Environmental Engineering Reference
In-Depth Information
4.7.1 Stochastic dynamics of two competing species
In this subsection we consider the case discussed by
May
(
1973
) of two species whose
populations,
A
1
(
t
)and
A
2
(
t
), compete for the same pool of resources according to the
dynamics
d
A
1
d
t
=
A
1
(
A
c
1
−
A
1
−
α
A
2
)
,
d
A
2
d
t
=
A
2
(
A
c
2
−
A
2
−
α
A
1
)
,
(4.53)
where
A
c
1
and
A
c
2
are the carrying capacities of
A
1
and
A
2
, respectively. The
steady state [
A
1
,
st
)] is stable
when the highest eigenvalue of the community matrix (see Box 4.3) has a nega-
tive real part. If
A
c
1
=
=
(
A
c
1
−
α
A
c
2
)
/
(1
+
α
)
,
A
2
,
st
=
(
A
c
2
−
α
A
c
1
)
/
(1
+
α
A
c
2
=
A
c
, the steady state
A
1
,
st
=
A
2
,
st
=
A
st
is stable when
=
A
st
(1
−
α
)
>
0, i.e., when
α<
1, which is the Gause-Lotka-Volterra criterion
(
May
,
1973
).
We can now consider the case in which the parameters
A
c
1
and
A
c
2
are two white-
Gaussian-noise processes with mean
A
0
and intensity
s
gn
. The randomization of
these parameters turns (
4.53
) into a set of nonlinear stochastic differential equations,
which does not lend itself to exact solutions. However, we can obtain an approximate
solution by expressing the state variables in (
4.53
) in terms of normalized deviation
from the steady state,
A
st
and linearizing
these equations (i.e., neglecting all the quadratic terms). In this case, if
ν
1
=
(
A
1
−
A
st
)
/
A
st
and
ν
2
=
(
A
2
−
A
st
)
/
α<
1, the
steady-state probability distributions of
ν
1
and
ν
2
are both Gaussian, with zero mean
and variance (
May
,
1973
):
2
s
gn
A
0
(1
2
ν
1
σ
=
)
.
(4.54)
−
α
,
2
Thus the effect of environmental variability (i.e.,
s
gn
=
0) on the fluctuations of
the state variables is enhanced as
1. It is intuitive to expect that extinction
occurs when
s
gn
exceeds a threshold value, similar to the case of the univariate
system discussed in Section 4.3. Similar to the univariate case, random environmental
fluctuations in the carrying capacity are able to cause noise-induced extinctions. Thus,
using Eq. (
4.54
),
May
(
1973
) suggested that extinction does not occur if
s
gn
α
→
A
0
(1
−
α
increases from 0 to 1 (i.e., the strength of the competitive interactions
between species increases), weaker environmental fluctuations are needed to cause
noise-induced extinctions. A similar analysis can be applied to predator-prey systems
with two or more species (May, 1973). More recently, the role of noise in predator-
prey systems was investigated in detail with respect to the emergence of stochastic
resonance and spatial patterns (
Spagnolo et al.
,
2004
). Some of these studies are
reviewed in this chapter and in Chapter 6.
)
/
2. As
α
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