Biological invasions often have outsized consequences for the invaded ecosystem and represent an interesting challenge to model mathematically. Landscape heterogeneity, non-local or time-dependent spreading mechanisms, coarse data, and air or water flow transport are but a few of the complications that can greatly affect our understanding of small organism movement – a critical component of both invasion success and the ability of native organisms to persist at a location. In this talk, I will look at dispersal and swarm behavior from a multi-scale, mathematical perspective in order to address some of these challenges.
Considering the problem of long-distance dispersal, I will discuss a method for modeling invasive spread over large, heterogeneous landscapes by interpreting the quantity of interest as the probability of species occurrence rather than population size. On large scales, one can also take advantage of ecological niche modeling approaches in order to reduce the dimensionality of data quantifying landscape heterogeneity. I will then shift focus to the initial stages of an invasion and concentrate on the local- and meso-scale by considering the intentional release of a parasitoid wasp biocontrol agent. In this case, we can utilize a Bayesian framework and maximum likelihood estimation to parameterize the model based on proxy time-series data collected in the field. Finally, I will describe some of my current work close to the microscale examining the dynamics of organism movement and behavior with respect to a surrounding fluid environment.