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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.

MSC:
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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