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.
I have determined the two equilibrium points of this system which are,
$$N^*=0$$
$$N^* = K\left(1-\frac{1}{\lambda}\right)$$
Now I am trying to find the conditions for each of the equilibria that I found above to be unstable.
Here is my explanation below,
Computing the derivative $f'(x)$ and evaluating it in the fixed points. If $|f'(x^*)|>1$ then $x^*$ is an unstable fixed point.
$$f'(x) = \lambda -\frac{2x\lambda}{K}$$
Using $x=0$
$$f'(0) = \lambda$$
So it is unstable when $\lambda>1$
But then how do I use $x = K\left(1-\frac{1}{\lambda}\right)$? Am I proceeding correctly?
Plug in and obtain
$$f'\left(K\left(1- \frac{1}{\lambda} \right)\right) = \lambda -2\lambda\left(1- \frac{1}{\lambda}\right)=-\lambda+2$$