The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system
Fengqi Yi Eamonn A. Gaffney Sungrim Seirin-Lee
Discrete & Continuous Dynamical Systems - B 2017, 22(2): 647-668 doi: 10.3934/dcdsb.2017031

A delayed reaction-diffusion Schnakenberg system with Neumann boundary conditions is considered in the context of long range biological self-organisation dynamics incorporating gene expression delays. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for determining the direction of bifurcation and the stability of the bifurcating periodic solution. The delay-diffusion driven instability of the unique spatially homogeneous steady state solution and the diffusion-driven instability of the spatially homogeneous periodic solution are investigated, with limited simulations to support our theoretical analysis. These studies analytically demonstrate that the modelling of gene expression time delays in Turing systems can eliminate or disrupt the formation of a stationary heterogeneous pattern in the Schnakenberg system.

keywords: Schnakenberg model delay and diffusion normal form method Hopf bifurcation diffusion driven instability delay-diffusion driven instability

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