The spread of invasive species is a key applied problem in ecology. In North America, invasive exotic species are widespread, ranging from gypsy moth to Asian longhorn beetle to weedy plants. The associated costs are immense, by some estimates exceeding $100 billion US per year. While many invasive species are introduced from Asia or Europe, others, like mountain pine beetle, are simply spreading into new areas of North America, due to processes such as climatic change.
Early models for invasive species were nonlinear reaction diffusion equations such as Fisher's equation, which describes quadratic growth coupled to Brownian motion. Here the analysis of traveling waves and of the convergence of initial data to wave solutions has been a fruitful area of classical mathematical research. The traveling wave speed, interpreted biologically as the rate of spread of the introduced population, has successfully predicted spread rates of many introduced species, but has failed dramatically with others. Modifications of these equations to include long-distance dispersal, stage structure, spatial heterogeneity, stochasticity, Allee effects, and nonlinear interactions with resident species (eg, competition or predation) have driven new advances in the theory of nonlinear dynamical systems, while, at the same time, providing a more realistic framework for the study of invasions.
The nonlinear dynamical systems models are not simply mathematical abstractions of key processes. They are the quantitative formulation of underlying hypotheses, and they provide the means for testing the hypotheses against data.
In parallel with the development of new mathematical models, has been increasing availability of detailed spatio-temporal datasets that can be used to track actual invasion processes. These datasets can be accessed via Geographic Information Systems (GIS), and, in some cases, they show yearly changes in the extent of invaders. Classic data sets include those for mountain pine beetle in western Canada and US, gypsy moth in the eastern US, and Spartina in coastal California.
New powerful statistical methods based on intensive computational algorithms such as the Markov Chain Monte Carlo methods, data cloning, profile likelihood based on cascading parameters, composite likelihood and estimating functions make it possible to interface these detailed data sets with the new realistic dynamical system models. This interface allows the models to be assessed, tested and validated against the real data for the invasions. Hypotheses regarding key factors governing invasions can be evaluated, and the means for controlling the invasions/adapting to the invasions can be investigated. This interface between nonlinear dynamical systems, large datasets and statistical and computer methods has only become possible recently, with the growth of large data sets via remote sensing, with the advent of new powerful computers, and with the development of new statistical methods. This interface provides fertile ground for new mathematical, statistical and scientific advances.
The purpose of the MBI workshop on invasive species is to bring together researchers from different groups: mathematicians, biologists and statisticians to develop the new interdisciplinary approaches to biological invasions described above. Possible participants are given below.