  In this paper we study a broad class of non-local advection-diffusion models
describing the behaviour of an arbitrary number of interacting species, each
moving in response to the non-local presence of others. Our model allows for
different non-local interaction kernels for each species and arbitrarily many
spatial dimensions. We prove the global existence of both non-negative weak
solutions in any spatial dimension and positive classical solutions in one
spatial dimension. These results generalise and unify various existing results
regarding existence of non-local advection-diffusion equations. We also prove
that solutions can blow up in finite time when the detection radius becomes
zero, i.e. when the system is local, thus showing that nonlocality is essential
for the global existence of solutions. We verify our results with some
numerical simulations on 2D spatial domains.
