Journal Title
Title of Journal: Probab Theory Relat Fields
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Abbravation: Probability Theory and Related Fields
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Publisher
Springer-Verlag
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Authors: Pierre Collet Servet Martínez Sylvie Méléard Jaime San Martín
Publish Date: 2010/05/05
Volume: 151, Issue: 1-2, Pages: 191-231
Abstract
We study the probabilistic evolution of a birth and death continuous time measurevalued 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 the quasistationary distributions of the process conditioned on nonextinction We first show the existence of quasistationary 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 quasistationary distribution with maximal exponential decay rate The proof of uniqueness is based on an absolute continuity property with respect to a reference measure
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