  We present a systematic comparison and analysis of four discrete-time,
host--parasitoid models. For each model, we specify that density-dependent
effects occur prior to parasitism in the life cycle of the host. We compare
density-dependent growth functions arising from the Beverton--Holt and Ricker
maps, as well as parasitism functions assuming either a Poisson or negative
binomial distribution for parasitoid attacks. We show that overcompensatory
density-dependence leads to period-doubling bifurcations, which may be
supercritical or subcritical. Stronger parasitism from the Poisson distribution
leads to loss of stability of the coexistence equilibrium through a
Neimark--Sacker bifurcation, resulting in population cycles. Our analytic
results also revealed dynamics for one of our models that were previously
undetected by authors who conducted a numerical investigation. Finally, we
emphasize the importance of clearly presenting biological assumptions that are
inherent to the structure of a discrete-time model in order to promote
communication and broader understanding.
