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Numerical approach to existence and stability of stationary solutions to a SKT cross-diffusion equation

The SKT cross-diffusion equation is proposed by N. Shigesada, K. Kawasaki and E. Teramoto in 1979 to investigate segregation phenomena of two competing species with each other in the same habitat area. The effect of cross-diffusion affects the population pressure between two different species. Lou and Ni derived limiting systems to see whether this effect may give rise to a spatial segregation or not, and to clarify its mechanism. In this paper, we introduce some new representation of solutions to a stationary limiting problem modified from representation by Lou, Ni and Yotsutani. We apply it to the numerical investigation of existence, non-existence, multiplicity and stability.