Coexistence conditions in generalized discrete-time models of insect population dynamics

Motivated by the annual life histories of insects residing in the temperate-regions of the world, there is a rich tradition of modeling their population dynamics using a discrete-time formalism. Here, we consider an antagonistic interaction between two insect species – a parasitoid wasp that lays an egg within its host insect species. The egg hatches into juvenile parasitoid that develops within the body of the host by using it as resource and this ultimately results in host death. The parasitoid emerges from the host as a free-living insect that goes on to attack other hosts. Such interaction are ubiquitous with over 65,000 known species of parasitoid wasps and such wasps have potential in biologic control of pest insect species. We introduce a general class of discrete-time models for capturing the population dynamics of two competing parasitoid species that attack the same vulnerable stage of the host species. These models are characterized by two density-dependent functions: an escape response defined by the fraction of hosts escaping parasitism; and a competition response defined by the fraction of parasitized hosts that develop into adult parasitoids of either species. Model analysis reveals remarkably simple stability conditions for the coexistence of competing parasitoids. Coexistence occurs, if and only if, the adult host population increases with host reproduction rate, and the log sensitivity of the competition response is less than half. The latter condition implies that any increase in the adult parasitoid population will result in a sufficiently slow increase in the fraction of parasitized hosts that develop into parasitoids of that type. We also consider a semi-discrete formulation of a competing species model that incorporates host density-dependence in the attack rate. This model expands the class of models for which the analytical results apply, and we systematically compare the numerical results of this model to the established coexistence conditions