A simplified model of coral density on a reef bed uses the dimensionless…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Pattern formation: instabilities that choose a wavelength

Problem

A simplified model of coral density on a reef bed uses the dimensionless Swift-Hohenberg equation

\partial_tu = ru - (1+\partial_x^2)^2u - u^3

, with control parameter

r = 0.36

set by nutrient supply. In this model the fraction of the sea floor covered by coral in the patterned state is

\phi = 0.5 + 0.5A^*

, where

A^*

is the saturated pattern amplitude (so

A^*=0

gives a bare

\phi=0.5

cover and

A^*=1

gives full cover,

\phi=1

). ((a)) Find the unstable band of wavenumbers. ((b)) Find the saturated amplitude

A^*

and the resulting coverage fraction

\phi

. ((c)) Site percolation on a square lattice, with threshold

p_{\rm c} = 0.5927

, assumes that each site is occupied independently at random. Explain why comparing

\phi

with

p_{\rm c}

cannot decide whether the coral in this pattern forms a connected network across the reef.

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