Question

This exercise is to be completed as a binary exercise. It is taken from Chapra Section 28.2. Note that exercises like these make good components of examination questions Predator-Prey models were developed independently in the early part of the twentieth century by the Italian mathemati cian Vito Volterra and the American biologist Alfred . Lotka These equations are commonly called Lotka Volterra equations. One example is the following pairs of ODEs Figur%2: Examples of Prey. d.r dt dy In these equations y is the number of predators, z is the num- ber of prey-a is the prey growth rate, s the predator death rate. The effect of the predator-prey interaction on prey deatlh and predator growth is represented by the rates b and d respec- tively. Assuming that a 12.b 0.6,c 0.s,d 0.3, your task is to estimate (t) and y(t) for 0 < t < 30. The initial conditions are (0)2 and y()-1 Follow the instructions MAN below 1. Use the Matlab function ode45 to solve this pair of ODEs You will have to use help ode45 to figure out how to 20 30 Teme t Fgue 3 Solution aing Mattab's soler use the function. Note that Matlab expects as an argument your function which returns the differentials of r and y given the current values of r andy 2. Plot r(f) and y(t) on the same plot. Use RED for r and BLUE for y. Your graph should look similar to that shown irn Figure 3 3. What is the maximum number of predators that the popu lation of prey can support? (Ans3). When the predators are at a maximum, how many prey are there? (Ans2)

Answer 1

% First write this code and save in same folder as other code

function dydt = myode(t,y)
%UNTITLED4 Summary of this function goes here
% Detailed explanation goes here
a=1.2;
b=0.6;
c=0.8;
d=0.3;
dydt = zeros(2,1);
dydt(1) = (a*y(1)) -(b*y(1)*y(2));
dydt(2) =(-c*y(2)) +(d*y(1)*y(2));

clc;clear all;close all;
%%
ft = [0 20];
x0=2;
y0=1;
[t,f] = ode45(@(t,y) myode(t,y), ft,[x0,y0]);
x=f(:, 1) ;
y=f(:, 2);
plot(t,x,'r',t,y,'b')
xlabel('Time t');
ylabel('Solution x and y');
legend('Prey','Predator')

for n=1: length(x)
if (x(n)-y(n)>=0)&&(x(n)-y(n)<=0.06)
disp('when Preys support predators is ')
disp(x(n))
end
end
for n=1: length (x)
if y(n) ==max(y)
disp('Preys when predators are at max ')
disp(x(n))
end
end


5.5 Prey Predator 5 4.5 4 3.5 3 2 1.5 0.5 0 2 468 10 12 1416 18 20 Time t

Command Window New to MATLAB? See resources for Getting Started when Preys support predators is 3.5458 Preys when predators are at max 2.6693 fx >>