Modeling impulsive resource inputs in host–parasitoid interactions with time delays

Abstract

For the interaction of parasitoids and their insect hosts in the laboratory environment, a novel mathematical model with impulsive resource inputs, stage-structure, maturation delays and negative binomial distribution is proposed. Based on the adaptability of the insect host to the environment, we study the permanence of the system in two cases and gain conditions under which the host and parasitoid species can coexist with impulsive resource inputs. We also discuss the existence of the positive periodic solution when the system is permanent by applying a fixed point theory. Besides, we perform numerical simulations which not only confirm but also further enhance our theoretical results. The simulations show that when total input of resource is fixed, smaller input amounts with shorter periods of impulsive delivery produce smaller oscillation amplitudes for both the host and parasitoid populations at the juvenile stage. However, both the densities of adult host and adult parasitoid are not affected by the resource management strategy. Furthermore, we also reconfirm that larger maturation delays, either the host or the parasitoid’s delay, lead to any more individuals staying at the inmature stage of the species, while the adult populations decline dramatically at the same time. On the other hand, larger host maturation delays promote the parasitoid’s population growth at both stages, and the impact of parasitoid maturation delay on the host population is almost the same but not as dramatic. These findings give us a deeper understanding about the host–parasitoid interaction in laboratory environment.