This work explores mechanisms for pattern formation through coupled bulksurface 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. In this framework we study three different systems as follows (i) non-linear reactions in the bulk and surface respectively, (ii) non-linear reactions in the bulk and linear reactions on the surface and (iii) linear reactions in the bulk and non-linear reactions on the surface. In all cases, the systems are 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 developed and implemented. We implement the numerical algorithm by using an open source software package known as deal.II and show computational results on spherical and cuboid domains. 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, while non-linear reactions in the bulk and linear reactions on the surface generate patterns in the bulk and on the surface with a pattern-less thin boundary layer. However, linear reactions in the bulk do not generate patterns on the surface even when the surface reactions are non-linear. The generality, robustness and applicability of our theoretical computational framework for coupled system of bulk-surface reaction-diffusion equations set premises to study experimentally driven models where coupling of bulk and surface chemical species is prevalent. Examples of such applications include cell motility, pattern formation in developmental biology, material science and cancer biology.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:751909 |
| Date | January 2018 |
| Creators | Alhazmi, Muflih |
| Publisher | University of Sussex |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://sro.sussex.ac.uk/id/eprint/78232/ |
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