How to add restrictions to our group of differential equation? For example, in predator-prey model I want the sum of predator and prey to be fixed.

%Here is a code of sample predator-prey model.

%y0=20,20 means 20 predators and 20 preys.

%How can I let the sum of predators and preys to be 40 all the time, and see their population change with time?

t0 = 0;

tfinal = 15;

y0 = [20; 20];

[t,y] = ode23(@lotka,[t0 tfinal],y0);

t0 = 0;

tf = 15;

nsteps = 50;

y0 = [20; 20];

[t,y] = ode23(@lotka,linspace(t0,tf,nsteps),y0);

plot(t,y)

title('Predator/Prey Populations Over Time')

xlabel('t')

ylabel('Population')

legend('Prey','Predators','Location','North')

function yp = lotka(t,y)

%LOTKA Lotka-Volterra predator-prey model.

% Copyright 1984-2014 The MathWorks, Inc.

yp = diag([1 - .01*y(2), -1 + .02*y(1)])*y;

end

Answer 1

[Mrutyunjaya Hiremath]

Here we go,

t0 = 0;
tfinal = 15;
y0 = [20; 20];  % Initial conditions [prey, predator]
[t, y] = ode23(@lotka, [t0 tfinal], y0);

plot(t, y)
title('Predator/Prey Populations Over Time with Fixed Sum')
xlabel('Time')
ylabel('Population')
legend('Prey', 'Predator', 'Location', 'North')

function yp = lotka(t, y)
    a = 1;  % Growth rate of prey when there are no predators
    b = 0.01;  % Rate at which predators consume prey
    c = 1;  % Decay rate of predators when there are no prey
    d = 0.02;  % Rate at which predators increase by consuming prey

    dpdt = a * y(1) - b * y(1) * y(2);
    dqdt = -dpdt;  % Making sure that the sum remains constant
    
    yp = [dpdt; dqdt];
end

This will give you a plot where the sum of predators and prey remains constant at 40 (or whatever you set y0 to sum to). The populations will still oscillate, but in a manner constrained by this sum.