2(0), Uz (0)) = x(r), a point of the unstable manifold of El, (x* (t), Qi (t), x2 (t), Q2 (t)) -*El as t-- -oo. This concludes
our proof.
We can determine the asymptotic behavior of solutions of (4.3) for which
x;(0) > 0, i = 1, 2, by considering solutions of (5.2) in A. We begin with
the following preliminary result.
R=((x1iU1,x2,U2)E[IB+:x;,U1>0(i=1,2) and
(0, 0, x2, U2) :5K (XI, U1, X2, U2) K (XI, UI, 0, 0)1
(5.8)
and
Q, (t) = U1(t)/xI(t)-01,
Q2(t) = U2(t)/x2(t)
02
(5.9)
as t -* oo.
Variable-Yield Models
202
for all t > 0. This assertion also follows from the comparison arguments
(5.5) and (5.6). Noting that y(r) satisfies y(r) <K (1], U1, 0, 0) and that
y(r) -- (Xl, U1, 0, 0) as r -* 0, it follows that we can find r > 0 such that
(x1(1), U(1), x2(l), U2(1)) <K y(r).
(x1(t+1), Ul(t+1), x2(t+1), U2(t+1)) <_K (xl (t), Ul (t), xZ(t), Uz (t))
for all t ? 0, where (x* (t), UU (t), xz(t), Uz (t)) is the solution of (5.2)
starting at y(r) and described in Proposition 5.3. This implies
x1(t+1)<xj'(t)-0
and
U2,
we conclude that
lim inf,
x2(t) >_ x2
and
lim inf1
On the other hand, the comparison arguments (5.5) and (5.6) imply that
limsup1
Therefore, (5.8) follows; (5.9) follows from the second assertion of Lemma 5.1.
Proof. This is obvious if the solution ever meets the positively invariant set R, which by Lemma 5.4 lies in the basin of attraction of E2. On
the other hand, the omega limit set L consists of points x satisfying
(0, 0, x2, U2) <_K X :SK ((1 , U1, 0, 0), by the comparison arguments (5.5)
and (5.6). We may suppose that the solution remains outside R for all t,
and that therefore an omega limit point x = (x1, U1, x2i U2) must satisfy
one or more of x1 = .xl, x2 = x2, U2 = U2, or U1 = U.
If x1 = zl then xi = 0 at (x1, U1, X2, U2), since L is invariant and zl < Xl
for all points (RD U1, 92, U2) e L. Therefore, U1 = U1.
If U1 = U1, then U(= 0 at (x1, U1, X2, U2) and so
U1 = P1(1-
Competitive Exclusion
203
Therefore, xl = zl and U2 = 0. By Lemma 5.1, U2 = 0 implies x2 = 0. It follows that if either xl = zl or U1 = Ul then (x1, U1, X2, U2) = (1 , (J1, 0, 0).
Similarly, if x2 = x2 or U2 = 02 then (x1, U1, X2, U2) _ (0, 0, x2, U2). Con-
6. Competitive Exclusion
In Section 5, the global behavior for the reduced system (4.3) was determined. It remains to show that the results obtained for this system carry
over to the original model system (3.4). This will be done by making a
change of variables in (3.4) and using the results of Appendix F.
Under the conventions and assumptions described previously, our main
result is the following.
THEOREM 6.1. Assume that the steady states of (3.4) are nondegenerate.
Then the following assertions hold.
(i) If (3.6) and (3.8) do not hold, then Eo is the only steady state and
every solution of (3.4) satisfies
(xl(t), Q1(t), x2(t), Q2(t), S(t)) --> Eo
as t
(ii) If (3.6) holds and (3.8) does not hold then Eo and El are the only
steady states and every solution for which x1(0) > 0 satisfies
(xl(t),Ql(t),x2(t),Q2(t),S(t))-*E1 as too.
(iii) If (3.8) holds and (3.6) does not hold then Eo and E2 are the only
steady states and every solution for which x2(0) > 0 satisfies
(x1(t), Ql(t), x2(t), Q2(t), S(t)) --E2 as t
cc.
(iv) If (3.6) and (3.8) hold then E0, El, E2 exist; if also (3.10) holds then
every solution for which x2(0) > 0 satisfies
(xl (t), Ql (t), x2(t), Q2(t), S(t)) --* E2
as t -> ao.
The first three assertions of the theorem describe outcomes in which one
or both populations are eliminated from the chemostat - owing not to
variable-Yield Models
204
generacy holds for Eo if and only if ,(Q) * D, i = 1, 2. For the second (third) assertion, only the single condition 2(Q2) # D (,(Q) # D)
is needed to ensure that the nondegeneracy assumption holds for both
steady states. As all eigenvalues are real, nondegeneracy is equivalent to
hyperbolicity.
The argument for Theorem 6.1 follows the familiar pattern, using Appendix F. The main ideas are briefly sketched here. Set
Z= 1-S-QIxI-Q2x2
in (3.4) and note that Z'= -Z. Replace S by 1-Z-Q1x1-Q2x2 in (3.4)
to obtain the new system
xi = xi(IAI(Q1) -1),
x2 =x2(t'2(Q2)-1),
(6.1)
Z' = -Z.
It suffices to determine the global behavior of solutions of (6.1). Clearly
Competitive Exclusion
205
Table 6.1
Equilibria
present
Conditions for
hyperbolicity
Eo
Vi(Q10) # 1,
i= 1,2
Eo, E,
2(Q2) #
E0, E2
I (Q) # 1
E,, E,, E2
S<S
Therefore, we conclude from Theorem F.1 that every solution of (6.1), and
The form of (6.1) implies that the dimension of the stable manifold for
each equilibrium of (3.4) is one more than the dimension of the stable
manifold for the corresponding equilibrium of (4.3). Consequently, assuming the hyperbolicity assumptions of Table 6.1, the dimension of the
stable manifold of each equilibrium is as follows:
dim M+(Eo) = 5 when Eo is the only equilibrium;
dim M+(Eo) = 4 when exactly one of the single-population equilibria
E,, E2 exists;
dim M+(E0) = 3 when both E, and E2 exist;
dim M+(E,) = 5 if only Eo and E, exist;
dim M+(E,) = 4 if E0, E,, and E2 exist;
dim M+(E2) = 4 if only Eo and E2 exist;
dim M+(E2) = 5 if E0, E,, E2 exist.
The arguments in the four cases of Theorem 6.1 are very similar, so we
present only one case, the last and most interesting one. When Eo, E,,
and E2 exist and 9 < S, then E2 is a local attractor for (3.4) and E, is
206
unstable, with a four-dimensional stable manifold consisting of that portion of the x2 = 0 invariant hyperplane for (3.4) with x, > 0. The onedimensional unstable manifold of El connects to E2, by Proposition 5.3.
The equilibrium E0 has a three-dimensional stable manifold contained
in the region where x1 = 0 and x2 = 0. Since every solution of (3.4) converges, we conclude that E2 attracts all solutions corresponding to initial
data satisfying x2(0) > 0. This establishes the last case of the main theorem. The other cases follow similar arguments.
7. Discussion
The conclusions of Theorem 6.1 correspond precisely to those of Theorem 5.1 of Chapter 1 and Theorem 3.2 of Chapter 2. In fact, following
Grover [G2], a constant-yield model can be associated with (3.1) in such
a way that both models give the same predictions (this is not proved in
[G2]). Consider the case where both El and E2 exist. Omit from (3.1) the
equations for Q, and substitute
t(Qi) = Qi 'P1(S, Qi),
122(Q2) = Q2'P2(S, 02)
(7.1)
S'=D(S-S)-xlp1(S,Q1)-x2P2(S,Q2),
which can be viewed as the constant-yield model corresponding to (3.1). Its
global behavior is determined by the break-even nutrient concentrations
for each population, that is, the value of S at which x, = 0. By (3.5) and
(3.7) these are S = S for i = 1 and S = S for i = 2. The main results of Chapters 1 and 2 show that the winner is the population with the smaller breakeven concentration - provided, of course, that it is smaller than S. This
is precisely the conclusion of Theorem 6.1. Furthermore, the equilibria
of (7.1) are obtained from those of (3.1) by deleting the Q; components.
The predictions of the variable-yield model (3.1) and the corresponding
constant-yield model (7.1) are identical. Typical solutions of each model
approach the corresponding equilibrium in a monotone fashion (see Proposition 5.3).
Discussion
207
1. Introduction
The models considered in previous chapters of this monograph have ignored the fact that populations of microorganisms contain individuals
with differing body size and that individuals of different size have different
characteristics. Body size is clearly an important factor in determining an
organism's energy requirements and its ability to uptake resources. Furthermore, if an organism can grow as well as reproduce, then it becomes
model, and the particular characteristics allowed to vary are called the
structure variables. In this chapter, a size-structured population model is
presented. There is a large and rapidly developing literature on structured
208
209
210
is the primary example. The factor 12 in the uptake rate reflects the assumption that, for fixed nutrient concentration, uptake is proportional
to surface area. We suppose that a fraction K of the energy derived from
the ingested nutrient is used for growth of the organism and the remaining fraction, 1- K, is channeled into reproduction. It is assumed that the
amount of energy required for maintenance of the organism can be neglected. If ,t is the conversion factor relating nutrient to biomass, then
the rate of growth of the organism is given by
d(13)
= 77
-1
Kl2 f(S)
dt
K
377
f(s)
Finally, the washout rate D in the chemostat and the population death
rate d are assumed to be constant, independent of 1. Therefore, the removal rate of the organism is given by D1 = D + d.
Having described the behavior of individuals, we now focus attention
at the population level. Let p(t,1) be the density of individuals of size 1 at
time t, so that
b
f p(t,l)dl
a
is the number of individuals with lengths l satisfying a:5 1:5 b. Fix a time
to > 0 and consider the fate of the cohort of individuals with length be-
tween a and b at time to. At time t > to, this cohort occupies the size
range a(t) <-1 <- b(t), where
211
3rd
Here, it is assumed that the nutrient S(t) is given for t > to. The number
of individuals in this cohort can change only owing to mortality and washout, so
d
dt
b(t)
p(t,1)dl=-D,
a(t)
b(t)
p(t,1)dl.
a(t)
Using the Leibniz rule for the derivative on the left, this expression becomes
fa(t)
f
-(t,1)dl+b'(t)p(t,b(t))-a'(t)p(t,a(t))=-D,
at
b(t) a p
b(t)
p(t,1)dl.
a(t)
Since a'(t) = b'(t) = (K/3j) f(S(t)), the fundamental theorem of calculus can be applied to the previous equality, resulting in
b(t) ap
fa(t)
ap
at
(t,l)+D,p(t,l)Jdl=O.
This holds at t = to, so the limits of integration can be taken to be arbitrary numbers a, b satisfying lb < a < b. If the integrand (the term in
brackets) is not identically zero as a function of 1, for arbitrary fixed to,
then there is a point l0 where it is (say) positive. It would then be positive
in some interval containing 10, since the integrand is continuous in 1. Taking a and b to be the endpoints of such an interval in the integral would
ap
f(S(t)) at = -D,p
at + K
3rd
holds for lb < I and t > 0. This equation describes how p changes with
time. It must, of course, be supplemented with appropriate boundary
and initial conditions. For example, it is necessary to specify the initial
density po of the population at time t = 0:
p(O,I)=po(1),
I>Ib-
The rate at which offspring of size lb are added to the population must
also be determined. The number of offspring born in the time interval
212
1-3
Wlb
f(S(t))fl"12p(t,1)dlzt.
