 
1
Spatial Dynamics of Invasion: The Geometry of Introduced Species  
Gyorgy Kornissa and Thomas Caracob 
a Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic 
Institute, Troy, New York 12180, U.S.A., email: korniss@rpi.edu 
b Department of Biological Sciences, University at Albany, Albany, New York 12222, 
U.S.A., email: caraco@albany.edu 
 
Abstract 
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. 
 
Keywords: invasion criteria, nucleation theory, spatial competition  

 
2
1. Introduction 
The breakdown of biogeographic barriers allows some introduced species to reshape communities 
(Drake et al., 1989; Hengeveld, 1989; Rosenzweig, 2001) and threaten local biodiversity (Kolar 
and Lodge, 2002; Pimm, 1987), especially in nature reserves (Usher et al., 1988).  Most 
introductions fail to initiate invasion (Lonsdale, 1999; Simberloff, 2000).  However, an exotic’s 
abundance often increases rapidly once introduction does succeed (Christian and Wilson, 1999; 
Sax and Brown, 2000; Shigesada and Kawasaki, 1997), particularly when an established exotic 
has an ecological advantage promoting its growth (Callaway and Aschehoug, 2000; Mack et al., 
2000; Pimentel et al., 2000).   
 
Veltman et al. (1996) analyze 496 documented, intentional introductions of 79 bird 
species to New Zealand.  The strongest predictor of establishment is repeated introduction.  Most 
species introduced four or fewer times never became established.  Despite multiple introduction 
attempts for most species, only 20% of the birds ever became established (Veltman et al., 1996).  
Repeated failure of introduction, followed by ecological success once established, appears 
characteristic of both natural dispersal and human-mediated introduction (Sax and Brown, 2000).  
The seeming inconsistency between repeated failure of the introduction process and an invader’s 
rapid growth once established motivates our study. 
 
Despite the observed uncertainty of introduction, spatial models for invasion processes 
typically yield deterministic criteria for growth when rare (Andow et al., 1990; Caraco et al., 
2002; Chesson, 2000; Kot et al., 1996; cf. Lewis and Pacala, 2000).  An invader’s growth or 
decline will ordinarily depend on locally structured interactions (Ellner et al., 1998; Wilson, 
1998), so that chance mechanisms should often govern the population dynamics of rarity (Durrett 
and Levin, 1994a).  Our results show how introduction rates and the size of the environment can 
generate random variation in an invader’s success or failure, through effects on the invader’s 
spatial clustering. 
Our analysis specifically distinguishes single-cluster growth from multi-cluster growth of a rare 
exotic competing with a resident plant species through clonal propagation (Harada and Iwasa, 
1994; Inghe, 1989).  Simulating the model reveals interesting variation in the waiting time for 
successful introduction and subsequent spread of the exotic.  To characterize our particular results 
and, more importantly, to offer a new perspective on the population dynamics of invasion, we 
invoke the physical theory for homogeneous nucleation of spatial systems (Avrami, 1940; 
Johnson and Mehl, 1939; Kolmogorov, 1937).  Originally formulated to model processes such as 
crystallization, nucleation theory readily addresses ecological clustering generated by local 
propagation in viscous populations (Gandhi et al., 1999).  We emphasize that under multi-cluster 
growth of the exotic species, the dynamics of competition for space, specifically the time-
dependent decay of the resident’s density, follows a powerful analytic approximation referred to 
as Avrami’s law (Duiker and Beale, 1990; Ishibashi and Takagi, 1971; Ramos et al., 1999; 
Rikvold et al., 1994). 
 
2. Spatial model for invader -resident competition 
Individual plants typically interact more with nearby than with distant individuals (Rees et al., 
1996; Tilman et al., 1997).  Consequently, an introduction’s success or failure can depend on 
effects regulated by neighborhood, rather than global, densities (Higgins et al., 1996; Wilson, 
1998).  We model two clonal species, a resident and an invader, competing for space in a lattice 
environment.  Each of the L2 lattice sites is either empty or occupied by a single plant; a site 
represents the minimal space an individual (ramet) requires.  Competition for space is pre-
emptive; a site already occupied cannot be colonized by either species until the occupant’s 
mortality opens the site.  Table 1 defines the model’s symbols. 

 
3
The elementary state of any site x belongs to the set σ = {0, i, r}.  The states indicate, 
respectively, an empty site, occupation by an individual invader, and occupation by an individual 
of the resident species. 
First, we describe (0 → i) and (0 → r) transitions.  An empty site may become occupied as a 
result of introduction from outside the environment, or through local clonal propagation (Cook, 
1983; Iwasa, 2000).  The invader occupies each empty site via dispersal at constant probabilistic 
introduction rate βi.  The introduction rate for the resident species is βr.  The introduction process 
does not depend on an open site’s local neighborhood; introduction corresponds to a spatially 
uniform, typically weak, background process, modeling long-distance propagule dispersal. 
 
