发表机构
Université Le Havre Normandie（勒阿弗尔诺曼底大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究奇异摄动三物种Lotka--Volterra系统，证明循环优势在强不对称下持续，给出优势模式定量描述，并证明长时间行为为循环非周期，由三条异宿轨道组成。
AI 中文摘要
我们研究一个具有循环优势的三物种竞争Lotka--Volterra系统，其动机来自生态群落中的石头--剪刀--布相互作用。该模型包含一个数量级为$1/\epsilon$的奇异强相互作用系数，代表一种物种对另一种物种施加更强竞争压力的情形。我们证明，尽管存在这种强不对称性，循环优势机制仍然持续存在。我们的主要结果是对奇异区域$0<\epsilon\ll 1$中一个完整优势模式的定量描述。从第三物种占优势的状态附近出发，我们证明解依次经过第二和第一物种的优势状态附近，然后返回到一个可比较的构型。我们获得了返回时间和返回截面上种群密度的对数估计。特别地，尽管一个物种在每个模式的开始和结束时变得极其微小，但它在循环期间恢复到一阶密度。然后我们证明该模式无限重复，并且omega极限集由正象限边界上的三条异宿轨道组成，连接三个单物种平衡点。因此，长时间行为是循环但非周期的，在连续的单物种状态附近有越来越长的逗留。
英文摘要
We study a three-species competitive Lotka--Volterra system with cyclic dominance, motivated by rock--paper--scissors interactions in ecological communities. The model contains a singularly strong interaction coefficient of order $1/ε$, representing a regime in which one species exerts a much stronger competitive pressure on another. We show that, despite this strong asymmetry, the cyclic dominance mechanism persists. Our main result is a quantitative description of one complete dominance pattern in the singular regime $0<ε\ll 1$. Starting near the state where the third species dominates, we prove that the solution successively passes near the dominance states of the second and first species before returning to a comparable configuration. We obtain logarithmic estimates for the return time and for the population densities at the return section. In particular, although one species becomes extremely small at the beginning and at the end of each pattern, it recovers to an order-one density during the cycle. We then prove that this pattern repeats indefinitely and that the omega-limit set is composed of three heteroclinic orbits on the boundary of the positive orthant, connecting the three single-species equilibria. Thus the long-time behavior is cyclic but aperiodic, with increasingly long excursions near successive single-species states.
Comments36 pages, 3 figures, submitted
