Dynamics of fixation probability in a population with fluctuating size

In many biological processes, the size of a population changes stochastically with time, and recent work in the context of cancer and bacterial growth have focused on the situation when the mean population size grows exponentially. Here, motivated by the evolutionary process of genetic hitchhiking in a selectively neutral population, we consider a model in which the mean size of the population increases linearly. We are interested in understanding how the fluctuations in the population size impact the first passage statistics and study the fixation probability that a mutant reaches frequency one by a given time in a population whose size follows a conditional Wright-Fisher process. We find that at sufficiently short and long times, the fixation probability can be approximated by a model that ignores temporal correlations between the inverse of the population size, but at intermediate times, it is significantly smaller than that obtained by neglecting the correlations. We also study the distribution of the population size at which the mutant fixes; we find that it is qualitatively different in the full model and the uncorrelated model and decays algebraically with different exponents in the two models. Our analytical and numerical study of the correlation functions show that the conditional Wright-Fisher process of interest is neither a stationary nor a Gaussian process; we also find that the variance of the inverse population size initially increases linearly with time t and then decreases as t^{-2} at intermediate times followed by an exponential decay at longer times. Our work emphasizes the importance of temporal correlations in populations with fluctuating size that are often ignored in population-genetic studies of biological evolution.