- To simulate the isocline-induced step function differential equations, we can use MATLAB's ODE solver `ode45`.
- First define the system of differential equations as a function and then solve it numerically.
- 1. Define the system of differential equations:
function dydt = isocline_system(t, y, r, k, a, e, m, F, s, w, b, M, sx)
x = y(1);
u = isocline_input(x, F, s, w, b, M, sx);
dxdt = r * x * (1 - x / k) - a * x * y / (1 + q(u) * u * M);
dydt = (e * a * x * y) / (1 + q(u) * u * M) - m * y;
dydt = [dxdt; dydt];
end
function u = isocline_input(x, F, s, w, b, M, sx)
pi0 = (w * F) / (s * (w + b * M));
pi1 = F * (w + b * M) / (s * w);
if x <= pi0
u = 0;
elseif x >= pi1
u = 1;
else
u = sx / (sx + F) + w * (sx - F) / (b * (sx + F) * M);
end
end
function q_u = q(u)
q_u = 1 / (1 - u);
end
- 2. Solve the differential equations numerically using `ode45`:
% Parameters
r = ...; % Define the value of r
k = ...; % Define the value of k
a = ...; % Define the value of a
e = ...; % Define the value of e
m = ...; % Define the value of m
F = ...; % Define the value of F
s = ...; % Define the value of s
w = ...; % Define the value of w
b = ...; % Define the value of b
M = ...; % Define the value of M
sx = ...; % Define the value of sx
% Initial conditions
x0 = ...; % Define the initial value of x
y0 = ...; % Define the initial value of y
initial_conditions = [x0; y0];
% Time span for simulation
tspan = ...; % Define the time span (e.g., [0, 10])
% Solve the system of differential equations
[t, y] = ode45(@(t, y) isocline_system(t, y, r, k, a, e, m, F, s, w, b, M, sx), tspan, initial_conditions);
% Extract the solution
x_sol = y(:, 1);
y_sol = y(:, 2);
- Replace the `...` with the specific values of the parameters and initial conditions you want to use in your simulation. The `ode45` solver will numerically integrate the system of differential equations over the given time span `tspan`, and the solutions for `x` and `y` will be stored in `x_sol` and `y_sol`, respectively.
