The PredatorPrey Equations
An application of the nonlinear system of differential equations in
mathematical biology
/
ecology: to model the
predatorprey relationship
of a
simple ecosystem.
Suppose in a
closed
ecosystem (i.e. no migration is allowed into or out of
the system) there are only 2 types of animals: the predator and the prey.
They form a simple foodchain where the predator species hunts the prey
species, while the prey grazes vegetation.
The size of the 2 populations can
be described by a simple system of 2 nonlinear first order differential
equations (a.k.a. the
LotkaVolterra equations
, which originated in the study
of fish populations of the Mediterranean during and immediately after WW
I).
Let
x
(
t
) denotes the population of the prey species, and
y
(
t
) denotes
the population of the predator species.
Then
x
′ =
a
x

α
xy
y
′ = 
c
y
+
γ
xy
a
,
c
,
α
, and
γ
are positive constants.
Note that in the absence of the predators (when
y
= 0), the prey population
would grow exponentially.
If the preys are absence (when
x
= 0), the
predator population would decay exponentially to zero due to starvation.
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This system has two critical points.
One is the origin, and the other is in the
first quadrant.
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 Spring '08
 CHEZHONGYUAN
 Differential Equations, Equations, Partial Differential Equations, prey species, sole food source

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