Noise-induced patterns in environmental systems - Noise-induced Phenomena in the Environmental Sciences - page 258

−1

x

Figure 6.8. Spatiotemporal patterns of φ obtained as solutions of the 1D version of

( 6.11 ) calculated for A 1 = A 2 = 1. Figure taken from Buceta et al. ( 2002a ).

respect to the rate of convergence to equilibrium in each of the two states, we can
replace the noise term in ( 6.13 ) with its average value. We obtain the (deterministic)
equation

∂φ ( r , t )

2

k 0 ) 2 = f 1 [ φ ( r , t )] P + f 2 [ φ ( r , t )](1 − P ) − D ( ∇ + φ ( r , t ) . (6.14)

∂ t

Thus, in this case, we can investigate pattern formation by referring to the aver-
age dynamics, using the methods presented for the deterministic case [Eqs. ( 5.18 )
and ( 5.19 )]. Although for P = 0and P = 1 the potential function defined as

f (

V (

)] has only one stable state, for inter-
mediate values of P bistability may emerge. When A 1 =

φ ) =− φ ) =− [ Pf 1 ( φ ) + (1 − P ) f 2 ( φ

A 2 = 1and P = 0 . 5, the

local dynamics are expressed by the function f

3 . In this system, stable
patterns emerge, as shown in Fig. 6.8 for the case of a 1D domain.
We need to stress again that, even though in this example patterns emerge as a result
of the random switching between two deterministic dynamics, these patterns are not
necessarily noise induced. In fact, Buceta and Lindenberg ( 2002a ) showed that the
same patterns would emerge even when the random forcing

= φ − φ

ξ dn in ( 6.13 ) is replaced
with a deterministic periodic function driving the switching between the two states,
1 and 2. Thus, in this system, dichotomous noise is used as a random mechanism to
drive the repeated alternation between two deterministic states. However, deterministic