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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)
.
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