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 ≥ 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.
Related
- Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules
- Existence and Asymptotic Stability of Traveling Waves of Two-species Lotka-Volterra Competition-Diffusion Systems via Geometric Singular Perturbations
- Sharp Thresholds for Linear Determinacy and Zero Wave Speed in Lotka-Volterra Competition-Diffusion Systems
- Spreading--vanishing dynamics and speed selection in a free-boundary competition model under a shifting climate
- Strong solutions of SDEs with critical discontinuities in diffusion coefficients
BibTeX
@misc{shu2026fastestspecies,
title = {{Fastest-Species Selection in Critical Multispecies Competition-Diffusion Systems via an Entropy Approach}},
author = {Haozhe Shu and Dongyuan Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.24253},
eprint = {2608.24253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24253},
}
RIS
TY - UNPB
TI - Fastest-Species Selection in Critical Multispecies Competition-Diffusion Systems via an Entropy Approach
AU - Haozhe Shu
AU - Dongyuan Xiao
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2608.24253
UR - https://arxiv.org/abs/2608.24253
AB - 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 ≥ 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.
ER -