A domain-dependent stability analysis of reaction–diffusion systems with linear cross-diffusion on circular domains

Abstract

In this study, we present theoretical considerations of, and analyse, the effects of circular geometry on the stability analysis of semi-linear parabolic PDEs of reaction–diffusion type with linear cross-diffusion for a two-component system on circular domains. The highlights of our theoretical and computational findings are: (i) By employing linear stability analysis for a two-component reaction–diffusion system with linear cross-diffusion on circular disc domains, we derive necessary and sufficient conditions for the system to exhibit cross-diffusion driven-instability, dependent on the length scale of the geometry. These analytical studies involve cross-diffusion and circular geometry to unravel analytical conditions for the full computational classification of the parameter spaces that allow the system to exhibit Turing, Hopf and transcritical patterns. (ii) We compute parameter spaces on which patterns are formed only due to linear cross-diffusion as well as due to a critical domain length. These spaces do not exist in the absence of cross-diffusion nor when the conditions on the domain length are violated. (iii) To support our theoretical findings, finite element simulations illustrating the formation of spot patterns on circular domains are presented. Model parameter values are selected from parameter spaces that are induced by cross-diffusion, thereby supporting linear cross-diffusion coupled with reaction–diffusion theory as a candidate mechanism for pattern formation. (iv) A by-product of this study, is that an activator-depleted reaction–diffusion system with linear cross-diffusion on circular domains, appears to favour the formation of spot patterns for most of the parameter values chosen. Such patterns are reminiscent of those observed on stingrays, which form on approximately circular domains during growth development.