How to do Matlab code the Impulsive DE for Lotka Volterra Model with a impulsive prey culling function of H(t+) = -k*H(t-) where k is culling rate for prey

Below, is the Matlab code for original Lotka Volterra Model

%Step 1: Define the parameters
alpha = 0.1;    % Prey growth rate
beta = 0.02;    % Prey death rate due to predation
gamma = 0.3;    % Predator death rate
delta = 0.01;   % Predator growth rate due to prey consumption

%Note that y(1) = H and y(2) = K


%Step 2: Definie initial condition values
H0 = 60;
K0 = 50;
Ti = T/4;

%Step 3: Define the ODE Function
 Lotkamodel = @(t,y)[alpha*y(1) - beta*y(1)*y(2);...
                   delta*y(1)*y(2)-gamma*y(2);
                   ];

 %Step 4: Solve the ODEs
 [t,y] = ode45(Lotkamodel, [0,300], [H0; K0]);

 %Step 5: Plot the results
plot(t, y(:, 1), 'b', 'LineWidth', 2); % plot S(t)
hold on;
plot(t, y(:, 2), 'Color', "#FF00FF", 'LineWidth', 2); % plot I(t)
hold on;
xlabel('Days');
ylabel('Population');
legend('Prey', 'Predator');
title({'SIR Model', 'Coded by: Myself'});

Answer 1

[Ayush]

Hi John,

I understand that you want to add the impulsive DE for Lotka Volterra Model with an “impulsive prey culling” function.

The “impulsive prey culling” function represents a sudden, instantaneous reduction in the prey population at a specific time point. At a specific time, point “t = Ti “, where “Ti” is the impulsive time thus we have “H(t^+) = -k*H(t^-) “

Where:

• “H(t^+)” represents the prey population immediately after the impulsive event.
• “H(t^-)” represents the prey population immediately before the impulsive event.
• “k” represents the culling rate for prey.

Keeping these points into consideration, you can include the impulsive prey culling term “-k*(t >Ti)*y(1)”, which evaluates to “-k*H(t-)” when “t” crosses the impulsive time “Ti”. Finally pass the function through “ode45”. You can refer to the modified code below for better understanding:

% Step 1: Define the parameters
alpha = 0.1;    % Prey growth rate
beta = 0.02;    % Prey death rate due to predation
gamma = 0.3;    % Predator death rate
delta = 0.01;   % Predator growth rate due to prey consumption
k = 0.05;       % Culling rate for prey

% Step 2: Define initial condition values
H0 = 60;
K0 = 50;
Ti = 300/4;     % Assuming T = 300 (total time)

% Step 3: Define the ODE Function with impulsive function
Lotkamodel = @(t,y) [
    alpha*y(1) - beta*y(1)*y(2) - k*(t > Ti)*y(1);  % Impulsive prey culling
    delta*y(1)*y(2) - gamma*y(2)
];

% Step 4: Solve the ODEs
[t,y] = ode45(Lotkamodel, [0, 300], [H0; K0]);

% Step 5: Plot the results
plot(t, y(:, 1), 'b', 'LineWidth', 2); % plot Prey
hold on;
plot(t, y(:, 2), 'Color', "#FF00FF", 'LineWidth', 2); % plot Predator
xlabel('Days');
ylabel('Population');
legend('Prey', 'Predator');
title({'Lotka-Volterra Model with Impulsive Prey Culling', 'Coded by: Myself'});

Refer the link below to explore more on “ode45” function:

https:/www.mathworks.com/help/matlab/ref/ode45.html

Regards,

Ayush