Spatially Explicit Models of Turelli-Hoffmann Wolbachia Invasive Wave Fronts

Abstract

This paper examines different mathematical models of insect dispersal and infection spread and compares these with field data. Reaction–diffusion and integro-difference equation models are used to model the spatio-temporal spread of Wolbachia in Drosophila simulans populations. The models include cytoplasmic incompatibility between infected females and uninfected males that creates a threshold density, similar to an Allee effect, preventing increase from low incidence of infection in the host population. The model builds on an earlier model (Turelli & Hoffmann, 1991) by incorporating imperfect maternal transmission. The results of simulations of the models using the same parameter values produce different dynamics for each model. These differences become very marked in the integro-difference equation models when insect dispersal patterns are assumed to be non-Gaussian. The success or failure of invasion by Wolbachia in the simulations may be attributed to the insect dispersal mechanism used in the model rather than the parameter values. As the models predict very different outcomes for the integro-difference models depending on the underlying assumptions of insect dispersal patterns, this emphasizes that good field data on real (rather than idealized) dispersal patterns need to be collected before models such as these can be used for predictive purposes.