, = lb+ 3n f(S(t))At;
hence, at time t+Ot, the newborns accounted for in the preceding expression occupy the size range Ib s 1:5 la,. It therefore follows that
p(t+zt, l)dl =
rb
_
I
mlb3
or
Cb o,
377(1-K)
p(t+Ot,l)dl=
Kf(S(t))Ot
JPp(t,1)dl.
p(t,1b)=
f(S(t)) f 12p(t,1)dl.
rb
S'=D(SrO>-S)-f(S(t))f 12p(t,I)dl.
rb
S'=D(S(O)-S)-f(S(t)) f
(2.1a)
b
ap
at
ap
K
f(S(t)) al -Dlp,
3)7
(2.1b)
lb) =
fl2 (t l)dl
p
KWIbs
213
(2 . 1c)
P(0,1) = PO(1),
(2.1d)
S(O) = S0.
(2.1e)
lm(t) = lm+
ft
0
/>_ Im(t),
(2.2)
- f(S(r)) dr.
3,q
One approach is to consider the ratio of the size of the daughter to the
mother as a random variable and treat the problem in a stochastic way.
See for example the discussion in Harvey [Ha, sec. 4.2]. Metz and Diekmann [MD, p. 237] analyze just such a model, although they scale uptake
in a different way (see Section 8). Despite the obvious deficiencies in the
model, it has the important advantage of being mathematically tractable,
as we shall see. It is highly likely that a model that more accurately accounts for reproduction will be much more complicated or impenetrable
214
Systems containing coupled partial differential equations and integrodifferential equations, such as (2.1), present significant challenges to mathematical analysis. Much progress on these difficult equations is presented
in [MD]. Following Cushing [Cu2] with only minor differences, we as-
sume that (2.1) defines a unique solution S(t) and p(t, t) for t > 0 and
introduce the moment functions:
A(t)=li 2
f(t
,1)12dl,
L(t)=l6'
(3.1)
tb
e0
P(t)=
fib
In [Cu2], the factors 162 and 16' of A(t) and L(t) are omitted. We introduce them to make for cleaner expressions. Ignoring a scaling factor,
A(t) is the total surface area of the population, L(t) is the total length of
the population, and P(t) is the total number of individuals in the population at time t.
The immediate goal is to obtain differential equations for these new
variables. Multiplying (2.1b) by (1/lb)2 and integrating from lb to infinity
results in
A'(t)+Kf(S(t))
'0 12a(t,1)dl=-DI A(t).
3Ib 2
l
fib
Integrating by parts in the integral, and requiring that 12p (t, 1) -> 0 (see
(2.2) for justification) as 1- co, leads to
A'(t) = 1016'f(S)L+alb'f(S)A-DMA,
where
215
In a similar way, multiplying (2.1b) by (I11b) (or by 1), and using the
boundary condition (2.1c) and requiring 1'p (t, 1)
0 as I moo, leads to
S'=D(S
0)-S)-.f(S)lbA,
A'=-DIA+alb1f(S)A+3i3lb 1f(S)L,
(3.2)
P'= -DIP+al61f(S)A.
Initial conditions for (3.2) are obtained from (2.1e) and by putting t = 0
into (3.1) and using (2.1d).
In [Cu2] a is called the reproductive efficiency of the organism, since
it is a ratio of the fraction of energy derived from uptake that is allocated
to reproduction to the conversion factor relating food units to weight
for reproduction (wlb is the amount of nutrient needed to produce one
offspring). For similar reasons, 0 is called the growth efficiency of the
organism.
It will be useful to write the last three equations of (3.2) in vector form.
Let p = col(A, L, P) and let
M =
(3.3)
q'= -D1q+I61.f(S)(T-'MT)'q.
The problem is to determine the matrix T in such a way that the new vari-
216
LEMMA 3.1. The matrix M has a positive eigenvalue a and a corresponding eigen vector
v - col(z(//3)2, 3(//3),1).
93 +6a/3+4a12
The remaining eigenvalues of M are y iv, where y < 0 and v > 0. There
is a nonsingular matrix T such that
T -'MT =
-v y
The first column of T is the eigenvector v, and the first row of T-1 is the
transpose of the eigenvector w.
Proof. Both M and M` are irreducible, nonnegative matrices, so Theorem
A.4 implies that the spectral radius it = (M) is a positive eigenvalue and
that there is a corresponding positive eigenvector v. As (M) = (M),
it is an eigenvector of M' with a corresponding positive eigenvector w.
The eigenvectors v and w are easily calculated in terms of .
The characteristic polynomial of M is
-A3+aA2+3a(3A+9a,62 = 0.
(3.4)
It is easy to show that (3.4) has only one real root, which must be . If we
denote the eigenvalues of M by Al = , A2 = y+iv, and A3 = A2, then
A1A2+A1A3+A2A3 = -3a1
or y2+v2+2y = -3a13.
217
Despite its slightly different scaling, our it agrees with the in [Cu2].
Cushing calls the "physiological efficiency coefficient" of the population, since it reflects both the reproductive efficiency and the growth efficiency of the organism.
In the new variables q = col(x, y, z),
x=v p=2(/6)2A+3(/,6)L+P>0
is a weighted average of A, L, and P that will serve as a measure of population size. Furthermore,
A=
c = col(w,, c2, c3) is the first column of T containing the first
component w, > 0 of the positive eigenvector w. Consequently, the equations for S, x, y, z take the form
(3.5)
z'= -D,z+Ib'f(S)(vy+-yz).
Introducing the complex variable
71 = Y+ iz,
Consequently,
d
I_712 = 2[-D,+-ylb'f(S(t))]1,n12.
dt
Since y < 0, it follows that 71(t) 0 as t -> oo at an exponential rate. Therefore, it suffices to consider the system (3.5) with y = z = 0:
218
S'= D(S0)-S)-w,lbf(S)x,
x'=-DIx+1216 If(S)x.
To compare this equation with the standard model of Chapter 1, let
S'=D(S 0)-5)-y-1f(S)x,
(3.6)
x'= (f(S)-DI)x.
(3.7)
wlb
D
DI
Whether or not the population described by the model (2.1) can survive
in the chemostat is determined solely by the break-even concentration A.
It is evident from (3.7) that A increases with increasing 1b and decreases
with increasing 12. Recall that smaller is "better" when it comes to A. A
population that can grow at low nutrient levels is more likely to survive
and be a strong competitor. Decreasing Ib has the effect of making offspring cheaper to produce - since each costs Wlb in nutrient units - so
decreasing lb should have a positive effect on a population's ability to
Competition
219
9,n22+67(2K-1)+2K(3K-2)
-27W,122+18,12(l -K)+6qK(1-K)
aK(0+)=X11(3-l.
Consequently, if ,1 < i W then an organism that devotes nearly all its energy
S'=D(S(
-f2(S(t))
12p2(t, l) d
12
(4.1a)
220
at = 3 fi(S(t)) a -Dial,
aP;
(4.1b)
(4.1c)
Pi(O'1)=Pio(l),
(4.1d)
S(0) = S.
(4.1e)
(4.2)
and
i = (1-K;)/W;
Ni = Ki/?7;.
The quantity a; is the reproductive efficiency and (3; is the growth efficiency of the ith organism.
In the same way that (3.2) was reduced to (3.6) by a change of variables, (4.2) can be reduced to
S'=
Xi =
(4.4)
a,
Mi
3a;
;,a;
cient" of the ith organism; w, > 0 is the first component of the positive
eigenvector W of M;` and is given by
w=
2iRi
9? + 61(3i; + 4a1 f3i
(4.5)
Competition
221
'Y l = w,lr/,
(4.6)
x2 = (f2(S)-D2)x2
Define the break-even nutrient concentration Ai to be the solution of
Dt
r_W
lim x2(t) = 0,
(4.7)
l-oo
Proof. If (i) holds then this is Theorem 4.1 of Chapter 2. If (ii) holds
then this is Theorem 3.2 of Chapter 2.
f(A) =1rDr/i
it is evident that the length at birth or the physiological efficiency coefficient can decide the winner under certain conditions. All else (i.e. f , Di,
i) being equal, the winner is the population with the smaller length at
222
birth; if f , D,, l; are equal then the winner is the population with the larger
physiological efficiency coefficient. As , depends on the growth efficiency
a and the reproductive efficiency i3, it follows that either can be of decisive importance under suitable conditions. In fact, it is possible that the
winner could be the population with a smaller uptake function for all
values of S in the range 0 < S < 5(0), provided its length at birth is suitably smaller - or its physiological efficiency coefficient suitably larger than that of its rival. This is not possible for the models of Chapters 1
and 2. Typically, however, all things are not equal and the winner will be
decided by a complicated weighting of the form of the uptake functions
(Michaelis-Menten parameters), the death rates, length at birth, and the
growth and reproductive efficiencies of the organisms.
In Section 5 we show that, under the hypotheses of Theorem 4.1,
li mP1(t) =
t-W
w-
(5(0) -A,)
Di
(4.8)
A(t) =A,(t)/P,(t),
L(t) = Li(t)/P,(t)
denote the (scaled) averages corresponding to the ith population at time
t. Direct calculation gives
(' `
dr
a 0 f(S(r))'
223
dA =A+ 3L-A2,
(5.1)
d A+ 3a -AL.
Analysis of (5.1) leads to the following result.
THEOREM $.I. For any solution of (4.2),
lim
r- 0
AL(t)
Pi(t)
__
\lim
; =b;(R`
a;
L`(t) = 1+
(5.2a)
a;
R`
(3f1()'
Proof It suffices to show that (A, L) _ (/a,1 +/3/3) is a globally attracting equilibrium for (5.1). Setting the derivatives to zero and solving
for L in terms of A in the second equation leads to the equation for A:
z
-A3+A2+ 3a A+ 9a2 = 0.
Comparing this with (3.4), it is clear that aA must satisfy (3.4). Therefore aA = , since is the only real root of (3.4). It follows that there is
only one equilibrium of (5.1) and it is given in the first line of the proof.
A direct calculation shows that the trace of the variational matrix at this
equilibrium is negative and the determinant is positive. Therefore, the
equilibrium is locally asymptotically stable.
The Dulac criterion of Chapter 1 can be used to rule out periodic orbits
for (5.1). In fact, setting 0 (A, L) = (AL) and computing the divergence
gives
aA
3aA-2-L-1_L-2_
1L z<0
224
Keeping in mind the scale factors lit in A, and li in L,, Theorem 5.1 implies that the asymptotic value of the average individual surface area is
li2i/ai and of the average individual length li(1+e3/(3i)). Since Si is
monotone increasing, the asymptotic average length increases with Iii and
decreases with increasing ai. Similarly, since ei is monotone increasing, the
asymptotic average surface area increases with /3i and decreases with ai.
The reader might wonder why the average individual volume has been
ignored. As in (3.1), it can be defined by
V'= (a+0)16'f(S)A-D, V.
(5.3)
(5.4)
JI
1`(t) =1+R'
(5.5)
ai
'_. Pi(t)
As before, this value should be multiplied by li3 to account for our scaling.
From (5.5) and (5.1) we may conclude that any reasonable measure of
average individual size for the ith population, whether it be the asymptotic value of Li(t)/Pi(t), Ai(t)/Pi(t), V (t)/Pi(t) or some weighted average of these, is increasing in (3i and decreasing in a,.
Finally, we aim to verify that (4.8) holds under the hypotheses of Theorem 4.1. From (4.4) it follows that
X, (t)
P1(t) =
The limit (4.8) results from taking the limit as t -> oo and using (4.5), (4.7),
and (5.1).
j-.co
1131
3l
225
DI=D2=D.
