Stability Analysis of Turing Patterns Generated by the Schnakenberg Model

Abstract

We consider the following Schnakenberg model on the interval (−1, 1):   ut = D1u − u + vu2 in (−1, 1), vt = D2v + B − vu2 in (−1, 1), u (−1) = u (1) = v (−1) = v (1) = 0, where D1 > 0, D2 > 0, B>0. We rigorously show that the stability of symmetric N−peaked steady-states can be reduced to computing two matrices in terms of the diffusion coefficients D1,D2 and the number N of peaks. These matrices and their spectra are calculated explicitly and sharp conditions for linear stability are derived. The results are verified by some numerical simulations.