find the equilibrium point of this equation

Question

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 am trying to find the equilibrium point of this equation.

I know that the equilibrium point are values of $N^*$ such that $N_{t+1} = N_t = N^*$. But how do I do that? thanks!

Answer 1

Let $x=N^*$ and solve: $$x=\lambda x(1-x/K)$$ so $x=0$ or $$1=\lambda (1-x/K)$$ $$1/\lambda=1-x/K$$ $$x/K=1-1/\lambda$$ $$x=K(1-1/\lambda)$$

So two solutions, $N^*=0,K(1-1/\lambda)$