  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, such a characterization must be based on specific features of this
quasi-ergodic regime.
