Chaos and Dynamical Systems

• May 22nd 2011, 03:06 AM
Chaos and Dynamical Systems
The equation$\displaystyle P_n_+_1 = rP_n(1-P_n/C) -k$ may be considered a density - dependent population model with constant harvesting. If the growth rate is r = 3 and the carrying capacity is C = 6000, what is the largest number k that could be harvested each generation so that the population has a stable positive equilibrium, and so may not become extinct. I thought you needed to differntaite it but this would get rid of k?
• May 22nd 2011, 07:05 AM
chisigma
Quote:

The equation$\displaystyle P_n_+_1 = rP_n(1-P_n/C) -k$ may be considered a density - dependent population model with constant harvesting. If the growth rate is r = 3 and the carrying capacity is C = 6000, what is the largest number k that could be harvested each generation so that the population has a stable positive equilibrium, and so may not become extinct. I thought you needed to differntaite it but this would get rid of k?

If r=3 and c=6000 the difference equation can be written in the form...

$\displaystyle \Delta_{n} = p_{n+1}-p_{n}= -\frac{p^{2}_{n}}{3000}+2\ p_{n}-k = f(p_{n})$ (1)

Necessary condition for convergence of $\displaystyle p_{n}$ is the existence of an 'attractive fixed point', i.e. a solution $\displaystyle x_{0}$ of the equation $\displaystyle f(x)=0$ with the condition $\displaystyle f'(x_{0})<0$ and that is true for $\displaystyle k<3000$. In that case the solutions are...

$\displaystyle x= 3000\ \pm \sqrt{3000}\ \sqrt{3000-k}$ (2)

... and $\displaystyle x_{0}$ corresponds to the sign '+'...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$