  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.
