Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusion

Madzvamuse, Anotida, Ndakwo, Hussaini S and Barreira, Raquel (2016) Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusion. Discrete and Continuous Dynamical Systems - Series A, 36 (4). pp. 2133-2170. ISSN 1078-0947

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Abstract

This article presents stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a short-range inhibition, long-range activation reaction-diffusion model with linear cross-diffusion in the presence of domain evolution.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0297 Numerical analysis
Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
Depositing User: Anotida Madzvamuse
Date Deposited: 30 Sep 2015 07:28
Last Modified: 06 Mar 2017 21:20
URI: http://sro.sussex.ac.uk/id/eprint/56954

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Project NameSussex Project NumberFunderFunder Ref
Mathematical Modelling and Analysis of Spatial Patterning on Evolving SurfacesG0872EPSRC-ENGINEERING & PHYSICAL SCIENCES RESEARCH COUNCILEP/J016780/1
Unravelling new mathematics for 3D cell migrationG1438LEVERHULME TRUSTRPG-2014-149
Research Training Network on Integrated Component Cycling in Epithelial Cell MotilityG1546European Union, Horizon 2020MSC-ITN