What is required is an expression for the amount of nutrient that is stored
by each population in the form of offspring and in the form of biomass
derived from growth. Such an expression for the ith population is given
by
3
U,(t)=
1,
V(t)=
W0,
(1Ki)ji+KjWj
00
fl3P(t,l)dl.
;
The total amount of nutrient (in all its forms) in the chemostat at time t is
T(t) = S(t) + U, (t) + U2 (t).
T'=D(S 0)-T).
(6.1)
(6.2)
as t -*oo.
If A, < A2 and A, < S(0) then (by Theorem 4.1) S(ao) = A, and U2(oo) = 0,
since the second population is eliminated from the chemostat. Therefore,
226
The fraction 1-K1 of this is in the form of offspring, each worth W,11
nutrient units, so
Pl(y) - (1-KI)(S(0)-A1)
3
W111
The steady-state size distribution for a single population obeying equations (2.1), or of the surviving population for two competing populations
obeying (4.1), can be readily computed. The case of a single population
will be treated, with an appropriate subscript on the result yielding the
distribution of the surviving population in the case of competition. Assume that 0 < A < S(0). Then set S' and ap/at equal to zero in (2.1a,b);
using the fact that the steady-state value of S is A, we have f(S) =lbD,/
in (2.Ib). This leads to
p(1) = p(lb) exp[-K1 (1-lb)J,
b
D
S w Ib
Klb
b
(7.2)
D1
The data in [ Wi] are in terms of cell volume, so for comparison purposes it is necessary to convert the distribution by length, (7.1), to one by
volume. Let R(v) be the steady-state cell-volume distribution corresponding to the distribution (7.1). Then the number of individuals with cell
volume in the range v, to v2, where Vb = lb < v1 < v2, is given by
fV2
R(v)dv.
V]
wi th respect to 1 leads to
R(v) = s v-2isp(vli3)
or
R(v) =
Kb
-'
(1- K ) vb
((vb
v-2/3 ex p
[
1/3
227
)].
-1
(7 . 3 )
V12 (o)
V1
6.18
6.09
(a)
Vli (b)
VIZ (b)
4.28
4.26
-T
0
Cell volume
Figure 7.1. Eight steady-state size distributions observed under different experimental conditions (flow rate, temperature, CO2), scaled for equal means and
areas. The mean cell size for each graph is indicated next to the graph. (From [ Wi,
fig. 19], Copyright 1971, Academic Press. Reproduced by permission.)
228
length at birth of the competitors can be of decisive importance in determining the outcome of competition under suitable circumstances. It
would seem to be theoretically possible and of considerable interest to
test the predictions of Theorem 4.1, especially those relating to the role
of physiological efficiency coefficients in determining the winner of competition in the chemostat.
Like all models in science, the one treated in this chapter makes many
unrealistic assumptions. These were pointed out at the end of Section 2.
The most notable deficiencies are that the model inadequately reflects
the cell division process and neglects the energy required for cell maintenance. It should be pointed out, however, that the main predictions of
the simple (even less realistic) models of Chapters 1 and 2 survive intact
in the more complex model treated in this chapter. Therefore, it is not
unreasonable to expect that many of the predictions made on the basis of
Theorem 4.1 will continue to hold for more realistic models.
Discussion
229
munities, Brooks and Dodson proposed that (1) larger individuals are
more efficient at exploiting resources, which provides the potential for
competitive exclusion of the smaller individuals in the population, and
(2) size-selective predation by predators, which falls more heavily upon
the larger individuals, can allow for the survival of smaller individuals or
in some cases can even result in the elimination of larger individuals. To
test hypothesis (1), it is natural to ask whether, in the model treated in
Section 4, the superior competitor is necessarily the larger competitor. To
answer this question, some measure of population size must be chosen.
In Section 5, it was argued that suitable choices are the asymptotic values
of average individual length, surface area, volume, or some weighted average of these. Each of these measures of average individual size shares
the common feature that it is an increasing function of growth efficiency
Ki/rli and the length 1i at birth, and a decreasing function of reproductive
efficiency (1-Ki)/wi.
On the other hand, competitive success is determined solely by having
the smaller break-even concentration Ai. Since Ai decreases (and so the ith
230
need not have the larger average size, by any of the measures of size.
Consequently, the model considered here does not appear to support the
size-efficiency hypothesis. For more on this interesting subject the reader
is referred to [Cu2] and the references therein.
10
New Directions
1. Introduction
In this chapter, several recent models that make use of the chemostat are
described. The situations that occur are not as fully understood as those
in the previous chapters, and no attempt will be made to present a detailed analysis. The mathematical results are only partial and, in fact, in
some cases the modeling is clearly inadequate at this time. The title of
the chapter is intended to suggest that further work is needed. The problems are important and interesting, and it is hoped that the reader might
find something of interest here and contribute to the development of the
theory.
Three types of new directions are discussed. In two of these, ordinary
differential equations are not an adequate model to describe the phenomenon of interest; functional differential equations and partial differential
equations provide the appropriate setting. In the remaining case ordinary
differential equations are appropriate but the modeling is not complete.
Improving the model would result in a larger system for which the techniques of monotone dynamical systems are inappropriate. The problems
will be described and results indicated, but no proofs are given. In all
cases, much more work needs to be done before the problem is appropriately modeled and analyzed.
New Directions
232
nutrient is added to the vessel at one point, allowed to diffuse, and then
removed at a different point, a gradient will occur. The nutrient concentration will become spatially dependent, and the organisms will compete
at different nutrient levels at different locations. (There is a tacit assumption that the turnover of the chemostat - typically 12-24 hours - is so
slow that there is no relevant transport.)
Reformulating the chemostat with diffusion introduces a new level of
difficulty into the modeling. First of all, the equations will become nonlinear partial differential equations with all of their attendant complexity.
The input and the output now occur at the boundary, so the boundary
conditions for the system of partial differential equations must be formulated with care. If nutrient and organisms are diffusing, a new constant
occurs: the diffusion coefficient. To simplify matters, we will assume that
all the quantities diffuse with the same constant, an assumption that is
mathematically convenient but not biologically rigorous. The analysis is
limited to only one space dimension, so one must perform a thought experiment to visualize a one-dimensional chemostat when the real one is
three-dimensional. A tubular reactor is an approximation. This will have
the mathematical consequence that the rest states will be solutions of a
boundary value problem for ordinary differential equations. With these
limiting assumptions, the problem was considered in [HSW], [HW2], and
[SoW] with an attempt to recover the standard gradostat results in this
setting. The resulting boundary value problems were also considered in
[BT].
We do not give the derivation here, but the equations take the form
as _
at -
au _
at
av
at
a2S
ax2
m1Su
m2Sv
al+S
a2+S'
a2u
m1Su
ax2 + ai+S'
a2v
ax2
(2.1)
+ m2Sv
a2+S'
0<x<1,
with boundary conditions
ax (t, 0) _
au(t'0)=
-S(o)
ax(t,0)=0,
ax
ax
(t,
1)+rS(t,1) = 0,
233
(2.2)
(t, 1)+ru(t,1) = 0,
(t,1)+rv(t,1) = 0
u(0, x) = u(x)
0,
u(x) 0 0;
v(0, x) = v(x) ? 0,
v(x) 0- 0.
(2.3)
(the rate of the pump operating the chemostat) then the parameter D is
defined as F/V. Rewriting the equation just displayed for the mass of the
substrate in the vessel yields
(2.4)
(2.5)
If one integrates over the interval [0, 1], an equation for the total mass
of the nutrient is obtained:
New Directions
234
d fo
S(t,x)dx=dSX(t,1)-dSX(t,0).
dt
(2.6)
The two terms on the right-hand side of the equation represent the flux
at the right and left endpoints, so equation (2.6) is the counterpart of
(2.4). These quantities must be determined from the boundary conditions. The flux at the left end is given by S(0)F, where S(0) corresponds to
the S(0) of the basic chemostat (as a density; i.e., the units are m/l ). The
condition at the left endpoint can be written
dSS(t, 0) = -S(0)F;
if one defines
S(O) = S(O)F/d
then the first boundary condition in (2.2) is obtained. Similarly, the flux
at the right-hand end is given by
dSX(t,1)=-FS(t,1).
Thus, if r is defined by r = F/d then the second boundary condition holds.
Equation (2.6) states that the rate of change of the mass of the nutrient
in the vessel is proportional to the difference between the input nutrient
flux and the output nutrient flux, as in the basic chemostat. The diffusion
coefficient d has units of length squared over time, 12/t; thus the units are
appropriate.
The following basic lemma will allow the problem to be simplified.
(Note that this simplification depends upon the fact that the diffusion
coefficients are the same.)
LEMMA 2.1. The solutions S(t, x), u(t, x), v(t, x) of (2.1)-(2.3) exist for
all t > 0 with 0 < x < 1. The solutions are nonnegative and bounded, and
(2.7)
=fi(x)=s() C1 + r
r
- x1
0 <x<1.
The function 4(x) represents the distribution of nutrient for the case of
no consumption (uo(x) = 0, vo(x) = 0). The lemma reflects the fact that
the total nutrient and equivalent organism biomass equilibrate to this
function as well. As noted frequently in this work, this is essentially a
definition of the chemostat if all variables are taken into account. The
235
parameters s 0) and r are reflected in the function O(x); these are the
operating parameters of the apparatus.
Solutions of (2.1)-(2.3) generate a semidynamical system on
C+ x C+ x C+,
au =
at
8x2
+,(0-u-v)u,
(2.8)
8t
d 8x2
+f2(0-u-v)v;
a-(t,1)+ru(t,1)=0,
ax(t,0)=0,
(2.9)
8x
aX (t,
(t, 0) = 0,
l)+rv(t, l) = 0;
u0(x) 0 0,
v0(x) 0 0,
where
.f (s) _
m,S/(a;+S) if S?0,
0
if S<--0
New Directions
236
_ a2u
d axe +
mO(x)-u)
a,+q(x)-u u
(2.11)
x(t,1)+ru(t,1)=0.
ax(t,0)=0,
V+AI
O(x)
\\a,+cb(x)
C(0) = 0,
0,
>G'(1) +r>'(1) = 0.
This lemma states that if the maximum growth rate is small then the
organism will tend to extinction as time becomes large.
LEMMA 2.3. If m, > and and u(t, x) is the solution of (2.11), then
a,+4(x)-u
u'(0) = 0,
u'(1)+ru(1) = 0.
Ap(x) =
Aq(x) =
237
I(a,+-u)2
2+
a1+0-u
(a1+0-u)2
q,
uq
P'(0) = 0,
p'(1)+rp(l) = 0,
q'(0) = 0,
q'(1)+rq(1) = 0.
Since the second equation is independent of the first, the set of eigenvalues for the full system is a subset of the set of eigenvalues of the second
equation and therefore they are real.
Think of m2 as a parameter and let A(m2) be the largest eigenvalue of
A(m2) = ao
One can proceed further and show the existence of an interior order
interval to which all solutions converge. In fact, from monotonicity, almost all solutions must converge to a rest point in this order interval
[HSW]. Finally, we note that the system of equations in this section is
New Directions
238
Table 2.1
Eo = (0, 0)
E, = (u, 0)
E, = (0, u)
Existence
Instability
Always
m, > .hod
m2 > pod
E, or E2 exists
m2 > mz
m, > m;
239
growth rate; that is, the quantity Ti is the time delay in nutrient conversion. The specific growth rate is assumed to be a function of the nutrient
level at time t-7-i. The model takes the form of a system of differential
difference equations:
x2(t) =x2(t)[f2(S(t-T2))-1]
with
f1(S) =
m'S
ai+S'
where 'TI, T2 ? 0, S(t) _ 0(t) ? 0 on [-T, 0], T = max(7-1, T2), and xi(0) _
xio >0 (i = 1, 2).
