Population Dynamic Models of the Spread of Wolbachia

Wolbachia are endosymbionts that are found in many insect species and can spread rapidly when introduced into a naive host population. Most Wolbachia spread when their infection frequency exceeds a threshold normally calculated using purely population genetic models. However, spread may also depend on the population dynamics of the insect host. We develop models to explore interactions between host population dynamics and Wolbachia infection frequency for an age-structured insect population regulated by larval density dependence. We first derive a new expression for the threshold frequency that extends existing theory to incorporate important details of the insect's life history. In the presence of immigration and emigration, the threshold also depends on the form of density-dependent regulation. We show how the type of immigration (constant or pulsed) and the temporal dynamics of the host population can strongly affect the spread of Wolbachia. The results help understand the natural dynamics of Wolbachia infections and aid the design of programs to introduce Wolbachia to control insects that are disease vectors or pests.