I am just starting dynamical systems and came across the following problem in my textbook.
Considering the discrete time logistic growth model,
$$N_{t+1} = \lambda N_t\left(1-\frac{N_t}{K}\right)$$
where $\lambda = 1+ b -d >0$ is the net reproductive rate and $K>0$ is a parameter that affects how the population grows when the population is large.
With some help I have determined the two equilibrium points of this system which are,
$$N^*=0$$
$$N^* = K\left(1-\frac{1}{\lambda}\right)$$
My question now is I am trying to find the conditions for each equilibria to be unstable. How do I got about this? Thanks
The Floquet multiplier must be larger than 1 for the fixed point to be unstable.
In other terms, your map is $$f(x)=\lambda x(1-x/K)$$ So now you compute the derivative $f'(x)$ and you evaluate it in the fixed points you worked out. If $|f'(x^*)|>1$ then $x^*$ is an unstable fixed point.