Open AccessThe bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system
Author(s) -
Fengqi Yi,
Eamonn A. Gaffney,
Sungrim Seirin-Lee
Publication year - 2017
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2017031
Subject(s) - hopf bifurcation , bifurcation , context (archaeology) , instability , neumann boundary condition , homogeneous , turing , diffusion , biological applications of bifurcation theory , stability (learning theory) , mathematics , statistical physics , pitchfork bifurcation , pattern formation , mathematical analysis , boundary (topology) , physics , computer science , mechanics , nonlinear system , quantum mechanics , programming language , thermodynamics , paleontology , genetics , machine learning , biology
A delayed reaction-diffusion Schnakenberg system with Neumann boundary conditions is considered in the context of long range biological selforganisation 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