The last two equations can be written in integral form as
xi (t) = xi (0) exp [ fo
f(S(B-Ti)-1)de].
Th is illustrates that the proper initial value problem is the one indicated
by the initial conditions just listed. The theory for such delay differential
equations is much more complicated than that for ordinary differential
equations, and is not so widely known among nonspecialists. The basic
reference is Hale [H1]; see also Kuang [K2].
Let C denote the space of continuous functions on [-T2, 0] equipped
with the sup norm. We will tacitly assume the labeling is such that 7,2 > T1.
Using our integral representation and a simple inequality argument for
S', it is not difficult to show that solutions of the system (3.1) are nonnegative for all positive time; (3.1) defines a semidynamical system on
C+ X C+ X C+ (C+ was defined in Section 2). The "conservation" argument used previously to obtain boundedness (and to reduce the complexity of the problem) is no longer valid, since the uptake and the consumption terms do not cancel. This fact alone casts suspicion on the model as
a description of the chemostat. The model does, however, produce oscillations, which makes it very interesting. The boundedness and the continuability of solutions of the system (3.1) can be established, but it is not
quite as easy as with the previous chemostat problems.
The investigation of solutions takes the following form. First, one population growing on the nutrient is analyzed (after some scaling) and a
bifurcation (with the delay as parameter) is shown to exist, establishing
the existence of a periodic solution (S(t), zi(t), 0). For one population of
microorganisms, the two-dimensional system governing growth is
New Directions
240
X] (t) =x,(t)[f,(S(t-T))-l].
(3.2)
XI(t) =TX,(t)[f,(S(t-l))-1].
(3.3)
Although stability may in principle be computed, the calculation is extremely complicated. Numerical calculations suggest the asymptotic stability of the limit cycle, but the stability has not been rigorously established. Assuming that the solution is asymptotically stable, a secondary
bifurcation can be shown to occur. The argument is quite technical and
requires a form of a Poincare map in the appropriate function space; it
is analogous to the bifurcation theorem used in Chapter 3 for bifurcation from a simple eigenvalue. The principal theorem takes the form of a
bifurcation statement.
THEOREM 3.2. Suppose that the parameters a,, m,, and T, are chosen so
that (3.2) has a (linearly) asymptotically orbitally stable periodic solution (S(t), z(t)) with period T > 0. Fix a2 and T2 > T,. Then there exist a
critical value m* and a branch of periodic orbits of (3.1), with positive x2
component, bifurcating from the hypothesized orbit for m2 near m2.
All of the comments raised in Chapter 3 in connection with a similar bifurcation apply. The computations needed to determine the direction of
bifurcation (which side of m*) and the stability are formidable. However, for particular parameter values one can solve the differential equations numerically and exhibit the periodic orbit. Figure 3.1 shows the time
course of a sample problem, and Figure 3.2 shows the projection of the
periodic or the coexisting periodic orbits onto each of the possible pairs
of variables.
Ellermeyer [ E ] takes a different approach to modeling the internal storage of nutrient. With our preceding notation letyi, i = 1, 2, be the nutrient
241
Figure 3.1. Plot of 100 time steps in the case of oscillatory coexistence. Parameters are a, = 1.0, a2 = 1.0, m, = 3.1, m2 = 3.09, r, = 3.0, r2 = 4.0. (From [FSWI],
reprinted with permission from the SIAM Journal on Applied Mathematics, volume 49, number 3, pp. 859-70. Copyright 1989 by the Society for Industrial and
Applied Mathematics, Philadelphia, Pennsylvania. All rights reserved.)
stored internally by population i. Taking the input and washout into consideration, this quantity is given by
Yi(t) =
i=1,2.
The exponential accounts for the stored nutrient, which washes out of
the vessel - along with the cells containing it - during the storage period.
Balancing input and output with consumption and washout yields integral equations of the form
2
xi(t) = -Dx,(t)+e-DT'fl(S(t-rl))xl(t-rl),
x'(t) = -Dx2(t)+e-DT2f2(S(t-r2))x2(t-r2).
The initial conditions now take the form
(3.4)
New Directions
242
10
a
X1
0.5
1.0
0.5
S
1.0
X2 0.5
11
0
1.0
0.5
S
10
C
X2 0.5
01
0
0.5
10
X1
Figure 3.2. Plot of projections onto two dimensions of the solution given in Figure 3.1: a S-x,; b S-x2; c x1-x2. (From [FSW1J, reprinted with permission from the
SIAM Journal on Applied Mathematics, volume 49, number 3, pp. 859-70. Copyright 1989 by the Society for Industrial and Applied Mathematics, Philadelphia,
Pennsylvania. All rights reserved.)
243
S(t) = 40(t),
xl(t) = 01(t),
x2(t) = 02(t),
-TZ<t<_O,
where again one assumes that T2 ? Tl and that the initial conditions are
continuous functions. These same equations occurred also in [FSW2].
Although the equations look quite similar to (3.1), the behavior of solutions is quite different. There is a conservation principle but, as we would
expect owing to the delay, it takes a rather different form.
LEMMA 3.3. For any fixed initial condition cb = (00, 01, 02), the corresponding trajectory satisfies
limeo[S(t)+eDnIxl(t+Tl)+eDr2x2(t+T2)] = S(0) .
of the boundary rest points is locally stable and the other unstable, the
locally stable one is globally stable [HWE]. In particular, the oscillation
observed in the case of system (3.2) does not occur with (3.4). Indeed, the
delayed system seems to behave much like the simple chemostat.
The totally different behavior of the two models illustrates the importance of the modeling process. Both models appear reasonable. Neither
uses any cell physiology in deriving the delay term. The cell cycle does
not appear in either. Clearly, more work in this direction is needed, with
particular emphasis on more careful modeling of the delay.
244
New Directions
Figure 4.1. A plasmid. Photograph courtesy of Mervyn Bibb, John Innes Institute, Norwich, England. Reproduced by permission.
and Macken, Levin, and Waldstatter [MLW]. The survey article of Simonsen [Si] contains a discussion of the experiments and the theory.
The model of Stephanopoulos and Lapidus takes the form
245
'V
xi = x1(fi(S)(1-q) -D),
(4.1)
xz = x2(f2(S) -D)+gxif,(S),
S(0) = S > 0,
_
f`(S)
m; S
a;+S
and 0<q<1.
The variables and the units are those which have been used since Chapter 1: S(t) is the nutrient concentration at time t, x,(t) is the concentration
of plasmid-bearing organisms at time t, and x2(t) is the concentration of
plasmid-free organisms at time t; 5(0) is the input concentration of the
nutrient, and D is the washout rate of the chemostat. These are the operating parameters. The m; term is the maximal growth rate of x,, and a; is
the Michaelis-Menten (or half-saturation) constant of x;. These are assumed to be known (measurable) properties of the organism that characterize its growth and reproduction. A plasmid is lost in reproduction with
probability q, and ry is the yield constant.
We proceed as before to obtain dimensionless variables by measuring
xi=x,(f,(S)(1-q)-1),
xi =x2(f2(S)-1)+gxjfi(S)
(4.2)
Although the system (4.2) looks similar to the equations for the chemostat in Chapter 1, the analysis is more difficult because the system is no
longer competitive. Stephanopoulos and Lapidus used a very clever index
argument to generate phase portraits. However, such arguments are only
local; [HWW] determined the global asymptotic behavior.
New Directions
246
Table 4.1
[
f,(1)(1 - q) < 1
f2(1) < 1
II
f1(I)(1 - q) > 1
f2(1)<1
III
f,(1)(1 - q) < 1
f2(I) > 1
IV
f1(1)(1-q)>1
f2(1) > 1
Table 4.2
I
(E,1
II
(EI,E*1
III
IE1i E21
IV
(a) IEI,E21
(b) 1El,E2,E*1
Clearly limr . E(t) = 0 and trajectories on the omega limit set satisfy
E = 0. If all of the rest points of the limiting system are hyperbolic (which
will be implied by the conditions stated) and if there are no periodic orbits (which needs to be proved), then the results of Appendix F apply.
The limiting system is
x[=xl[f1(1-x1-x2)(1-q)-11,
xz=x2[f2(1-x1-x2)-1]+gxlfl(I-x1-x2).
(4.3)
=I(x1,x2)I x1?0,x1+x2<11;
0 is a positively invariant region. Note that the system is not a competitive one.
The analysis breaks conveniently into four cases, which are given in
Table 4.1. The rest point sets are shown in Table 4.2. The rest points are
defined as E, = (0, 0), E2 = (0, 1 -A2), and E* = (xr , xZ) where the quantities A2, x*, x2 are defined next.
The A2 term is defined to be the unique value such that f2(A2) = I and
A* is defined by fl(A*) = 1/(1-q), if such a A* exists. A necessary and suf-
ficient condition for there to be such a A* is that f1(1) > 1/(1-q). If one
247
(1-A*)(1-.f2(A*))
.fi(A*)-f2(A*)
xz = 1-A*-xi*
(1-A*)(f,(A*) -1)
.ff(A*)-.f2(A*)
11
Open Questions
In this brief final chapter we collect in one place the main questions that
remain unanswered concerning the models explored in this work. We
proceed more or less in the order of the chapters. In many cases, the open
problems mentioned here have already been identified and discussed in
the discussion section of the corresponding chapter, so the reader may
wish to check there as well.
As noted in the discussion section of Chapter 2, there remains a gap in
our knowledge of the basic chemostat model in the case of differing removal rates for the competitors. The principal open problem is to extend the result of [Hsul], described in Section 4 of Chapter 2, to general
the cycle "lifts off" from the cycle in the x-y plane of the octant (one
248
Open Questions
249
food chain) and moves toward and eventually coalesces with the cycle in
the x-z plane (second food chain). This was established in [ Ke2] under
an additional condition. The limit cycle continues, so long as hyperbolicity is maintained, since there is strong dissipation (solutions are bounded
by a uniform constant). The problem seems to be bounding the period of
the limit cycle from above. Establishing the global bifurcation rigorously
would be of interest and, along with uniqueness, would help to complete
the theory. The proof of Lemma 5.1 in Chapter 3 is very long and inelegant. A simpler proof would be of interest.
250
Open Questions
was given. This analysis relied on two calculations which established that
if a positive coexistence rest point exists then it is both unique and asymptotically stable. As noted in Section 4 of Chapter 6, more general conditions are known [HSo] for two monotone uptake functions in order that
these two results hold. Furthermore, counterexamples are given in [ HSoJ
where these conditions fail and there exists an unstable positive rest point
for some two-vessel gradostat (not necessarily the same one considered in
Chapter 5). As a result of our ignorance of general sufficient conditions
for these two results to hold for monotone uptake functions and for more
Open Questions
251
into two daughter cells, one of size px and one of size (1- p)x, with
probability d(p), 0 < p < 1. One must assume that d(p) = d(1 - p) and
fo d(p) dp = 1. The unit of size x - whether length, area, or volume - is
not specified in [MD], and this makes their assumption that the growth
rate of a cell of size x is proportional to x (and to f(S)) subject to different interpretations. Their model also can be reduced to the equations
considered in Chapter 1.
An interesting open problem is to construct and analyze a model which,
following [Cu2], treats growth and consumption as in Chapter 9 (i.e., as
proportional to surface area) and which treats cell division as in [MD].
