Semi-discrete host–parasitoid models

Arthropod host–parasitoid interactions constitute a very important class of consumer resource dynamics. Discrete-time models are a tradition for such interactions and are characterized by an updating function, which relates the population densities at a fixed date in one year to those at the same date in the previous year. Previous workers have investigated the effects of functional response and density dependence on the stability of the host–parasitoid interaction by heuristically incorporating them in the updating function. Such an approach ignores the effects of population changing continuously within a year due to different processes (for example intraspecific competition, mortality from parasitism) that may act simultaneously. Their cumulative effect on the updating function is not obvious and a more systematic methodology is needed. This paper uses a hybrid approach to formulate the updating function. This is done by modeling the dynamics of various within-year processes in continuous-time, and reproduction as a discrete event. Using this formalism we derive results connecting the stability of the host–parasitoid interaction with different forms of density dependence and the form of the functional response. The latter results contradict previous conclusions from heuristically formulated models, and illustrate the need for such a hybrid approach in discrete-time host–parasitoid theory.