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We present a systematic comparison and analysis of four discrete-time,host--parasitoid models. For each model, we specify that density-dependenteffects occur prior to parasitism in the life cycle of the host. We comparedensity-dependent growth functions arising from the Beverton--Holt and Rickermaps, as well as parasitism functions assuming either a Poisson or negativebinomial distribution for parasitoid attacks. We show that overcompensatorydensity-dependence leads to period-doubling bifurcations, which may besupercritical or subcritical. Stronger parasitism from the Poisson distributionleads to loss of stability of the coexistence equilibrium through aNeimark--Sacker bifurcation, resulting in population cycles. Our analyticresults also revealed dynamics for one of our models that were previouslyundetected by authors who conducted a numerical investigation. Finally, weemphasize the importance of clearly presenting biological assumptions that areinherent to the structure of a discrete-time model in order to promotecommunication and broader understanding.

A Comparative Analysis of Host--Parasitoid Models with Density Dependence Preceding Parasitism