Competing Species Models
The general equations for competing species are of the form
x
′
=
x
(
ǫ
1
−
σ
1
x
−
α
1
y
)
y
′
=
y
(
ǫ
2
−
σ
2
y
−
α
2
x
)
,
where
x
and
y
are the populations of two species in some environ
ment,
ǫ
1
and
ǫ
2
are positive constants representing the growth rates
of these species when there is no competition,
σ
1
and
σ
2
are positive
constants representing intraspecies competition for resurces, and
α
1
and
α
2
are positive constants representing interspecies competition.
Depending on the values of the parameters, these models may be
have in various ways. We will concentrate on two of these:
strong
competition
, in which one or the other species eventually dies out,
with the winner determined by the initial conditions, and
weak com
petition
, in which both species can survive. We will make choose
values for the parameters of these two types of models which keep
our computations simple, not values corresponding to realistic mod
els.
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 Fall '07
 SPEER
 Derivative, phase plane

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