Local clonal propagation depends on neighborhood composition.  A plant occupying a 
site x may propagate locally if at least one of the δ sites neighboring x is open.  An invader at site 
x attempts to colonize neighboring sites at total probabilistic rate αi; the propagation rate per 
neighboring site is αj/δ.  The resident’s total propagation rate is αr, so the rate per site is αr/δ.  
The chance of successful clonal growth declines with local density.  ηi(x, t) counts invaders 
neighboring an open site x at time t.  ηr(x, t) counts individuals of the resident species on the same 
neighborhood; ηi(x, t) + ηr(x, t) ≤ δ. 
We assume density-independent mortality.  If a site is occupied by an invader, that site becomes 
open at constant probabilistic rate µi.  The mortality rate for individuals of the resident species is 
µr. 
Next we specify the model’s transition rates.  Introduction and local propagation occur 
independently, so that an open site becomes occupied by an invader, a (0 → i) transition, at total 
rate βi + ηi(x, t) αi/δ.  An open site becomes occupied by a resident, a (0 → r) transition, at total 
rate βr + ηr(x, t) αr/δ.  The (i → 0) transition, where invader mortality opens a site, occurs at rate 
µi; the (r → 0) transition occurs at rate µr.  
 
Throughout, we set βi = βr = β, and µi = µr = µ, so that asymmetry between the invader 
and resident is due solely to the difference in the local propagation rates αi and αr.  We keep αi > 
αr, and ask how the invader’s clonal-growth advantage affects the dynamics of invasion as we 
vary lattice size and the introduction rate.  We keep β much smaller than the other rates, so that 
neighborhood competition drives the dynamics, but trapping states are avoided during simulation. 
 
3. Mean-field approximation 
The mean-field approximation offers a clear picture of the basic biological problem, and so 
guides analysis of the spatial model.  The mean-field model assumes homogeneous mixing, and 
so ignores effects of spatial clustering on the dynamics. 
Let ρj represent the global density of sites with state j.  Under the mean-field approximation, 
(
)(
)
i
r
j
dt
d
j
i
r
j
j
j
,
;
1
=
−
−
−
+
=
µρ
ρ
ρ
ρ
α
β
ρ
 
   
 
 
        (1) 
To focus on spatial competition, we temporarily suppress immigration by setting 
0
=
β
 (see 
Lehman and Tilman, 1997).  Then there are three equilibrium fixed points (ρ*
r, ρ*
i); none allows 
coexistence.  Mutual extinction, (ρ*
r, ρ*
i) = (0, 0), is stable when each αj  < µ.  Competitive 
exclusion of the invader by the resident, where (ρ*
r, ρ*
i) = (1 - µ/αr, 0), is stable when αr  > µ and 
αr  > αi.  Finally, exclusion of the resident species by the exotic, where (ρ*
r, ρ*
i)  = (0, 1 - µ/αi), is 
stable when αi  > µ, and αi  > αr.  We assume αi > αr > µ, so that each species can grow in an 
empty environment, and the invader has an individual-level advantage.  Given this ordering of 
parameters, only the last stability criterion can hold.  Therefore, under homogeneous mixing, the 
exotic is introduced at rate β and the invasion condition, i.e. the condition for the exotic’s 
advance when rare against the resident at positive equilibrium, is simply αi > αr.  Since the 
invader has this advantage, it will proceed to exclude the resident (to order β with introduction 
allowed) under homogeneous mixing.  

 
4
Integrating Eqs. (1) numerically yields time dependent global densities. Figure 1 shows the 
results for different values of β (β << µ, αr, αi).  Initializing the environment as fully occupied by 
resident, the system very rapidly relaxes to a phase dominated by the resident species, where the 
global densities are simply (ρ*
r, ρ*
i) = (1 - µ/αr, 0) up to order β.  This fixed point of Eqs. (1) is 
not stable, but the densities stay very close to these values for some time before the invader’s 
growth becomes noticeable.  In this sense, we refer to this phase where the resident resists 
invasion as “meta-stable.”  We define the characteristic time τ, the lifetime of the resident 
species, as the time until the resident’s density decays to half of its meta-stable value.  As β is 
decreased, the global densities as functions of time shift to the right (Fig. 1); note that the increase 
of the lifetime as β decreases is extremely slow, proportional to – log(β). 
As noted above, approximating the spatial dynamics by mean-field or pair correlation methods 
(Bolker and Pacala, 1999; Matsuda et al., 1992), yields deterministic criteria for advance when 
rare; these methods, by construction, cannot address the observed stochastic variation in success 
or failure of introduction.  Furthermore, for parameter values we use in simulations of the 
spatially detailed model, the standard approximations fail to describe the dynamics following 
invasion; see Discussion.  After describing our simulations, we apply nucleation theory to 
characterize both the uncertainty of the introduction process and the advance of the invader. 
 
