turkmath.org

In this talk, a domain-dependent mathematical analysis of reaction-diffusion systems is presented to understand the role of geometry and cross-diffusion in pattern formation. Linear stability analysis is employed to derive the constraints which are necessary in understanding the dual roles of linear cross-diffusion and domain-size for studying the instability of a reaction-diffusion system. Theorems are stated for the conditions for Turing, Hopf and transcritical instabilities are proven in terms of lower and upper bounds of the domain-size together with the reaction, self- and cross-diffusion coefficients. These bounds allow for the full parameter classification of the model system, which is presented in terms of the relationship between the domain size, self and cross-diffusion parameters. Regions showing Turing instability, Hopf and transcritical types of bifurcations are demonstrated using the parameter values of the system. To support theoretical findings, a state-of-the-art finite element method is employed. The finite element method is a numerical method that solves highly nonlinear systems of partial differential equations on complex geometries. The finite element numerical solutions reveal spatial and spatiotemporal patterns on rectangular, circular, and annular geometries, with no flux boundary conditions. Observed patterns on non-convex geometries, for example, resemble ring-shaped cross-sectional scans of hypoxic tumours. Specifically, the cross-section of an actively invasive region in a hypoxic tumour can be effectively approximated by an annulus.