Global bifurcation diagram of steady states of systems of PDEs via rigorous numerics: a 3-component reaction-diffusion system

TL;DR

This paper employs rigorous numerical methods to construct a comprehensive global bifurcation diagram for steady states in a three-component reaction-diffusion PDE system, revealing detailed insights into the effects of cross-diffusion.

Contribution

It introduces a novel gluing-free approach and new analytic estimates for global bifurcation analysis of PDE systems, enhancing computational rigor and efficiency.

Findings

Successful computation of global smooth branches of steady states

Explicit bifurcation diagram construction with rigorous proofs

Analysis of parameter choices to optimize computational success

Abstract

In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol., {\bf 53}, 617--641 (2006)] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comp., {\bf 79}, 1565--1584 (2010)], introduces new analytic estimates, a new {\em gluing-free} approach for the construction of global smooth branches and provides a detailed analysis of the choice of the parameters to be made in order to maximize the chances of performing successfully the computational proofs.