diffusion model (Fisher 1937, Liebhold et al. 1992, Skellam 1951). Range expands because of the combined effects of diffusion and growth, and the change in population density over time is approximated by an asymptotic rate of spread. The advancing front of the population will appear to advance at a constant velocity and shape–a wave front. As described by Shigesada and Kawasaki (1997), such a scenario occurs when the density of the initial colonists invading the origin decreases rapidly because of diffusion and there is little competition. As the population gradually recovers through reproduction, competition effects start to be manifested, so the density in the occupied range approaches the carrying capacity of the habitat. The range front forms a sigmoidal pattern and spreads at a constant speed–a traveling frontal wave. At the edge of the expanding population, competition is low, so expansion by diffusion predominates and density-dependent effects on growth rate are insignificant. Despite its relative simplicity, the growth-diffusion model has been used successfully to simulate the spread of a variety of organisms (Hastings 1996, Levin 1989, Liebhold et al. 1992, Long 1977, Willamson and Brown 1986).

More-sophisticated models with additional variables may be needed when an organism is transported by wind or water or tends to orient to a stimulus. For example, in a convection model, a velocity term is applied to the reaction-diffusion equation, which may better describe an insect, plant, or plant-pathogen species that is transported by wind (Lewis and Kareiva 1993). Insects may select their flight direction or respond to host or conspecific volatiles, whereas the spread of seed or pollen may be affected such variables as settling velocity, height of release, wind speed, and turbulence (Okubo and Levin 1989). Spread of populations can be linear or streamlike if dispersing propagules are channeled by topographic or climatic factors (Carey 1996).

If the population is influenced by an Allee effect, success or failure of an invasion may be determined before the asymptotic form of expansion is achieved (Lewis and Kareiva 1993). Presence of an Allee effect or an increase in the diffusion coefficient will require the initial “beachhead” to be larger if the population is to avoid extinction. Even if an immigrant population initially arrives in a new area at a density above its critical Allee effect threshold, an Allee effect will increase the period before expansion begins (Veit and Lewis 1996) and can substantially reduce the traveling-wave speed, resulting in a lower asymptotic rate of spread. Presence of an Allee effect will affect the curvature of the boundary between invaded and uninvaded regions and affect the rate of spread. When “fingers” of an uninvaded area protrude into the invaded area, the population will spread faster than when occupied territory curves sharply into unoccupied territory (Lewis and Kareiva 1993).

Environmental heterogeneity—for example, the patchiness of the environment—can also influence the rate of spread. Diffusion coefficients may vary widely, depending on the degree of environmental, habitat, or resource heterogeneity and how the nonindigenous species responds to the heterogeneity. When



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