Local and Global Existence for Nonlocal Multispecies Advection-Diffusion Models

Nonlocal 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 modeling non-local advection mathematically. These often take the form of partial differential equations, with integral terms modeling the nonlocality. One common formalism is the aggregation-diffusion equation, a class of advection-diffusion models with nonlocal advection. This was originally used to model a single population but has recently been extended to the multispecies 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 nonlocal multispecies advection-diffusion models, with an arbitrary number of coexistent 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 an example simulation output. Overall, this helps provide a solid mathematical foundation for studying the effect of interspecies interactions on movement and space use.