Environmental Engineering Reference

In-Depth Information

4.7.2 Phase transitions in multivariate systems driven by dichotomous noise

We now consider the case of transitions that are not associated with the emergence of

new ordered states (modes) in the probability distribution of the state variables, but

with the noise-induced coexistence of two alternative stable phases represented by

two different steady-state probability distributions. In this case the dynamics select

either one of these phases, depending on the initial condition.

We follow the framework by
Mankin et al.
(
2004
), who investigated the dynamics of

an
N
-species symbiotic ecological system with a generalized Verhulst self-regulation

mechanism (i.e., density-dependent population growth) and symbiotic interspecies

interactions. The abundance
A
i
(
A
i
>

0) of species
i
varies with time as

⎧

⎨

⎩
α
i

⎫

⎬

⎭
,

1

β

A
i

K
i

d
A
i

d
t

=

A
i

−

+

J
i
,
j
A
j

(4.55)

j

=

i

where

α
i
is the growth-rate parameter and
K
i
is the carrying

capacity for species
i
.(
J
i
,
j
) is the interaction matrix, with coefficients
J
i
,
j
expressing

the effect of species
j
on the dynamics of species
i
. Following
Mankin et al.
(
2004
),

we concentrate on the case of symbiotic interactions, i.e., with
J
i
,
j
>

β
≥

0. In Eq. (
4.55
)

0 both for

i

i
. Moreover, to simplify the notation and the mathematical analysis of

Eq. (
4.55
) we refer to the case of a system with species-independent parameters. Thus

we take

>

j
and
j

>

N
.

Environmental variability (e.g., climate fluctuations) determines random changes

in the availability of the limiting resources, which translates into fluctuations in

the carrying capacity. To investigate the effect of environmental fluctuations on the

system's dynamics, we treat
K
i
(i.e., the carrying capacities) as an autocorrelated

random variable:

α
i
=
α

and
J
i
,
j
=

J

/

K
i
=

K
[1

+

a
0
ξ
dn
,
i
(
t
)]

,

(4.56)

where
a
0
<

1and

ξ
dn
,
i
(
t
) is a zero-mean DMN, with

ξ
=
1
,
2
=±

1 with autocor-

relation scale

τ
c
(see Chapter 2). Following
Mankin et al.
(
2004
), we can rewrite the

term
K
−
β

in Eq. (
4.55
)as

i

K
−
β

K
−
β
γ

=

+

ξ

,

[1

a

i
(
t
)]

(4.57)

dn

,

i

with

a
0
)
(1

−
a
0
)
β
,

1

+
a
0
)
β
+

γ
=

(1

(4.58)

2(1

−

a
0
)
β
−

a
0
)
β

(1

+

(1

−

a

=

a
0
)
β
.

(4.59)

(1

+

a
0
)
β
+

(1

−

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