4. Simulation results 
We implemented standard dynamic Monte Carlo (MC) simulations on an L×L lattice with 
periodic boundaries.  Neighborhood size was δ  = 4.  The time unit was one MC step per site 
(MCSS), during which L×L sites were picked randomly and updated probabilistically.  This 
procedure simulates continuous-time dynamics in the large L limit (Durrett and Levin, 1994b; 
Korniss et al., 1999).  From the mean-field computations, one might not expect the competitively 
inferior resident to resist invasion and induce slow dynamics, since the lifetime of the resident 
species increased only logarithmically with decreasing β.  However, simulation of the spatial 
model revealed slow “meta-stable” decay of the resident.  That is, spatially structured interactions 
slow the timescale of competitive systems (Hurtt and Pacala, 1995; Lehman and Tilman, 1997). 
Fixing µ = 0.10, αr = 0.70, and αi = 0.80 (the same values used in the mean-field approximation), 
we found qualitative variation in the dynamics for introduction rates 10-8 ≤ β ≤ 10-4, and similarly 
for system sizes 32 ≤ L ≤ 512.  At t = 0 we initialize the environment with all sites occupied by 
the resident.  After about (typically less than) 10 MCSS the system relaxes to a “meta-stable” 
configuration dominated by the resident with a small density of empty sites; we designate the 
meta-stable density of the resident as ρr = rms.  Individuals of the exotic species occasionally 
occupy empty sites via introduction.  An immigrant invader may die without propagating.  If a 
site opens up in the neighborhood of a rare invader, the empty site is likely surrounded by more 
than one resident.  The resident’s greater local density more than compensates for its lower 
propagation rate per individual, so the resident has the better chance of colonizing the empty site; 
see Discussion.  Consequently, small clusters of invaders usually shrink and disappear.  
Introductions fail since preemptive competition imposes a strong constraint on the exotic’s 
growth.  Introduction succeeds only if the exotic can generate a spatial cluster large enough that it 
statistically tends to grow at its periphery.  On coarse-grained length scales, nucleation theory 
suggests that there exists a critical radius for invader clusters; smaller clusters decline in size, 
while clusters with a radius larger than the critical value will more likely grow than decline 
(Gandhi et al., 1999).  Simulations confirm this picture, and suggest an interesting distinction 
between single and multi-cluster invasion.  For β = 10-6, a 128×128 system exhibits single-cluster 
invasion (Fig 2).  Increasing β to 10-4 produces multi-cluster invasion (Fig. 3).  Holding β 
constant and increasing the system size L would also generate multi-cluster invasion.  Since the 
spatial structure of the introduction process exhibits nucleation and growth of invader clusters, we 
apply nucleation theory as described in the next section. 

 
5
 
5. Nucleation theory 
Nucleation theory’s ecological significance lies in its prediction of population dynamics 
according to an invader’s spatial-clustering pattern.  Our application of nucleation theory rests on 
the conceptual similarity between biological invasion and meta-stable decay in physical systems.  
In particular, our analysis parallels the theory developed by Rikvold et al. (1994), Richards et al. 
(1995), and Korniss et al. (1999) for magnetization switching in ferromagnetic materials. 
When the resident species’ density ρr(t) first falls to rms/2, we consider the resident competitively 
dominated by the exotic (defining competitive dominance in terms of any particular global 
fraction of sites which the resident occupies does not affect the dynamics).  The non-negative 
random variable τ will represent the first-passage time of the resident’s density to rms/2, in 
accordance with earlier definition of the lifetime for the mean-field approximation.  The mean 
waiting time 〈τ〉 is called the meta-stable lifetime; the expected time elapsing until the resident’s 
global density is ½ its meta-stable value.  In single-cluster invasion τ sums the random waiting 
time until successful introduction and a subsequent period of invasive spread.  Nucleation is 
equivalent to successful introduction; invasive spread begins when a critically sized cluster of 
invaders first forms.  In multi-cluster invasion, clusters of the critical size continuously form and 
grow, leading to many invading species’ clusters of various sizes; see snapshots of the 
environment in Fig. 3.  In this section we identify how the probability distribution of τ differs 
between single-cluster and multi-cluster invasion, and interpret this difference ecologically. 
 
5.1. Single-cluster invasion 
In single-cluster invasion, random variation in the time until a cluster exceeds critical size 
contributes more to the variance of τ than does variation in the period of invasive growth 
following successful introduction.  Different realizations of the introduction and invasion 
dynamics (simulations with the same initial configuration of the resident, but using different 
random numbers) show that the average time for meta-stable decay has little predictive 
significance for single-cluster dynamics.  In fact, the standard deviation is comparable to the 
average (Fig. 4a); the ecological implication is that uncertainty of the success or failure of 
introduction makes single-cluster invasion inherently stochastic. 
When the typical cluster separation (of a hypothetical infinite system) exceeds the finite 
environment size L, the spatial dynamics of the latter (finite) system almost always exhibits meta-
stable escape through nucleation and growth of a single cluster of the superior species (Rikvold et 
al., 1994).  Ecologically, if the introduction rate or environment size is sufficiently small, the 
invasion geometry will always involve single-cluster dynamics.  While the introduction of an 
invader at an empty site is a Poisson process by the construction of our stochastic spatial model, it 
is not known a priori whether the nucleation of a successful invading cluster (one which has just 
reached the critical size) will also be Poisson.  However, we shall suppose, as a working 
assumption, that nucleation of a single invader cluster is a Poisson process as well, and verify this 
assumption using simulations.  Following this assumption, single-cluster invasion (advance of the 
first critical cluster) should be described by an exponential distribution of time until successful 
introduction of the invader.  Consider the cumulative probability distribution of competitive 
waiting times, i.e., the probability that the resident’s global density has not decayed to rms/2 by 
time t, Pnot(t)  = P(τ > t).  For single-cluster invasion, analysis by Richards et al. (1995) shows us 
that 






>










−
−
<
=
g
t
t
it
g
t
t
g
t
t
t
P
for
exp
for
1
)
(
not
  
 
 
 
        (2) 

