The margins within the geographic range of species are often specific in terms of ecological and evolutionary processes, and can strongly influence the species' reaction to climate change. One of the frequently observed features at range margins is fragmentation, caused internally by population dynamics or externally by the limited availability of suitable habitat sites. We study both causes, and describe the transition from a connected to a fragmented state across space by means of a gradient metapopulation model. The main features of our approach are the following. 1) Inhomogeneities can occur at two spatial scales: there is a broad-scale gradient, which can be patterned by fine-scale heterogeneities. The latter is implemented by dispersing a variable number of small obstacles over the terrain, which can be penetrable or unpenetrable by the spreading species. 2) We study the occupancy of this terrain in a steady-state on two temporal scales: in snapshots and by long-term averages. The simulations reveal some general scaling laws that are applicable in various environments, independently of the mechanism of fragmentation. The edge of the connected region (the hull) is a fractal with dimension 7/4. Its width and length changes with the gradient according to universal scaling laws, that are characteristic for percolation transitions. The results suggest that percolation theory is a powerful tool for understanding the structure of range margins in a broad variety of real-life scenarios, including those in which the environmental gradient is combined with fine-scale heterogeneity. This provides a new method for comparing the range margins of different species in various geographic regions, and monitoring range shifts under climate change.

中文翻译：

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渗流理论提出了跨环境梯度的范围边缘的一些一般特征

物种地理范围内的边缘在生态和进化过程方面通常是特定的，并且可以强烈影响物种对气候变化的反应。在范围边缘经常观察到的特征之一是碎片化，它是由人口动态内部引起的，或者是外部由合适栖息地的有限可用性引起的。我们研究了这两个原因，并通过梯度元种群模型描述了跨空间从连接状态到碎片状态的转变。我们的方法的主要特点如下。1) 不均匀性可以发生在两个空间尺度上：存在一个大尺度梯度，可以通过精细尺度的不均匀性来模式化。后者是通过在地形上散布可变数量的小障碍物来实现的，它可以被传播的物种穿透或不可穿透。2）我们在两个时间尺度上研究该地形在稳态中的占用情况：快照和长期平均值。模拟揭示了一些适用于各种环境的一般缩放定律，独立于碎片机制。连接区域（船体）的边缘是一个维度为 7/4 的分形。它的宽度和长度根据通用缩放定律随梯度变化，这是渗透过渡的特征。结果表明，渗透理论是一种强大的工具，可用于了解各种现实生活场景中的范围边缘结构，包括环境梯度与细尺度异质性相结合的场景。