Sunday, April 21, 2013

Using Differential Equations for Population Ecology

In class, we covered exponential and logistic growth; however, systems of differential equations is an approach to model population growth with two species, often referred to as Lotka Volterra Predator-Prey models.

You start with two equations:

  • the rate of change in population of the predator with respect to time as a function of the populations 
  • the rate of change in population of the prey with respect to time as a function of the populations

You then form a matrix representation of the functions and find the eigenvalues and eigenvectors, which are used to plot a phase diagram (a graphical representation of the solution to the differential equations). The magnitude and sign of the eigenvalues determine the stability of the fluctuations between the populations of the two species and the rate of growth of each population. Of the several different possibilities for the phase diagrams, there are two relevant situations: concentric circles, or saddle point. 
The concentric circles represent populations in a stable relationship, as shown below.
A saddle point is unstable and suggests the extinction of one of the populations, as shown below.

This is a topic that was covered in Math 4B, but you can also read about it here

2 comments:

John Latto said...

You can simulate Lotka-Volterra competition and Lotka-Volterra predator prey models in the Populus program we used last quarter. It's not the most elegant program but it does allow you to play around with these models very easily.

Verena said...

This is cool!