Mathematical Biology seminar: Multistability and time-periodic spatial patterns in the cross-diffusion SKT model

Abstract: The Shigesada-Kawasaki-Teramoto model (SKT) was proposed to account for stable inhomogeneous steady states exhibiting spatial segregation, which describes a situation of coexistence of two competing species. Even though the reaction part does not present the activator-inhibitor structure, the cross-diffusion terms are the key ingredient for the appearance of spatial patterns. We provide a deeper understanding of the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearised and weakly non-linear analysis with advanced numerical bifurcation methods via the continuation software pde2path. We study the role of the additional cross-diffusion term in pattern formation, focusing on multistability regions and on the presence of time-periodic spatial patterns appearing via Hopf bifurcation points.