Competing Species Model with Two Species

parameters

ϵ

1

0.9

σ

1

0.5

α

1

0.4

ϵ

2

1

σ

2

0.5

α

2

0.25

end time

t

max

10

This Demonstration illustrates the competing species model with two species, foxes and hawks. Foxes and hawks both prey on rabbits, of which there is a limited supply. The fox population is

x

and the hawk population is

y

; both depend on time

t

.

1. In the absence of hawks, the foxes would grow logistically; thus

dx

dt

=x(

ϵ

1

-

σ

1

x)

when

y=0

, where

ϵ

1

is the growth rate and

ϵ

1

σ

1

is the carrying capacity for the fox population.

2. In the absence of foxes, the hawks would grow logistically; thus

dy

dt

=y(

ϵ

2

-

σ

2

y)

when

x=0

, where

ϵ

2

is the growth rate and

ϵ

2

σ

2

is the carrying capacity for the hawk population.

3. When both foxes and hawks are present, a sufficiently large number in either population will cause a decrease in the growth of both populations. Thus the model for competing species requires additional parameters

α

1

and

α

2

that determine the effect of large populations:

dx

dt

=x(

ϵ

1

-

σ

1

x-

α

1

y)

,

dy

dt

=y(

ϵ

2

-

σ

2

y-

α

2

x)

.