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This paper studies the Lotka-Volterra competition model with cross-diffusionterms under homogeneous Dirichlet boundary conditions. We consider theasymptotic behavior of positive steady-states as equal two cross-diffusioncoefficients tend to infinity (so-called the full cross-diffusion limit). Thefirst result shows that, at the full cross-diffusion limit, the set of positivesteady-state solutions can be classified into two types: the small coexistenceor the complete segregation. The small coexistence can be characterized by theset of positive solutions of a nonlinear elliptic equation, while the completesegregation can be characterized by the set of nodal solutions of anothernonlinear elliptic equation. The second result concerns the global bifurcationstructure of the 1d model with large cross-diffusion terms to show that thebranch of small coexistence bifurcates from the trivial solution and manybranches of complete segregation bifurcate from solutions on the branch ofsmall coexistence. Finally, in order to verify the theoretical results, wechase some bifurcation branches numerically via the continuation softwarepde2path.

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model