Questions of existence, uniqueness, and continuous dependence of solutions on initial data were not considered for the model discussed in Chapter 9. This problem seems to be an open one. The analysis should follow
along the lines of arguments given in [MD, p. 238].
All of the open problems for the standard gradostat system of Chapter 6 are open problems for the unstirred chemostat model discussed in
Chapter 10. It can be shown [HSW] that the dynamics of the unstirred
chemostat system mirror those of the gradostat in the sense that there
is an order interval, bounded by two (possibly identical) positive rest
points, that attracts all solutions. Furthermore, an open and dense set of
initial data generates solutions that converge to a stable rest point. The
question of the uniqueness of the interior rest point is a major open problem. Another is how to handle the case where the diffusion coefficients of
the competitors and nutrient are distinct. Although there must still be
conservation of total nutrient, it is no longer a pointwise conservation
relation and the reduction to two equations is not clear. Even if accomplished, it may be difficult to exploit. If one is forced to analyze the full
252
Open Questions
Appendices
B Differential Inequalities
C Monotone Systems
D Persistence
E Some Techniques in Nonlinear Analysis
F A Convergence Theorem
253
A
Matrices and Their Eigenvalues
Matrices are encountered on a number of occasions in the text, particularly in arguments for the linearization about rest points or periodic orbits.
On some occasions we encounter matrices of dimension large enough that
direct computation of the eigenvalues may not be feasible. Fortunately,
the matrices are often of a special form and there are theorems to cover
such cases. In this appendix we list some of the useful theorems in the
analysis of these special systems, along with appropriate references.
For stability at a rest point one wishes to show that the eigenvalues of
the linearization lie in the left half of the complex plane. There is a totally
general result, the Routh-Hurwitz criterion, that can determine this. It
is an algorithm for determining the signs of the real parts of the zeros
of a polynomial. Since the eigenvalues of a matrix A are the roots of a
polynomial
f(z)
obtained from
f(z) = det(A-zI),
this theory applies. Unfortunately, the explanation of the algorithm depends on describing a certain index for quotients of polynomials, and the
explanation of the necessary computations is also very complex. Thus,
although the question of the signs of the real parts of the roots can be
answered theoretically, practical application of the theory is very difficult.
A complete explanation of the theory can be found in the appendix to
[Co]. When the degree of the polynomial is small, however, the computations can be carried out. If
aoz3+a1z2+a2z+a3 = 0
with ao > 0, then the relevant condition is
255
256
(zlJz-arJ
pr},
i=1,2,...,n,
Bl
I 0
P,Pj connecting Pi to Pp The resulting graph is said to be strongly connected if, for each pair (P,, Pj), there is a directed path P,Pk,, Pk, Pk2, ...,
P,_,Pj. A square matrix is irreducible if and only if its directed graph is
strongly connected [LT, p. 529].
257
This test reflects the preceding interpretation of "separated compartments." When there is no path between two vertices of the graph, no
material may pass from the first to the second vertex. Hence there is no
influence of the first on the second.
For special matrices there are theorems that give information about the
stability modulus. A matrix is said to be positive if all of the entries are
positive; this is written A > 0. (Similarly, a matrix is nonnegative if all of
the entries are nonnegative.) The very elegant Perron-Frobenius theory
applies to such matrices.
THEOREM A.3. If then x n matrix A is nonnegative, then
258
The following theorem is a consequence of Theorem A.4 and the observation that ifA is a matrix such that A+cI >_ 0, then (A+cI) = s(A)+c.
THEOREM A.5. If A is irreducible and has nonnegative off-diagonal elements, then s(A) is an eigenvalue with algebraic multiplicity 1, and Re A <
IL'
DE
A=(B C1
JJ
Another concept that is important is that of "positive (or negative) definite." For this, it is required that A be a symmetric matrix, that is a,j = aj,.
An important theorem is the following.
THEOREM A.8. The spectrum of a symmetric matrix is real.
A symmetric matrix A is said to be positive definite if all of the eigenvalues are positive; it is said to be negative definite if all of the eigenvalues are negative. Semidefinite is similarly defined. There is a simple
test to determine if a symmetric matrix is positive or negative definite.
259
The principal minors are those determinants whose upper left element is
all and have contiguous rows and columns. For example, the first principal minor, dl, is just all. The second is given by
d2 = detl all
a21
a21
a22
all
a12
...
aln
a21
a22
...
a2n
and
ant
...
ann
do = det
d1>0,d2>0,...,dn>0.
THEOREM A.10. A symmetric matrix is negative definite if and only if
d1<0,d2>0,...,(-1)ndn>0.
Theorem A.10 allows us to conclude stability if the matrix is the variational matrix evaluated at a rest point. An important result is that the
test in Theorem A.10 will work for some matrices that are not symmetric,
not in the sense of being negative definite but in the sense of yielding stability based on the sign of the real parts of the eigenvalues. The type of
matrix is closely associated with the orderings and monotone flow discussed in Appendices B and C.
THEOREM A.11 [S6, thm. 2.7; BP, chap. 6]. Let A be as in Theorem A.6.
Let A be defined by
A=
E
J
Then s(A) < 0 if and only if (-1)kdk > 0, k = 1, 2, ..., n, where dk is the
kth principal minor of A.
Finally, we will have need of the following.
260
Differential Inequalities
The same notation will be used for matrices with a similar meaning.
Let f : IIB X D - E", where D is an open subset of W", be a vector-valued
function, f = (fl, fZ, ..., f"). We first give the general form of the needed
condition. The function f is said to be of type K in D if, for each i and
all t, f (t, a) <- f (t, b) for any two points a and b in D satisfying a:5 b
and a,=b;.
The object is to compare solutions of the system of differential equations
x'=f(t,x),
(B. 1)
(B.2)
Y' ? f(t,Y)
(B.3)
or
262
Differential Inequalities
x'=f(t,x)+(1/m)e
satisfying xm(a) = x(a), where e = (1, 1, ..., 1). Then [H2, chap. 1, lemma
3.1], xm(t) is defined on [a, b] for all sufficiently large m and xm(t) ->x(t)
as m oo, uniformly on [a, b]. We show that z(t) <xm(t), a < t < b, for
all large m, from which the first assertion of the theorem follows by taking limits as m -+ oo. The second assertion is proved in a similar manner.
Let m >- 1 be fixed such that xm(t) is defined on [a, b]. As zi(a) = xmi(a)
(the latter is the ith component of xm(a)) and z,(a) < xmi(a) for 1 <_ i <_ n,
it follows that zi (t) < xmi (t) for t > a and t -a small. Consequently, if
z(t) < xm(t) does not hold for some t e (a, b) then there exist j and to C
(a, b) such that zi(t) < x,,,i(t) for a < t < to and 1 <_ i:5 n and such that
zj(to) =xmj(to). Therefore,
f (to, z(to)) ? zj' (to) ? x,,,-(to) = f (to, xm(to))+(1/m) > f (to, xm(to))
But z(to) <_ xm(to) and zj(to) = xmj(to) implies, by the type-K condition,
that fj(to, z(to)) <_ fj(to, xm(to)). This contradiction proves the theorem.
See [Co] for a more general result. The theorem is traditionally used when
compare two solutions of the same equation which have related initial
values.
i#j, (t,x)eD,
(B.4)
hold. If y(t) and z(t) are two solutions of (B.1) defined for t >_ to satisfying y(to) <_ z(to), then y(t) <_ z(t) for all t >_ to.
Differential Inequalities
263
Proof. Conditions (B.4), together with the fundamental theorem of calculus, imply that f is of type K in 118 x D. In fact, if a <- b and a, = bi then
of (t,a+r(b-a))(bj-aj)dr>0,
f(t,b)-f(t,a)= fo
X(t) = ai (t, s,
satisfies X(s) = I (the identity matrix), and X(t) is the fundamental matrix solution of
z'(t) =
aX (t,
x(t))z(t),
(B.5)
0,
t?s.
(B.6)
Recall that the inequality (B.6) means that each entry in the matrix is
nonnegative. A stronger conclusion holds if (B.1) is cooperative and irreducible.
THEOREM B.3. Let (B.1) be a cooperative and irreducible system in R x D.
Then
Differential Inequalities
264
ax(t,s,
(B.7)
< 2,
then
<x(t,s, 2)
(B.8)
for t > s.
Proof. Inequality (B.8) follows from (B.7) and the formula
i)_ J
Let
X(t) = ax (t, s, E)
and
0=xkj(t)akl(t)x1j(t)=
!
akl(t)xlj(t)
But, by the definition of S, it follows that xlj(t) > 0 for t near eij, 1 E Sc.
Consequently there exists t, < eij such that xlj(t) > 0 for t, < t and all
1 e S`. It follows from the equality just displayed that akl(t) = 0 for t, <
t <_ eij and for all l E S`. As k e S was arbitrary, we have that akl(t) = 0
for t, < t :s eij for all k e S and all 1 e S'. This contradicts the irreducibility
of A(t) and so completes the proof.
x'=F(t,x,y),
y'=G(t,x,y),
(B.9)
where x, y e 118", H = (F, G): R x D -> ff 2n, and D C l 2 " is open. The func-
Differential Inequalities
265
bi; and (b) for each j (n+1 <- j< 2n), Gj(t, a, c) <- Gj(t, b, d) whenever
a >: b, c:!5 d, t E 118, and cj = d;. It is assumed that (a, c) and (b, d) belong
alized type Kin D, let D'= PD. The change of variables (u, v) = P(x, y)
in (B.9) yields the system
(B.10)
type K in D', and conversely. Also observe that (b, d) <-K (a, c) if and
only if P(b, d):5 P(a, c). These simple observations allow us to state analogs of each of the preceding results.
Consider the differential inequality
u'<- F(t, u, v),
(B.11)
Observe that if z = (u, v) then (B.11) is just z' <-K H(t, z).
The following result is the analog of Theorem B.I. There is, of course
an analogous result for the reverse set of inequalities in (B.11).
THEOREM B.4. Let H be continuous on 118 x D and suppose that H is of
generalized type K in D. Let (x(t), y(t)) be a solution of (B.9) on an interval [a, b]. If (u(t), v(t)) is continuous on [a, b] and satisfies (B.11) on
(a, b) and if (u(a), v(a)) <-K (x(a),y(a)), then (u(t), v(t)) <-K (x(t), y(t))
266
Differential Inequalities
aF;
ax;
> 0,
8G`
_0'
ay;
#j;
(B.12)
aF; < 0,
aGi < 0,
aye
axe
all i, j.
Let (x(t),y(t)) and (u(t), v(t)) be solutions of (B.9) defined for t >_ to
satisfying (x(to), y(to)) <_K (u(to), v(to)). Then (x(t), y(t)) <_K (u(t), v(t))
for all t >_ to.
(t,
act
s,a)=
ax
ax
aE
an
ay
ay
La
a,,
satisfies
ax
ay
a ' 0, 2: 0,
17
ax<0,
a?,
(B.13)
ay<_0.
THEOREM B.6. Let (B.9) satisfy (B.12) and suppose that (aH/az)(t, z) is
irreducible for each (t, z) e R x D, where D is a convex subset of R2 ' and
H = (F, G), z = (x, y). Then strict inequality (> or <) holds in each of
the four inequalities of (B.13). Furthermore, if (i,, n;) for i = 1, 2 are distinct points of D, se Q8, and ( 1, n1) <K ( 2,17 2), then
z(t, S, E1, 171) <K z(t, s, 2, 112)
(B.14)
for t > s.
