Global Bifurcation Diagrams of Steady States of Systems of PDEs via Rigorous Numerics: a 3-Component Reaction-Diffusion System

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. 53(4):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. Comput. 79(271):1565–1584, 2010], introduces new analytic estimates, a new 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.