Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof

Abstract

In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.

Numerical continuation for a fast-reaction system and its cross-diffusion limit

“…In the OOPDE setting, we can employ the routine assema to compute those matrices, but this needs c(v) andc(u) on each element center, which is obtained interpolating u and v from the nodes to the element centers, as it can be seen in Listings 1 and 2, which show the main files implementing the triangular cross-diffusion system (1.5) in pde2path. Remark : Note that the parameter set used in [7,9,30,32] and reported in Table 1 gives α > 0. In Table 2 we report the bifurcation values d B (λ k ) corresponding to the eigenvalue λ k of the Laplacian for the 1D domain (0, 1) and obtained by formula (B.3), for different values of n, compared with the numerical values estimated by the software pde2path.…”

“…The software setup required for system (1.5) can be found in Appendix A. For the numerical results we use the set of parameter values widely used in literature for the weak competition regime [7,9,30,32], here reported in Table 1. We set d 1 = d 2 =: d and use d as one main bifurcation parameter.…”

“…which is one possible starting point for the continuation. In this section we consider a 1D domain (interval) Ω = (0, 1), as in [7,8,9,30,32]. We provide in Section 2.1 a detailed characterization of different steady state types of the cross-diffusion system (1.5).…”

“…Numerical continuation techniques are going to allow us to obtain a more global picture far from the homogeneous branch. The bifurcation diagram of the triangular cross-diffusion system on a 1D domain was presented in [30]; the existence of these non-homogeneous steady states significantly far from being perturbations of the homogeneous solutions, was proved in [7] applying a computer assisted method. A mathematically rigorous construction of the bifurcation structure of the three-component system was obtained in [9], while the convergence of the bifurcation structure on a 1D domain was also investigated in [32] with respect to two different bifurcation parameters appearing in the diffusion part, both in the weak and strong competition case.…”

A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

“…Such cases have been studied extensively, see e.g. [4,11,10,24,28]. Furthermore, we do not split into real and imaginary parts (cf.…”