  In $1993$, Holt and Lawton introduced a stochastic model of two host species
parasitized by a common parasitoid species. We introduce and analyze a
generalization of these stochastic difference equations with any number of host
species, stochastically varying parasitism rates, stochastically varying host
intrinsic fitnesses, and stochastic immigration of parasitoids. Despite the
lack of direct, host density-dependence, we show that this system is
dissipative i.e. enters a compact set in finite time for all initial
conditions. When there is a single host species, stochastic persistence and
extinction of the host is characterized using external Lyapunov exponents
corresponding to the average per-capita growth rates of the host when rare.
When a single host persists, say species $i$, a explicit expression is derived
for the average density, $P_i^*$, of the parasitoid at the stationary
distributions supporting both species. When there are multiple host species, we
prove that the host species with the largest $P_i^*$ value stochastically
persists, while the other host species are asymptotically driven to extinction.
A review of the main mathematical methods used to prove the results and future
challenges are given.
