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

Author： Breden Maxime Lessard Jean-Philippe Vanicat Matthieu

Publisher： Springer Publishing Company

ISSN： 0167-8019

Source： Acta Applicandae Mathematicae, Vol.128, Iss.1, 2013-12, pp. : 113-152

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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. 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.