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We are interested in a microscopic stochastic description of a
population of discrete individuals characterized by one adaptive
trait. The population is modeled 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. Once this point process is in place, the quest
for tractable approximations can follow different mathematical paths,
which differ in the normalization they assume (taking limit on
population size , rescaling time) and in the nature of the
corresponding approximation models: integro or integro-differential
equations, superprocesses. In particular cases, we consider the long
time behaviour for the stochastic or deterministic models.