 
6
where 〈ti〉 is the mean time until successful introduction of the exotic (mean nucleation time) and 
tg is the duration of invasive spread; we take the latter as constant for single-cluster invasion, 
since the growth of a single, supercritical cluster can be treated deterministically.  Note that tg 
does not depend on β, but only on µ, αi, αr, and system size L.  Given the symmetry of resident 
decline and invader growth (Fig. 4a), we approximate tg as the time for the supercritical cluster of 
the invading species to grow and fill half the environment.  For fixed µ, αi, and αr, the 
characteristic time scale (the mean nucleation time 〈ti〉), depends only on β and the system size L.  
More precisely, nucleation of a critical cluster is a Poisson process with nucleation rate per unit 
area I(β).  Thus, for systems in the single-cluster regime 
1
2
)]
(
[
−
∝
〉
〈
β
I
L
ti
                           
 
              
 
        (3) 
in two dimensions (Rikvold et al., 1994).  The distribution of first-passage time τ has a two-
parameter exponential density (see Bury, 1975) with mean 〈τ〉 = (〈ti〉 + tg), variance 〈ti〉2, and skew 
2〈ti〉3 > 0. 
To verify that nucleation of a cluster of critical size is a Poisson process, we simulated 103 (104 
for some parameter values) independent realizations of introduction and invasion, and constructed 
the cumulative distribution Pnot(t).  Figure 5, (a) and (b), compares the results with Eq. (2) for 
fixed β and various L values, and for fixed L and various β values, respectively.  Fitting an 
exponential to the data (a straight line on log-linear scales), we estimated the mean time to 
successful introduction 〈ti〉 (i.e., the nucleation time of the first critical cluster) as the inverse of 
the slope in Fig. 5.  Then one can read the invasive-spread time tg from the figure; from 
expression (2) tg is equal to the maximal time at which
)
(
not t
P
 is unity. 
We also confirmed the dependence of 〈ti〉 on L, as indicated in Eq. (3), and found that 
β
β ∝
)
(I
; 
the rate of successful introduction is proportional to the introduction rate.  Since the typical 
separation between invading species’ clusters increases with decreasing β (Richards et al., 1995), 
the fundamental consequence of these findings is that given an arbitrarily large but finite and 
fixed system size L, for sufficiently small β the system will exhibit single-cluster growth of the 
invader with 
1
−
∝
〉
〈
β
it
.  Note that this behavior of the time elapsing until successful 
introduction follows directly follows from the microscopic dynamics for small β .  However, 
〉
〈it
 remains orders of magnitude larger than one would estimate based simply on the system size 
and the density of open sites, since many exotic clusters fail to grow before introduction 
succeeds.  Most importantly, the average lifetime for decay of the resident’s density increases 
rather quickly as β decreases, in stark contrast with the weak logarithmic divergence predicted by 
the mean-field model (and pair approximation models; see Discussion).  Also note that for small 
β, 
〉
〈
≈
〉
〈
it
τ
, (since 
g
i
t
t
>>
〉
〈
), and 〈ti〉 equals to the standard deviation of the exponential 
lifetime distribution, Eq. (2). 
When introduction rates or the numbers of habitable sites in an environment are sufficiently 
small, ecological invasion of a locally propagating species should occur as the growth of a single 
invader cluster.  The inherent stochasticity of the single-cluster process renders invasion time 
highly unpredictable.  The standard deviation of the meta-stable lifetime τ is as large as the mean, 
and standard ecological methods for spatial systems do not address the uncertainty of invasion 
dynamics.  Nucleation theory, as a first step, provides a functional form characterizing the 
random waiting time of single-cluster invasion process. 
 
5.2. Multi-cluster invasion 
In multi-cluster invasion, realizations of introduction and invasion processes are quite different 
from those of single-cluster growth.  Species’ global densities exhibit only small fluctuations 
about their time-dependent averages (Fig. 4b; cf. Fig. 4a).  Therefore, the first-passage time of the 

 
7
resident’s density to half its initial value has a standard deviation much smaller than its average 
〈τ〉.  Nucleation and subsequent expansion of many clusters implies that global density is a sum of 
random variables, and spatial averaging of local densities within the multi-cluster process reduces 
the variability of the time-dependent global densities among different realizations of the process.  
This reduction in variability of the dynamics of the global species’ densities, resulting from 
averaging local behavior of a large number of clusters, is sometimes termed “self-averaging.” 
To model decay of the resident’s density under multi-cluster invasion, we again invoke nucleation 
theory.  Increasing the introduction rate β or system size L2 leads to the nucleation and 
subsequent growth of many invader clusters (Ramos et al., 1999).  The meta-stable lifetime 
〉
〈τ , 
the resident’s mean first-passage time to rms/2, becomes independent of the size of the 
environment; the standard deviation is proportional to the inverse of the square root of system 
size, i.e., 
L
/
1
.  Further, since global densities reflect spatial averaging of local densities during 
the nucleation and growth of many invader clusters, the distribution of the first-passage time τ is 
normal (Richards et al., 1995).  We express the corresponding cumulative probability distribution 
Pnot(t) as an error function (Fig. 6a); the point where the Pnot(t) functions for different system sizes 
cross corresponds to the system-size independent meta-stable lifetime for the resident species’ 
decay (Korniss et al., 1999).  Compared to the single-cluster mode, the mean first-passage time 
〉
〈τ  is decreased (and becomes system-size independent in the large-L limit).  Importantly, as the 
size of the environment invaded increases, the variance and skew of τ go to zero. 
The preceding observations imply that, for multi-cluster invasion, global densities ρj(t) converge 
to deterministic functions in large environments.  Hence we tested KJMA theory, so named for its 
developers: Kolmogorov (1937), Johnson & Mehl (1939) and Avrami (1940).  In particular, we 
tested Avrami’s law, which predicts the time-dependent global density of the resident species 
when the superior competitor invades through the nucleation and growth of many clusters.  The 
Avrami picture of meta-stable decay works accurately until invader clusters begin to coalesce, 
and the invader becomes the more abundant species; Ramos et al. (1999) offer a visual 
characterization of multi-cluster dynamics.  According to Avrami’s law the global density of the 
meta-stable resident species decays as 
( )














