Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity

Abstract

In this paper, we consider the following one-dimensional Schnakenberg model with periodic heterogeneity:        ut − ε 2 uxx = dε − u + g(x)u 2 v, x ∈ (−1, 1), t > 0, εvt − Dvxx = 1 2 − c ε g(x)u 2 v, x ∈ (−1, 1), t > 0, ux(±1) = vx(±1) = 0. where d, c, D > 0 are given constants, ε > 0 is sufficiently small, and g(x) is a given positive function. Let N ≥ 1 be an arbitrary natural number. We assume that g(x) is a periodic and symmetric function, namely g(x) = g(−x) and g(x) = g(x + 2N −1). We study the stability of N-peak stationary symmetric solutions. In particular, we are interested in the effect of the periodic heterogeneity g(x) above on their stability. For the standard Schnakenberg model, namely the case of g(x) = 1, with d = 0, the stability of N-peak solutions was established by Iron, Wei, and Winter in 2004. In this paper, we rigorously give a linear stability analysis and reveal the effect of the periodic heterogeneity on the stability of N-peak solution. In particular, we investigate how N-peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case g(x) = 1.

Cross-diffusion induced spatiotemporal patterns in Schnakenberg reaction–diffusion model

“…Pattern dynamics generated from Schnakenberg model have been recognized by [3][4][5][6][7][8][9][10][11] and so on. Different solutions as peaked steady states, pulses, spiky solutions and semi-analytical solutions have been widely studied by many authors and they have found much richer dynamics exhibited by the Schnakenberg model (see, for example, [12][13][14][15][16][17][18][19]). Local dynamics and bifurcation problems in Schnakenberg system have been investigated [20][21][22].…”

Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs

“…[24] and the reference therein.) Recently, there are the study of the existence and stability of spike patterns for the Schnakenberg model on a one-dimensional interval with heterogeneity [14,10,11,1]. In these works, the effect of heterogeneity on the existence of spiky solutions and their stability were investigated.…”