An activator and an inhibitor obey a reaction-diffusion system whose reaction…

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Genetic circuits, regulation and biological information · Genetic circuits, regulation and biological information

Forces, Machines and Signals, Ch Genetic circuits and biological pattern

Problem

An activator

u

and an inhibitor

v

obey a reaction-diffusion system whose reaction Jacobian at the uniform steady state, in scaled dimensionless units, is

\mathbf{J} = \begin{pmatrix} 0.5 & -1 \\ 2 & -3\end{pmatrix},

with

u

the first variable and

v

the second. The activator diffuses with diffusion coefficient

D_u=1

(in the same scaled units) and the inhibitor with

D_v=d

. ((a)) Show that the uniform state is stable in the absence of diffusion. ((b)) Linearise about the uniform state with a perturbation

\propto e^{iqx}

and write

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

, with

\mathbf D=\mathrm{diag}(1,d)

, as a quadratic function of

q^2

. ((c)) Derive the condition on

d

for this determinant to go negative for some real

q

, and evaluate the critical value

d_c

. ((d)) At

d=20

, find the range of unstable wavenumbers

q

and the wavenumber of the fastest-growing mode.

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