−
≈
3
2
ln
exp
τ
ρ
t
r
t
ms
r
          
 
 
 
         
        (4) 
where 〈τ〉 is the asymptotically system-size independent mean lifetime of the resident’s decay.  
The Appendix presents a derivation of Avrami’s law based on the assumptions of multi-cluster 
nucleation and applies the general result to our ecological-invasion problem to obtain expression 
(4).  Figure 6(b) shows convincing agreement between simulation averages and Avrami’s law; 
deviations are noticeable only for very large times [t3 ≥ 1010 (t ≥ 2154), inset of Fig. 6b] when 
invader clusters begin to coalesce (Korniss et al., 1999) and percolation effects become important 
(Gandhi et al., 1999). 
Nucleation theory also relates the system-size independent lifetime 〈τ〉 of the multi-cluster regime 
to the inherent nucleation rate per unit area I(β) through 
[
]
3
/
1
)
(
−
∝
〉
〈
β
τ
I
                    
 
                           
 
        (5) 
[see Eq. (A.6) of the Appendix]; note that this proportionality is quite different than that obtained 
above for single-cluster dynamics.  Eq. (5) enables us to predict the β-dependence of the lifetime 
in the multi-cluster regime based on measuring the mean nucleation time in single-cluster 
invasions; see Eq. (3).  Combining that result with Eq. (5) implies that 
3
/
1
−
∝
〉
〈
β
τ
 in the multi-
cluster regime. Thus, for an arbitrary small β, in the limit as 
∞
→
L
, the lifetime of the resident 
species’ decline eventually approaches an asymptotic system-size independent value, proportional 

 
8
to
3
/
1
−
β
.  While the global densities become deterministic in the multi cluster regime, the β-
dependence of the lifetime is very different from the weak logarithmic divergence yielded by the 
mean-field approximation (pair approximation, not shown, is similar to the mean field).  That is, 
invader density may vary locally across the environment, but the multi-cluster dynamics will 
maintain the spatially averaged, i.e. the global, density with little variability about a time-
dependent mean.  Nevertheless, despite this lack of random variation in the global densities, the 
multi-cluster dynamics with nearest-neighbor interaction may be poorly predicted by standard 
approximations to spatial processes.  But Avrami’s law, Eq. (4), accurately predicts the simulated 
multi-cluster invasion dynamics. 
 
6. Discussion 
Theory in population biology relies heavily on invasion analyses.  That is, predictions commonly 
are inferred from conditions promoting the initial increase of a rare type (allele, phenotype, or 
species) among resident types, usually at dynamic equilibrium (see Ferriere and Gatto, 1995).  
Invasion criteria formalize conditions for local stability of the rare type’s extinction; when 
extinction is unstable, invasion succeeds.  Models for spatial population processes may be 
deterministic or stochastic, but the associated invasion criteria typically are deduced from linear, 
deterministic approximations to a rare type’s dynamics.  However, most introductions in nature 
presumably begin with a small number of colonists, implying that a more realistic approach to 
invasion analysis would model discrete individuals and include a random component in the 
dynamics of rarity (Caraco et al., 1998; Durrett and Levin, 1994a).  Nucleation theory 
characterizes stochastic properties of introduction and invasion that must often result from 
spatially structured interactions and locally clustered growth. 
Our analysis associates random variation in the spatial dynamics of an introduced species with the 
contrast between the high probability that a given introduction will fail and the ecological 
dominance of exotics once introduction succeeds (Kolar and Lodge, 2002; Sax and Brown, 2000).  
If the exotic’s introduction rate or the system size is small, invasion occurs almost always through 
a single successful invading cluster.  Before that successful event, invader clusters fail to achieve 
critical size and decline to extinction.  The ultimate decline of the resident occurs only after a 
number of stochastic introductions of the exotic species fail; when the exotic invades 
successfully, it grows as a single super-critical cluster (Rikvold et al., 1994). 
Multi-cluster invasion has distinct features.  In larger systems, or with higher introduction rates, 
invasive growth of the exotic begins (almost) as soon as biogeographic barriers break down, and 
dynamics of the global densities becomes nearly deterministic.  For multi-cluster invasion, 
nucleation theory approximates the decay of the resident by Avrami’s law (Ramos et al., 1999; 
Richards et al., 1995).  The global densities become self-averaging with an asymptotically 
system-size independent mean lifetime 〈τ〉 for the resident. Quantitative or qualitative variation in 
local interactions, through effects on cluster formation and dissolution (Gandhi et al., 1999; van 
Baalen and Rand, 1998), influences 〈τ〉 and consequently should exert predictable effects on 
global dynamics; see the Appendix. 
Ecological implications of nucleation theory extend beyond the neighborhood-level competition 
we model.  The two limiting processes we emphasize suggest general significance for spatial 
ecologies. Consider a hypothetically infinite system (i.e., when we take the limit 
∞
→
L
 first).  
Then, for all β much smaller than αi, αr, and µ (so that competition drives the dynamics), the 
system must be in the multi-cluster regime.  Spatial averaging of local dynamics generates 
(almost) deterministic behavior of the global densities, and Avrami’s law accurately describes the 
resident’s decay.  Although the global densities are deterministic functions of time for large 
systems, recall that their dynamics qualitatively differ from the results of the mean-field 

