% This is the main function to solve the logistic growth
% dN/dt = r(1-N/K)N in p.212, Mathematical Models in Biology


global r K

K = 10; % carrying capacity
r = 1; % reproduction rate

options = odeset('RelTol',1e-4,'AbsTol',1e-4);

t0 = 0;
tfinal = 3;

for N_ini = 0:2.5:20;
[T,N] = ode45(@logisticgrowth,[t0 tfinal],N_ini,options);

% plot N at different T
figure(1); hold on; plot(T,N)
xlabel t, ylabel N

% plot exact solution

Nexact = N_ini*K./(N_ini+(K-N_ini)*exp(-r*T));

figure(2); hold on
plot(T,Nexact,'r')
xlabel t, ylabel('exact solution of N') 

pause(3)
end