Noise-induced patterns in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 265

where 0 is a function of P [see Eq. ( 6.25 )]. Because in Fig. 6.9 (b) (bottom) the most

unstable mode k max is greater than zero (i.e.,

v

1
4 ), the instability (i.e.,

χ> − ζ>ζ ∗ )of

the homogeneous stable state v 0 is associated with the emergence of spatial patterns.

6.7 Spatiotemporal stochastic resonance in predator-prey systems
Some of the early stochastic resonance models of noise-induced pattern formation
in ecosystems were developed within the context of spatially explicit predator-prey
systems ( Spagnolo et al. , 2004 ). We present here the case of three interacting species:
two preys, A 1 , 2 ( r , t ). Following Spagnolo et al. ( 2004 ), we

consider the discrete representation of the dynamics of these interacting species; for
each point r i in a 2D square lattice, the biomass of the three species at time t

t ), and one predator, A 3 ( r ,

1 can
be related to their values at time t through the difference equations ( Spagnolo et al. ,
2004 ):

+

A 1 ( r i , t + 1) = μ A 1 ( r i , t ) [1 − ν A 1 ( r i , t ) − β ( t ) A 2 ( r i , t ) − γ A 3 ( r i , t )]

D
δ

+ A 1 ( r i , t ) ξ 1 ( r i , t ) + [ A 1 ( r δ , t ) − A 1 ( r i , t )] ,

, + = μ , − ν , − β , − γ ,

A 2 ( r i t 1) A 2 ( r i t ) [1 A 2 ( r i t ) ( t ) A 1 ( r i t ) A 3 ( r i t )]

+ D
δ

+ A 2 ( r i , t ) ξ 2 ( r i , t )

[ A 2 ( r δ , t )

− A 2 ( r i , t ) ] ,

t ) − β + γ [ A 1 ( r i , t )] = μ A 3 ( r i , A 3 ( r i , t + 1) t ) + A 2 ( r i ,

D
δ

+ A 3 ( r i , t ) ξ 3 ( r i , t ) + [ A 3 ( r δ , t ) − A 3 ( r i , t )] (6.28)

γ express the predator-prey interactions, and μ are the

where the parameters γ and

ν

reproduction rate of the logistic growth, is a parameter determining the resources

used by the same species, D is a diffusion coefficient, ξ 1 , 2 , 3 ( r i , t ) are independent

zero-mean white-Gaussian-noise terms with intensities s 1 , 2 , 3 , respectively. The sum

δ is calculated over the four nearest neighbors of r i . The interaction between the two
preys is expressed by the time-dependent coefficient β ( t ), which undergoes periodical

fluctuations with amplitude α and angular frequency ω 0 :

β ( t ) = 1 + + α cos( ω 0 , t ) ,

where is a parameter expressing the displacement of the mean from the unit value.

Thus , thereby inducing the periodic al-

ternation between conditions of coexistence of the two preys (

β ( t ) fluctuates around the value β = 1 +

β< 1) and exclusion

of one of them ( 1). Spagnolo et al. ( 2004 ) showed that in this system spatial

patterns emerge as an effect of the random forcing, as shown in Fig. 6.12 . If homoge-
neous initial conditions are used the two preys exhibit anticorrelated spatial patterns
(i.e., mutual exclusion of A 1 and A 2 ), whereas the predator has a spatial distribution

β>