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.

译文

物种栖息地的空间范围通常取决于物种应对空间变化的生物和非生物变量的能力。因此,物种范围本身就是一种可进化的属性。实际上,环境梯度允许一种进化模式,在这种模式下,范围扩展和适应齐头并进。这个过程可以促进耐药细菌和病毒的快速进化,因为用抗生素治疗的人和牲畜的药物浓度远非均匀。在这里,我们使用在连续空间中进化的离散相互作用生物的最小随机模型来研究数量性状的适应速度如何取决于梯度的陡度和各种种群参数。我们讨论了平均场极限的分析结果以及广泛的随机模拟。这些模拟是使用精确的,事件驱动的模拟方案进行的,该方案可以处理连续的时间,密度和坐标相关的反应速率,并且可以用于各种随机系统。结果揭示了两种定性制度。如果梯度较浅,则适应率受色散限制,并随梯度斜率线性增加。如果梯度陡峭,则适应率受到突变的限制。在这种情况下,平均场结果具有很大的误导性: 它预测适应率随梯度斜率的增加而继续增加,而随机模拟表明,它实际上随斜率的平方根而减小。即使在高人口密度下,这种差异也强调了离散性和随机性的重要性; 平均场结果,包括定量遗传学中常规使用的结果,应谨慎解释。

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