A reaction-diffusion system has 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 reaction-diffusion system has kinetics Jacobian, at the uniform steady state,J=(fufvgugv)=(0.5−2.01.0−0.7),\mathbf{J} = \begin{pmatrix} f_u & f_v \\ g_u & g_v \end{pmatrix} = \begin{pmatrix} 0.5 & -2.0 \\ 1.0 & -0.7 \end{pmatrix},with tr J=−0.2<0\mathrm{tr}\,\mathbf{J} = -0.2 < 0 and det⁡J=1.65>0\det\mathbf{J} = 1.65 > 0, so the uniform state is stable without diffusion. Diffusion coefficients are Du=1D_u = 1 and Dv=1.2D_v = 1.2. Someone argues:"The inhibitor vv diffuses faster than the activator uu (Dv/Du=1.2>1D_v/D_u = 1.2 > 1), so the activator-inhibitor mechanism applies and a Turing pattern must form." ((a)) Find the error in this reasoning. ((b)) Find the minimum ratio Dv/DuD_v/D_u for which a Turing instability actually occurs in this system, keeping Du=1D_u = 1 fixed.

Browse the problem library