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In a previous paper(2021), the author studied the asymptotic behavior ofcoexistence steady-states to the Shigesada-Kawasaki-Teramoto model as bothcross-diffusion coefficients tend to infinity at the same rate. As a result, heproved that the asymptotic behavior can be characterized by a limiting systemthat consists of a semilinear elliptic equation and an integral constraint.This paper studies the set of solutions of the limiting system. The first mainresult gives sufficient conditions for the existence/nonexistence ofnonconstant solutions to the limiting system by a topological approach usingthe Leray-Schauder degree. The second main result exhibits a bifurcationdiagram of nonconstant solutions to the one-dimensional limiting system byanalysis of a weighted time-map and a nonlocal constraint.

Global structure of steady-states to the full cross-diffusion limit in the Shigesada-Kawasaki-Teramoto model