In the systems treated in this work, each component represents the concentration of a nutrient or microbial population and hence must be nonnegative. Therefore, in order to be biologically meaningful, the systems
Differential Inequalities
267
treated here must have the property that R is positively invariant. The
next result supplies sufficient conditions for this basic property to hold.
PROPOSITION B.7. Suppose that f in (B.1) has the property that solutions of initial value problems x(to) = xo ? 0 are unique and, for all i,
f (t, x) ? 0 whenever x >- 0 satisfies xi = 0. Then x(t) > 0 for all t >- to
for which it is defined, provided x(to) > 0.
Proof. The assertion is obvious when f satisfies the stronger condition that
f (t, x) > 0 whenever x ? 0 satisfies xi = 0. The general case can be treated
Let H = (F, G) satisfy the hypotheses of Corollary B.5, and suppose there
exists zo ED such that 0 <-K H(t, zo) for all t ? to. Then the solution of
(B.9) and z(to) = zo satisfies zo <-K z(t) for all t ? to. A similar conclusion
holds if the inequalities between vectors are reversed.
Proof. Only the first assertion regarding (B.1) will be treated in detail.
Following the change of variables y = x-xo, this assertion is equivalent
to showing that solutions of y'= f(t, xo+y) which begin nonnegative remain nonnegative. Because f is type K, we have that if y > 0 and yi = 0
for some i then
f(t,xo+Y)?fi(t,xo)?0.
Therefore, Proposition B.7 implies the desired conclusion.
C
Monotone Systems
x'= AX),
(C.1)
x # y implies that lr(x, t) <K ir(y, t) for all t > 0. A dynamical system
is a monotone (resp. strongly monotone) dynamical system with respect
to <_ (<) when these conditions hold with <_ replacing <_K (< replacing <K). In order to simplify the statement of results, we assume that D
is convex throughout this appendix.
Sufficient conditions for (C.1) to generate a dynamical system which is
monotone (strongly monotone) are given in Appendix B. Corollary B.2,
Theorem B.3, Corollary B.5, and Theorem B.6 imply the following result.
THEOREM C.1. If (C.1) is cooperative in D, then ir is a monotone dynamical system with respect to <_ in D. If (C.1) is cooperative and irreducible
Monotone Systems
269
(C.1) has the form (B.9) where F and G are independent oft and (B.12)
holds in D, then it is a monotone dynamical system with respect to <K.
If in addition, the Jacobian matrix off is irreducible at every point of D,
then 7r is strongly monotone with respect to <_K.
Hereafter, results will be stated only for the partial order <_K; it will be
understood that they hold as well for the order :5.
Although the most important and well-known results of the theory hold
only for strongly monotone dynamical systems, there are some significant results that hold when it is merely a monotone dynamical system.
For example, Theorem 4.2 of Chapter 4 implies that if the dimension of
the state space is 2 (xE R2) then every bounded forward (or backward)
orbit of a monotone dynamical system converges to an equilibrium. This
result no longer holds for higher-dimensional monotone dynamical systems. The next result gives two different sufficient conditions for a bounded
THEOREM C.2. Let y (x) be an orbit of the monotone dynamical system (C.1) which has compact closure in D. Then either of the following
conditions is sufficient for w(x) to be a rest point:
(a) [S] 0 :5K f(x) (f(x) :5K 0);
(b) [Hi3] X <K lr(x, T) (lr(x, T) <K x) for some T > 0.
Proof If 0:5K f(x) then, by Corollary B.8, X :5K lr(x, t) for all t >_ 0.
By monotonicity, lr(x, s) <_K 7r(x, t+s) for t, s >_ 0. Therefore, 9r(x, t) is
monotone nondecreasing in t and lim, lr(x, t) = e exists. The hypotheses ensure that e is a rest point.
If X <K 7r(x, T) then monotonicity implies that
which is closed under addition and subtraction (it's a subgroup of (R, +))
and which contains nT for every integer n. Since the strict inequality
x <K lr(x, T) holds, the continuity of rr implies that for some e > 0, x <K
lr(x, T+s) holds for all s satisfying Isl < E. Arguing as before, this implies that w(x) is a periodic orbit generated by a solution having period
270
Monotone Systems
T+s. But w(x) is the orbit of the point p, so P must contain the interval
(T-e, T+e) and consequently it must contain the interval
Since
P is closed under addition, it must contain an open interval of length
2e centered on each of its points. Therefore, P is open. Since it is also
closed, P = R. This implies that p is a rest point and that ir(x, t) -> p
ast---.
Theorem C.2(b) immediately leads to the following result, first proved by
Hadeler and Glas [HG]. See also [Hi2].
the stable manifold cannot contain two distinct points that are related
by <-K. In other words, the stable manifold is unordered.
Monotone Systems
271
xt
Proof. If x0 is a hyperbolic rest point then B = M+(xo), the stable manifold of x0. Since x0 is hyperbolic and unstable, M+(xo) has empty interior (see the proof of Theorem F.1). It follows that M+(xo) cannot contain two points xl and x2 satisfying x, <K x2, since then M+(xo) would
contain all points of the open set {z: x1 <K Z <K x2). The second assertion
of the theorem follows from strong monotonicity and the positive invariance of the stable manifold.
Figure C.1 depicts a two-dimensional stable manifold in 083. The reader
should sketch a one-dimensional stable manifold in 082.
A similar assertion to that of Theorem C.4 holds for a compact limit
set of a monotone dynamical system.
272
Monotone Systems
such that 7r(x0, t)) <K lr(xo, t2) = lr(lr(xo, t,), t2 - ti). By Theorem C.2(b),
L is a rest point and the proof is complete in this case.
Consider now the case that L is the alpha limit set of -y-(xo). Let x(t) _
lr(xo, t) for t - 0. Arguing as before, there exists t, < 0 such that x, <K
x(t,) and t2 < t, such that x(t2) <K XU,). Continue by choosing t3 <
t2 such that x(t3) <K x2 and t4 < t3 such that x(t3) <K x(t4). Therefore,
the interval I = [t4i t,] contains the interval [t4, t3] on which x(t) "falls"
and the interval [t2, t,] on which x(t) "rises," and these intervals are disjoint. This contradicts the following lemma, proving the theorem in this
case.
Proof. The most important observation is that if [a, b] is a rising (falling) interval contained in I and if s> 0 is such that [a+s, b+s] is contained in I, then the latter is also a rising (falling) interval in I. In fact, if
x(a) <_K x(b) and equality does not hold, then by monotonicity x(a+s) =
lr(x(a), s) <_K 7r(x(b), s) = x(b+s) and equality does not hold. Consequently, rising and falling intervals remain such under right translation.
Suppose that I contains the falling interval [a, r] and the rising interval
[s, b], and suppose that a < r < s < b. The other cases can be treated similarly. Let A = It e [s, b]: x(t) <-K x(s)] and s'= sup A. Then s:5 s'< b
and [s', b] is a rising interval that contains no falling interval [s', r] for
any r e (s', b]. Redefines = s'so that the rising interval [s, b] has the same
property as just described for [s', b]. A contradiction will be reached in
each of the two cases r-a <- b-s and r-a> b-s.
Monotone Systems
273
either inequality. Let c = sup[te [a+b-r,s]: x(b) <-K x(t)]. Then c <
s < b and x(b) <-K x(c), so [c, s] is a falling interval adjacent to the rising
interval [s, b]. Ifs-c <- b-s then [s, 2s-c] is a translate to the right of
the falling interval [c, s], contained in [s, b], and therefore is a falling
interval. But this contradicts the earlier argument that [s, b] contains no
such interval. If s-c> b-s then c < c+b-s <s and [c+b-s, b] is a
right translate of the falling interval [c,s], so it too is a falling interval.
Therefore x(b) <-K x(c+b-s), with equality not holding, and this contradicts the definition of c.
Theorem C.5 places strong restrictions on how a limit set is imbedded in
W. Note in particular that a periodic orbit may always be considered as a
limit set of one of its points and hence Theorem C.5 applies to a periodic
orbit. This should convince the reader that periodic orbits are ruled out
for two-dimensional monotone systems.
The following result of Hirsch [Hill exploits the strong restrictions on
how a limit set is imbedded in space, described in the previous theorem.
THEOREM C.6. A compact limit set of a monotone dynamical system in
Il8" can be deformed by a Lipschitz homeomorphism (with a Lipschitz
inverse) to a compact invariant set of a Lipschitz system in 118"-I in such
a way that trajectories are mapped to trajectories and such that the parameterization of solutions is respected.
274
Monotone Systems
F(y) = QL(.f(QL'(Y)))
on Q(L). Since a Lipschitz vector field on an arbitrary subset of R"-1 can
be extended to a Lipschitz vector field on all of R"-1 while preserving the
Lipschitz constant [McS], it follows that F can be extended to all of H
Theorem C.6 establishes that the dynamical system generated by the vector field (C.1) on R", restricted to the limit set L, is topologically equivalent to the dynamical system generated by a Lipschitz vector field on
Monotone Systems
275
that the limit set is, in fact, a single periodic orbit (see the references listed
just prior to Theorem C.7).
but which vanishes on the boundary of S. Smale shows that this latter
vector field can be extended to a smooth vector field on R having the
form x, = x,M; (x) where 8M;/8xj < 0; that is, the system is competitive.
Furthermore, all solutions corresponding to positive initial data approach
the invariant set S as t - oo. Therefore, the competitive system can have
essentially arbitrary dynamics on S, consistent of course with the fact
that S is an (n -1)-dimensional manifold! For example, if n = 3 then one
could choose h so that in the interior of S there is a single equilibrium
surrounded by a single periodic orbit. Several periodic orbits could easily
1r: E=fx:f(x)=01.
276
Monotone Systems
closure in D for every xe D. Then the set of points x for which 7r(x, t)
does not converge to an equilibrium has Lebesgue measure zero.
Recall that x e D is an accumulation point of E if every neighborhood of
x contains a point of E different from x. The hypotheses of Theorem C.8
exclude such points. In most applications, E is a finite set and so this
hypothesis holds.
Hirsch proves (a more general result than) Theorem C.8 by first obtaining the following result, which is useful in its own right, and then
appealing to Fubini's theorem of integration theory.
THEOREM C.9. Let the hypotheses of Theorem C.8 hold, except that E
is assumed to be a finite set, and let J be a compact, totally ordered arc
contained in D. Then w(x) is an equilibrium point for all except possibly
a finite subset of points of J.
IC
Persistence
As we have seen in the text, the equations governing interacting populations in a chemostat-like environment eventually take the form
xi = xif (xl, x2, ..., xn),
(D.1)
i=1,2,...,n
for every trajectory with positive initial conditions. The system (D.1) is
said to be uniformly persistent if there exists a positive number e such that
i=1,2,...,n
for every trajectory with positive initial conditions. The term "persistent"
was first (?) used in this context by Freedman and Waltman, [FW1], with
lim sup instead of lim inf. Other definitions are relevant; see Freedman
277
278
Persistence
and there exists a compact set G in E such that the invariant set Sl =
UXEEW(x) lies in G. A nonempty invariant subset M of X is called an
isolated invariant set if it is the maximal invariant set in some neighborhood of itself. Such a neighborhood is called an isolating neighborhood.
