Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system

Pages: 647 - 668, Volume 22, Issue 2, March 2017      doi:10.3934/dcdsb.2017031

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Fengqi Yi - Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China (email)
Eamonn A. Gaffney - Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcli e Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom (email)
Sungrim Seirin-Lee - Department of Mathematical and Life Sciences, Hiroshima University, Kagamiyama 1-3-1, Higashi-hiroshima 739-0046, Japan (email)

Abstract: 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.
Mathematics Subject Classification:  Primary: 35B32, 37G15, 35B36, 92C15; Secondary: 37L10.

Received: January 2016;      Revised: June 2016;      Available Online: December 2016.