LAB 1.

3 Logistic Population Models with Harvesting

Abstract

The laboratory experiment deals with the logistic equation presenting two harvesting
cases, which are the constant harvesting and the periodic harvesting, where the nature of
equilibrium points and solutions behaviour has been analyzed for both of the cases by finding
the first integral and solution curve using solving applications. Based in the solved equation,
in constant harvesting the maximum limit where the harvesting of fish will occur is at 0.25
which means that exceeding this value will make the population undefined or the population
will become extinct as the harvesting be done often making the population smaller. While in
the periodic harvesting, since the equation is non-autonomous, the effect of the initial condition
when a=a1 is a decrease in the population of the fish per season of harvesting while a rapid drop
when a=a2.
I. Introduction

Philippine seas supply food for the whole country and livelihood for millions of people.
Fisheries are economically, culturally, socially and ecologically important to all Filipinos.
These resources are in crisis as evidenced by the declining fish catch, size and species
composition around the country. The current condition of fisheries in the Philippines and
worldwide is bleak. Overfishing, illegal fishing and habitat destruction combined with
increased demand for fish and population growth continue to drive fisheries production into a
deeper abyss. Seemingly impossible just 20 years ago, protein deficiency among fishing
communities is now increasing at an alarming rate.

It is possible to model the behaviour of a given fish population under certain conditions.
There are many factors that can be considered: the conditions of their habitat, such as water
quality and temperature, the starting population, how frequently they are harvested, and the
probability that the fish are eaten by the predator. Using a model, we can figure out when and
how many of the fish can be harvested to maximize the amount of food obtained without
completely depleting the population.

This report will explore the logistic growth model with a fish population and how fish
are harvested safely.

II. Objective

The main concern of the population is to investigate the effect of harvesting on a logistic
population model with harvesting.

The purpose of this study is to;

1. Determine the effect on the fish population for various initial condition if a = a1.

2. Determine the effect on the fish population for various initial condition if a = a2.

3. Describe the parameter used in the model using various initial conditions.
IV. Methodology

In this lab, we consider logistic models of population growth that have been modified
to include terms that account for “harvesting.” In particular, you should imagine a fish
population subject to various degrees and types of fishing. The differential equation models are
given below. (Your instructor will indicate the values of the parameters k, N, a1, and a2 you
should use. Several possible choices are listed in Table 1.10.) In your report, you should include
a discussion of the meaning of each variable and parameter and an explanation of why the
equation is written the way it is. We have discussed three general approaches that can be
employed to study a differential equation: Numerical techniques yield graphs of approximate
solutions, geometric/qualitative techniques provide predictions of the long-term behavior of
the solution and in special cases analytic techniques provide explicit formulas for the solution.
In your report, you should employ as many of these techniques as is appropriate to help
understand the models, and you should consider the following equations:

1. (Logistic growth with constant harvesting) The equation

dp p
 kp (1 - ) - a
dt N

represents a logistic model of population growth with constant harvesting at a rate a. For a =
a1, what will happen to the fish population for various initial conditions? (Note: This equation
is autonomous, so you can take advantage of the special techniques that are available for
autonomous equations.)

2. . (Logistic growth with periodic harvesting) The equation

dp  p
 kp1    a 1  sinbt
dt  N

is a non-autonomous equation that considers periodic harvesting. What do the parameters a and
b represent? Let b = 1. If a = a1, what will happen to the fish population for various initial
conditions?

3. Consider the same equation as in Part 2 above, but let a = a2. What will happen to the fish
population for various initial conditions with this value of a?

In your report you should address these three questions, one at a time, in the form of a
short essay. Begin Questions 1 and 2 with a description of the meaning of each of the variables
and parameters and an explanation of why the differential equation is the way it is. You should
include pictures and graphs of data and of solutions of your models as appropriate. (Remember
that one carefully chosen picture can be worth a thousand words, but a thousand pictures aren’t
worth anything.)

III. Results and Discussion

A. Logistic Growth with Constant Harvesting

The equation

dp p
 kp (1 - ) - a eq (1)
dt N

Where:

k= growth capacity; describes the rate of growth of population p.

N= carrying capacity; describes the total amount of p the resource can support.

a= harvesting rate; describes the rate the p will be taken from the system.

p(t)= number of fish at time t.

Represent a logistic model of population growth with constant harvesting at a rate a.

For a = a1, what will happen to the fish population for various initial condition? (Note: The
equation is autonomous, so the group can take advantage of the special techniques that are
available for autonomous equations.

Derivation;

dp  p
 kp1    a
dt  N

dp
  p
 dt
kp1    a
 N

 A B 
   dp
 p - p1 p  p2 

 Alnp  p1  Bln p  p2
  ln(p  p1 )A(p  p2 )B

Which implies,

(p  p1 )A(p  p2 )B  Ce t eq(2)

Where p1 and p2 are roots to the quadratic polynomial in p and A= (p1−p2)-1, B= (p2 –
p1)-1 are found by partial fractions. If we assume the initial population p(0) = p0 is given then
we find that C= (p0−p1)A(p0 –p2)B. It is not, in general, clear how we should solve for p
explicitly. To do this, the values of for N, k, and a are need to find p1 and p2 and thus A and B.
If these numbers were known then polynomial root finding would give explicit formula for p.

