Most introductions of non-indigenous species occur as a result of commerce, travel, agriculture
or other human activity. The majority (between 80 and 95 percent) of non-native species never become
established in their new environment [Williamson, 1996]. Once established, however, the spread of an
invasive species is typically characterized by three distinct phases [Shigeshada and Kawasaki, 1997, pp
26-27]. The first is an initial establishment phase during which little or no expansion occurs. This is
followed by an expansion phase where the population and range of the invasive species increases.
Finally, there is a saturation phase as the invasion approaches geographical, climatic, or ecological limits
to its range.2
The classic ecological model of the spread of an invasion is the reaction-diffusion model of
Kolmogorov, Petrovsky, and Piscounov [1937], Fisher [1937] and Skellam [1951]. The model of a
biological invasion developed in this paper is an aggregate model that abstracts from spatial
considerations inherent in reaction-diffusion models. In our model, the state of the invasion at each point
in time is defined by its size. Depending on the context, the size of an invasion may be either the area
contained within the frontal boundary of the invasion, or the population or biomass of the invasive
species. This aggregation is analogous to the way that standard bioeconomic models of natural resource
harvesting aggregate over spatial and other life history characteristics that influence resource growth. At
the same time, the model of this paper accommodates invasions that exhibit almost any pattern of growth
2In the reaction-diffusion model growth and spread jointly determine the density, n(y,z,t), of a species at
location (y,z) at time t according to the partial differential equation ðn/ðt = G(n) + D(c2nΛy2 + ∂1n∕∂z1). Local
growth is governed by the growth function G(n) while the coefficient D that determines how fast the species
disperses in space. Fisher [1937] analyzed the case of logistic growth, G(n) = rn(1-n/K), while Skellam [1951]
considered Malthusian growth, G(n) = rn. Kolmogorov, Petrovsky, and Piscounov [1937] considered the general
class of growth functions that satisfy G(0) = G(1) = 0, G(n) > 0 for 0 < n < 1, G '(0) > 0, G '(n) < G '(0) for 0 < n ≤
1. The frontal boundary of an invasion is defined to be the radial distance at which the species density equals a
critical threshold, n*. For example, n* may define a detection threshold below which the species density is low
enough to avoid detection. For our purposes it useful to think of n* as an economic threshold below which the
species density is not sufficient to cause damage. In what has become a classic result in the ecological theory of
biological invasions Kolmogorov, Petrovsky, and Picounouv [1937] showed that the frontal boundary of an invasion
governed by (3.1) expands asymptotically (as t^∞) at a constant rate 2^∕θ'(0)D . For the case of Maltusian
growth this was proved by Kendall [1948]. Mollison [1977] provides a useful review.