A population that adapts to gradual environmental change will typically experience temporal variation in its population size and the selection pressure. On the basis of the mathematical theory of inhomogeneous branching processes, we present a framework to describe the fixation process of a single beneficial allele under these conditions. The approach allows for arbitrary time-dependence of the selection coefficient s(t) and the population size N(t), as may result from an underlying ecological model. We derive compact analytical approximations for the fixation probability and the distribution of passage times for the beneficial allele to reach a given intermediate frequency. We apply the formalism to several biologically relevant scenarios, such as linear or cyclic changes in the selection coefficient, and logistic population growth. Comparison with computer simulations shows that the analytical results are accurate for a large parameter range, as long as selection is not very weak. FOR adaptive evolution to proceed, it is not enough that new beneficial mutations enter a population. To complete an adaptive step, these mutations also need to escape stochastic loss due to genetic drift, get established, and finally rise to fixation. The fixation process of beneficial (or neutral or deleterious) alleles is one of the building blocks of population genetic theory and many of the key results on fixation probabilities and times date back to its early days. Two alternative mathematical frameworks have been developed to derive analytical expressions for these quantities: branching processes (Fisher 1922(Fisher , 1930Haldane 1927) and diffusion theory (Kimura 1962;. Today, a large body of literature exists to study fixation under various ecological scenarios and genetic conditions (reviewed in Patwa and Wahl 2008), such as the effects of population structure (Whitlock 2003) and spatial heterogeneity (Whitlock and Gomulkiewicz 2005) and interference due to selection on linked loci (Barton 1995) or due to epistatic interaction (Takahasi and Tajima 2005).In this article, we consider the fixation process in a variable environment, leading to time-dependent selection coefficients and population sizes. Aspects of this problem have already been studied in previous work: In particular, the impact of various scenarios of demographic change (growth, decline, cycles) on the fixation probability has been treated in a series of articles (Ewens 1967;Chia 1968;Otto and Whitlock 1997;Pollak 2000;Parsons and Quince 2007a;Orr and Unckless 2008). Studies on time-dependent selection mostly concentrate on stochastic fluctuations of the selection coefficient (Jensen 1973;Karlin and Levikson 1974;Takahata and Ishii 1975;Huillet 2011). Since the distribution of the selection coefficients is constant across generations, these models are still time-homogeneous in a probabilistic sense. In contrast, surprisingly little is known when the changes of the selection coefficient s = s(t) follow an explicit trend. Ohta and Kojima (1968) derive an...

Environmental change, if severe, can drive a population extinct unless the population succeeds in adapting to the new conditions. How likely is a population to win the race between population decline and adaptive evolution? Assuming that environmental degradation progresses across a habitat, we analyze the impact of several ecological factors on the probability of evolutionary rescue. Specifically, we study the influence of population structure and density-dependent competition as well as the speed and severity of environmental change. We also determine the relative contribution of standing genetic variation and new mutations to evolutionary rescue. To describe population structure, we use a generalized island model, where islands are affected by environmental deterioration one after the other. Our analysis is based on the mathematical theory of time-inhomogeneous branching processes and complemented by computer simulations. We find that in the interplay of various, partially antagonistic effects, the probability of evolutionary rescue can show nontrivial and unexpected dependence on ecological characteristics. In particular, we generally observe a nonmonotonic dependence on the migration rate between islands. Counterintuitively, under some circumstances, evolutionary rescue can occur more readily in the face of harsher environmental shifts, because of the reduced competition experienced by mutant individuals. Similarly, rescue sometimes occurs more readily when the entire habitat degrades rapidly, rather than progressively over time, particularly when migration is high and competition strong.

Multiple treatment strategies are available for empiric antibiotic therapy in hospitals, but neither clinical studies nor theoretical investigations have yielded a clear picture when which strategy is optimal and why. Extending earlier work of others and us, we present a mathematical model capturing treatment strategies using two drugs, i.e the multi-drug therapies referred to as cycling, mixing, and combination therapy, as well as monotherapy with either drug. We randomly sample a large parameter space to determine the conditions determining success or failure of these strategies. We find that combination therapy tends to outperform the other treatment strategies. By using linear discriminant analysis and particle swarm optimization, we find that the most important parameters determining success or failure of combination therapy relative to the other treatment strategies are the de novo rate of emergence of double resistance in patients infected with sensitive bacteria and the fitness costs associated with double resistance. The rate at which double resistance is imported into the hospital via patients admitted from the outside community has little influence, as all treatment strategies are affected equally. The parameter sets for which combination therapy fails tend to fall into areas with low biological plausibility as they are characterised by very high rates of de novo emergence of resistance to both drugs compared to a single drug, and the cost of double resistance is considerably smaller than the sum of the costs of single resistance.