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.