A reaction-diffusion system has kinetics with Jacobian entries , , , (dimensionless…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Linear algebra and vector spaces

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

Problem

A reaction-diffusion system has kinetics with Jacobian entries

f_u=0.8

f_v=-1.2

g_u=1.5

g_v=-1.0

(dimensionless model units) at a uniform stationary state, with diffusion coefficients

D_u=1

for the activator

u

and

D_v

for the inhibitor

v

(same units). ((a)) Show that the uniform state is stable in the absence of diffusion. ((b)) Derive the general Turing condition from

h(q^2)=\det(\mathbf J - q^2\mathbf D)

and find the minimum ratio

D_v/D_u

at which a Turing instability first appears. ((c)) Find the critical wavenumber

q_{\rm c}

and the pattern wavelength

\Lambda_{\rm c}

at that ratio.

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