  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., {\bf 53}, 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. 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 detailed
analysis of the choice of the parameters to be made in order to maximize the
chances of performing successfully the computational proofs.