The stable (or attracting) set of a compact invariant set A is denoted
by W+ and is defined as
W+(A) = (x I xe X, w(x) # 0, w(x) C A
The unstable set W- is defined by
Persistence
279
where a,y(x) is the alpha limit set of the orbit y through x. The weakly
stable and unstable sets are defined as
LEMMA D.I. Let M be a compact isolated invariant set for the dynamical system 7r, defined on a locally compact metric space. Then for any
xe Ww (M)\W+(M) it follows that
Persistence
280
isolated invariant set for 7r. This is a sort of "hyperbolicity" assumption; for example, it prevents interior rest points (or other invariant sets)
from accumulating on the boundary. In this case M is called an isolated
covering. The boundary flow ira is called acyclic if there exists some isolated covering M = U I M/ of ira such that no subset of the M,s forms
a cycle. An isolated covering satisfying this condition is also said to be
acyclic.
(H)
Persistence
281
of points inside the positive cone must eventually be inside a ball (the
dissipative hypothesis) and "outside" a strip along the boundary (uniform
persistence). This region is homeomorphic to a ball in I18". Thus it has the
fixed-point property.
E
Some Techniques in Nonlinear Analysis
In this section, we compile some results from nonlinear analysis that are
used in the text. The implicit function theorem and Sard's theorem are
stated. A brief overview of degree theory is given and applied to prove
some results stated in Chapters 5 and 6. The section ends with an outline
of the construction of a Poincare map for a periodic solution of an autonomous system of ordinary differential equations and the calculation of its
Jacobian (Lemma 6.2 of Chapter 3 is proved).
In working with nonlinear differential equations, one often faces the
problem of solving nonlinear equations of the form
f(x) = 0,
(E.1)
where f is a map from one Euclidean space lJ 'to another one 118'. When
p > m and a particular solution can be found, the implicit function theo-
rem then gives a method of finding all nearby solutions. Because this
result is used so frequently, we state it here (following [H2]).
IMPLICIT FUNCTION THEOREM. Suppose that F: 1 m x R"- 118m has con-
det az (0, 0) * 0,
then there exist neighborhoods U of 0 E 118' and V of 0 E R" such that for
each fixed y c- V the equation F(x, y) = 0 has a unique solution x e U. Furthermore, this solution can be given as a junction x = g(y), where g(0) _
0 and g has continuous first partial derivatives.
More generally [Smo], g is as smooth as F. For example, if F has continuous second partial derivatives then so does g.
282
283
Recall that the rank of the Jacobian matrix Df(x) at x is the dimension
of its column space (or range). When p = n, r must be at least 1 and the
rank condition becomes det Df(x) = 0.
We now return to the problem of solving (E.1). When the Euclidean
spaces have the same dimension, a more topological approach to solving
equation (E.1) is often appropriate. In what follows we describe such an
approach, called (topological) degree theory. Degree theory is useful because it gives an algebraic count of the solutions of the equation (E.1)
that is stable under small perturbations of the function f. We will often
refer to a solution of (E.1) as a zero of f. For example, if f is the righthand side of a differential equation then we will be concerned with the set
of rest points of that equation. A full account of degree theory is beyond
the scope of this brief appendix. Instead, we will simply state some of its
important definitions and properties and then go ahead and use it for our
purposes. The reader may consult [Smo] for a more thorough account
of the theory.
Let f be continuously differentiable on an open bounded subset 0 of
R' and continuous on the closure, O, and suppose that f has no zeros on
the boundary of 0. If it is assumed that every solution of (E.1) is nondegenerate - in the sense that the Jacobian J(x) of f at the zero xe 0 is
deg(f, 0) =
(E.2)
f(X)=o
where sgn denotes the sign (+1 or -1), det denotes determinant, and the
sum is taken over all zeros of f in 0. This sum is finite by the inverse function theorem [Smo] and because O is compact and f does not vanish on
284
Sard's theorem - and the fact that continuous functions can be uniformly approximated by continuously differentiable ones on O - allows
one to extend this definition to continuous functions that do not vanish
on the boundary of O. So defined, the degree of a mapping has many
useful properties, among which the following will be particularly important for our purposes.
HOMOTOPY INVARIANCE. If H(x, t) = O has no solutions (x, t) for which
t), 0) is
points of certain vector fields that have the monotonicity properties described in Appendix C. Consequently, we will use the notation from that
appendix. As the applications of our results come from Chapters 5 and
6 and it will frequently be necessary to refer to numbered equations from
one of these chapters, we use the following convention: equation number
(5.2.4) will refer to equation (2.4) in Chapter 5.
In order for the degree of a mapping to be stable to small perturbations
of the mapping, the restriction that f not vanish on the boundary of 0
is obviously necessary. However, for the problems we have in mind, it
turns out that f does vanish on the boundary of the appropriate open set
0; our goal is to show that there must exist solutions in O. This dilemma
occurs frequently in the applications. One often has quite a bit of knowledge about the zeros off on the boundary of 0 and very little information about those inside O. For example, in Chapters 5 and 6, we may
know that rest points E0, El, E2 lie on the boundary of the region F and
we may also know the stability properties of these rest points. We would
like to know if there is a positive rest point in the interior of F. Of course,
one is free to choose the open set 0 in a different way so that no zeros
285
cannot belong to the boundary of [x1, x2]. In fact, if x1 <- xo <- x2 and
equality does not hold, then strong monotonicity of the dynamical system
induced by f implies that x1 < xo < x2.
REMARK 3. Generically, one expects x0 to be nondegenerate, which implies that s(Df(xo)) > 0 or that x0 is unstable.
Proof of Theorem E.1. Let 0 denote the interior of [x1, x2], and let y
be a point of the line segment joining x1 and x2 so that y-x1 > 0 and
286
X2-Y> 0. Define F(x,s) = f(x) -s(x-y) and note that F(x,s) is cooperative in [x1, x2] for s? 0. Since F(x1, s) = s(y-x1) > 0 and F(x2, s) =
-s(x2-y) < 0 for s>0, Corollary B.8 implies that [x1, x21 is positively
invariant for the autonomous differential equation x' = F(x, s) with fixed
positive parameter s. Let lr(x, t, s) denote the solution operator corresponding to this one-parameter system. Theorem C.2 and the inequalities
just listed imply that lr(x1i t, s) monotonically increases to a rest point in
O as t increases. Similarly, 7r(x2i t, s) monotonically decreases to a rest
point in 0 as t increases. We use the implicit function theorem to study
these two rest points.
Since DF(x1, 0) = Df(x1) is nonsingular by hypothesis, the implicit
function theorem implies that there exists a smooth branch of rest points
x=X1(s),X1(0) = x1, F(X1(s),s) =0. Implicit differentiation of the last
identity at s = 0 gives
dX1
ds
(0)
-Df(x1)-1(y-x1).
.
s, 0), 0) =
s, 1), 0) =
s), 0).
287
0) =
03)
s), 03).
Consequently,
03) _ (-1)"+1
and therefore F(x, s) = 0 has a solution x = X0(s) E 03(5) for small positive s, by the solution property of the degree. Clearly, X0(s) # X,(s) for
i = 1, 2, and the comparison argument (using the differential equation)
shows that Xo e [X1, X2].
If every zero x of F(., s) in 03(s) satisfies s(DXF(x, s)) < 0 then, using
the domain decomposition property as before, we would have
s), 03) = p(-1)",
where p >- 1 is the number of such zeros. Since this contradicts the previous
displayed formula, X0(s) may be chosen to satisfy s(DXF(Xo(s), s)) >- 0.
Since [x1, x2] is compact, we can select a sequence s" such that s" -> 0
and Xo(s") xo as n -> oo. Continuity of F implies that f(xo) = 0. Furthermore, xo * x; for i = 1, 2, since if (for example) xo = x1 then (Xo(sn), sn)
would be a branch of solutions of F = 0 distinct from the branch (XI (s), s),
both converging to (x1, 0). This violates the uniqueness of the latter branch,
as provided by the implicit function theorem. This establishes the existence of a zero xo for f distinct from the xi. Finally, s(Df(xo)) >- 0 by
continuity of DXF(x, s) and the fact that s(DXF(Xo(s), s)) >- 0. This completes the proof.
We now turn our attention to the zeros of the map F: F --> R 2n, F(x) =
(F,(u, v), F2 (u, v)) where x = (u, v), r = [(u, v) e I18+": u+v <_ z], z > 0,
and
F1(u,v) = [A + Fu(z - u - v)]u,
F2(u, v) = [A+FU(z-a-v)]v.
See Chapter 6, and in particular (6.2.1), for further details. In the case of
the simple gradostat model (5.2.4), we have
288
2
1
Fu(Z-u-v)
-2
z-(3,3),
z
=Cfu(3-u,-v,)
fu(3
u2 - v2)
289
Lemma 6.2.2 lead to the following vector field F(x, e), the analog of
(6.2.1):
F1(u, v, e) =[A
+":
u + v <- z(e)1. The vector field F(x, e) possesses all the monotonicity prop-
de (0) +e = 0.
Therefore
d(0)
e> 0,
where the inequality follows from Theorem A.12 and the fact that s(J1) =
s(A+FU(z-u)) < 0. It follows immediately that El (E) belongs to the interior of F(e) for small positive E. By continuity, s(J1(e)) < 0 where J, (E)
is the Jacobian of F(., c) at El (c).
A similar analysis applies to E2 and we obtain E2(e), a smooth branch
of zeros of F(., e) belonging to the interior of F(e) with s(J2(e)) < 0 for
small positive e, where J2(e) is the Jacobian of F(., e) at E2(e). Clearly,
for small positive E.
E2 (E)
The Jacobian JO of F(., e) at E0, with e = 0, is described in the text following (6.4.1). It is nonsingular, so the implicit function theorem implies
that there exists a zero E0(e) of F(., E) which is smooth in e and satisfies
E0(0) = E. We must determine whether E0(e) belongs to F(e) for small
positive E. Letting E0(e) = (u(e), v(e)), implicit differentiation yields
290
(A+FF(z)) d (0)+e = 0,
(0)+e = 0.
It is easy to see that both derivatives have at least one negative component. For example, if the derivative of u were nonnegative then one easily
sees, by writing the equations in component form, that it is necessarily
positive. But this violates Theorem A.12 since s(A+Fu(z)) > 0. Consequently, E0(E) does not belong to IF(E) for small positive E.
We will now use degree theory to show that there exist zeros of F in
r(E) distinct from the E,(E). Let O(E) denote the interior of F(E). Define
H(x, c, t) = (Au+Ee,Av+Ee)+t(Fu(S(E))u,FF(S(E))v)
for xe r(E) with E > 0 and 0 <- t <- 1. Observe that if u; = 0 then
H(x,E,t), - E>0,
so H has no zeros x with ui = 0. Similarly, it has no zeros with v, = 0.
Suppose that H vanishes at a point xer(E) such that u;+v, =z,(E) for
some i. Let I be the set of all such indices i and let J be the complementary
set of indices. If j e J, then uJ+ vj < zi(E). Taking components of the identity H = 0 we find that, for i e I,
0= Z A,JZJ+ F, Ai(ui+vi)+2E
jeI
jE
<-EA,jzj+2E=-(eo)r:5 0.
J
Since eo * 0, it follows that J is non-empty. Furthermore, the inequality implies that Aij = 0 for every i E I and j e J since otherwise a strict
inequality would occur, a contradiction. But this contradicts the irreducibility of A. We conclude that no zero of H can belong to the boundary of
r(E). The homotopy property of degree implies that
E), O(E)) =
E,1), O(E))
= sgn(det A)2
= +1,
291
Let O,(e) be an open subset of O(e) such that E,(e) e O,(e) and F(., e)
has no zeros in O, (e) other than E, (e). Then
e), Oi(E)) = (_1)2n = +1,
a zero of F. We claim that E. # E0, El, E2. Indeed, this follows by the
uniqueness part of the implicit function theorem and the fact that xn #
E,(1/n) for i = 0, 1, 2. Therefore, by Lemma 6.4.1(d), E. > 0. Since every
solution of (6.2.1) is attracted to [E2, El]K, by a comparison argument
(see e.g. the last part of the proof of Theorem 6.4.4 or Lemma 5.6.1), the
inequality E2 <K E* <K El is easy to see.
