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

Non-local advection is a key process in a range of biological systems, from cells within individuals to the movement of whole organisms. Consequently, in recent years, there has been increasing attention on modelling non-local advection mathematically. These often take the form of partial differential equations, with integral terms modelling the non-locality. One common formalism is the aggregation-diffusion equation, a class of advection diffusion models with non-local advection. This was originally used to model a single population, but has recently been extended to the multi-species case to model the way organisms may alter their movement in the presence of coexistent species. Here we prove existence theorems for a class of non-local multi-species advection-diffusion models, with an arbitrary number of co-existent species. We prove global existence for models in n=1 spatial dimension and local existence for n>1. We describe an efficient spectral method for numerically solving these models and provide example simulation output. Overall, this helps provide a solid mathematical foundation for studying the effect of inter-species interactions on movement and space use.