How to draw phase space of two ODEs

Hello, I'm having troubles trying to recreate the phase space:

Parameters are alpha=1, beta=1, k= 0.1, K=1, and mu= 0.1 ; initial conditions: (u0, s0) = (0.75, 0.01K)

Also u is a fraction between 0 and 1.

The system that generates it is the following:

I'm very new to matlab and trying to adapt different codes that worked for some other problems didn't accomplish anything. The most I could do was to create a graph with all the little arrows, but it wasn't even close to resemble the one above.

Even some advices would be awesome: how do I set conditions? Such as u being in [0,1]; how do I change the y units to be s/K and not just s?

Thanks

Answer 1

[Ayush]

Hi,

To recreate the phase space for the system described in the above equations, you can use the "ode45" function. This will help you to integrate the above equations over time. To change the "y" units to be "s/K" instead of just "s", I scaled the "s" values by "1/K" in the plotting function. You can refer to the example code below for better understanding:

function phase_space_plot
    % Parameters
    alpha = 1;
    beta = 1;
    k = 0.1;
    K = 1;
    mu = 0.1;
    u0 = 0.75;
    s0 = 0.01 * K; % Initial condition for s
    
    % Time span
    tspan = [0 100]; % Adjust as necessary
    
    % Initial conditions
    y0 = [u0, s0];
    
    % Solve the ODE system
    [~, Y] = ode45(@(t, y) systemODEs(t, y, k, K, alpha, beta, mu), tspan, y0);
    
    % Extract solutions
    u = Y(:, 1);
    s = Y(:, 2) / K; % Scale s by K for plotting
    
    % Plotting
    figure;
    hold on;
    
    % Plotting the trajectory
    plot(u, s, 'b', 'LineWidth', 2);
    
    % Generating a grid of points for the direction field
    [U, S] = meshgrid(0:0.05:1, 0:0.05:1.2); % Adjust grid density and range as needed
    Udot = -k.*U.*(1-U).*(1-S/K);
    Sdot = -k.*S.*U.*(1-S/K) + alpha.*(1+beta.*U).*S.*(1-S/K) - mu.*S;
    
    % Normalizing vectors for uniform arrow size
    Vnorm = sqrt(Udot.^2 + Sdot.^2);
    Udot = Udot ./ Vnorm;
    Sdot = Sdot ./ Vnorm;
    
    % Adding the directional arrows to the plot
    quiver(U, S, Udot, Sdot, 0.3, 'k'); % Adjust scaling factor as necessary
    
    xlabel('u');
    ylabel('s/K');
    title('Combined Phase Space and Directional Plot');
    grid on;
    hold off;
end

function dydt = systemODEs(~, y, k, K, alpha, beta, mu)
    u = y(1);
    s = y(2);
    
    % System of equations
    du_dt = -k*u*(1-u)*(1-s/K);
    ds_dt = -k*s*u*(1-s/K) + alpha*(1+beta*u)*s*(1-s/K) - mu*s;
    
    dydt = [du_dt; ds_dt];
end

phase_space_plot

I defined a function "phase_space_plot" that solves the system of the given differential equations using "ode45" and plots the phase space trajectory with directional arrows, where "u" is on the x-axis and "s/K" on the y-axis. You can adjust the code to your requirements.

For more information on the "ode45" function, you can refer to below documentation:

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