Cross-diffusion systems in population dynamics: derivation, bifurcations and pattern formation

Abstract:

In population dynamics, cross-diffusion describes the influence of one species on the diffusion of another. The cross-diffusion SKT model was proposed to account for stable inhomogeneous steady states exhibiting spatial segregation 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 for non-homogeneous steady states to exist, focusing on multistability regions and on the presence of time-periodic spatial patterns, by combining a detailed linearised and weakly non-linear analysis with advanced numerical bifurcation methods via the continuation software pde2path [1,2]. Recent results on the fractional cross-diffusion SKT model will be also discussed.

Even though the particular form of cross-diffusion terms in the SKT model may seem artificial, they naturally incorporates processes occurring at different time scales. It can be easily seen, at least at a formal level, that cross-diffusion appears in the fast-reaction limit of a "microscopic" model (in terms of time scales) presenting only standard diffusion and fast-reaction terms. The same approach can also be exploited in other contexts, e.g. predator-prey interactions, plant ecology and epidemiology.

[1] M. Breden, C. Kuehn, C. Soresina, On the influence of cross-diffusion in pattern formation, Journal of Computational Dynamics, 8(2):21, 2021.
[2] C. Soresina, Hopf bifurcations in the full SKT model and where to find them, Discrete and Continuous Dynamical Systems - S, 15(9):2673-2693, 2022.