Abstract
In this paper we investigate the bifurcation structure of the cross-diffusion Shigesada–Kawasaki–Teramoto model (SKT) in the triangular form and in the weak competition regime, and of a corresponding fast-reaction system in 1D and 2D domains via numerical continuation methods. We show that the software pde2path can be exploited to treat cross-diffusion systems, reproducing the already computed bifurcation diagrams on 1D domains. We show the convergence of the bifurcation structure obtained selecting the growth rate as bifurcation parameter. Then, we compute the bifurcation diagram on a 2D rectangular domain providing the shape of the solutions along the branches and linking the results with the linearized analysis. In 1D and 2D, we pay particular attention to the fast-reaction limit by always 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 limits onto the cross-diffusion singular limit. Furthermore, 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 in 2D for the SKT model.
Highlights
Systems with multiple time-scales appear in a wide variety of mathematical areas and in many applications [38]
– We show a link between the computed bifurcation diagrams in 1D and 2D domains, and the linearized analysis as a tool to fully understand and validate the numerical continuation calculations
In this paper we have investigated the bifurcation structure of the triangular Shigesada– Kawasaki–Teramoto model (SKT) model and of the corresponding fast-reaction system in 1D and 2D domains in the weak competition case via numerical continuation
Summary
Systems with multiple time-scales appear in a wide variety of mathematical areas and in many applications [38]. It turns out that when the fast-reaction system has three components, the limiting system are often of two types: free boundary problems [42] and cross-diffusion systems [6], which arise in population dynamics [11, 17, 30] In this framework, individuals of one or more species exist in different states and the small parameter represents the average switching time. The model (1.5) is known as Shigesada–Kawasaki–Teramoto (SKT) model as it was initially proposed in [45] in 1979 to account for stable non-homogeneous steady states in certain ecological systems These states describe, for suitable parameters sets, spatial segregation that is a situation of coexistence of two competitive species on a bounded domain. With standard diffusion in a convex domain and with zero-flux boundary conditions, any non-negative solution generically converges to homogeneous
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
