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

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 reactiondiffusion model with linear cross-diffusion in the presence of domain evolution.

Analysis and Simulations of Coupled Bulk-surface Reaction-Diffusion Systems on Exponentially Evolving Volumes

“…For k << 1 (in particular when k gets closer to zero), the Coupled bulk-surface reaction-diffusion systems parameter spaces become almost identical and these represent parameter spaces in the absence of domain growth. As k becomes larger and larger, the parameter spaces become more distinct and larger in size (including the circular subregions), reinforcing earlier results for standard reaction-diffusion systems on growing domains [21,23,25,27]. The disconnected parameter spaces emerge due to the increase in the growth rate k as well as due to the dimension m.…”

“…Equally, ρ(t) could be an interpolation function fitted to a discrete sequence of experimental observations. To proceed, we map for all time t, the model equations posed on evolving volumes and surfaces to models posed on stationary reference domains and surfaces as shown below [12,21,27]. …”

Domain-Dependent Stability Analysis of a Reaction–Diffusion Model on Compact Circular Geometries

“…Systems of reaction-diffusion equations that model the evolution of pattern formation in nature are often a set of non-linear parabolic equations [5,7,9,10,17,18,21,23], whose solution is seldom analytically retrievable. The nature and complexity of these equations make numerical approaches [3,6,10,12,14,15] a necessary tool to investigate these systems [6,8,9,10,12,15,16,20,21,22]. Numerical approaches in their own right provide a partial insight to obtain an empirical understanding of the spatiotemporal behaviour of the dynamics governed by RDS, since it requires a verified analysis and classification of the parameter spaces [38,47] from which the values of the relevant parameters of a particular RDS are to be chosen such that these parameter values are within the bifurcation region of a particular expected behaviour in the evolving dynamics.…”

“…From the literature review on the topic one can easily notice that analytical approaches to study Hopf and transcritical bifurcation of RDS are often conducted on particular cases using one spatial dimension. Few examples in the literature where they derive parameter spaces in the context of bifurcation analysis is the work of Madzvamuse et al, [20,18] where they compute regions of parameter space, corresponding to diffusion-driven instability with and without cross-diffusion respectively for activator-depleted RDS. Their approach to computationally find the unstable spaces is restricted to Turing spaces.…”

Projected Finite Elements for Systems of Reaction-Diffusion Equations on Closed Evolving Spheroidal Surfaces

“…However, by introducing domain growth, an evolving Turing parameter space (i.e. the Turing diffusion-driven instability parameter space is time-dependent [27,28]) emerges from which patterning can arise only due to surface evolution. Our results confirm theoretical predictions published on evolving domains (see [27] for details).…”