Why do species not adapt to ever-wider ranges of conditions, gradually expanding their ecological niche and geographic range? Gene flow across environments has two conflicting effects: although it increases genetic variation, which is a prerequisite for adaptation, gene flow may swamp adaptation to local conditions. In 1956, Haldane proposed that, when the environment varies across space, "swamping" by gene flow creates a positive feedback between low population size and maladaptation, leading to a sharp range margin. However, current deterministic theory shows that, when variance can evolve, there is no such limit. Using simple analytical tools and simulations, we show that genetic drift can generate a sharp margin to a species' range, by reducing genetic variance below the level needed for adaptation to spatially variable conditions. Aided by separation of ecological and evolutionary timescales, the identified effective dimensionless parameters reveal a simple threshold that predicts when adaptation at the range margin fails. Two observable parameters determine the threshold: (i) the effective environmental gradient, which can be measured by the loss of fitness due to dispersal to a different environment; and (ii) the efficacy of selection relative to genetic drift. The theory predicts sharp range margins even in the absence of abrupt changes in the environment. Furthermore, it implies that gradual worsening of conditions across a species' habitat may lead to a sudden range fragmentation, when adaptation to a wide span of conditions within a single species becomes impossible.W hy a species' range sometimes ends abruptly, even when the environment changes smoothly across space, has interested ecologists and evolutionary biologists for many decades (1-8). Haldane (2) proposed that, when the environment is spatially heterogeneous, a species may be unable to adapt and expand its range because gene flow from the center swamps the populations at the range margins, preventing their adaptation. Theory showed that, when genetic variance is fixed, adaptation indeed fails if the environment changes too steeply across space (9), and a sharp margin to the species' range forms. The population remains well adapted only in the center of the range, and gene flow swamps variants adapted to the margins, preventing range expansion. This result also elucidates range margins in the presence of competitors: then, interspecific competition in effect steepens the environmental gradient (10). However, this limit to adaptation assumes that local genetic variation is fixed. Current deterministic theory states that, when genetic variance can evolve, there is no sharp limit to a species' range (11). The genetic mixing caused by gene flow inflates the genetic variance and facilitates further divergence. Gene flow across a phenotypic gradient maintained by the environment can generate much more variance than would be maintained by mutation alone (12,13). This rise of genetic variance with environmental gradient can allow species to adapt to ...

Populations living in a spatially and temporally changing environment can adapt to the changing optimum and/or migrate toward favorable habitats. Here we extend previous analyses with a static optimum to allow the environment to vary in time as well as in space. The model follows both population dynamics and the trait mean under stabilizing selection, and the outcomes can be understood by comparing the loads due to genetic variance, dispersal, and temporal change. With fixed genetic variance, we obtain two regimes:(1) adaptation that is uniform along the environmental gradient and that responds to the moving optimum as expected for panmictic populations and when the spatial gradient is sufficiently steep, and (2) a population with limited range that adapts more slowly than the environmental optimum changes in both time and space; the population therefore becomes locally extinct and migrates toward suitable habitat. We also use a population-genetic model with many loci to allow genetic variance to evolve, and we show that the only solution now has uniform adaptation.

All species are restricted in their distribution. Currently, ecological models can only explain such limits if patches vary in quality, leading to asymmetrical dispersal, or if genetic variation is too low at the margins for adaptation. However, population genetic models suggest that the increase in genetic variance resulting from dispersal should allow adaptation to almost any ecological gradient. Clearly therefore, these models miss something that prevents evolution in natural populations. We developed an individual-based simulation to explore stochastic effects in these models. At high carrying capacities, our simulations largely agree with deterministic predictions. However, when carrying capacity is low, the population fails to establish for a wide range of parameter values where adaptation was expected from previous models. Stochastic or transient effects appear critical around the boundaries in parameter space between simulation behaviours. Dispersal, gradient steepness, and population density emerge as key factors determining adaptation on an ecological gradient.

More than a hundred years after Grigg's influential analysis of species' borders, research into the causes of limits to species' ranges is more active than ever, fuelled by our need to understand their dynamics in the changing environments. Current predictions are either very specific, requiring measurements of many interrelated parameters, or make restrictive assumptions such as fixing the genetic variance or neglecting the two-dimensional spatial structure of most natural habitats. I show that the range margin can be understood based on just two measurable parameters: i) the fitness cost of dispersal -a measure of environmental heterogeneity -and ii) the strength of genetic drift, which reduces genetic diversity. Together, these two parameters define an expansion threshold : adaptation fails when the neighbourhood size is so small that genetic drift reduces diversity below the level required for adaptation to environmental heterogeneity. When the key parameters drop below this expansion threshold locally, a sharp range margin forms. When they drop below this threshold throughout the species' range, adaptation collapses everywhere, resulting in either extinction, or formation of a fragmented meta-population. Below the expansion threshold, increased dispersal is beneficial, because the reduction of both genetic and demographic stochasticity has a stronger effect than is its cost through increased maladaptation. Because the effects of dispersal differ fundamentally with dimension, the predictions are qualitatively different from those in a linear habitat. The expansion threshold provides a novel, theoretically justified and testable prediction for formation of the range margin and collapse of the species' range in two-dimensional habitats.

We examined causes of speciation in asexual populations in both sympatry and parapatry, providing an alternative explanation for the speciation patterns reported by Dieckmann and Doebeli (1999) and Doebeli and Dieckmann (2003). Both in sympatry and parapatry, they find that speciation occurs relatively easily. We reveal that in the sympatric clonal model, the equilibrium distribution is continuous and the disruptive selection driving evolution of discrete clusters is only transient. Hence, if discrete phenotypes are to remain stable in the sympatric sexual model, there should be some source of nontransient disruptive selection that will drive evolution of assortment. We analyze sexually reproducing populations using the Bulmer's infinitesimal model and show that cost-free assortment alone leads to speciation and disruptive selection only arises when the optimal distribution cannot be matched--in this example, because the phenotypic range is limited. In addition, Doebeli and Dieckmann's analyses assumed a high genetic variance and a high mutation rate. Thus, these theoretical models do not support the conclusion that sympatric speciation is a likely outcome of competition for resources. In their parapatric model (Doebeli and Dieckmann 2003), clustering into distinct phenotypes is driven by edge effects, rather than by frequency-dependent competition.