 
9
approximation (Figs. 7a and b).  Further, the lifetime 〈τ〉 increases as 
3
/
1
−
β
 for decreasing β, 
much faster than the weak logarithmic increase predicted by the mean-field model (Fig. 7c).   
We also developed a pair approximation to the spatially detailed model (after Iwasa et al., 1998).  
Pair approximation incorporates short-range spatial correlations, and so evaluates both global 
densities and the conditional densities of the states of paired, neighboring sites (Rand, 1999).  The 
pair approximation marginally improves the estimation of the resident species’ lifetime over the 
mean-field model; pair approximation might better predict global densities for interaction 
neighborhoods extending beyond nearest neighbors (e.g., Caraco et al., 2001).   Importantly, both 
mean-field and pair approximations fail to capture the actual behavior of the time-dependent 
global densities of the spatial model, well described by nucleation theory and Avrami’s law. 
Now consider the second limiting process.  For any finite system, there is a sufficiently small β 
where the typical cluster separation increases beyond the system size, and invasion crosses over 
to the single-cluster mode.  Below this β, we found that 
1
−
∝
〉
〈
β
τ
; this is the 
0
→
β
 limiting 
behavior of the meta-stable lifetime for a finite system.  Illustrating this scenario for an L = 128 
system (Fig 7c), we confirmed that for 
3
5
10
10
−
−
≤
≤β
 the system is in the multi-cluster regime 
and the β-dependence of the lifetime follows 
3
/
1
−
β
.  At around 
6
10−
≈
β
 the crossover occurs, 
and for 
6
10−
≤
β
 the system exhibits single-cluster invasion and indeed approaches the 
1
−
∝
〉
〈
β
τ
 behavior (Fig. 7c).  Here the invasion is inherently stochastic, and the mean of the 
lifetime becomes equal to its standard deviation.  Note that in the large-β region (
2
10−
≥
β
) 
nucleation theory breaks down as invading clusters coalesce almost immediately after 
introduction.  In fact, the mean-field and pair approximations begin to work much better here as a 
result of the almost immediate mixing of small clusters. 
We set βi =βr for simplicity; the difference between species was due solely to different rates of 
local propagation.  However, equal introduction rates might imply that the competitors’ 
populations outside the environment are the same size (or equally distant), which is unlikely.  For 
the initial conditions we consider, however, the dynamics is insensitive to βr  (as long as it 
remains small), since the system is initially occupied densely by the resident species.  Assigning 
separate introduction rates to the species, in particular, 
i
r
i
α
α
µ
β
,
,
<<
 as previously, but 
with
0
=
r
β
, does not alter the dynamics qualitatively; the ecological distinction between single-
cluster and multi-cluster invasion processes remains important. 
 
6.1.  Local dynamics 
Our results, in particular the slow decay of the resident species’ density, depend on the 
discreteness of space.  Bolker et al. (2000) suggest that lattice-based models overemphasize 
effects of local clumping on dynamics, but we can regard a lattice site as the minimal amount of 
space (hence, minimal access to resources) necessary to sustain an individual (ramet).  At some 
scales, this may imply greater realism for the discrete-space approach (Neuhauser and Pacala, 
1999).  Of course, our detailed simulation results further depend on choices of parameter values, 
neighborhood size, and system sizes (Filipe and Maule, 2003; McCauley et al., 1993). 
 
The local dynamics help explain the resistance to invasion.  In the absence of the invader, 
the resident’s global density will be approximated by ρ*
r = (1 - µ/αr).  Suppose that an individual 
of the invasive species is introduced, via dispersal, at an open site y at time t.  Approximating the 
local state frequencies by global densities (e.g., Duryea et al., 1999), the probability that no site is 
open (for local propagation) among the neighbors of site y is close to (ρ*
r)δ.  If no site 
neighboring the single invader is open, the chance that the immigrant invader dies before a 
neighboring site opens is just (δ + 1) -1, since each individual has the same exponentially 
distributed waiting time for mortality.  Larger neighborhoods increase the probability that the 

 
10
immigrant will find a neighboring site open, but do not necessarily increase the chance that the 
invader will propagate into an open site.  To examine the latter probability, we consider a simple 
example. 
 
Suppose that neighboring sites (x, y) form an (open, invader) pair.  The open site (x) 
becomes occupied by the invading species at constant probabilistic rate (β + αi /δ).  The same 
open site becomes occupied by the resident at rate (β + ηr(x, t) αr/δ ), which depends on the 
number of resident individuals neighboring site x.  Approximating that number ηr(x, t) with a 
binomial random variable, the resident species occupies the open site at constant probabilistic rate 
(β + (αr - µ) (δ - 1)/δ).  Although αi > αr, that resident is more likely to acquire the open site as 
long as 
(
)(
)
i
r
α
δ
µ
α
>
−
−
1
 
 
 
 
 
 
 
        (6) 
This expression holds in our simulations; more generally, the invader is less likely to acquire the 
open site as neighborhood size δ increases. 
The preceding emphasizes that it is the discreteness of the introduction process and the 
preemptive nature of the competition that, in combination, allow the resident species to repel 
small, rare clusters of invaders, before the resident declines.  Recall that this is the sense in which 
we refer to the possibly lengthy domination of the environment by the resident species as “meta-
stable.”  In contrast to Gandhi et al. (1999; see below), we do not assume an underlying bistability 
(such that either species, when common, tends to repel invasion by the other).  That is, our initial 
global densities are unstable, and the system eventually reaches the stable stationary state 
dominated by the competitively superior invader. 
 
