Abstract
This work explores mechanisms for pattern forma-tion through coupled bulk-surface partial differential equations of reaction-diffusion type. Reaction-diffusion systems posed both in the bulk and on the surface on stationary volumes are coupled through linear Robin-type boundary conditions. The presented work in this paper studies the case of non-linear reactions in the bulk and surface, respectively. For the investigated system is non-dimensionalised and rigorous linear stability analysis is carried out to determine the necessary and sufficient conditions for pattern formation. Appropriate parameter spaces are generated from which model parameters are selected. To exhibit pattern formation, a coupled bulk-surface finite element method is devel-oped and implemented. The numerical algorithm is implemented using an open source software package known as deal.II and show computational results on spherical and cuboid domains. Also, theoretical predictions of the linear stability analysis are verified and supported by numerical simulations. The results show that non-linear reactions in the bulk and surface generate patterns everywhere.
Highlights
Most biological and chemical processes that can be explored through reaction and diffusion of chemical species are often modelled by systems of partial differential equations (PDEs) [1]–[3]
In order to quantify the evolution of chemical reaction kinetics associated to biological processes, it is a usual approach to employ a system of partial differential equations describing the chemical reactions, which is investigated through mathematical techniques to reveal the long-term behaviour of the evolving kinetics [8], [9]
We find bulk-surface reaction-diffusion equations that model a particular aspect of cellular functions with relevance to chemical signalling
Summary
Most biological and chemical processes that can be explored through reaction and diffusion of chemical species are often modelled by systems of partial differential equations (PDEs) [1]–[3]. Stability and bifurcation analysis are two other usual analytical approaches to understanding the dynamical properties of reaction-diffusion system near a uniform steady state [17]– [21] It is evident from the literature on the subject of stability analysis that a very limited amount of work is done on stability analysis in a coupled bulk-surface set-up. This is done through investigating the necessary conditions for diffusion-driven instability for the system.
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