Question

MATLAB MATLAB MATLAB

Model description The logistic map is a function that is often used to model population growth. It is defined by P(t+1) = rP(t) (1 -P(t)/K)

Here, P(t) represents the density of a population at year 1, the parameter r is a growth rate and the parameter K is the maximum possible population density (known as the carrying capacity). This equation says that if we know the density at one year, we can substitute it into the right-hand side to find the population at the next year. For instance, if we knew the population at year 0 (given by P(0)), we could then calculate

P(1) = rPO (1-P(0)/K)

Once we had the population density at year 1, we could then find the density at year 2 using

P(2) = rP(1)(1-P/K).

We can continue in this manner for as many steps as we want. (It is important to notice that we started counting time at t=0, not t= 1. This is a common choice in mathematical modeling, but it is not the way Matlab indexes vectors.)

For the rest of this problem, we will assume that P(0) = 8 and K = 20, but we will use various values of r.

Calculating population densities Suppose that r = 2.5. Calculate P(498), P(499) and P(500) and save them in that order) in a 1x3 row vector called ans1. Try plotting P VS t. You should see that the population density approaches a single value and stays there forever. This value of P is called a stable equilibrium or stable steady state of the logistic map. We say that the logistic map has a stable equilibrium when r = 2.5.

Repeat the same process with r = 3.2 and save the vector P 498), P (499), P(500) in a 1x3 row vector called ans2. Try plotting P vs t.

Similarly, repeat the same process with r= 3.5 and save the vector P 498), P (499),P(500) in 1x3 vector called ans3. Try plotting P vs t. In the latter two cases, you should see very different behavior. In particular, the population density should not approach a single value, so we say that the logistic map does not have a stable equilibrium for these values of r.

Answer 1

PLEASE REFER BELOW CODE

close all
clear all
clc

%given data
P(1) = 8; %P(0) = 8
K = 20;

%at r=2.5
r = 2.5;

%given expression
%P(t+1) = rP(t)(1-P(t)/K)

%finding all values of P(t+1) for t=1 to 499
for t = 1:499
    P(t+1) = r * P(t) * (1-(P(t)/K));
end
%size(P)
%collecting P(498), P(499) and P(500)
ans1 = [P(498) P(499) P(500)]
t = 0:499; %t vector
figure();
plot(t,P); %plotting P vs t
xlabel('t');
ylabel('P(t)');
title('population density when r = 2.5');

r = 3.2;
for t = 1:499
    P(t+1) = r * P(t) * (1-(P(t)/K));
end
%size(P)
ans2 = [P(498) P(499) P(500)]
t = 0:499;
figure();
plot(t,P);
xlabel('t');
ylabel('P(t)');
title('population density when r = 3.2');

r = 3.5;
for t = 1:499
    P(t+1) = r * P(t) * (1-(P(t)/K));
end
%size(P)
ans3 = [P(498) P(499) P(500)]
t = 0:499;
figure();
plot(t,P);
xlabel('t');
ylabel('P(t)');
title('population density when r = 3.5');

PLEASE REFER BELOW OUTPUT

ans1 =

    12    12    12

ans2 =

   15.9891   10.2609   15.9891

ans3 =

   16.5388   10.0177   17.4999

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