Deterministic mechanisms of pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 288

Figure B.6. Spatial pattern emerging for the variable u using model ( B.22 ). The
parameters are η = 6 . 10, δ = 5 . 14, ϕ = 0 . 72, p =− 1 . 315, d 1 = 0, and d 2 = 1. The

simulation is carried out over a domain of 256 × 256 cells, each cell representing a

spatial step x = y = 0 . 8.

The plot of this relation (see Fig. B.5 ) shows that in this case the interval of the
unstable wave numbers has an upper bound and the most unstable mode has a finite
wave number. When a diffusive term is also added to the second equation (i.e., d 1 = 0,

d 2 = 0) as in Eqs. ( B.27 ), the dispersion relation becomes

2

( d 1 k 2 d 2 k 2 γ + + − f u − g v − i pk x ) γ (B.29)

i d 2 pk x k 2 d 1 k 2 g v − d 2 k 2 f u d 1 d 2 k 4 ) + ( f u g v − f v g u + i pk x g v − − + = 0 ,

with no substantial differences in the amplitude of the interval of unstable modes
(see Fig. B.5 ).
An example of spatial patterns emerging with this model is shown in Fig. B.6 .