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

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posted on 2023-06-08, 22:39 authored by Anotida MadzvamuseAnotida Madzvamuse, Hussaini S Ndakwo, Raquel Barreira
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.

Funding

Mathematical Modelling and Analysis of Spatial Patterning on Evolving Surfaces; G0872; EPSRC-ENGINEERING & PHYSICAL SCIENCES RESEARCH COUNCIL; EP/J016780/1

Research Training Network on Integrated Component Cycling in Epithelial Cell Motility; G1546; European Union, Horizon 2020; MSC-ITN

Unravelling new mathematics for 3D cell migration; G1438; LEVERHULME TRUST; RPG-2014-149

History

Publication status

  • Published

File Version

  • Published version

Journal

Discrete and Continuous Dynamical Systems - Series A

ISSN

1078-0947

Publisher

American Institute of Mathematical Sciences

Issue

4

Volume

36

Page range

2133-2170

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2015-09-30

First Open Access (FOA) Date

2015-09-30

First Compliant Deposit (FCD) Date

2015-09-29

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