6.2.  Nucleation theory in biology 
Gandhi et al. (1998, 1999) model two species competing for space, and consider how cluster 
growth/decay can influence time to extinction.  The authors assume a somewhat elaborate locally 
structured dynamics.  The mortality rate for individuals of each species increases as the total 
density in the local neighborhood increases.  The birth rate for individuals of either species 
increases as the relative frequency of conspecifics increase locally.  Finally, individuals may 
move diffusively (Gandhi et al., 1998). 
As an initial condition, Gandhi et al. (1998, 1999) distribute large numbers of individuals of each 
species uniformly across the environment.  Gandhi et al. (1998) assume competitively identical 
species.  If initial densities differ sufficiently, a mean-field model approximates the time elapsing 
until the less numerous species reaches extinction.  But when both competitors initially occur at 
high density, clusters quickly form in each species.  Thereafter, the dynamics are driven by 
interactions at cluster interfaces, and the expected time to extinction increases beyond the mean-
field prediction.  Gandhi et al. (1999) conduct a similar analysis with asymmetric species, and 
they invoke nucleation theory’s concept of a cluster size critical for growth in a competitive 
environment.  The model’s positive frequency-dependent birth rates likely accentuate the rapid 
decay of small clusters; smaller clusters have greater perimeter curvature, so that individuals on 
the perimeter have relatively few conspecific neighbors.  Again, in their models nucleation is 
driven by an underlying bistability (two stable fixed points in terms of the global densities). 
Our analysis complements the results in Gandhi et al. (1998, 1999).  Each model addresses 
competition; we assume clonal propagation, while Gandhi et al. model social interactions where 
paired conspecifics do better than heterospecific pairs (see Giraldeau and Caraco, 2000).  Our 
analysis focuses on ecological invasion, a process where one species begins at zero density, enters 
the environment as rare individuals, and advances through single or multi-cluster growth 
(depending on parameter values).  Gandhi et al. (1998, 1999) initially disperse each species 
uniformly.  Both species form multiple clusters, and then the sometimes lengthy process of 
competitive exclusion begins.  Our model’s biological assumptions differ considerably from the 
model by Gandhi et al. (1998, 1999), but both analyses point to the importance of cluster 

 
11
geometry as a basis for understanding relationships between individual-based interactions and 
global dynamics. 
The term “nucleation” has been applied, metaphorically, in studies of community succession 
(Franks, 2003; Moody and Mack, 1988) and ecological restoration (Robinson and Handel, 2000).  
These applications refer to an interspecific facilitation, where a plant of one species modifies 
local sites in a manner promoting the germination and survival of a second species.  Yarranton 
and Morrison (1974) describe an interesting example by tracing primary succession in a sand-
dune community.  Persistent vegetation (oak-pine forest) replaces colonizing grassland during 
succession.  Persistent species begin as local clusters that expand and eventually coalesce, 
replacing the colonizing species in the process.  But many of the persistent-species clusters are 
initiated through seedling establishment under individual junipers (Juniperus virginiana), where 
microclimate is moderated and soil nutrients are concentrated (Yarranton and Morrison, 1974).  
Our model assumes equivalent sites, but logical extensions of the theory would examine effects 
of spatial heterogeneity in site quality and exogenous temporal variation in demographic 
parameters (Korniss et al., 2001). 
 
Finally, we suggest that nucleation theory offers a quantitative context for addressing a 
diverse series of fundamental questions in biology.  For example, Herrick et al. (2002) analyze 
DNA replication by invoking formal equivalence between KJMA theory and their stochastic 
model for replication kinetics of Xenopus DNA.  Initiation of replication forks along the linear 
DNA molecule is equivalent to nucleation events, and replication fork velocity is equivalent to 
the rate of cluster growth.  Herrick et al. (2002) present the first reliable description of temporal 
organization in a higher eukaryote’s DNA replication; nucleation theory provides the study’s 
conceptual perspective and so guides the interpretation of data. 
 
Acknowledgements 
Discussion with M. A. Novotny and Z. Toroczkai is gratefully acknowledged.  We also thank Z. 
Toroczkai for helping us implement a numerical integrator routine for the mean-field and pair 
approximations. We appreciate the support of NSF Grant DEB-0342689.  G. K. is also supported 
by NSF through Grant DMR-0113049 and by the Research Corporation through Grant No. 
RI0761. 

