J. Theoret. Biol. (1966) 12, 119-129
Optimizing Reproduction in a Randomly Varying
Environment
DAN COHEN t
Electrical
Biological Computer Laboratory,
Engineering Research Laboratory, University
Urbana, Illinois, U.S. A.
(Received 6 August 1965, and in revisedform
of Illinois,
8 November 1965)
A model is constructed of optimizing long-term growth rate in a randomly
varying environment. Specifically, the model is of an annual plant, the
seeds of which can either germinate and yield more seeds in numbers
which depend on environmental conditions, or remain dormant in the
soil and undergo some decay according to their viability. The optimal
germination fraction is derived for any combination of probabilities of
the yield of seeds per germinating seed, and of various values of the
viability of the seeds. The relevance of the model is discussed.
1. rntmduction
Most living organisms are faced with a considerable risk of failure when
attempting to reproduce. One obvious way to survive and reproduce in a
risky environment is to spread the risk so that one failure will not be decisively
harmful.
2. The Model
Consider an organism such as an annual plant that reproduces only once
in its lifetime, each growth and reproduction cycle being completed within a
discrete time interval (a year). The average number of seeds per germinated
seedling, Y, is a random variable depending on environmental conditions,
and is assumed to be independent of the population density.
Of the total number of seeds present, a fraction G germinates every year.
Of the seeds that do not germinate, a fraction D decays every year. Our
aim is to find that value of G which maximizes the long-term expectation
of growth, i.e. the optimal strategy.
In mathematical terms, the one-step transformation is
St+l = St-S,.G-D.(S,-S,.G)+G.Y,.S,
(1)
t Present address: Research Laboratory
Technology, Cambridge, Mass., U.S.A.
of Electronics,
119
Massachusetts Institute
of
120
D.
COHEN
where S is the number of seeds present. For a sequence of N steps,
s, = s,.n [(I-~).(i-~)+~.yi]n~
(2)
I
where ni is the number of times that a particular Yi occurred in the sequence.
Taking the logarithms and dividing by N gives
log
----=SN
N
logs,
N
~ ~ ~log[(l-G).(l-D)+G.
When N approaches infinity,
1%
7 so= 0, and 2
= Pi,
lim !%S? = C Pilog[(l-G).(l-D)+G.
N
I
N+m
Yi].
SO
(3)
that
YJ
(4)
1%SN is equal to the
associated with Yi. Lim ___
N
long-term average, or the mathematical expectation, of the specific growth
rate of the seed population.
It is clear from equation (4) that decreasing the decay constant D, and
increasing the yield of seeds per germinating seed, Yi, will always increase
the growth rate expectation. We would like to show how the value of the
germination factor G influences the expectation of growth at any particular
combination of D, Yi and P, and what values of G give the maximal growth
rate.
where Pi is the probability
3. The Case of Two Outcomes
Considering first the case where Y, can assume only two values, 0 and Y.
Equation (4) then takes the form
lim 1~=(1-P,)log[(1-G).(1-D)]+P,log[(1-G).(1-D)+G.Y](5)
N
N-rm
where P, is the probability
of lim 1%
of having Yj = Y. We then computed the value
for various combinations
of P,, Y, D and G. Some charac-
teristic results are shown in Figs 1 to 4.
Examining equation (5), it can be seen that when Y or D are large enough
so that G. Y is large relative to (1 - G)(l - D), equation (5) can be simplified
to give
lim &g?!
N-rm
iv
= P,log[G.
Y]+(l-P,)log[(l-@(l-G)]
(6)
FIG. 1. Long-termexpectationof growth rate, GRLT, plotted againstthe germination
fraction G for variouscombinationsof Py, Y and D, with i = 2, Y0 = 0, Y, = Y. All
the cmvesgo from log (1 - D) for G = 0 to f Pg log Y, for G = 1, whichgoesto -co
in this casesincewehave Y0 = 0.
RG.
2. GRLT plotted againstlog (1 - D), showingthe almostlinear relationships
betweenthemexceptwhenPy, Y andG areall low.
GRLT=f(ln
GRLT = f(P,) for varying
Oc5
OQ
x D. and G
Y) ot P=@5
REPRODUCTION
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ENVIRONMENT
123
which is linear with respect to log (1- 0) and log Y. These relationships
can be seen in the linear portions of Figs 2 and 3, respectively.
Equation (5) is also linear with respect to P, over all possible values of
the other parameters, as can be seen in Fig. 4.
Differentiating equation (5) with respect to G gives
Y+D-1
l-’
=-l-G+P,.
