  In this paper, we consider the Shigesada-Kawasaki-Teramoto (SKT) model, which
presents cross-diffusion terms describing competition pressure effects. Even
though the reaction part does not present the activator-inhibitor structure,
cross-diffusion can destabilise the homogeneous equilibrium. However, in the
full cross-diffusion system and weak competition regime, the cross-diffusion
terms have an opposite effect and the bifurcation structure of the system
modifies increasing the interspecific competition pressure. The major changes
in the bifurcation structure, the type of pitchfork bifurcations on the
homogeneous branch, as well as the presence of Hopf bifurcation points are here
investigated. Through weakly nonlinear analysis, we can predict the type of
pitchfork bifurcation. Increasing the additional cross-diffusion coefficients,
the first two pitchfork bifurcation points from super-critical become
sub-critical, leading to the appearance of a multi-stability region. The
interspecific competition pressure also influences the possible appearance of
stable time-period spatial patterns appearing through a Hopf bifurcation point.