Addressing now the qualitative information that is given by the differential equation
itself. To do this, the equilibrium solutions should be first found by solving;

dp  p
 kp1    a
dt  N

dp
If  0;
dt

 p
kp1 -   a  0
 N

-N  kp2 
kp   a  0
K  N 

aN
p2  Np  0 eq(3)
k

Where ;
a 1
b  -N
aN
c
k

To get the equilibrium solutions,

 b  b2  4ac
pt 
2a
  aN 
( N)  N2  4(1)  
 k 
p(t)
1 
2(1)

aN
N  N2  4
p(t)  k eq(4)
1
2

aN
N  N2  4
p2(t)  k eq(5)
2

Assuming that kN ≠ 4a. Otherwise, p1 = p2. The group also noted that p2 is not physically for

aN a
N2  4  N or N  4  0 and that for physically relevant cases p1(t) > p2(t) for all t. To
k k
classify these equilibria we define p1=p+ and p1=p- apply linearization to get,

df  2p  
p p   k1  
dp  N 
 4a 
 k  1 - 

 kN 

Which implies that p1=p+ is a sink and p1=p- is a source.

Table 1.3 Chosen data and Solved p1 and p2 using Excel

Choice k N a1 p1 p2
0
5 0
.21
3.5 1.5
.22
3 0.20 5 3.366025 1.633975
.23
3.207107 1.792893
.24
3 2
.25
2.5 2.5
In the table 1.3, we consider choice 3 were k-value and N-value are constant in solving
p1 and p2. The value for a1 is set from 0 to 0.25, were 0.25 represent as the maximum limit.
That when you extend the value of it the equation will obtain undefined answer. In the table
we use the equation (4) and (5) to solve for the value of p1 and p2 respectively
 6

4
p1,p2

0
0 0.05 0.1 0.15 0.2 0.25 0.3
a
p1 p2

Figure 1.3 Graph of Table 1.3

B. Logistic Growth with Periodic Harvesting

dp  p
 kp1    a 1  sinbt eq (6)
dt  N
In this case it is found out that b is a parameter, which controls the frequency of the
periodic harvesting and a represents the overall amplitude of the periodic harvesting. It is also
noted that in this case the harvesting can be as much as 2a and as little as 0. It also implies that
it is not separable. So there is no clear way to solve this differential equation analytically.

The group also noticed similar behaviour of this experiment in part (1) when assumed
that t = 0. However, in this case, since the system is non-autonomous there are no classical
equilibrium solutions. Instead, for this system, those trajectories, which do not go extinct tend
to a steady long-term oscillatory behaviour.

Though there is no clear analytic solutions to this differential equation, using qualitative
and numerical techniques to the problem is possible. So with the aid of an online graphing tool
for differential equations by Bluffton University, the solutions of this study were graphed.

For initial condition a=a1, values of k and N be of choice 3, then equation eq (6) will be
 dp  p
 0.20p1    0.211  sin(t)  eq (7)
dt  5

Figure 3.2 Slope Field of equation eq (7)

Figure 3.2 shows the slope field of equation eq (7) above with its corresponding initial
condition values.

Figure 3.3 Phase Line for the Equation eq (7)

Figure 3.3 shows the phase line for the solution of the given differential equation and
its different values, as shown these phase lines are wave functions which means that the fish
population is rapidly decreasing.

At a=a2 the same chosen parameters, the differential equation is

Below are the results upon substituting the equation in the said technology.

Figure 3.4 Slope Field of Equation eq (8)

Figure 3.4 shows the slope field of equation the differential equation above with the value of
a=a2.

Figure 3.5 shows the phase lines for the solution of equation eq (8) at different values.
The effect of the sin function are apparent in the slope field. Additionally, the sin function
shifts the equilibrium state below the carrying capacity.
 Figure 3.6 shows that the values after the 0.25, both the equilibrium states disappear
and the population drops off rapidly for all initial values

IV. Conclusion

In this laboratory experiment two possible models for population harvesting were
presented. With these models one can conclude that there is a relationship between harvesting
rate and the long-term population. The group, in general, noted that increased harvesting leads
to ‘more’ extinction-trajectories. Moreover, it is possible to use the autonomous system to infer
behaviours of the non-autonomous system. While the autonomous system has equilibrium
solutions, the non-autonomous seems to have what could be ‘steady-state solutions’. These
steady-state can be used, like equilibrium solutions, as reference point to describe the behaviour
of neighbouring solutions. If this long term patterns were to be thought as steady-state solution
then it can be concluded that the effect of harvesting is to shift these steady-state solution,
which creates initial populations destined for extinction similar to the autonomous system.
Equilibrium solutions as well as steady-state solutions make it possible to determine the long-
term behaviour of a particular solution satisfying an initial condition.
V. Appendix
A. Possible Choices for the Parameters
Choices K N a1 a2
1 0.25 4 0.16 0.25
2 0.50 2 0.21 0.25
3 0.20 5 0.21 0.25
4 0.20 5 0.16 0.25
5 0.25 4 0.09 0.25
6 0.20 5 0.09 0.25
7 0.50 2 0.16 0.25
8 0.20 5 0.24 0.25
9 0.25 4 0.21 0.25
10 0.50 2 0.09 0.25

B. Excel Files
VI. References
1. Differential Equations, Fourth Edition Paul Blanchard, Robert L. Devaney, Glen R.
Hall
2. Fundamentals of differential equations. -- 8th ed. R. Kent Nagle, Edward B. Saff,
David Snider
3. A First Course in Differential Equations with Modeling Applications, 9th ed. Dennis
G. Zill
4. Slope and Direction Fields-Bluffton University.
https://www.bluffton.edu/homepages/facstaff/nesterd/java/slopefields.html