Modelling predator-prey interactions

Introduction

The classic, textbook predator-prey model is that proposed by Lotka and Volterra in 1927. In words, the model states that:

  • Each prey gives rise to a constant number of offspring per year;
    In other words, there are no other factors limiting prey population growth apart from predation.
  • Each predator eats a constant proportion of the prey population per year;
    In other words, doubling the prey population will double the number eaten per predator, regardless of how big the prey population is.
  • Predator reproduction is directly proportional to prey consumed;
    Another way of expressing this is that a certain number of prey consumed results in one new predator; or that one prey consumed produces some fraction of a new predator.
  • A constant proportion of the predator population dies per year.
    In other words, the predator death rate is independent of the amount of food available.

Mathematically, the model is conventionally expressed as:

dX/dt = aX - bXY
dY/dt = cbXY - dY

where:

X = size of the prey population (set to an initial value of 5000 in this tutorial);
Y = size of the predator population (45);
a = number of offspring per prey per year (set to 0.5 in this tutorial);
b = proportion of the prey population consumed by one predator per year (0.01);
c = conversion of one prey consumed into new predators (0.01, i.e. 100 rabbits eaten gives rise to one new fox);
d = proportion of predator population dying per year (0.2).

At the end of this tutorial, we will see how to implement the conventional form of the model, exactly as given. However, we will begin by building up a population model for a single species in System Dynamics terms, then add the predator population, then show how one of the relationships can be made more biologically realistic, before returning to the conventional form of the model.