基于个体的适应性进化概率模型和各种尺度近似
Individual-based probabilistic models of adaptive evolution and various scaling approximations
摘要：
我们有兴趣在模拟达尔文的进化，通过生态相互作用的表型变异和自然选择的相互作用。我们的模型是植根于微观的，随机描述的离散个体的人口特征的一个或几个适应性特征。人口建模为一个随机点过程，其发生器捕获的概率动态连续时间的出生，突变和死亡，每个人的特质值的影响，和个人之间的相互作用。后代通常继承其祖先的性状值，除非突变导致后代在出生时采取瞬时突变步骤以获得新的性状值。我们寻找易于处理的大总体近似。通过结合总体规模，出生率和死亡率，突变率，突变步骤，或时间的各种缩放，一个单一的微观模型，导致对比宏观限制，不同的性质：确定性，在形式的普通，积分，或偏微分方程，或概率，如随机偏微分方程或超过程。在罕见突变的限制，我们表明，一个可能的近似是一个跳跃过程，严格证明所谓的性状替代序列。因此，我们统一不同的观点突变选择进化模型。
英文摘要：
We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models are rooted in the microscopic, stochastic description of a population of discrete individuals characterized by one or several adaptive traits. The population is modelled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. We look for tractable large population approximations. By combining various scalings on population size, birth and death rates, mutation rate, mutation step, or time, a single microscopic model is shown to lead to contrasting macroscopic limits, of different nature: deterministic, in the form of ordinary, integro-, or partial differential equations, or probabilistic, like stochastic partial differential equations or superprocesses. In the limit of rare mutations, we show that a possible approximation is a jump process, justifying rigorously the so-called trait substitution sequence. We thus unify different points of view concerning mutation-selection evolutionary models.
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