Cross-diffusion induced Turing instability for a three species food chain model
type: Item
instance of: scholarly article
title: Cross-diffusion induced Turing instability for a three species food chain model
zbMATH Open document ID: 1243.35093
author: Zhifu Xie
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-02-11T00:00:00Z
review text: The author studies a three species food chain model in which the third species preys on the second one and the second species preys on the first one. This is modeled by the following strongly coupled reaction-diffusion system \[ \begin{cases} u_{1t}=\Delta [(k_{11}+k_{12}u_2)u_1]+u_1(d_1-b_{11}u_1-b_{12}u_2)],\\ u_{2t}=\Delta [(k_{21}u_1+k_{22}+k_{23}u_3)u_2]+u_2(d_2+b_{21}u_1-b_{22}u_2-b_{23}u_3)],\\ u_{3t}=\Delta [(k_{32}u_2+k_{33})u_3]+u_3(d_3+b_{32}u_2-b_{33}u_3)] \end{cases} \] on \(\Omega\times (0,\infty)\) under the Neumann boundary condition \(\displaystyle {\partial u_1\over \partial n}={\partial u_2\over \partial n}={\partial u_3\over \partial n}=0\) in \(\partial \Omega\times (0,\infty)\) and the initial condition \(u_i(x,0)=u_{i0}(x)\) in \(\Omega\) for \(i=1,2,3\). Here \(n\) is the unit outward normal to \(\partial \Omega\), \(b_{ij}, d_i, k_{ii}>0\), and \(k_{ij}\geq 0\) for \(i\neq j\). It is shown that the system has a unique positive equilibrium that is globally asymptotically stable for the corresponding ODE system. It remains linearly stable for the reaction-diffusion system without cross-diffusion (\(k_{ij}=0\) for \(i\neq j\)). But with increasing cross-diffusion rates \(k_{21}\) and \(k_{32}\), the equilibrium can become unstable. This shows that the instability is not of the classical Turing type, but solely driven by cross-diffusion.
reviewed by: Yaping Liu
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35Q92
zbMATH DE Number: 6005711
zbMATH Keywords: predator-prey models
zbMATH Keywords: reaction-diffusion system
zbMATH Keywords: cross-diffusions
zbMATH Keywords: Turing instability
zbMATH Keywords: stationary patterns
zbMATH Keywords: food chain
zbMATH Keywords: positive equilibrium
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.10.054
OpenAlex ID: W1984409514
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DOI: 10.1016/J.JMAA.2011.10.054
description: scientific article; zbMATH DE number 6005711