## 13.2 Predator - Prey Systems

The population of species F changes with time from natural growth and from being eaten.

The natural growth term is proportional to its population, and we model it by a term in the expression for the derivative of F of the form +aF, for some constant a. (Obviously when the population of F grows tremendously, other factors will have to be considered, but in a normal range of values for F this term seems reasonable.)

The attrition from being eaten term should, to a first approximation be proportional to the number of encounters of P's with F's which should be proportional to the product of the population of each, and hence of the form -bPF, for some constant b.

Without any F's to eat, the popululation of specie P will probably decline, (through starvation or moving away) which can be modeled by a term in the derivative of P(t) of the form -cP for some constant c, since the decline would probably be probably proportional to the population present.

Finally the presence of F allows the P's to survive and perhaps grow, again by a factor proportional to the encounters between them, and hence of the form +dPF.

The coupled equations of the standard predator-prey model are therefore

We can explore the consequences of this model for varying constants a,b,c and d and initial values, P(0) and F(0) with the following applet which allows us to deduce the consequences of this model.

Without doing so we can notice that there is a static solution at which P and F don't change; this will occur when the right hand sides here are both 0. This will occur when and . When these identities hold, the populations stay fixed.

It is interesting to examine what happens when we stray from this "equilibrium". Suppose, for example, we decrease the initial population of F, below . Try this on the applet and see what happens.