moth and locust species, for example, long-distance dispersers become more common as population density increases. Long-distance dispersal can also occur when plants’ propagules are carried by insects, wind, or water or by birds or mammals. In addition, our ability to detect a newly established population is typically limited when the density of a population is very low. The population must exceed a threshold density before it can be detected; this threshold will depend on traits or behavior of the organism, including the extent of the damage it causes.

Dispersal information about a species is often anecdotal and usually lacks quantification (Ridley 1930). Comparisons of biotic and abiotic dispersal among woody invaders and noninvaders suggested that the dispersal mechanisms of a species were not predictive of its likelihood of becoming invasive (Reichard and Hamilton 1997). Data from mark-recapture studies of 12 plant-feeding insects indicated that variation in diffusion was considerable among ecologically similar species and even within the same species (Kareiva 1983). In addition, inadvertent transportation of a nonindigenous organism by humans can establish new foci at substantially greater distances than would occur by natural dispersal mechanisms of the species. Such transportation has been shown to have substantially increased the spread rate of such species as gypsy moth (Liebhold et al. 1992) and cereal leaf beetle (Andow et al. 1993). Differences in dispersal, however, have been recognized among plant community types. Wind dispersal, for example, is more common among arid treeless ecosystems, and bird dispersal is more common in forest systems. Special attention should be given to the detection of newly established species with large, fleshy fruits in habitats that support an array of frugivorous birds. Such birds have contributed substantially to the spread of naturalized species, such as Clidemia hirta (Koster’s curse) and Hedychium gardnerianum (kahili ginger), in Hawaii (Cuddihy and Stone 1990).

The economic and ecological importance of invasive species has given rise to numerous models that seek to describe how invading species spread and increase in abundance (Liebhold et al. 1992). In a simple diffusion model, the range of an immigrant expands solely by diffusion without population growth. In other words, the population expands in a concentric manner around the origin, and the density of the population decreases rapidly away from the origin.

Logistic expansion occurs when a population spreads by growth alone without diffusion (Skellam 1951). In that case, the rate of spread will depend on the reproductive rate of the organism and the degree of competition it encounters. If competition does not occur, the relationship between rate, distance, and time approaches a Malthusian curve. When competition increases with density, however, a logistic curve results because growth slows as population density increases. The spread rate becomes asymptotic as the population approaches the carrying capacity of the habitat.

Perhaps the most widely used model to describe the spread of an invader is a growth-diffusion model that links an exponential population growth term with a



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