  In this paper we investigate the bifurcation structure of the triangular SKT
model in the weak competition regime and of the corresponding fast-reaction
system in 1D and 2D domains via numerical continuation methods. We show that
the software pde2path can be adapted to treat cross-diffusion systems,
reproducing the already computed bifurcation diagrams on 1D domains. We show
the convergence of the bifurcation structure obtained selecting the growth rate
as bifurcation parameter. Then, we compute the bifurcation diagram on a 2D
rectangular domain providing the shape of the solutions along the branches and
linking the results with the Turing instability analysis. In 1D and 2D, we pay
particular attention to the fast-reaction limit by always computing sequences
of bifurcation diagrams as the time-scale separation parameter tends to zero.
We show that the bifurcation diagram undergoes major deformations once the
fast-reaction systems limits onto the cross-diffusion singular limit.
Furthermore, we find evidence for time-periodic solutions by detecting Hopf
bifurcations, we characterize several regions of multi-stability, and improve
our understanding of the shape of patterns in 2D for the SKT model.
