This Demonstration illustrates the predator-prey model with two species, foxes and rabbits. Foxes prey on rabbits that live on vegetation. The rabbit population is and the fox population is ; both depend on time .

1. In the absence of foxes, the rabbit population grows at a rate proportional to its current population; thus when with .

2. In the absence of rabbits, the foxes die out; thus when with .

3. The number of encounters between the species is proportional to the product of their populations. Each encounter tends to increase and decrease . Thus the growth rate of includes a term of the form and that of includes a term of the form , where and positive. The parameters , , , and are independent of . These assumptions lead to the equations:

(United States Military Academy West Point, Department of Mathematics)

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DETAILS

This example comes from [1], Section 7.4, Predator Prey Equations.

Reference:

[1] J. R. Brannan and W. E. Boyce, Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications, New York: John Wiley and Sons, 2010.