Parameter Spaces for Cross-Diffusive-Driven Instability in a Reaction–Diffusion System on an Annular Domain

In this work, the influences of geometry and domain size on spatiotemporal pattern formation are investigated to establish the parameter spaces for a cross-diffusive reaction–diffusion model on an annulus. By applying the linear stability theory, we derive the conditions which can give rise to Turing, Hopf and transcritical types of diffusion-driven instabilities. We explore whether the selection of a sufficiently large domain size, together with the appropriate selection of parameters, can give rise to the development of patterns on nonconvex geometries, e.g. annulus. Hence, the key research methodology and outcomes of our studies include a complete analytical exploration of the spatiotemporal dynamics in an activator-depleted reaction–diffusion system; a linear stability analysis to characterize the dual roles of cross-diffusion and domain size of pattern formation on an annulus region; the derivation of the instability conditions through the lower and upper bounds of the domain size; the full classification of the model parameters; and a demonstration of how cross-diffusion relaxes the general conditions for the reaction–diffusion system to exhibit pattern formation. To validate the theoretical findings and predictions, we employ the finite element method to reveal the spatial and spatiotemporal patterns in the dynamics of the cross-diffusive reaction–diffusion system within a two-dimensional annular domain. These observed patterns resemble those found in ring-shaped cross-sectional scans of hypoxic tumors. Specifically, the cross-section of an actively invasive region in a hypoxic tumor can be effectively approximated by an annulus.