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

生态系统如何在太空中自我安排？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然能够提供人类所依赖的服务（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移动的各种大型哺乳动物，或者珊瑚礁上的无数动物。当它们移动并相互作用（以及更静态的植物物种）时，它们在空间中形成排列。它们可以是共生种群的聚集、竞争者的分离，或者是在时间和空间上波动的更复杂的模式，当然，这些空间安排不是以“自上而下”的方式规划的。相反，它们是个体动物日常生活中运动和相互作用的自然结果。通过建立这些运动和相互作用的数学模型，我们可以理解和预测不同相互作用场景下的空间分布。例如，假设来自一个物种的个体倾向于向有另一个互惠物种的成员的地区移动，而同时来自后一个物种的个体喜欢向前一个物种居住的地区移动。然后数学模型可以回答这个问题：这些吸引力的倾向必须有多强，才能使两个物种聚集在比它们可能占据的空间更小的空间中？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）。各种物种的空间分布会出现什么样的情况？它是否会在时间上稳定下来，使某些物种占据一部分空间，而其他物种占据不同的区域？或者分布会永远变化，随着时间的推移不断变化？这项建议旨在提供一个通用的理论来回答这些问题，使用一个数学形式主义称为“多物种聚集方程”。目前对动物物种分布的理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。在这里，相反，我们将展示如何人口之间的运动反应可以导致自发形成的广泛的时空分布。我们的目标是对这些进行分类，将涌现模式的定性特征与潜在的运动和互动过程联系起来。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基本过程中出现，取决于最近的历史时空模式。这项工作有可能改变科学界认为动物物种在空间中的分布方式，通过将重点从静态环境协变量转移到动物运动机制的非线性反馈。如果成功的话，这可能会导致在空间保护和维持生物多样性的干预措施方面做出更明智的决定。该项目提供了一个核心例子，说明机械的数学方法在理解生态现象方面的至关重要性。

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.