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In this paper, we use rigorous numerics to compute several global smoothbranches of steady states for a system of three reaction-diffusion PDEsintroduced by Iida et al. [J. Math. Biol., {\bf 53}, 617--641 (2006)] to studythe effect of cross-diffusion in competitive interactions. An explicit andmathematically rigorous construction of a global bifurcation diagram is done,except in small neighborhoods of the bifurcations. The proposed method, eventhough 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 detailedanalysis of the choice of the parameters to be made in order to maximize thechances of performing successfully the computational proofs.

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