Hybrid systems modeling of ecological population dynamics

Discrete-time models are the traditional approach for capturing population dynamics of insects living in the temperate regions of the world. These models are characterized by an update function that connects the population densities from one year to the next. We revisit classical discrete-time models used for modeling interactions between two insect species (a host and a parasitoid), and provide novel result on the stability of the population dynamics. In particular, for a class of models we show that the fixed point is stable, if and only if, the host equilibrium density is an increasing function of the host’s reproduction rate. We also 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 by suppressed beyond a certain point without making the population dynamics unstable. In contrast, when the parasitoid’s attack rate increases with increasing host density, then the host population density can be suppressed to arbitrarily low levels. These results have important implications for biological control where a natural enemy, such as a parasitoid wasp, is introduced to eliminate a pest that is the host species for the parasitoid.