Noise-induced phenomena in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 150

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 −