A two-species reaction-diffusion system has the local kinetics Jacobian

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 two-species reaction-diffusion system has the local kinetics Jacobian, evaluated at the uniform steady state,

\mathbf{J} = \begin{pmatrix} f_u & f_v \\ g_u & g_v \end{pmatrix} = \begin{pmatrix} 1.0 & -2.0 \\ 3.0 & -0.5 \end{pmatrix},

with diffusion coefficients

D_u = 1

and

D_v = 5

(in the model's length-squared over time unit). Someone argues:"Since

f_u = 1 > 0

(self-activation) and

g_v = -0.5 < 0

(self-inhibition), and

D_v/D_u = 5 > 1

, the activator-inhibitor conditions are satisfied. Computing

\det\mathbf{J} = f_ug_v - f_vg_u = (1)(-0.5) - (-2)(3) = 5.5

, the critical wavenumber is

q_{\rm c}^2 = \sqrt{\det\mathbf{J}/(D_uD_v)} = \sqrt{5.5/5} = 1.049

, so

q_{\rm c} = 1.024

, and a Turing pattern of this wavenumber appears." ((a)) Find the error in this reasoning. ((b)) Determine what actually happens to the uniform state in the absence of diffusion, and say whether the resulting instability, if any, can be called a Turing instability.

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