Bifurcations in reaction-diffusion systems for competing species: fast-slow and cross-diffusion
Università degli Studi Trento
38123 Povo (TN)
Tel +39 04 61/281508-1625-1701-3898-1980.
dept.math [at] unitn.it
Venue:The event will take place online through the ZOOM platform. To get the access codes please contact the secretary office (dept.math [at] unitn.it)
At: 11.30
Speaker:
- Cinzia Soresina (Institut für Mathematik und Wissenschaftliches Rechnen - Universität Graz)
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 linearized analysis with advanced numerical bifurcation methods via the continuation software pde2path. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. In 1D and 2D, we pay particular attention to the fast-reaction limit by computing sequences of bifurcation diagrams as the time-scale separation parameter tends to zero. We show that the bifurcation diagram undergoes major deformations once the fast-reaction systems limit onto the cross-diffusion singular limit. Furthermore, in 2D we find evidence for time-periodic solutions by detecting Hopf bifurcations, we characterize several regions of multi-stability and improve our understanding of the shape of patterns.
Contact person: Andrea Pugliese