强竞争下对称Lotka-Volterra系统中波速符号的完整刻画
Complete characterization of the sign of the wave speed in the symmetric Lotka-Volterra system under strong competition
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中文总结 AI 辅助
本文刻画了强竞争下对称两物种Lotka-Volterra竞争扩散模型的行波前速度符号，证明扩散率不同时波速非零，扩散快的物种会扩张，d=1时波速为零，确立了“团结并非力量”定理。
中文摘要 AI 辅助
本文对强竞争条件下的对称两物种Lotka-Volterra竞争扩散模型中的传播速度符号进行了完整刻画。该系统存在唯一的双稳行波前，其速度符号决定了两个物种中哪一个会入侵另一个。我们证明，对于每个大于1的强竞争强度和每个扩散比率d≠1，行波前会传播以扩大扩散速度更快的物种所占的领地；当d=1时，波速恰好为零。我们还建立了波速和行波前对模型参数的光滑依赖性。符号刻画的主要步骤是证明当扩散率不同时，单调驻波前不存在。结合波速对参数的连续性、系统的物种交换对称性以及某一特定参数值下的显式行波前，我们得到了整个参数区域内传播速度的符号。这确立了此前仅在受限参数区域内已知的“团结并非力量”定理。
英文摘要
This paper provides a complete characterization of the sign of the propagation speed in the symmetric two-species Lotka-Volterra competition-diffusion model under strong competition. The system admits a unique bistable travelling front and the sign of its speed determines which of the two species invades the other. We prove that for every strong competition intensity greater than $1$ and every diffusion ratio $d\neq1$, the travelling front propagates so as to expand the territory occupied by the faster-diffusing species. The front has a zero speed exactly when $d=1$. We also establish the smooth dependence of the wave speed and the travelling front on the model parameters. The main step in the sign characterization is to prove that a monotone standing front cannot exist when the diffusion rates are different. Combined with continuity of the wave speed with respect to the parameters, the species-exchange symmetry of the system, and an explicit travelling front at a particular parameter value, we obtain the sign of the propagation speed throughout the entire parameter region. This establishes the ``Unity is not strength'' theorem, which was previously known only in restricted parameter regimes.
