表型可塑性群体的多态进化序列
The Polymorphic Evolution Sequence for Populations with Phenotypic Plasticity
摘要：
在本文中，我们研究了一类随机个体为基础的模型，描述了单倍体种群的演变，每个人的特点是由一个表型和一个基因型。个体的表型决定其自然出生率和死亡率以及竞争核$c(x,y)$，其描述了由于个体或$y$型的存在而导致的$x$型个体经历的诱导死亡率。当一个新的个体出生时，发生突变的概率很小，即后代具有与父母不同的基因型。我们研究的模型的新颖之处在于，具有给定基因型的个体可以表达一组不同的表型，并且在其一生中，它可以在不同的表型之间切换，其速率比突变率大得多，而且可能取决于整个群体的状态。种群的演化是一个连续时间的、测度值的马尔可夫过程。在我们的上一篇论文[4]中，提出了这样一个模型来描述免疫治疗下的肿瘤演变。在本文中，我们考虑了一个大类的模型，其中包括在[4]中研究的例子，并分析了它们的标度极限的总体规模趋于无穷大和突变率趋于零。在适当的假设下，我们证明了收敛到马尔可夫跳跃过程，这是Champagnat和Méléard [8，10]分析的多态演化序列（PES）的推广。
英文摘要：
In this paper we study a class of stochastic individual-based models that describe the evolution of haploid populations where each individual is characterised by a phenotype and a genotype. The phenotype of an individual determines its natural birth- and death rates as well as the competition kernel, $c(x,y)$ which describes the induced death rate that an individual of type $x$ experiences due to the presence of an individual or type $y$. When a new individual is born, with a small probability a mutation occurs, i.e. the offspring has different genotype as the parent. The novel aspect of the models we study is that an individual with a given genotype may express a certain set of different phenotypes, and during its lifetime it may switch between different phenotypes, with rates that are much larger then the mutation rates and that, moreover, may depend on the state of the entire population. The evolution of the population is described by a continuous-time, measure-valued Markov process. In our last paper [4], such a model was proposed to describe tumor evolution under immunotherapy. In the present paper we consider a large class of models which comprises the example studied in [4] and analyse their scaling limits as the population size tends to infinity and the mutation rate tends to zero. Under suitable assumptions, we prove convergence to a Markov jump process that is a generalisation of the polymorphic evolution sequence (PES) as analysed by Champagnat and Méléard [8, 10].
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