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We analyze the long-term stability of a stochastic model designed toillustrate the adaptation of a population to variation in its environment. Apiecewise-deterministic process modeling adaptation is coupled to a Fellerlogistic diffusion modeling population size. As the individual features in thepopulation become further away from the optimal ones, the growth rate declines,making population extinction more likely. Assuming that the environment changesdeterministically and steadily in a constant direction, we obtain the existenceand uniqueness of the quasi-stationary distribution, the associated survivalcapacity and the Q-process. Our approach also provides several exponentialconvergence results (in total variation for the measures). From this syntheticinformation, we can characterize the efficiency of internal adaptation (i.e.population turnover from mutant invasions). When the latter is lacking, thereis still stability, but because of the high level of population extinction.Therefore, such a characterization must be based on specific features of thisquasi-ergodic regime.

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