% Stability of fixed points and long term behavior in 
% discrete single species population dynamics models.

function bifurcation

init = .6;
wait = 1;
iter = 50;
range = 5:.02:6.6;

figure;
for c = range
  n(1) = init;
  for i=1:wait
    n(i+1) = c * n(i)^2 * ( 1 - n(i) );
  end
  for i=(wait+1):(iter+wait)
    n(i+1) = c * n(i)^2 * ( 1 - n(i) );
    plot(c,n(i),'.'); hold on;
  end
end

figure;
ctr = 0;
for c = range
  liap = 0;
  n(1)=init;
  for i=1:(wait+iter)
    n(i+1) = c * n(i)^2 * ( 1 - n(i) );
    liap = liap + log(abs(c*n(i)*(2-3*n(i)))+1e-5);
  end
  ctr = ctr + 1;
  liapunov(ctr) = liap / (wait+iter);
end
plot(range,liapunov,'.');
hold on;
plot(range,zeros(size(range)),'k');