The effect of heterogeneity on one-peak stationary solutions to the Schnakenberg model

Abstract

In this paper, we consider the Schnakenberg model with heterogeneity on the interval (−1,1). We first construct stationary solutions which concentrate at a suitable point by using the Liapunov-Schmidt reduction method. Moreover, by investigating the associated linearized eigenvalue problem, we establish the linear stability of the solutions above. Iron, Wei, and Winter (2004) established the existence and stability of multi-peak symmetric stationary solutions in non-heterogeneity case. In their work, the one-peak solution is always stable. For the symmetric heterogeneity case, Ishii and Kurata (2019) gave the analysis of one-peak symmetric solutions in details and revealed a destabilization effect of the heterogeneity. In this paper, we reveal that the mechanism which the location of the concentration point and the stability with respect to eigenvalues of order o(1) are determined by the interaction of the heterogeneity with the associated Green's function. In particular, we not suppose that the heterogeneity is symmetric. Also, by a typical example, we performed several numerical simulations to illustrate our main results.