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

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

Pages: 2133 - 2170, Volume 36, Issue 4, April 2016      doi:10.3934/dcds.2016.36.2133

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Anotida Madzvamuse - University of Sussex, School of Mathematical and Physical Sciences, Pevensey III, 5C15, Brighton, BN1 9QH, United Kingdom (email)
Hussaini Ndakwo - School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, England, United Kingdom (email)
Raquel Barreira - Polytechnic Institute of Setubal, Barreiro School of Technology, Rua Américo da Silva Marinho-Lavradio, 2839-001 Barreiro, Portugal (email)

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.

Keywords:  Reaction-diffusion systems, cross-diffusion, evolving domains, growing domains, cross-diffusion driven instability, activator-activator model, long-range activation, short-range inhibition, evolving finite elements.
Mathematics Subject Classification:  Primary: 35K55, 35K57, 35K58; Secondary: 37B55, 37C60, 37C75.

Received: January 2015;      Revised: August 2015;      Available Online: September 2015.