Predicting the rates at which introduced species can spread in their new ranges and at which native species can shift their current ranges in response to climate change has captivated mathematical biologists for a long time [e.g. Fisher 1937; Kot et al 1996; Potapov and Lewis, 2004; Hastings et al 2005]. Much progress has been made with respect to determining the speed of spread from various assumptions about the ecological processes of dispersal and population growth in spatially homogeneous or heterogeneous environments [e.g. Shigesada et al 1986; Lewis and Pacala, 2000; Berestycki et al 2005; Maciel and Lutscher 2013]. From a mathematical viewpoint, many different modelling frameworks have been used, e.g., reaction-diffusion equations, integrodifference and integrodifferential equations, and stochastic processes. Few models within these classes of equations consider evolutionary aspects [Pease et al 1989].

 

More recently, ecologists have explored the role of rapid evolution on range-expansion processes via individual-based simulation models [Travis and Dytham, 2002; Burton et al 2010; Shaw and Kokko, 2015]. One prediction from such models is that various evolutionary processes act to accelerate spread rates. Within the last three years, empirical work has confirmed these predictions using tightly controlled laboratory systems [e.g. Fronhofer and Altermatt 2015]. These studies, which appeared in high-profile journals (e.g. Science, Nature Communications, PNAS, Ecology Letters), revealed that not only the speed increased on average but also the variability in spread rate between different replicates was affected by evolutionary processes [Phillips 2015]. Surprisingly, across studies, the evolutionary effects on invasion variability were themselves variable in direction and magnitude. That is, in some systems evolution increased variability between replicates [Ochocki and Miller 2017, Weiss-Lehman et al 2017], in others, evolution decreased variability [Williams et al 2016], and in others, it had no effect [Van Petegem et al 2018]. In all cases, there were consistently positive effects of evolution on the mean velocity. Invasion variability is a key concept because it affects how well we can predict the trajectories of expanding populations [Melbourne and Hastings 2009; Phillips 2015], something biologists and modellers are increasingly called upon to do. Resolving when, why, and in what direction evolutionary mechanisms may influence the predictability of spread is therefore an urgent challenge. There is currently no mathematical framework in place to evaluate and predict the effects of evolutionary processes on the variability of spread rates or to tease apart the relative influences of ecological (demography and dispersal) and evolutionary (selection and drift) processes and environmental conditions (e.g., landscape structure) on the variability of spread rates. The goal of this workshop is to bring together mathematicians, modellers and empiricists to develop novel mathematical models and an analytical framework to address these questions.

 

Among the earlier approaches to integrate evolutionary processes into population spread models are quantitative genetics models that track population density and the mean of some quantitative trait related to reproduction [Pease et al 1989, Kirkpatrick and Barton 1997] but not to dispersal. More recently, mathematicians have begun to study the spatio-temporal evolution of an entire trait distribution (reproduction or dispersal) in space with traits under selection [Alfaro et al, 2013; Bouin and Calvez, 2014] or neutral traits [e.g. Roques et al 2012]. An alternative model for spatial spread is based on tracking the furthest-forward individual [e.g. Lewis et al 2016]. This approach can give information on the mean speed and higher moments in some simple cases of linear ecological dynamics, but nonlinear analogues are challenging to study and evolutionary processes have not been incorporated. A few individual-based simulation models exist for special cases [Shaw and Kokko 2015], but a systematic synthesis is not available. Stochastic process models together with various limiting equations and perturbation techniques could prove to be a fruitful direction of research.