  Spatial patterning can be crucially important for understanding the behavior
of interacting populations. Here we investigate a simple model of parasite and
host populations in which parasites are random walkers that must come into
contact with a host in order to reproduce. We focus on the spatial arrangement
of parasites around a single host, and we derive using analytics and numerical
simulations the necessary conditions placed on the parasite fecundity and
lifetime for the populations long-term survival. We also show that the parasite
population can be pushed to extinction by a large drift velocity, but,
counterintuitively, a small drift velocity generally increases the parasite
population.
