Abstract
In this article we present, for the first time, domain-growth induced pattern formation for reaction-diffusion systems with linear cross-diffusion on evolving domains and surfaces. Our major contribution is that by selecting parameter values from spaces induced by domain and surface evolution, patterns emerge only when domain growth is present. Such patterns do not exist in the absence of domain and surface evolution. In order to compute these domain-induced parameter spaces, linear stability theory is employed to establish the necessary conditions for domain-growth induced cross-diffusion-driven instability for reaction-diffusion systems with linear cross-diffusion. Model reaction-kinetic parameter values are then identified from parameter spaces induced by domain-growth only; these exist outside the classical standard Turing space on stationary domains and surfaces. To exhibit these patterns we employ the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces.
Highlights
Understanding of biological processes during growth development is an unresolved issue in developmental biology that is only starting to be addressed recently
In [32] we proved that in the presence of domain growth, it is no longer necessary to restrict reaction kinetics to an activatorinhibitor type; a long-range activation and short-range inhibition and/or activation chemical processes are all capable of giving rise to what we termed domain-growth induced diffusion-driven instability
Our paper is structured as follows: in Section 2 we present the model equations posed on evolving domains and surfaces and these consist of a system of reaction-diffusion equations with linear cross-diffusion
Summary
Understanding of biological processes during growth development is an unresolved issue in developmental biology that is only starting to be addressed recently. The focus of this paper is to study the role of domain growth in the generation of pattern formation in the absence and presence of cross-diffusion for systems of reaction-diffusion equations. Our paper is structured as follows: in Section 2 we present the model equations posed on evolving domains and surfaces and these consist of a system of reaction-diffusion equations with linear cross-diffusion. Such patterns do not exist in the absence of domain or surface evolution with or without linear cross-diffusion. We note that domains are a special case of evolving surfaces with appropriate boundary conditions
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