  Nonlocal interactions are ubiquitous in nature and play a central role in
many biological systems. In this paper, we perform a bifurcation analysis of a
widely-applicable advection-diffusion model with nonlocal advection terms
describing the species movements generated by inter-species interactions. We
use linear analysis to assess the stability of the constant steady state, then
weakly nonlinear analysis to recover the shape and stability of non-homogeneous
solutions. Since the system arises from a conservation law, the resulting
amplitude equations consist of a Ginzburg-Landau equation coupled with an
equation for the zero mode. In particular, this means that supercritical
branches from the Ginzburg-Landau equation need not be stable. Indeed, we find
that, depending on the parameters, bifurcations can be subcritical (always
unstable), stable supercritical, or unstable supercritical. We show numerically
that, when small amplitude patterns are unstable, the system exhibits large
amplitude patterns and hysteresis, even in supercritical regimes. Finally, we
construct bifurcation diagrams by combining our analysis with a previous study
of the minimisers of the associated energy functional. Through this approach we
reveal parameter regions in which stable small amplitude patterns coexist with
strongly modulated solutions.
