Generalized stability conditions for host–parasitoid population dynamics: Implications for biological control

Discrete-time models are the traditional approach for capturing population dynamics of insects living in the temperate regions of the world. We revisit classical discrete-time models of host–parasitoid population dynamics and provide novel results on the stability of the population dynamics. Discrete-time host–parasitoid models are characterized by update functions that connect the population densities from one year to the next, and a host escape response — the fraction of hosts escaping parasitism each year. For a general class of models we show that the stability can be simply characterized in terms of two quantities: the rate at which the host equilibrium changes with the host’s growth rate, and the sensitivity of the host’s escape response to the host density. Interestingly, stability is more likely to arise when the escape response is a decreasing function of the host density rather than an increasing function. Moreover, if the host’s escape response only depends on the parasitoid population density then the stability condition is further simplified to the host equilibrium density being an increasing function of the host’s reproduction rate. We interpret several mechanisms known for stabilizing host–parasitoid population dynamics in the context of these generalized stability conditions. Next, we introduce a hybrid approach for obtaining the update functions by solving ordinary differential equations that mechanistically capture the ecological interactions between the host and the parasitoid. This hybrid approach is used to study the suppression of host density by a parasitoid. Our analysis shows that when the parasitoid attacks the host at a constant rate, then the host density cannot be suppressed beyond a certain point without making the population dynamics unstable. In contrast, when the parasitoid’s attack rate increases with increasing host density (Type III functional response), then the host population density can be suppressed to arbitrarily low levels while maintaining system stability. These results have important implications for biological control where parasitoids are introduced to eliminate a pest that is the host species for the parasitoid.