Abstract
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 the quasi-stationary distributions of the process conditioned on non-extinction. We first show 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.
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Collet, P., Martínez, S., Méléard, S. et al. Quasi-stationary distributions for structured birth and death processes with mutations. Probab. Theory Relat. Fields 151, 191–231 (2011). https://doi.org/10.1007/s00440-010-0297-4
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DOI: https://doi.org/10.1007/s00440-010-0297-4