We study the probabilistic evolution of a birth and death continuous time
measure-valued process with mutations and ecological interactions. The
individuals are characterized by (phenotypic) traits that take values in a
compact metric space. Each individual can die or generate a new individual. The
birth and death rates may depend on the environment through the action of the
whole population. The offspring can have the same trait or can mutate to a
randomly distributed trait. We assume that the population will be extinct
almost surely. Our goal is the study, in this infinite dimensional framework,
of quasi-stationary distributions when the process is conditioned on
non-extinction. We firstly show in this general setting, the existence of
quasi-stationary distributions. This result is based on an abstract theorem
proving the existence of finite eigenmeasures for some positive operators. We
then consider a population with constant birth and death rates per individual
and prove that there exists a unique quasi-stationary distribution with maximal
exponential decay rate. The proof of uniqueness is based on an absolute
continuity property with respect to a reference measure.Comment: 39 page