(7)
(l-D)(l-G)+G.Y’
Setting equation (7) equal to zero and solving for G,,, gives
03)
Examining equation (8) it can be seen that G,, is a linear function of P,,
and becomes approximately equal to P, when Y is large enough or when
D or P, are close enough to one so that the second term in the equation can
be neglected. The value of G,,, is then independent of D and Y. These
relationships are illustrated in Fig. 5, where, except for a combination of
low Y and low D, the GmaXpoints all fall on the identity line G,,, = Py.
I.c
of )-
uE
C)’
-0.:
,
FIO. 5. G,, plotted against Py, demonstrating
the linearity of the relationshipunder
all conditionsand the approachto the identity G,, = Pr for large Y and D.
Gmax cannot assume negative values. A negative G,,, according to
equation (8), means that G,,,,, = 0, i.e. that in such conditions no positive
growth rate is possible.
124
The conditions
D.
for a positive
COHEN
long-term
growth expectation,
i.e. for
lim log SN > 0, as a function of P,, Y and D, after optimization had already
N
taken place, i.e. when G = G,,,,, are illustrated in Fig. 6. In this figure the
N-m
FIG. 6. A plot of the relationship between D and Pr for various values of Y, after
optimization had already taken place, i.e. G = G,,,, when GRLT = 0. The dotted lines
are approximate extrapolations to the two boundary conditions of Pu = 0, D = 0 and
D = 1, Py = 1.
values of Pr and D for zero long-term growth rate have been plotted for
several values of Y. Each one of the curves divides the unit square into an
area of positive growth towards higher Py values to the right and lower D
values at the bottom, and an area of negative growth in the opposite directions. The two limiting conditions of P, = 0, D = 0 and D = 1, P, = 1
have been calculated by setting equation (5) equal to zero, with G =
G*ax = PY.
4. The Case of Maoy Outcomes
In the case of a finite set of Yi values it is possible to take the derivative
of equation (4)
Y,fD-1
slim, 1% SN aG=CPi
(9)
i
(1~G).(l-D)+G.
x
I
to set it equal to zero, and to compute G,,, for any values of the other
parameters. For the sum to equal zero it is necessary for at least one of the
iN
REPRODUCTION
RAN~WLY VARYING mviR6mim-r
125
terms to be negative, i.e. Yr+ D < 1, and for at least one to be positive,
Yi+ D > 1. That means that at least one Yi > 1 and one Y, 1 < are
necessary for 0 < G,, < 1.
Since the expansion of equation (9) results in a polynomial of G to the
(i- 1)th degree, it is impossible to obtain analytically the value of G,,
for i > 4.
However, since each single term in (9) is a decreasing function of G
for 0 I G 5 1, 0 I; D 5 1, and 0 I Yi, when Yr- (1-D) is either positive
or negative, their sum must also be a decreasing function of G under these
conditions. From this is follows that the derivative can be equal to zero
at most at only one value of G between zero and one, which means that the
long-term growth expectation as a function of G has at most a single maximum in this range.
For 0 < G,, < 1, the derivative must be greater than zero for G = 0,
and smaller than zero for G = 1. For G = 0, equation (9) gives
a Iirn
log
7
&
ac=pi.+$I
1
i
(10)
and for G = 1
Equation (10) is greater than zero for c P, Yi > 1, which is also a necessary
condition
for having a positive 1ong:term growth rate. Equation (11) is
1-D
negative when C Pi 7
> 1, from which it follows that the conditions
i
1
for Max c 1 are given by the inequality
(12)
The expression on the right-hand side of (12) is the harmonic mean of Y1,
symbolized by Hr. We thus have that for G,,, < 1 it is a sufficient condition
that 1-D > Hr.
The harmonic mean bears the following relationship to the arithmetic
mean Pand the variance oi, of Yi,
(13)
126
D.
COHEN
provided that (Y- y)3/H3 is small, compared with (Y- P)/y Using (12)
and (13) we get the necessary condition for Gmax < 1 as
P l-7 4 <l-D.
(14)
(
>
Thus, it is only when the variance of the yield becomes large enough in
relation to the mean yield and to the viability of the seeds that it becomes
advantageous to postpone the germination of some fraction of the seeds.
Even if we do not use the approximate relationship between the harmonic
mean and the variance, as in equation (13), it is possible to make use of the
fact that the harmonic mean is always less than the geometric mean. A
suficient condition that G,,,,, < 1 is thus given by
1 -D
Taking logarithms
> geometric mean of Yi.
(15)
of both sides, we get
lo~w~qp-%Y,
(16)
so that if the mean log seed yield per germinating seed is less than the log
of seed viability, G,,, will be less than one.
