2017/08/04 by Martina Baar, Baar, Martina, Anton Bovier +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Evolution and Genetic Dynamics #FOS: Biological sciences #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Populations and Evolution (q-bio.PE) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1708.01528
openalex publication_date 2017/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study a class of stochastic individual-based models that\ndescribe the evolution of haploid populations where each individual is\ncharacterised by a phenotype and a genotype. The phenotype of an individual\ndetermines its natural birth- and death rates as well as the competition\nkernel, c(x,y) which describes the induced death rate that an individual of\ntype x experiences due to the presence of an individual or type y. When a\nnew individual is born, with a small probability a mutation occurs, i.e. the\noffspring has different genotype as the parent. The novel aspect of the models\nwe study is that an individual with a given genotype may express a certain set\nof different phenotypes, and during its lifetime it may switch between\ndifferent phenotypes, with rates that are much larger then the mutation rates\nand that, moreover, may depend on the state of the entire population. The\nevolution of the population is described by a continuous-time, measure-valued\nMarkov process. In our last paper [4], such a model was proposed to describe\ntumor evolution under immunotherapy. In the present paper we consider a large\nclass of models which comprises the example studied in [4] and analyse their\nscaling limits as the population size tends to infinity and the mutation rate\ntends to zero. Under suitable assumptions, we prove convergence to a Markov\njump process that is a generalisation of the polymorphic evolution sequence\n(PES) as analysed by Champagnat and M 'el 'eard [8, 10].\n