Individual-based probabilistic models of adaptive evolution and various scaling approximations

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 in uenced by each individual's trait values, and interactions between individuals. An o spring usually inherits the trait values of her progenitor, except when a mutation causes the o spring 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 di erent nature: deterministic, in the form of ordinary, integro-, or partial di erential equations, or probabilistic, like stochastic partial di erential 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 di erent points of view concerning mutation-selection evolutionary models. Key-words: Darwinian evolution, birth-death-mutation-competition point process, mutationselection dynamics, nonlinear integro-di erential equations, nonlinear partial di erential equations, nonlinear superprocesses, tness, adaptive dynamics, trait substitution sequence.