AI 中文总结
该研究揭示空间结构可诱导后向分岔，使空间异质性的两物种复制子模型中，空间入侵率无法完全预测共存，推导了共存状态出现的显式局部条件，改变了经典复制子动力学的结论。
AI 中文摘要
复制子方程是研究生态学和进化博弈理论中频率依赖选择的核心框架。在均匀混合种群中，两物种系统的长期结果由成对入侵适合度的符号决定，导致优势、共存或双稳态。然而，许多生态系统具有空间结构，环境异质性和扩散塑造着局部相互作用。这些空间效应如何改变经典复制子机制仍未被完全理解。本研究中，我们探讨两物种复制子方程的空间异质性扩展，其中成对入侵适合度随空间变化，频率在扩散和平流下演化。在此设定下，经典入侵适合度被相关线性化算子主特征值给出的空间入侵率取代。利用分岔理论，我们表明这些空间入侵率无法完全确定系统的定性动力学。具体而言，空间异质性可诱导后向分岔，即使在均匀混合复制子预测竞争排斥的参数区域，也能产生稳定共存状态。我们推导了该机制的显式局部条件，刻画此类共存状态的出现时机。这些结果表明，空间结构可从根本上改变经典复制子动力学，并为共存的产生提供了具体机制。
英文摘要
The replicator equation is a central framework for studying frequency--dependent selection in ecology and evolutionary game theory. In well-mixed populations, the long-term outcome of a two-species system is determined by the signs of the pairwise invasion fitnesses, leading to dominance, coexistence, or bistability. However, many ecological systems are spatially structured, with environmental heterogeneity and dispersal shaping local interactions. How these spatial effects modify the classical replicator regimes remains incompletely understood. In this work, we study a spatially heterogeneous extension of the two-species replicator equation in which pairwise invasion fitnesses vary across space, and frequencies evolve under diffusion and advection. In this setting, the classical invasion fitnesses are replaced by spatial invasion rates given by the principal eigenvalues of the associated linearized operators. Using bifurcation theory, we show that these spatial invasion rates do not fully determine the qualitative dynamics of the system. In particular, spatial heterogeneity can induce a backward bifurcation, generating stable coexistence states even in parameter regimes where the well-mixed replicator predicts competitive exclusion. We derive an explicit local condition for this mechanism, characterizing when such coexistence states arise. These results show that spatial structure can fundamentally alter classical replicator dynamics and provide a concrete mechanism through which coexistence may emerge.