The values of G,,, for i > 4 were most conveniently computed from
numerical computations of the long-term growth expectation as a function
of G and of the other parameters. The value of G,,, was found by graphical
interpolation on the plotted curves, or by numerical curve fitting.
A most plausible distribution of Pi(Y) is of an approximately normal
distribution of P, as a function of log Yi. The growth rate expectation for
several such distributions, differing in their means and variances, have been
computed (Fig. 7(a)-(d)). The results are qualitatively similar to the case
of two-valued Yi, in that when there is some probability for high Y, values
(> lo), Cnax tends to equal the weighted probabilities of these high Y,
values.
It is clear that G,,,,, must be less than one if there is even a very small
probability of Yr = 0. Thus, 1 -G,,, ? C Pi for the very small values of
Yi(< 0.1). G,,, always increases with D, the effect decreasing with increased
c P, log Yi.
1
For any number of Yi values,
lim
logs,
-=log(l-D)
N
forG=O
(17)
lim~=FPl.logY,
N
forG=l.
(18)
N-a,
N-rm
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001 @IO
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005, O-20
100 001 040
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127
000
'I-
is plotted as a function of G and for several values of
of P(Y,). The discrete distributions are shown in block
diagrams at the bottom of the corresponding curves.
FIG.
7. (a) to (d) GRLT
D for 4 different distributions
It follows from equation (18) that with G = 1, any finite probability of
Y, = 0 will lead to a - co value for the long-term growth rate expectation,
i.e. population size will be zero.
The above relationships are illustrated very well in Fig. 7(a) to (d).
095
I.00
128
0.
COHEN
5. Discussion
Our model is formally similar to that of economic decision making under
risk. G is equivalent to the fraction of capital invested, D to the depreciation
of uninvested capital due to rise in prices, fees, taxation, etc., and Y is
the return on the investment. The results are formally exactly the same.
For the biological model the results show the necessity of having the
combination
of high yield when successful, the ability of ungerminated
seeds to survive for many generations in the soil and a low yearly germination
fraction, in order to survive in an environment in which there is a high
probability of total failure.
Conversely, with a high probability for successful reproduction, optimal
germination fraction is high and the ability to survive for a long time becomes
less important.
Of a particular interest is the relation between G and D. When G,,, is
close to one, due to high probability of high yield, the effect of D on the
growth rate is negligible (Fig. 1). Thus, in such an environment there will
be almost no selection for seeds with better ability to survive and D would
remain high.
Under conditions of increasing probability of no yields, selection will
act to decrease G, following which the advantage of decreasing D will become
greater, so that D will be also decreased.
The same relations operate in the opposite direction. Where D is very
high because of any combination of external or internal factors there would
be a strong selection to keep G close to P,. Where D is low, and specially
with a low Y, G,,,,, may be much lower than P,.
We can expect, therefore, to find in seeds and spores in their natural
environment a high positive correlation between the fraction which germinates
and the fraction of those that do not germinate which does not survive each
generation. Such a relationship is indeed well known in seeds (Mayer &
Poljakoff-Mayber,
1963). We can also expect to find the fraction that germinates every year to be approximately equal to the probability of producing
a high yield, and the fraction which does not germinate to be approximately
equal to the probability of total or near total failure to produce seeds.
The results of our model have nothing to say about one important characteristic of population growth, which is the prediction of the actual or the
most probable sequence of sizes as the population goes on growing in time.
Even the very important measure of the deviation from the expected longterm size, usually measured by the variance, is not given. It can definitely
be said that for each long-term growth rate there is an infinite number of
possible sequences and an infinite number of different variances, all leading to
the same limit of growth expectation. It is intuitively clear, for example,
REPRODUCTION
IN
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ENVIRONMENT
129
that in the case of two-valued Yi, a combination of low Py and high Y will
be expected to result in wider fluctuations and higher variance than a combination of high Pr and low Y having the same long-term growth rate expectation.
It is hoped that a model will be constructed from which it would be possible
to predict the variance of the growth expectation at any point in the time
sequence and the general form of the trajectory.
Supported by the United States Public Health Service grant GM-10718 (03).
I would like to thank Professor H. Von Foerster, head of the Biological Computer Laboratory, for his encouragement and stimulating discussions, and Dr
Klaus Witz of the Department of Mathematics, University of Illinois, for his
helpful suggestions and criticism. It is also a pleasure to thank Mr B. Lipnitzky
for his assistance.
REFERENCJSS
MAYER, A. M. & POUAKOFF-MAYBER, A. (1963).Chapter7 ia “The Germinationof Seeds”.
Oxford, New York: PergamonPress.