Math 334 Lecture #38
§
9.4: Competing Species
Outcome A: Apply Phase Plane Analysis to Population Dynamics
. Suppose in a closed
environment, there are two species whose populations at time
t
are
x
and
y
.
In the absence of competition, the two populations are assumed to be governed by logistic
growth (Section 2.5):
dx/dt
=
x
(
1

σ
1
x
)
,
dy/dt
=
y
(
2

σ
2
y
)
,
where
1
,
are the intrinsic growth rates and
1
/σ
1
,
2
/σ
2
) the saturation levels.
When the two species compete for the available food, they reduce each others’ growth
rates and saturations levels.
A modification of the uncoupled logistic equations above that accounts for this competi
tion is
dx/dt
=
x
(
1

σ
1
x

α
1
y
)
,
dy/dt
=
y
(
2

σ
2
y

α
2
x
)
,
where
α
1
,
α
2
are quantities describing the interference the two species with each other.
The parameters
1
,
2
,
σ
1
,
σ
2
,
α
1
, and
α
2
depend on the particular species and are
estimated from empirical studies.
Since we are dealing with populations, we restrict attention to those solutions that lie in
the first quandrant, i.e.,
x
≥
0 and
y
≥
0.
The competing species model is an autonomous nonlinear system that is generally not
solvable for explicit solutions
x
(
t
),
y
(
t
).
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 Spring '11
 Smith
 Math, Fundamental physics concepts, Linear system, Stability theory, ﬁrst quadrant

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