find the conditions for each equilibria to be unstable

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.

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

@Akababa if all Eigenvalues are less than one in absolute value. would be able to show me this? — user123, Feb 02 '18 at 22:00

score 1 · Answer 1 · answered Feb 02 '18 at 21:59

1

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.

answered Feb 02 '18 at 21:59

Luke

141

okay so the deriviative is $$\lambda-\frac{2x\lambda}{K}$$ – user123 Feb 02 '18 at 22:04
so if I put in $x=0$ then its just $\lambda$ so would it be when $\lambda>1$ x is unstable? – user123 Feb 02 '18 at 22:05

find the conditions for each equilibria to be unstable

1 Answers1