  Many exotic species combine low probability of establishment at each
introduction with rapid population growth once introduction does succeed. To
analyze this phenomenon, we note that invaders often cluster spatially when
rare, and consequently an introduced exotic's population dynamics should depend
on locally structured interactions. Ecological theory for spatially structured
invasion relies on deterministic approximations, and determinism does not
address the observed uncertainty of the exotic-introduction process. We take a
new approach to the population dynamics of invasion and, by extension, to the
general question of invasibility in any spatial ecology. We apply the physical
theory for nucleation of spatial systems to a lattice-based model of
competition between plant species, a resident and an invader, and the analysis
reaches conclusions that differ qualitatively from the standard ecological
theories. Nucleation theory distinguishes between dynamics of single-cluster
and multi-cluster invasion. Low introduction rates and small system size
produce single-cluster dynamics, where success or failure of introduction is
inherently stochastic. Single-cluster invasion occurs only if the cluster
reaches a critical size, typically preceded by a number of failed attempts. For
this case, we identify the functional form of the probability distribution of
time elapsing until invasion succeeds. Although multi-cluster invasion for
sufficiently large systems exhibits spatial averaging and almost-deterministic
dynamics of the global densities, an analytical approximation from nucleation
theory, known as Avrami's law, describes our simulation results far better than
standard ecological approximations.
