Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme
type: Item
instance of: scholarly article
title: Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme
zbMATH Open document ID: 1270.35090
author: Rui Peng
author: Fengqi Yi
author: Xiao-Qiang Zhao
published in: Journal of Differential Equations
publication date: 2013-03-04T00:00:00Z
review text: The authors treat the reaction-diffusion model proposed in the article [\textit{H. Degn} and \textit{D. E. F. Harrison}, ``Theory of oscillations of respiration rate in continuous culture of Klebsiella aerogenes'', J. Theor. Biol. 22, 238--248 (1969)] to explain the observed oscillatory behavior of respiration rate in the continuous cultures of the bacteria Klebsiella aerogenes, described by the system in dimensionless parameters \[ u_{t}-\Delta u=a-u-\frac{uv}{1+ku^{2}} \] \[ v_{t}-d\Delta v=b-\frac{uv}{1+ku^{2}} \] in \(\Omega\subset \mathbf{R}^{n},\) \(n\geq 1\) with homogeneous Neumann boundary conditions \(\partial_{v}u=\partial_{v}v=0\) on \(\partial\Omega\) and nonnegative continuous initial values \(u(x,0)=u_{0}(x)\geq 0,\) \(v(x,0)=v_{0}(x)\geq 0\) in \(\Omega.\) The formation of spatial and temporal patterns in this system are investigated with special attention to Turing patterns. Following Turing's idea of ``diffusion driven instability'' the relevant steady state problem is investigated. The obtained results include the global attracting of the constant steady state, the existence and nonexistence conditions of positive nonconstant steady states, the Hopf and steady state bifurcations, generation of spatiotemporal patterns.
reviewed by: Svetlana A. Grishina
Mathematics Subject Classification ID: 35B36
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 92C40
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35B32
zbMATH DE Number: 6140825
zbMATH Keywords: Turing pattern
zbMATH Keywords: inhomogeneous periodic oscillatory pattern
zbMATH Keywords: global attractivity
MaRDI profile type: Publication
full work available at URL: j.jde.2012.12.009
OpenAlex ID: W1990042727
DOI: 10.1016/J.JDE.2012.12.009
description: scientific article; zbMATH DE number 6140825