Environmental Engineering Reference
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(a)
t
(b)
t
Figure 4.30. (a) Deterministic (i.e.,
s
gn
=
8) dynamics
of the modified Holt-McPeek system (
Lai and Liu
,
2005
) calculated with
r
0) and (b) stochastic (
s
gn
=
=
2
.
6,
K
1
=
100,
K
2
=
50,
e
1
=
0
.
5,
e
2
=
0
.
01,
m
=
1 (after Lai and Liu, 2005).
fraction undergoes mortality (the
cost of dispersal
)atarate1
−
m
(i.e., only a
fraction
m
of the migrators survive). The dynamics take place in such a way that
density-dependent growth precedes both dispersal and cost of dispersal. The overall
dynamics can be expressed as
N
1
,
1
(
t
+
1)
=
(1
−
e
1
)
W
1
(
t
)
N
1
,
1
(
t
)
+
me
1
N
1
,
2
(
t
)
,
N
1
,
2
(
t
+
1)
=
(1
−
e
1
)
W
2
(
t
)
N
1
,
2
(
t
)
+
me
1
N
1
,
1
(
t
)
,
N
2
,
1
(
t
+
1)
=
(1
−
e
2
)
W
1
(
t
)
N
2
,
1
(
t
)
+
me
2
N
2
,
2
(
t
)
,
N
2
,
2
(
t
+
1)
=
(1
−
e
2
)
W
2
(
t
)
N
2
,
2
(
t
)
+
me
2
N
2
,
1
(
t
)
.
(4.77)
Holt and McPeek
(
1996
) investigated the dynamical properties of this system when
one species has a much higher dispersal rate than the other (
e
1
01).
For relatively small values of
r
the dynamics are stable and converge to a state in
which species 1 goes extinct. Thus the frequency of the high-dispersal species,
=
0
.
5and
e
2
=
0
.
N
1
,
1
(
t
)
+
N
1
,
2
(
t
)
N
t
,
1
+
p
1
(
t
)
=
,
(4.78)
N
t
,
2
tends to zero [Figure
4.30
(a)]. This happens for
r
3 in the case shown in Fig.
4.30
.As
r
tends to 3, the dynamics exhibit a cyclic behavior. For larger values of
r
, a transition
to chaos occurs: Instead of tending to zero, the density of the high-dispersal species
undergoes episodic and abrupt increases. This behavior is completely deterministic
and does not depend on the existence of any external forcing (
Holt and McPeek
,
1996
).
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