Fastest-Species Selection in Critical Multispecies Competition-Diffusion Systems via an Entropy Approach

Abstract

We study an $N$-species competition--diffusion system with a common competition term, for which the associated ODE system admits infinitely many equilibria. For $N=2$, Alfaro and Xiao \cite{AlfaroXiao2023} studied the long-time behavior of solutions with compactly supported initial data by constructing precise super- and sub-solutions. For $N\ge 3$, the absence of a comparison principle for the entire system presents a major difficulty. Using a weighted entropy method, we prove that every bounded nonnegative traveling profile connecting nonzero equilibria is constant, without imposing any monotonicity assumption. For nontrivial compactly supported initial populations, we further show that the species with the strictly largest Fisher-KPP speed excludes all slower species and spreads at its scalar Fisher-KPP speed. Finally, we establish heat-type lower bounds for the slower species, revealing a bump phenomenon despite their uniform extinction.