# Introduction

The last session looked at modelling the growth of a single population. This class we will extend that into looking at two interacting populations, particularly predator-prey models. These types of model mainly look at the interaction between one species which is the prey for another species, the predators.

We have already seen a basic population model

$\frac{\text{dN}}{\text{dt}}=\text{\beta N}-\text{\delta N}$where β and δ are the birth and death rates respectively. Now we will introduce some predators denoted by P(t). Let us assume, naively, that the prey can only be die because they are being eaten by the predators but the rate at which they are eaten will depend on the number of prey and the number of predators. This gives us our new equation for the prey, our N from before

$\frac{\text{dN}}{\text{dt}}=\text{aN}-\text{bNP}$where a>0 is the birth rate of the prey and b>0 the rate at which they get eaten. Now we also need an equation for the predators. They die at some rate, again we assume naively that they only die of natural causes but their birth rate depends on how many prey are available. The more prey, the more food available and so the population will grow faster. Putting this together gives us an equation for the predators

$\frac{\text{dP}}{\text{dt}}=\text{cNP}-\text{dP}$Here c>0 is the birth rate of the predators and d>0 the death rate. As both N and P appear in both equations they need to be treated as a system of equations. This gives us what is known as the Lotka-Volterra system:

$\frac{\text{dN}}{\text{dt}}=\text{aN}-\text{bNP}$$\frac{\text{dP}}{\text{dt}}=\text{cNP}-\text{dP}$

## Information

This resource was created as part of the Educational Development Unit's ASP2 project, developing resources for the Applied Science degree at the University of the Highlands and Islands.

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