具有随机死亡的物种进化模型
Models for species evolution with random deaths
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中文总结 AI 辅助
研究三种物种进化离散时间模型，其在死亡机制上有差异。通过耦合论证和平均场极限的新方法，建立渐近结果，表明模型死亡机制微小变化会致渐近行为大不同。
中文摘要 AI 辅助
我们考虑三种物种进化的离散时间模型。在每个时间步\(n\)，以概率\(p\)一个物种诞生并具有独立的\(\text{均匀}[0,1]\)适应度值，以概率\(1 - p\)一个物种死亡。三种模型的区别在于死亡时选择杀死哪个物种的机制：第一种模型总是杀死最不适应的物种；第二种模型，以概率\(r\)杀死最不适应的物种，以概率\(1 - r\)从种群中均匀选择一个物种杀死；第三种模型，以概率\(r\)杀死最不适应的物种，以概率\(1 - r\)杀死适应度小于独立的\(\text{均匀}[0,1]\)结果的最大适应度物种。我们建立了三种模型当\(n \to \infty\)时的渐近结果。这些结果表明模型死亡机制的微小变化会导致截然不同的渐近行为。为证明结果，我们开发了一种依赖耦合论证和平均场极限的新方法。
英文摘要
We consider three discrete-time models for species evolution. In all three models, at each time step $n$, with probability $p$, a species is born with an independent $\text{Uniform}[0,1]$ fitness value and, with probability $1-p$, a species is killed. The mechanism for selecting which species to kill when a death occurs distinguishes the three models: in the first model, the least fit species is always killed; in the second model, with probability $r$, the least fit species is killed and, with probability $1-r$, a species chosen uniformly from the population is killed; in the third model, with probability $r$, the least fit species is killed and, with probability $1-r$, the species with the largest fitness less than an independent $\text{Uniform}[0,1]$ outcome is killed. We establish asymptotic results as $n \to \infty$ for the three models. These results demonstrate that small changes to the death mechanism of the model can lead to vastly different asymptotic behaviour. To prove our results we develop a novel approach that relies on coupling arguments and mean-field limits.
