Evolutionary dynamics under demographic fluctuations: beyond the effective population size

It has long been appreciated that changing population size can strongly affect the evolutionary dynamics and genetic diversity, and traditionally, the effects of demographic fluctuations are captured via an effective population size. However, as has been pointed out in previous studies, an effective population size does not always exist and can be defined only when the relevant time scales are well separated. A mechanistic understanding of the non-existence of effective population size is, however, missing, and a very few analytical results have been obtained when the relevant time scales overlap with each other. To address these issues, we consider a neutral, panmictic population whose size fluctuates in time, and for a general demography, we show that the correlations between the population size at different times preclude the existence of an effective population size. We then consider a specific demography in which the population size switches between two fixed values in a stochastic fashion, and formulate a diffusion theory that allows us to obtain exact results for fixation probability, fixation time and mean sojourn time of a mutant for arbitrary switching rates. We find that when the switching time and the time scale for the genetic drift are of the same order, these quantities differ substantially from the corresponding results predicted by a neutral model with an effective population size given by the harmonic mean of the population size, and discuss the implications of these results for neutral genetic diversity in changing environments.