图上Moran过程中替换者及其演化稳定性
Replacers and their evolutionary stability in the Moran process on graphs
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中文总结 AI 辅助
本研究探究图上Moran过程中替换者表型的演化稳定性，发现定居替换者对突变入侵有强抵御性，其固定概率在高度数图中指数级下降，低度数下依赖结构特征，整体呈现“先到者生存”而非“适者生存”的规律。
中文摘要 AI 辅助
有限结构化种群中的演化动力学通常采用Moran生死过程建模。核心指标之一是单个入侵者试图接管定居者种群的固定概率。近期研究提出了一种具备邻域感知能力的新表型，称为replacer（替换者）：替换者永远不会浪费繁殖回合，只要存在另一类型的个体，就总会替换该类型个体。本研究针对被突变型替换者入侵的定居型替换者种群，探究其演化稳定性。研究发现定居者对这类入侵具有极强的抵御能力，并通过三类结果量化了该效应的强度。第一，在规模为$N$的均匀混合种群中，即使入侵者的相对繁殖率为固定值$r>1$，其固定概率仍随$N$呈指数级下降，所有高度数图均满足这一规律。第二，针对有界度数图展开研究：证明在环图上，优势入侵者的固定概率仅下降常数倍；但同时提出了最大度数为4的图结构，当$r\not\rm le 1.9$时，入侵者固定概率随$N$呈指数级下降。由此可见，高度数是演化稳定性的充分条件，而低度数下的演化稳定性取决于底层空间结构的特定特征。第三，证明了任意图的通用界：对于任意图$G$，无论$r\not\rm approx 1$还是$r\not\rm ge 2$，入侵者的固定概率均低于自然基线——即均匀混合种群中无感知个体的标准Moran过程的固定概率。综上，研究结果表明替换者的演化动力学更符合“先到者生存”的规律，而非经典的“适者生存”。
英文摘要
Evolutionary dynamics in finite structured populations are commonly modeled by the Moran Birth-death process. A key quantity is the fixation probability of a single invader attempting to take over a population of residents. A recent work introduced a new neighborhood-aware phenotype called a replacer. A replacer never wastes their reproductive turn by always replacing an individual of the other type (if available). In this work, we study the evolutionary stability of resident replacers who are invaded by mutant replacers. We find that residents are strongly protected against such invasions, and we quantify the strength of this effect by showing three types of results. First, we show that on well-mixed populations of size $N$, the invader fixation probability is exponentially small in $N$, even when the invader has a fixed relative reproductive rate $r>1$, and the same holds for all high-degree graphs. Second, we study bounded-degree graphs. We prove that on cycles, the fixation probability of an advantageous invader decreases only by a constant factor. However, we also present graphs with maximum degree 4, where the invader fixation probability is exponentially small in $N$ whenever $r\le1.9$. Thus, high degrees are sufficient for evolutionary stability, whereas with low degrees the evolutionary stability depends on specific features of the underlying spatial structure. Third, we prove general bounds for arbitrary graphs. Namely, we show that for any graph $G$ the invader fixation probability drops below the natural baseline given by the standard Moran process with oblivious individuals on a well-mixed population, both for $r\approx 1$ and for $r\ge 2$. Together, our results establish that the evolutionary dynamics of replacers is better characterized by the phrase ``survival of the first'' rather than the classic ``survival of the fittest''.
