  We investigate the evolutionary rescue of a microbial population in a
gradually deteriorating environment, through a combination of analytical
calculations and stochastic simulations. We consider a population destined for
extinction in the absence of mutants, which can only survive if mutants
sufficiently adapted to the new environment arise and fix. We show that mutants
that appear later during the environment deterioration have a higher
probability to fix. The rescue probability of the population increases with a
sigmoidal shape when the product of the carrying capacity and of the mutation
probability increases. Furthermore, we find that rescue becomes more likely for
smaller population sizes and/or mutation probabilities if the environment
degradation is slower, which illustrates the key impact of the rapidity of
environment degradation on the fate of a population. We also show that our main
conclusions are robust across various types of adaptive mutants, including
specialist and generalist ones, as well as mutants modeling antimicrobial
resistance evolution. We further express the average time of appearance of the
mutants that do rescue the population and the average extinction time of those
that do not. Our methods can be applied to other situations with continuously
variable fitnesses and population sizes, and our analytical predictions are
valid in the weak-to-moderate mutation regime.
