  This paper studies the Lotka-Volterra competition model with cross-diffusion
terms under homogeneous Dirichlet boundary conditions. We consider the
asymptotic behavior of positive steady-states as equal two cross-diffusion
coefficients tend to infinity (so-called the full cross-diffusion limit). The
first result shows that, at the full cross-diffusion limit, the set of positive
steady-state solutions can be classified into two types: the small coexistence
or the complete segregation. The small coexistence can be characterized by the
set of positive solutions of a nonlinear elliptic equation, while the complete
segregation can be characterized by the set of nodal solutions of another
nonlinear elliptic equation. The second result concerns the global bifurcation
structure of the 1d model with large cross-diffusion terms to show that the
branch of small coexistence bifurcates from the trivial solution and many
branches of complete segregation bifurcate from solutions on the branch of
small coexistence. Finally, in order to verify the theoretical results, we
chase some bifurcation branches numerically via the continuation software
pde2path.
