Crossing a fitness valley in a changing environment: With and without pit stop

We consider a stochastic individual-based model of adaptive dynamics for an asexually reproducing population with mutation. Biologically motivated by the influence of seasons or the variation of drug concentration during medical treatment, the model parameters vary over time as piecewise constant and periodic functions. We study the typical evolutionary behavior of the population by looking at limits of large populations and rare mutations. An analysis of the crossing of valleys in the fitness landscape in a changing environment leads to various interesting phenomena on different time scales, which depend on the length of the valley. By carefully examining the influence of the changing environment on each time scale, we are able to determine the crossing rates of fit mutants and their ability to invade the resident population. In addition, we investigate the special scenario of pit stops, where single intermediate mutants within the valley have phases of positive fitness and can thus grow to a diverging size before going extinct again. This significantly accelerates the traversal of the valley and leads to an interesting new time scale.