02 - Stability

Steady State

Let discrete-time single specific population model be

$$N_{t+1}=f(N_t)$$

$N^{\ast }$ is a steady state if

$$N^{\ast }=f(N^{\ast })$$

Linearly Stable

Let $N^{\ast }$ be a steady state of

$$N_{t+1}=f(N_t)$$

$N^{\ast }$ is Linearly Stable if

$$|f^{\prime }(N^{\ast })|<1$$

Proof

Let

$$N_t=N^{\ast }+n_t$$

for some small change around $N^{\ast }$

Then

$$N_{t+1}=N^{\ast }+n_{t+1}\text{ and }{N}_{t+1}=f(N_t)=f(N^{\ast }+n_t)$$

Hence using Taylor Series Expansion then

$$N^{\ast }+n_{t+1}=f(N^{\ast })+n_t{f}^{\prime }(N^{\ast })+O(n_t^2)$$

As $N^{\ast }=f(N^{\ast })$ then

$$n_{t+1}\approx f^{\prime }(N^{\ast })n_t$$

where $f^{\prime }(N^{\ast })$ is constant

Hence

$$n_t\approx [f^{\prime }(N^{\ast }){]}^t{n}_0$$

So stable if $|f^{\prime }(N^{\ast })|<1$ (and unstable if $>1$)

Linearly Unstable

Let $N^{\ast }$ be a steady state of

$$N_{t+1}=f(N_t)$$

$N^{\ast }$ is Linearly Unstable if

$$|f^{\prime }(N^{\ast })|>1$$

Proof - Refer to Linearly Stable Proof

Bifurcation Point

Bifurcation Point is a point in parameter space where

1. Number of Steady State Changes

2. Stability of Steady State Changes

Bifurcation Diagram

Bifurcation Diagram plots values of the steady states as a function of model parameter value
where stable steady states are solid lines and unstable states are dotted