Mathematical modelling of spatial sorting and evolution in a host-parasite system

There have been numerous empirical and agent-based modelling studies on the spatial self-structuring of traits, particularly in regard to dispersal ability (termed spatial sorting) of cane toads in northern Australia, but few mathematical modelling studies. In this study, we formulate a reaction-diffusion based partial-integro-differential equation model based on an earlier model by Bouin et al. (2012) to examine this spatial self-structuring of traits in both a cane toad population and lungworm parasite population, which evolves with the cane toad population. In particular, the traits we focus on are dispersal ability for the cane toad population and both prepatent period and larval size for the lungworm parasite population. Apart from the spatial self-structuring of these traits, our results confirm a number of observations made in empirical and agent-based studies; particularly, that there is a noticeable lag between the host and parasite population which is critically dependent on the parasite functional response to host densities, that older populations regress back to lower dispersal speeds and that spatial sorting can still occur with a disadvantage in reproductivity and/or survival in more motile individuals. Moreover, we find that such a disadvantage in reproductivity and/or survival is unlikely to be large if spatial sorting is to have a noticeable effect on the rate of range expansion, as it has been observed to have over the last 60 years in northern Australia.