We present a systematic comparison and analysis of four discrete-time, hostparasitoid 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.
Climate change has created new and evolving environmental conditions, impacting all species, including hosts and parasitoids. I therefore present integrodifference equation (IDE) models of host--parasitoid systems to model population dynamics in the context of climate-driven shifts in habitats. I describe and analyze two IDE models of host--parasitoid systems to determine criteria for coexistence of the host and parasitoid. Specifically, I determine the critical habitat speed, beyond which the parasitoid cannot survive. By comparing the results from two IDE models, I investigate the impacts of assumptions that reduce the system to a single-species model. I also compare critical speeds predicted by a spatially-implicit difference-equation model with critical speeds determined from numerical simulations of the IDE system. The spatially-implicit model uses approximations for the dominant eigenvalue of an integral operator. The classic methods to approximate the dominant eigenvalue for IDE systems do not perform well for asymmetric kernels, including those that are present in shifting-habitat IDE models. Therefore, I compare several methods for approximating dominant eigenvalues and ultimately conclude that geometric symmetrization and iterated geometric symmetrization give the best estimates of the parasitoid critical speed.
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