An Application: A Predator-Prey Model

A Predator-Prey Model

Suppose we study the population of two species in a region - hawks and mice. Hawks are the predator and mice are the prey. Let

and

be the population of the hawks and mice in the region after

months respectively. We assume the following formulas for the transition of populations:

Interpretation:

The "" term: Without any mice to eat, the hawk population will decrease by half in a month.
The "" term: Without any hawk, the mice population will increase by a factor of in a month.
The "" term: On average, 100 mice can support an additional hawk in a month.
The "" term: On average, 4 hawks can kill 50 mice in a month.

Question: Suppose there are 50 hawks and 1600 mice in the region initially. Will the populations eventually become stabilized? First of all, it is given that

. And the transition of populations can be written as follows:

, where

. Then

If we can write down the entries of

explicitly in terms of

, then we can study the behaviour of

and

when

is large. We can diagonalize

Hence

When

. Hence

. Therefore,

An Application: A Predator-Prey Model

A Predator-Prey Model

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