Abstract
The spatial range of a species habitat is generally determined by the ability of the species to cope with biotic and abiotic variables that vary in space. Therefore, the species range is itself an evolvable property. Indeed, environmental gradients permit a mode of evolution in which range expansion and adaptation go hand in hand. This process can contribute to rapid evolution of drug resistant bacteria and viruses, because drug concentrations in humans and livestock treated with antibiotics are far from uniform. Here, we use a minimal stochastic model of discrete, interacting organisms evolving in continuous space to study how the rate of adaptation of a quantitative trait depends on the steepness of the gradient and various population parameters. We discuss analytical results for the mean-field limit as well as extensive stochastic simulations. These simulations were performed using an exact, event-driven simulation scheme that can deal with continuous time-, density- and coordinate-dependent reaction rates and could be used for a wide variety of stochastic systems. The results reveal two qualitative regimes. If the gradient is shallow, the rate of adaptation is limited by dispersion and increases linearly with the gradient slope. If the gradient is steep, the adaptation rate is limited by mutation. In this regime, the mean-field result is highly misleading: it predicts that the adaptation rate continues to increase with the gradient slope, whereas stochastic simulations show that it in fact decreases with the square root of the slope. This discrepancy underscores the importance of discreteness and stochasticity even at high population densities; mean-field results, including those routinely used in quantitative genetics, should be interpreted with care.
Highlights
In those classical models of evolving populations that include spatial degrees of freedom, the habitat typically has a fixed range and geometry [1]
This work has shown that, in such fronts, the convergence to the mean-field limit with increasing population density is anomalously slow, so that significant quantitative differences can be found between mean-field and finitedensity stochastic models even at large population densities
The large difference between the mean-field result and the stochastic simulations in the mutation-limited regime can be understood by considering the effects of scaling the spatial dimension
Summary
In those classical models of evolving populations that include spatial degrees of freedom, the habitat typically has a fixed range and geometry [1]. Many range boundaries observed in nature appear relatively stable, and yet cannot be explained by an obvious shift in environmental conditions [4] For this reason, a significant amount of theoretical work has been devoted to identifying the factors that determine the range limits of species and the mechanisms that contribute to their stability (e.g., see [2, 4,5,6,7,8,9] and references therein). A significant amount of theoretical work has been devoted to identifying the factors that determine the range limits of species and the mechanisms that contribute to their stability (e.g., see [2, 4,5,6,7,8,9] and references therein) These studies often rely on the framework of quantitative genetics. Perhaps because of this focus on understanding stable boundaries, less attention has been given to the dynamics of range expansion when the boundaries are not evolutionarily stable
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