Predator-Prey Model

parameters

a

1

α

0.5

c

0.75

γ

0.25

end time

t

max

10

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

x

and the fox population is

y

; both depend on time

t

.

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

dx/dt=ax

when

y=0

with

a>0

.

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

dy/dt=-cy

when

x=0

with

c>0

.

3. The number of encounters between the species is proportional to the product of their populations. Each encounter tends to increase

y

and decrease

x

. Thus the growth rate of

x

includes a term of the form

-αxy

and that of

y

includes a term of the form

γxy

, where

α

and

γ

positive. The parameters

a

,

α

,

c

, and

γ

are independent of

t

. These assumptions lead to the equations:

dx

dt

=ax-αxy=x(a-αy)

,

dy

dt

=-cy+γxy=y(-c+γx)

.