Although a large number of experimental and theoretical studies have been carried out in a constant environment, as natural environments vary in time, it is important to ask if and how these results are affected by a changing environment. Here, we study the properties of the conditional fixation time of a mutant in a finite, randomly mating diploid population which is evolving in a periodically changing environment. In a static environment, as the conditional mean fixation time of a co-dominant beneficial mutant is equal to that of a deleterious mutant with the same magnitude of selection coefficient, similar patterns for beneficial and deleterious sweeps may result. We find that this symmetry breaks even when the environment is changing slowly. Furthermore, for intermediate dominance, the conditional mean fixation time of a beneficial mutant in a slowly changing environment depends weakly on the dominance coefficient and is close to the corresponding results in the static environment; however, the fixation time for a deleterious mutant under moderate selection with a slowly varying selection coefficient differs substantially from that in the constant environment when the mutant is recessive. Our results thus suggest that the variability patterns and levels for beneficial sweeps are mildly affected by temporally varying environment but changing environment is likely to strongly impact those due to recessive deleterious sweeps.

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The site frequency spectrum (SFS) is an important statistic that summarizes the molecular variation in a population, and used to estimate population-genetic parameters and detect natural selection. While the equilibrium SFS in a constant environment is quite well studied, recent research has focused on nonequilibrium SFS to elucidate the role of demography when selection is constant in time and of fluctuating selection in a population of constant size. However, the joint effect of time-dependent selection and population size has not been investigated so far. Here, we study the SFS in a randomly mating, diploid population in which both the population size and selection coefficient vary periodically with time using a diffusion theory approach, and derive simple analytical expressions for the time-averaged SFS in slowly and rapidly changing environments. We show that for strong selection and in slowly changing environments, the time-averaged SFS differs significantly from the equilibrium SFS when the population experiences both positive and negative cycles of the selection coefficient. The deviation depends on the time spent by the population in the deleterious part of the selection cycle and the phase difference between the selection coefficient and population size. In particular, we find that the time-averaged SFS in slowly to moderately fast varying, on-average neutral environment has the same qualitative shape as the equilibrium SFS for positively selected mutant but differs quantitatively from it which can be captured by an effective population size.

Although many experimental and theoretical studies on natural selection have been carried out in a constant environment, as natural environments typically vary in time, it is important to ask if and how the results of these investigations are affected by a changing environment. Here, we study the properties of the conditional fixation time defined as the time to fixation of a new mutant that is destined to fix in a finite, randomly mating diploid population with intermediate dominance that is evolving in a periodically changing environment. It is known that in a static environment, the conditional mean fixation time of a co-dominant beneficial mutant is equal to that of a deleterious mutant with the same magnitude of selection coefficient. We find that this symmetry is not preserved, even when the environment is changing slowly. More generally, we find that the conditional mean fixation time of an initially beneficial mutant in a slowly changing environment depends weakly on the dominance coefficient and remains close to the corresponding result in the static environment. However, for an initially deleterious mutant under moderate and slowly varying selection, the fixation time differs substantially from that in a constant environment when the mutant is recessive. As fixation times are intimately related to the levels and patterns of genetic diversity, our results suggest that for beneficial sweeps, these quantities are only mildly affected by temporal variation in environment. In contrast, environmental change is likely to impact the patterns due to recessive deleterious sweeps strongly.