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Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: a 3-component reaction-diffusion system. (English) Zbl 1277.65088
Summary: 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 M. Iida et al. [J. Math. Biol. 53, No. 4, 617–641 (2006; Zbl 1113.92064)] 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 J. B. van den Berg et al. [Math. Comput. 79, No. 271, 1565–1584 (2010; Zbl 1206.37045)], 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.

65N15 Error bounds for boundary value problems involving PDEs
37M20 Computational methods for bifurcation problems in dynamical systems
35K55 Nonlinear parabolic equations
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