Lotka volterra model for 3 species (1 prey - 2 predators)

How can i adapt the following code and function to solve a three species lotka volterra model?

clc

clear clc

t0 = 0; %Condiciones iniciales.

tfinal = 15; %Condiciones finales.

y0 = [60.28; 2.043051]; %Poblaciones iniciales de depredadores y presas.

[t,y] = ode45(@lotka,[t0 tfinal],y0); %%Resuelve la ecuación de Lotka-Volterra

% con los parámetros anteriores gracias a la función "ode45 “ – (Fórmula Runge-Kutta (4,5)).

%Primer Gráfica.

nexttile % Graficar en una sola ventana.

plot(t,y) % Graficar función Lotka-Volterra.

title('Poblaciones de depredadores y presas a lo largo del tiempo')

xlabel('Tiempo t')

ylabel('Población')

legend('Presa','Depredador','Location','North')

%Segunda Gráfica.

nexttile % Graficar en una sola ventana.

plot(y(:,1),y(:,2))

title('Retrato de fase')

xlabel('Población Presa')

ylabel('Población Depredador')

function yp = lotka(t,y)

%LOTKA Lotka-Volterra predator-prey model.

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

Answer 1

[Sam Chak]

Hi @José Manuel

This isn't strictly a MATLAB-related question, but I believe animal behaviorists might be able to advise you. Alternatively, from a critical thinking perspective, you could consider modifying the Lotka-Volterra equations for a 2-predator-1-prey model.

In a particular phase, the prey population attains a level of abundance that initiates the growth of the predator communities. Over time, the heightened presence of predators induces a decline in the prey population. Subsequently, this reduction in prey numbers contributes to a decrease in the populations of the predators. Eventually, this cyclical pattern recurs.

tspan  = [0 20];
y0     = [5; 3; 2];
[t, y] = ode45(@lotka, tspan, y0);

plot(t, y), grid on, ylim([0 6])
legend('Prey', 'Predator 1', 'Predator 2', 'location', 'SE')
xlabel('t')
title('Numbers of prey & predators for the Lotka-Volterra model')

%% Modified Lotka-Volterra 2-predator-1-prey model
function dydt = lotka(t, y)
    dydt    =   zeros(3, 1);
    
    % Parameters
    a       =   2.4;    % maximum prey per capita growth rate
    b1      =   0.6;    % effect of the presence of predator 1 on the prey's growth rate
    b2      =   0.6;    % effect of the presence of predator 2 on the prey's growth rate
    c1      =   0.8;    % predator 1 per capita death rate
    c2      =   0.8;    % predator 2 per capita death rate
    d1      =   0.3;    % effect of the presence of preys on the predator 1's growth rate
    d2      =   0.3;    % effect of the presence of preys on the predator 2's growth rate
    
    % Lotka-Volterra equations for 2 predators and 1 prey
    dydt(1) =    a*y(1) - b1*y(1)*y(2) - b2*y(1)*y(3);  % Prey
    dydt(2) = - c1*y(2) + d1*y(1)*y(2);                 % Predator 1
    dydt(3) = - c2*y(3) + d2*y(1)*y(3);                 % Predator 2
end