## Introduction

The spatial spread of invasive species has been a subject of empirical and theoretical study for many decades (Fisher 1937; Skellam 1951). The earliest work seemed to point to a simple and robust prediction of a linear rate of spread, perhaps after some initial phase where spread was not linear (reviewed by Hengeveld 1989; Andow *et al.* 1990; Okubo & Levin 2002). The problem of spread has, however, turned out to be much richer and more interesting than was supposed 20 years ago. Some recent theoretical developments have called basic results into question, while others have clarified just when the earlier results should hold (Kot *et al.* 1996; Weinberger *et al.* 2002). Much more data are becoming available, and new statistical techniques are being developed to match data with theory. While the early models included only a single non-evolving species in a homogeneous habitat with random short range dispersal, newer theory is beginning to include the effects of changing all these assumptions.

The initial models of Fisher (1937) were phrased as partial differential equations that could be used to predict an asymptotic rate of spread. In the simplest form in one spatial dimension, with *p*(*x*,*t* ) a density function for the population level as a function of spatial location *x* at time *t,**f* [ *p*(*x*,*t*)] the per capita rate of increase of the population at spatial location *x* and *D* a measure of the mean squared displacement of individuals per unit time, the model can be written as

It is intriguing to note that rigorous results on the dynamics of eqn 1 are very difficult to obtain, and Bramson (1983) is one of the first papers where results describing the long-term dynamics of eqn 1 were fully justified.

Similar results for spread are obtained either with *f* as a constant, or with *f* decreasing monotonically as *p* increases (Okubo & Levin 2002), since at least at the initial stages of invasions population levels are small. The rate of spread in either case can be shown to be

with analogous results in more than one spatial dimension. This basic result has two implications. First the rate of spread is a linear function of time; second the rate of spread can be predicted quantitatively as a function of measurable life history parameters. However, these classical results have been shown to depend strongly on some of the explicit assumptions as well as the implicit assumptions inherent in the modelling framework of eqn 1.

Given the importance of invasive species in both basic and applied ecology, it is important to update and interpret these results in the light of recent empirical and theoretical developments. We begin with an overview of the empirical evidence that is available on spatial spread of invasions and how these data are collected. Then we review newer modelling and statistical approaches and indicate how these developments change the classic results. We then examine the potential to include, in both analytic and conceptual models, more biological information about long distance dispersal, temporal variability, spatial heterogeneity, interactions with other species, and evolution. The ultimate goal is a deeper understanding of spread, and improved ability to predict spread based on either determination of the underlying life history or on observations of initial rates of spread.