Logistic equation

This article shows how to solve the logistic equation using the Euler Method.

1 Logistic equation

Logistic equation is used to simulate a number of applications. It was first introduced by P.F. Verhulst to simulate population growth. It reads,

[math]y' = y\cdot (1-y), \quad t\in(t_0,\infty) [/math]

[math]y(t_0) = y_0[/math]

Here, t is the time, [math]y(t)[/math] represents the population size and [math]y_0[/math] the population size at initial time [math]t=t_0[/math].

2 Numerical scheme

We propose an Euler explicit method with time step h,

[math] y_0, \; t_0 [/math]

[math]y_{n+1} = y_{n} + h\cdot y_{n}\cdot(1 - y_{n})[/math]

3 MATLAB code

% Euler method to solve the logistic equation y'=y(1-y)
clear all;
t0=0; tN=4;           % initial and final time 
y0=1/10;              % value of y at time t=0
N=40;                 % Number of intervals 
h=(tN-t0)/40;         % Time step h  
yy=y0;                % yy -> variable with the solution at each time step
y(1)=yy;              % y -> vector where we store the solution
for n=1:N 
   yy=yy+h*yy*(1-yy);  % numerical scheme
   y(n+1)=yy;          % store the solution
end 
x=t0:h:tN;             % Draw the solution
plot(x,y,'x');

4 Results

Consider the particular case: [math] y_0=1/10, \; t_0=0, \; h=1/10 [/math] The exact solution can be computed in this case: [math] y(t)=\frac{e^t}{9+e^t} [/math]

Numerical approximation of the solution

Error = abs( exact solution - numerical approximation)

--Carlos Castro (discusión) 15:09 31 ene 2013 (CET)