Consider a population of size . We have

• birth rate: with ,

• death rate: , where

  • models isolated deaths
  • models deaths due to overcrowding.

Thus we have the ODE

To make things simpler, let and . Then,

This is called a differential logistic equation. It is an example of an autonomous system. The fixed points are and . We can carry out perturbation analysis to determine their stability.

For (), is an unstable fixed point, while is a stable fixed point.

We can sketch a 1D phase portrait to visualise the behaviour of solutions.

This equation can be solved exactly and sketched.