Multi-species aggregation equations: a bridge between movement ecology and spatial population dynamics

生态系统在太空中是如何安排自己的？这是理解如何保护物种、维持生物多样性并确保生态系统仍在运转以提供人类赖以生存的服务(如食物、水和空气)的核心问题。通常，生态系统包含了各种各样的移动和相互作用的动物。想想在塞伦盖蒂平原上活动的各种大型哺乳动物物种，或者珊瑚礁上的无数动物物种。当它们相互移动和相互作用时(以及更静态的植物物种)，它们在空间中形成排列。这些形式可以是共生种群的聚集、竞争者的隔离，也可以是在时间和空间上波动的更复杂的模式。当然，这些空间安排并不是以“自上而下”的方式规划的。相反，它们的出现是日常生活中个体动物运动和互动的自然结果。通过建立这些运动和交互的数学模型，我们可以了解和预测不同交互场景下应该出现的空间分布。例如，假设一个物种的个体倾向于向有另一个互惠物种成员的地区移动，而同时来自后一个物种的个体喜欢向前者居住的地区移动。然后，数学模型可以回答这个问题：这些吸引人的倾向必须有多强，才能使这两个物种聚集在比它们原本可能占据的空间更小的部分？或者假设我们有一个更复杂的系统，有多个物种，其中一些相互吸引，一些相互排斥，另一些具有不对称的运动趋势(例如，一个追逐另一个，后者撤退)。各种物种将出现什么样的空间分布？它会不会在时间上稳定下来，让某些物种占据一部分空间，而另一些物种占据不同的区域？或者，这些分布会一直在变化，随着时间的推移而不断变化吗？这一提议旨在通过一种名为“多物种聚集方程”的数学形式，为回答这些问题提供一个普遍的理论。目前对动物物种分布的了解通常集中在了解它们如何与相对静态的环境特征(如地形和植被覆盖)相关联。相反，在这里，我们将展示人口之间的流动反应如何导致广泛的时空分布的自发形成。我们的目标是将这些分类，将涌现模式的质量特征与潜在的运动和相互作用过程联系起来。我们还将研究所谓的“滞后”效应，即不同的模式可以从相同的潜在过程中出现，这取决于最近的时空模式的历史。这项工作有可能改变科学界思考动物物种在空间分布的方式，将重点从静态的环境协变量转移到动物运动机制中的非线性反馈。如果成功，这可能会产生关于空间保护和干预措施的更明智的决定，以维护生物多样性。该项目提供了一个核心例子，说明了机械的、数学的方法在理解生态现象方面的极端重要性。

How do ecosystems arrange themselves in space? This is a core question for understanding how to conserve species, maintain biodiversity, and ensure that ecosystems are still functioning to provide services on which humanity depends (such as food, water, and air). Often, ecosystems incorporate a large variety of moving and interacting animals. Think of the various large mammal species moving on the Serengeti Plains, or the myriad animal species on a coral reef. As they move and interact with one another (as well as the more static plant species) they form arrangements in space. These can take the form of aggregations of symbiotic populations, segregations of competitors, or more complex patterns that can fluctuate in time and space.These spatial arrangements are not, of course, planned in a "top down" fashion. Rather they emerge as a natural consequence of the movements and interactions of individual animals going about their daily lives. By building mathematical models of these movements and interactions, we can understand and predict the spatial distributions that ought to arise from different interaction scenarios. For example, suppose individuals from one species have a tendency to move towards areas where there are members of another, mutualist species, whilst at the same time individuals from the latter species like to move towards areas inhabited by the former. Then mathematical models can answer the question: how strong do these attractive tendencies have to be so that both species aggregate in a smaller part of space than they might otherwise occupy? Or suppose we have a more complicated system, with multiple species, some of which are attracted to one another, some of which repel each other, and others that have asymmetric movement tendencies (e.g. one chases the other and the latter retreats). What sort of spatial distribution of the various species will emerge? Will it stabilise in time, so that certain species occupy one part of space and others occupy different areas? Or will the distributions be in perpetual flux, continually changing over time?This proposal aims to provide a general theory for answering such questions, using a mathematical formalism called a "multi-species aggregation equation". Present understanding of animal species distributions typically centres around understanding how they are correlated with relatively static environmental features, such as topography and vegetation cover. Here, instead, we will show how between-population movement responses can lead to the spontaneous formation of a wide range of spatio-temporal distributions. We aim to classify these, relating qualitative features of the emergent patterns to underlying movement-and-interaction processes. We will also examine so-called "hysteresis" effects, whereby different patterns can emerge from the same underlying processes, dependent upon the recent history of spatio-temporal patterns.This work has the potential to change the way the scientific community thinks about how animal species are distributed in space, by shifting focus from static environmental covariates to non-linear feedbacks in animal movement mechanisms. If successful, this could give rise to much better-informed decisions regarding spatial conservation and interventions to maintain biodiversity. The project gives a core example of the vital importance of a mechanistic, mathematical approach in understanding ecological phenomena.