Equilibria and Stability

Logistic Model Example

Finding solutions

We can find a solution for this one. But what if we couldn't?

\frac{d N}{d T} = r (1 - N / K) N = f (N)

This is a continuous-time model over the time interval $d t$ short time interval

We need to find the values of $N$ where there is no change in $N$ (or $f (N) = 0$ ). These points are the equilibria.

Trivial Equilibrium

The first solution of this one is obvious:

r (1 - N / K) N = 0

\hat{N_{1}} = 0

This makes sense - there is no growth when the population is zero. THis is the Trivial Equilibrium

Non-trivial equilibrium

\hat{N_{2}} = K

This is the carrying capacity - there is no growth at this point

Stability Analysis

!mathematical-methods-lecturenotes, p.49
$K = 10, r = 1$

left = -0.5
bottom = -3
right = 15
---
(0,0)|open|label:Trivial equilibrium (unstable)
(K,0)|label:Carrying capacity: Globally stable
r = 1|hidden
K = 10
y=r(1-x/K)x

Any population lower than $N = K$ trends towards $N = K$ . Anything higher than $N = K$ trends to $K$

This shows us that $N = 0$ is an unstable equilibrium and $N = K$ is stable equilibrium

Allee Effect

See the linked page

Condition for stability

A point of equilibrium $x$ is stable when $f^{'} (x) < 0$ (and $f (x) = 0$ ) .
If the derivative is positive, it's a stable equilibrium, if it's negative, it's unstable.

Examples:

Harvested Logistic Model
Equilibria of reversible processes
Predator-prey dynamics with Holling type II functional resonse