plotting predator-prey model and its phase portrait

I have derived the eignevalues and eigenvectors of the predator prey model using symbolic notation. However, I am unable to plot the givn solution in the form it is given. Is it necessary to provide with values for the chosen variables in order to plot them?

syms A B D G t positive

syms x y x0 y0

assumeAlso([x0 y0], 'positive')

%The matrix dX/dt = f(X) is represented by F

F = [A*x - B*x*y; -G*y + D*x*y]

%The matrix v is used to denote the variables in focus

v = [x y]

%Now we find the Jacobian of F to approximate the given system about

% its stationary points. Through hand calculation, we know that the

% stationary points of the given system are: (0,0) and (G/D, A/B)

% This is represented by J1 and J2.

P1 = [0 0];

P2 = [G/D A/B];

j = jacobian(F, v)

J1 = subs(j,v,P1)

J2 = subs(j,v,P2)

%Now, we compute the eigenvalues and eigenvectors about the given points to

%determine the stability of the system. We know that, for a system to be

%stable, the eigenvalues of the system all have real part < 0

[T1, D1] = eig(J1)

[T2, D2] = eig(J2);

%T1 and T2 are eigenvectors while D1 and D2 are eigenvalues.

%In order to compute the phase portrait, we utilize x(t) = T*e^(Dt)*T^(-1)*x(0).

%For each case of stationary point, we get a different transformation

%matrix (i.e. A) and thus we find different x(t) for each case.

D1(t) = D1*t

D2(t) = D2*t

T1inv = inv(T1)

T2inv = inv(T2)

eD1(t) = exp(D1)

eD2(t) = exp(D2)

c1(t) = T1*eD1(t)*T1inv;

c2(t) = T2*eD2(t)*T2inv;

xt1(t) = c1(t)*[x0; y0]

xt2(t) = c2(t)*[x0; y0]

Can I plot it in terms of A,B,D and G?

Thanks

Answer 1

[Sam Chak]

Hi @Krtin Kala

If you want to plot the phase portrait, you can try using the numerical approach.

% Parameters
A = 2/3;
B = 4/3;
G = 1;
D = 1;

% Lotka–Volterra equations
f = @(t, x) [A*x(1) - B*x(1)*x(2);
	        -G*x(2) + D*x(1)*x(2)];

% Plotting multiple phase portraits
for x0 = 0.9:0.1:1.8
    tspan = [0 10];
    initc = [x0 x0];	% initial condition
    [t, x] = ode45(f, tspan, initc);
    plot(x(:,1), x(:,2), 'linewidth', 1), hold on
end
grid on, xlabel('x'), ylabel('y'), hold off

See Example at:

https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations#Phase-space_plot_of_a_further_example

[Krtin Kala]

Thanks a lot. Is there any way I can get a slider for A,B,G and D so that I can see the variation in plot with respect to the change in these values?

[Sam Chak]

You are welcome, @Krtin Kala. You can learn to design slider using the App Designer.

https://www.mathworks.com/discovery/matlab-gui.html

https://www.mathworks.com/products/matlab/app-designer.html

If you find the MATLAB code for the predator-prey model is helpful, please consider accepting ✔ and voting 👍 the Answer. Thanks!

[Krtin Kala]

Thanks a lot @Sam Chak. one final query. I have been trying to use quiver on the phase portrait graph. How should I initialize u and v for quiver(u,v,x,y)?