  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 M\'el\'eard [8, 10].

Probability;Populations and Evolution