  We investigate a nonlocal single-species reaction-diffusion-advection model
that integrates the spatial memory of previously visited locations and nonlocal
detection in space, resulting in a coupled PDE-ODE system reflective of several
existing models found in spatial ecology. We prove the existence and uniqueness
of a H\"older continuous weak solution in one spatial dimension under some
general conditions, allowing for discontinuous kernels such as the top-hat
detection kernel. A robust spectral and bifurcation analysis is also performed,
providing the rigorous analytical study not yet found in the existing
literature. In particular, the essential spectrum is shown to be entirely
negative, and we classify the nature of the bifurcation near the critical
values obtained via a linear stability analysis. A pseudo-spectral method is
used to solve and plot the steady states near and far away from these critical
values, complementing the analytical insights.