Since s(J(e)) >_ 0 for e > 0 and x, -> E* as n
cobian of F in (x, e) implies that s(J*) >_ 0.
We conclude this appendix by constructing the Poincare map corresponding to a nonconstant periodic solution x(t) = x(t + T) of the autonomous
system of differential equations x' = f(x) and by verifying Lemma 6.2 of
292
I(T)e, = e,.
Let the Floquet multipliers (eigenvalues of F(T)) of the variational equation be 1, p,, p2, ..., pn_,, where the terms are listed according to multiplicity and the first one corresponds to the eigenvector e,. Finally, recall from the fundamental theory of ordinary differential equations [H2,
chap. 1, thm. 3.3] that
E=(x:x,=0J=R"-'
as follows. Since we know that solutions starting at points on E near x = 0
remain near x(t) = ir(O, t), at least on bounded t intervals, they will return to E after approximately time T. Consequently, we look for solutions
of 7r,(x, t) = 0 for x near 0 and t near T. For the time being, we do not
restrict x to lie on E. Since
a 7r,
at
(0 T)=f,(0)=a>0
'
'
the implicit function theorem implies that we can solve for t = T(x) as a
function of x for x in some neighborhood N of zero: 7r,(x, T(x)) = 0 satisfying T(0) = T. Restricted to E, T is called the first-return-time map.
Finally, we define Q to be the orthogonal projection onto E along e, that is,
QX=(x2,x3,...,Xn)ER"-'
for x = (x,, ..., x") - and R to be the injection of Rn-' into R" defined by
R(x2, x3, ..., xn) = (0, x2, ..., Xn).
P = QoHoR,
where H(x) = lr(x, T(x)); or, in simpler terms,
P(X2i ..., xn) = Q1r(O, x2, x3, ..., X,, TO, X2, ..., xn))
for (x2, ..., x") E Nn E. By the chain rule and the fact that Q and R are
linear, the Jacobian of P at 0 E R"-' is given by
DP(0) = QDH(0)R.
293
T) +
axal-
at (0' T)
(0).
DP(0) = QP(T)R.
Note that the first column of F(T) is e1. Consequently, the eigenvalues
P1, P2, ",Pn-1 of l(T) are the eigenvalues of the (n-1)x(n-1) lower
right block B of 4(T). Using QR = Ii_1, we find that
DP(0) = B.
It follows that p1, ...,
are the eigenvalues of DP(0), as asserted in
Lemma 6.2 of Chapter 3.
F
A Convergence Theorem
z'=Az,
Y'=AY,z),
(F.1)
and
(F.2)
where
z E III',
xES2=[xI(x,O)ED)CR".
It will be assumed that f is continuously differentiable, D is positively
invariant for (F.1), and (F.1) is dissipative in the sense that there is a compact subset of D into which every solution eventually enters and remains.
The following additional hypotheses will be used.
294
A Convergence Theorem
295
j = r+1, .., p.
(H4) 12 = UP 1 M+(xi)
(H5) Equation (F.2) does not possess a cycle of rest points.
Note first that the only rest points of (F.1) are of the form (xi, 0), and
that each is hyperbolic for (F.1). To avoid confusion between (F.1) and
(F.2), denote the stable and unstable manifolds for (F.1) by A+ and A-,
respectively. Then
Note also that xi and (xi, 0) are locally asymptotically stable for (F.1) and
(F.2), respectively, i = 1, ..., r. By (H4), every point of 12 is attracted to
one of the rest points xi, i = 1, ..., p.
THEOREM F.I. Let (Hl)-(H5) hold and let (y(t), z(t)) be a solution of
(F.1). Then, for some i,
Proof. Let (y(t), z(t)) be a solution of (F.1), and assume that the first
assertion of the theorem is false. If 0 denotes the omega limit set of
this solution, then 0 # 1(xi, 0) for i = 1, 2, ..., p. Let (x, 0) E O. By (H4),
either x = xi for some i or the solution of (F.2) through x converges to
some xi. By the invariance of 0 we conclude that (xi, 0) belongs to 0 for
some i. Clearly, i ? r+1 because points of the form (xi, 0), 1 <_ i <_ r, are
asymptotically stable, and a limit set containing an asymptotically stable
rest point is that rest point, a contradiction to our hypothesis. Since 0 #
{(xi, 01), the Butler-McGehee theorem says that 0 must contain a point
(x, 0) with xE M-(xi) and x * {(xi, 0)1. By (H4), xE M+(xj) for some j,
A Convergence Theorem
296
and consequently x; is chained to x;, x, -> x;, in 0. Continuing the argument a finite number of steps leads to a cycle, contradicting (H5). This
proves the first assertion of the theorem, D C U f 1 A(x;, 0).
It is well known that the stable manifold A+(x;, 0) of a hyperbolic, unstable rest point (x1, 0) has Lebesgue measure zero. This follows from Sard's
theorem (see Appendix E) and the fact that the stable manifold is the image
of a smooth one-to-one map of 0811 into 118" x R', where 1, is the dimension
of the stable subspace of the linearization of (F.1) about (x1, 0) and consequently I, < n+m [Pi, p. 43]. It follows that Up ,+1 A+(x;, 0) has measure zero.
The theorem states that almost all trajectories converge to one of the
asymptotically stable rest points (x,, 0). In most of our applications, r =
1: there is exactly one asymptotically stable rest point. In this case, an
observer would conclude that all trajectories converge to (xi, 0), since
the probability of an initial condition being in the exceptional set is zero.
To illustrate the procedure, Theorem F.1 will be applied to the systems
(5.1) and (5.2) of Chapter 1. The system corresponding to (F.1) is
E' = -E,
X
'-x
'
(M10_E-x1-x2)
1+ai-E-xi-x2 _11,
)
m2(I-E-x1-x2)
X2' -x (
1+a2-E-xi-x2
D=((E,xi,x2)Ix;>O,xi+x2+E:51J.
(F.3)
Cmi(1-x1-x2) -1
I+a1-X1-x2
x -x
2
Cm2(1-x1-x2) -1\
z
(F.4)
I+a2-x1-x2
=f(x1,x2)IXi>0,x1+x2<-11,
which is the same as (5.2) of Chapter 1.
In this case, m = 1 and n = 2. The matrix A is 1 x 1, or A = [-1]. Clearly (Hl) is satisfied. System (F.4) has three rest points, which were labeled
E0, E1, E2 in Section 5 of Chapter 1. We restrict the analysis to the situation of Theorem 5.1 of Chapter 1. (The other cases can be done similarly.) The set 0 is just the portion of the nonnegative cone in 1182 which
A Convergence Theorem
297
lies on the origin side of the line xi +x2 = 1 (SZ is a positively invariant
set). All of the rest points are hyperbolic. Since El is a local attractor, the
dimension of its stable manifold is 2. The stability of El prevents it from
being a part of any chain of equilibria. Similarly, E0 is a repeller, has no
stable manifold, and cannot be part of any chain of equilibria. The rest
point E2 is not chained to itself, so (H4) is satisfied. The dimension of the
stable manifold of E2 is 1. This shows that (H3) is satisfied. The part of
the stable manifold of E2 which lies in 9 is just a portion of the x2 axis;
thus (H5) is satisfied. The proof given of Theorem 5.1 in Chapter 1 establishes (H4) by showing that every solution of (F.4) converges to a rest
point. Theorem F.1 then allows one to conclude that every trajectory of
(F.3) tends to a rest point of (F.4). Orbits through all initial conditions
excluding those in the exceptional set converge to El.
The exceptional set is just the E-x2 face of the positive cone in IR3. No
interesting initial conditions lie here.
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299
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Author index
Andronov, A. A., 9
Aris, R., 16, 159, 161
D'Ans, G., 2
Dawes, I. W., 5
Dean, A. M., 2
DeMottoni, P., 169, 180
251
309
310
Author index
Hunt, T., 6
Hutson, V., 278
Powell, E. 0., 16
Powell, G. E., 25, 27
Powell, T., 71
McCracken, M., 60
MacDonald, N., 238
McGehee, R., 32, 250
Macken, C. A., 244
McShane, E. J., 274
Maier, A. G. ,9
Marsden, J. E., 60
Markus, L., 294
Mass, P., 207
Mateles, R. T., 2
Mayer, H., 51
Metz, J. A. J., 44, 208, 209, 213, 214, 229,
251
Monod, J., 4
Moson, P., 278
Muller, M., 261
Murray, A., 6
Nisbet, R. M., 2, 48, 182, 185, 207, 238,
250
Novick, A., 2
Pavlou, S., 77
Pilyugin, S. Y., 296
Plemmons, R. J., 133, 260
Sutherland, I. W., 5
Szego, G. P., 280
Szilard, L., 2
Tang, B., 103, 126, 127, 128, 129, 138, 147,
157, 207, 250, 285
Taylor, P. A., 2
Telling, R. C., 4
Thieme, H. R., 12, 276, 278, 294
Thompson, H. B., 232, 237
Tilman, D., 27
Tismenetsky, M., 256, 257, 258
Tsuchiya, H. M., 48, 73, 76
Tumpson, D. B., 27
Veldcamp, H., 2
Waltman, P., 2, 3, 12, 16, 37, 43, 79, 99,
100, 103, 126, 127, 129, 147, 157, 161,
181, 183, 232, 237, 238, 241, 244, 251,
Plesser, T., 51
Subject index
contaminant, 41
continuous culture, 2, 4
continuous dynamical system, 7
cooperative system, 95, 120, 169, 263
culture vessel, 3
cycle: of equilibria, 11; of invariant sets,
279
consumption term, 4, 5
311
312
Subject index
Jacobian matrix, 10
Jordan canonical form, 217
knife-edge effect, 18, 24
negative orbit, 8
negative trajectory, 8
non-autonomous equations, 160
nondegenerate rest point, 283, 285
nutrient gradient, 101, 102, 129, 231
omega limit point, 8
omega limit set, 8
open system, 2
orbit, 8
oscillations, 85, 239, 243
oscillatory washout, 160
overflow vessel, 3, 130
p-convex set, 262
periodic orbit, 8
Perron-Frobenius theory, 95, 215, 257
persistent dynamical system, 96, 277, 279
physiological efficiency coefficient, 217,
221, 222
piecewise linear kinetics, 31
plasmid, 243, 244
Poincar6 map, 62, 63, 66, 67, 163, 169, 170,
172, 250, 292
Poincar6-Bendixson trichotomy, 9
pollutants, 78
positive definite matrix, 258
positive orbit, 7
positive trajectory, 7
positively invariant set, 8, 32
predator-prey equations, 43
principal minors, 117, 259
Pseudomonas aeruginosa, 21
quasipositive matrix, 134
repeller, 11
121, 268
Subject index
size distribution, 226
size-structured competitors, 228
size-efficiency hypothesis, 229
solution map, 62, 63
spectral radius, 257
spectrum, 257
stability modulus, 134, 257
stable limit cycle, 43
stable manifold, 11, 117, 163, 205
stable set, 278
stable solution, 10
standard n-vessel gradostat, 130, 131
steady state, 8
stored nutrient, 182, 241
strongly connected graph, 256
strongly monotone dynamical system, 137,
142, 197, 268
313