Ecologists frequently try to predict the future geographic distributions of species. Most studies assume that the current distribution of a species reflects its environmental requirements (i.e., the species' niche). However, the current distributions of many species are unlikely to be at equilibrium with the current distribution of environmental conditions, both because of ongoing invasions and because the distribution of suitable environmental conditions is always changing. This mismatch between the equilibrium assumptions inherent in many analyses and the disequilibrium conditions in the real world leads to inaccurate predictions of species' geographic distributions and suggests the need for theory and analytical tools that avoid equilibrium assumptions. Here, we develop a general theory of environmental associations during periods of transient dynamics. We show that time-invariant relationships between environmental conditions and rates of local colonization and extinction can produce substantial temporal variation in occupancy-environment relationships. We then estimate occupancy-environment relationships during three avian invasions. Changes in occupancy-environment relationships over time differ among species but are predicted by dynamic occupancy models. Since estimates of the occupancy-environment relationships themselves are frequently poor predictors of future occupancy patterns, research should increasingly focus on characterizing how rates of local colonization and extinction vary with environmental conditions.

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We analyze the dynamics of two alternative alleles in a simple model of a population that allows for large family sizes in the distribution of offspring number. This population model was first introduced by Eldon and Wakeley, who described the backward-time genealogical relationships among sampled individuals, assuming neutrality. We study the corresponding forward-time dynamics of allele frequencies, with or without selection. We derive a continuum approximation, analogous to Kimura's diffusion approximation, and we describe three distinct regimes of behavior that correspond to distinct regimes in the coalescent processes of Eldon and Wakeley. We demonstrate that the effect of selection is strongly amplified in the Eldon-Wakeley model, compared to the WrightFisher model with the same variance effective population size. Remarkably, an advantageous allele can even be guaranteed to fix in the Eldon-Wakeley model, despite the presence of genetic drift. We compute the selection coefficient required for such behavior in populations of Pacific oysters, based on estimates of their family sizes. Our analysis underscores that populations with the same effective population size may nevertheless experience radically different forms of genetic drift, depending on the reproductive mechanism, with significant consequences for the resulting allele dynamics. W HEREAS natural selection undoubtedly plays an important role in adaptation, genetic drift is recognized as an equally or even more important force in shaping the patterns of heritable variation in a population. At a qualitative level, genetic drift implies that variation will tend to be lost from a population over time, even in the absence of selection, and that, in the presence of selection, the fitter type in a population is not guaranteed to fix. Our understanding of genetic drift and its interplay with natural selection has been shaped primarily by the population models introduced by Wright and Fisher (Wright 1931;Fisher 1958) and Moran (Moran 1958). Most work in population genetics rests implicitly on the Wright-Fisher framework, including Kimura's work on fixation probabilities (Kimura 1955), Ewens' sampling formula (Ewens 1972;Lessard 2007), Kingman's coalescent (Kingman 1982), tests of neutrality (Hudson et al. 1987;Tajima 1989;McDonald and Kreitman 1991;Fu and Li 1992;Fay and Wu 2000), and techniques for inferring mutation rates and selection pressures (Sawyer and Hartl 1992;Yang and Bielawski 2000;Bustamante et al. 2001;Desai and Plotkin 2008).The behavior of the Wright-Fisher model is remarkably robust to modifications of the model's underlying assumptions. Indeed, many other population genetic models, including the Moran process (Moran 1958), and a number of Karlin-Taylor and Cannings processes (Karlin and McGregor 1964;Cannings 1974;Ewens 2004), share the same diffusion limit as the Wright-Fisher model (Möhle 2001) and therefore have very similar behavior in sufficiently large populations. Likewise, the Kingman coalescent-that is, the backwardtime d...

We study the population genetics of two neutral alleles under reversible mutation in a model that features a skewed offspring distribution, called the L-Fleming-Viot process. We describe the shape of the equilibrium allele frequency distribution as a function of the model parameters. We show that the mutation rates can be uniquely identified from this equilibrium distribution, but the form of the offspring distribution cannot itself always be so identified. We introduce an estimator for the mutation rate that is consistent, independent of the form of reproductive skew. We also introduce a two-allele infinite-sites version of the L-Fleming-Viot process, and we use it to study how reproductive skew influences standing genetic diversity in a population. We derive asymptotic formulas for the expected number of segregating sites as a function of sample size and offspring distribution. We find that the WrightFisher model minimizes the equilibrium genetic diversity, for a given mutation rate and variance effective population size, compared to all other L-processes. MANY questions in population genetics concern the role of demographic stochasticity and its interaction with mutation and selection in determining the fates of allelic types. The foundational work of Fisher, Wright, Haldane, Kimura (Wright 1931;Haldane 1932;Fisher 1958;Kimura 1994), and others has been instrumental in shaping our intuition about the powerful role that genetic drift plays in evolution and especially its role in maintaining diversity. This classical theory, which views genetic drift as a strong force, emanates from the Wright-Fisher model of replication and its large-population limit, the Kimura diffusion (Kimura 1955). The diffusion approximation has been particularly well studied, not only because it is mathematically tractable, but also because it is robust to variation in many of the underlying model details. Many discrete population-genetic models, including a large number of Karlin-Taylor and Cannings processes (Karlin and McGregor 1964;Cannings 1974;Ewens 2004), share the same diffusion limit as the Wright-Fisher model, and they therefore exhibit qualitatively similar behavior.Nevertheless, Kimura's classical diffusion is not appropriate in every circumstance. Its central assumption is the absence of skew in the reproduction process-that is, the assumption that no single individual can contribute a sizable proportion to the composition of the population in a single generation. Recent studies have suggested that this assumption is violated in several species, especially in marine taxa but also including many types of plants (Beckenbach 1994;Hedgecock 1994), whose mode of reproduction involves a heavy-tailed offspring distribution.While the number of empirical studies on heavy-tailed offspring distributions is limited, there is a rich mathematical theory to describe the dynamics of populations with heavy reproductive skew. Beginning with Cannings' (1974) paper on neutral exchangeable reproduction processes, this literature has le...

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