Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model

Abstract

In a previous paper [8], the author studied the asymptotic behavior of coexistence steady-states to the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. As a result, he proved that the asymptotic behavior can be characterized by a limiting system that consists of a semilinear elliptic equation and an integral constraint. This paper studies the set of solutions of the limiting system. The first main result gives sufficient conditions for the existence/nonexistence of nonconstant solutions to the limiting system by a topological approach using the Leray-Schauder degree. The second main result exhibits a bifurcation diagram of nonconstant solutions to the one-dimensional limiting system by analysis of a weighted time-map and a nonlocal constraint.

Nonconstant steady states and pattern formations of generalized 1D cross‐diffusion systems with prey‐taxis

“…Unfortunately, nearly all researches in biomathematics have a confusing hypothesis that 𝑑 > 0 for convenience of discussion or realistic principle in biology. [17][18][19][20][21][22][23] Therefore, their models cannot illustrate various phenomena in a precise way. If all creatures are profit and avoid loss, then why there exist suicide phenomena?…”

“…In the same way, the population moves toward higher density of another population if the cross‐diffusion rate is negative and the species tends to lower density of another species if the cross‐diffusion rate is positive. Unfortunately, nearly all researches in biomathematics have a confusing hypothesis that $$d>0$$ for convenience of discussion or realistic principle in biology 17–23 . Therefore, their models cannot illustrate various phenomena in a precise way.…”