Consider a chemical reaction

The number of molecules of at time are respectively.

The initial numbers are . Hence, we have the conservation laws

since one of and is consumed to produce one of and . Assume that the rate of reaction is proportional to the product of the numbers of and molecules (e.g. we considering dilute gases):

Therefore, we have an example of an autonomous non-linear first-order ODE.

Note that the fixed points are and (corresponding to the complete consumption of either or ).

Now assume . Then, is unphysical. We shall now carry out perturbation analysis to determine the stability of the fixed points.

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

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

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.