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Non-local advection is a key process in a range of biological systems, fromcells within individuals to the movement of whole organisms. Consequently, inrecent years, there has been increasing attention on modelling non-localadvection mathematically. These often take the form of partial differentialequations, with integral terms modelling the non-locality. One common formalismis the aggregation-diffusion equation, a class of advection diffusion modelswith non-local advection. This was originally used to model a singlepopulation, but has recently been extended to the multi-species case to modelthe way organisms may alter their movement in the presence of coexistentspecies. Here we prove existence theorems for a class of non-localmulti-species advection-diffusion models, with an arbitrary number ofco-existent species. We prove global existence for models in n=1 spatialdimension and local existence for n>1. We describe an efficient spectral methodfor numerically solving these models and provide example simulation output.Overall, this helps provide a solid mathematical foundation for studying theeffect of inter-species interactions on movement and space use.

Local and Global Existence for Non-local Multi-Species Advection-Diffusion Models