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Adaptation of a population to a changing environment in the light of quasi-stationarity

Part of: Markov processes Ergodic theory Stochastic processes

Published online by Cambridge University Press:  30 August 2023

Aurélien Velleret Open the ORCID record for Aurélien Velleret [Opens in a new window]
Aurélien Velleret*
Affiliation:
Université Paris-Saclay, INRAE
*
*Postal address: Université Paris-Saclay, INRAE, MaIAGE, F-78350 Jouy-en-Josas, France. Email address: aurelien.velleret@nsup.org
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Abstract

We analyze the long-term stability of a stochastic model designed to illustrate the adaptation of a population to variation in its environment. A piecewise deterministic process modeling adaptation is coupled to a Feller logistic diffusion modeling population size. As the individual features in the population become further away from the optimal ones, the growth rate declines, making population extinction more likely. Assuming that the environment changes deterministically and steadily in a constant direction, we obtain the existence and uniqueness of the quasi-stationary distribution, the associated survival capacity, and the Q-process. Our approach also provides several exponential convergence results (in total variation for the measures). From this synthetic information, we can characterize the efficiency of internal adaptation (i.e. population turnover from mutant invasions). When the latter is lacking, there is still stability, but because of the high level of population extinction. Therefore, any characterization of internal adaptation should be based on specific features of this quasi-ergodic regime rather than the mere existence of the regime itself.

Keywords

Mobile optimumquasi-stationary distributionevolutionecologyjump processesMarkov process in continuous time and continuous space

MSC classification

Primary: 92D15: Problems related to evolution 37A10: One-parameter continuous families of measure-preserving transformations 60J25: Continuous-time Markov processes on general state spaces
Secondary: 92D25: Population dynamics (general) 37A30: Ergodic theorems, spectral theory, Markov operators 60G57: Random measures
Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 1 , March 2024 , pp. 235 - 286
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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