 
12
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15
Appendix.  KJMA theory for homogeneous nucleation 
Here we present a brief derivation of KJMA theory (commonly referred to as Avrami’s law) for 
an infinite system.  Avrami’s law was originally formulated to describe how solids transform 
from one state of matter (“phase”) to another as they crystallize. Since then, nucleation theory and 
Avrami’s law have also been applied successfully to domain switching in ferromagnetic (Ramos 
et al., 1999; Rikvold et al., 1994) and ferroelectric (Duiker and Beale, 1990; Ishibashi and Takagi, 
1971) materials.  Our development follows Ishibashi and Takagi (1971).  See Duiker and Beale 
(1990) for discussion of large, but finite systems, in particular, for consideration of finite-size 
effects where clusters begin to coalesce.  The appendix first offers a description applicable to 
spatial systems in general, and then specifies application to the model analyzed in the text. 
The system is initialized in the meta-stable phase.  In homogeneous nucleation the decay of the 
meta-stable phase (or “switching” to the final equilibrium phase) occurs through random 
nucleation and subsequent growth of local clusters.  Consider an arbitrary point Q in the d-
dimensional, infinite “volume.”  The probability that this point is not in the switched volume by 
time t, 
)
(
not t
P
, equals to the volume fraction of the initial phase, 
)
(t
ϕ
. 
Recall that we expect an invading cluster to continue to grow only after its radius reaches a 
critical length rc.  We assume that nucleation of a successful invading cluster (with initial radius 
rc) is a Poisson process with constant nucleation rate I per unit volume (i.e., the probability of 
nucleation per unit volume per unit time is I).  Such a cluster, nucleated at time t′, will cover a 
volume 
d
c
d
t
t
v
r
C
t
t
S
)]
(
[
)
,
(
′
−
+
=
′
 
 
 
 
 
 
    (A.1) 
at later time t, where v, the radial velocity of a growing cluster, is approximated by a constant, 
and Cd defines the relationship between radius and volume in the d-dimensional volume  (i.e., 
π
=
2
C
 and 
3
/
4
3
π
=
C
). 
Now divide the 
)
,0
(
t  time interval into N infinitesimal intervals (
)t
j
t
j
∆
+
∆
)1
(,
 with 
N
t
t
/
=
∆
, 
1
,
,2
,1,0
−
=
N
j
K
.  For infinitesimal ∆t, under the assumption of a Poisson 
process, the probability that no cluster, nucleated in the infinitesimal interval (
)t
j
t
j
∆
+
∆
)1
(,
, 
will cover point Q at time t  is 
t
t
j
t
IS
∆
∆
−
)
,
(
1
.  Thus, the probability that point Q is not swept 
by any cluster (i.e., Q is not in the switched volume) at time t is given by  
.]
)
,
(
1[
]
,
(
1[
]
)
2,
(
1
][
)
,
(
1
][
)
0,
(
1[
)
(
1
0
not
∏
−
=
∆
∆
−
=
∆
−
∆
∆
−
∆
∆
−
∆
−
=
N
j
t
t
j
t
IS
t
N
t
IS
t
t
t
IS
t
t
t
IS
t
t
IS
t
P
K
 
    (A.2) 
Taking the logarithm of Eq. (A.2), letting 
0
→
∆t
, and letting
∞
→
N
(such that 
t
t
N
=
∆
) 
yields 
[
] .
)
(
)1
(
)]
(
[
)1
(
)]
(
[
)
,
(
]
)
,
(
1
ln[
)
(
ln
1
1
0
1
0
0
0
1
0
+
+
+
→
∆
−
=
−
+
+
−
=
′
−
+
+
=
′
′
−
+
−
=
′
′
−

→

∆
∆
−
=
∫
∫
∑
d
c
d
c
d
t
d
c
d
t
d
c
d
t
t
N
j
not
r
vt
r
d
v
IC
t
t
v
r
d
v
IC
td
t
t
v
r
IC
td
t
t
S
I
t
t
j
t
IS
t
P
  (A.3) 
In our case, as in many applications, the critical radius is much smaller than the typical cluster 
separation (which equals to the average diameter of the cluster when they begin to coalesce) and 
can be neglected. Then, for rc = 0, the volume fraction of the meta-stable phase becomes 
  
 
 






+
−
=
=
+1
not
1
exp
)
(
)
(
d
d
d
t
d
v
IC
t
P
t
ϕ
,  
 
 
    (A.4) 
which is the general form of Avrami’s law. 

 
16
For our application, 
2
=
d
.  The meta-stable phase corresponds to the competitively inferior, 
resident species with a small density of open sites in the background.  The introduced species 
advances through nucleation (successful introduction) and subsequent cluster growth.  Thus, for 
the decay of the resident we find  





−
=
=
3
2
2
3
exp
)
(
)
(
t
v
IC
r
t
r
t
ms
ms
r
ϕ
ρ
 , 
 
 
 
    (A.5) 
where 
ms
r
 is the meta-stable density of the resident.  Recall that we define the meta-stable 
lifetime 〈τ〉 as the time until the residents’ density decays to one half of its meta-stable value 
ms
r
, 
i.e., 
2
/
)
(
ms
r
r
=
〉
〈τ
ρ
.  Then from Eq. (A.5) we obtain 
3
1
2
2
)
2
ln(
3






=
〉
〈
v
IC
τ
, 
 
 
 
 
 
    (A.6) 
which explicitly shows the dependence of the lifetime on the nucleation rate per unit volume I  
and the radial growth velocity of the invading clusters v .  Using this expression for the lifetime, 
we can write Eq. (A.5) in the form of Eq. (4) of the text 
  
 
 














〉
〈
−
=
3
)
2
ln(
exp
)
(
τ
ρ
t
r
t
ms
r
. 
 
 
 
    (A.7) 
From the above derivation, Avrami’s law, Eq. (A.7), is generic when the switching mechanism is 
governed by homogeneous nucleation.  Parameters of the ecological dynamics (αj, β, µ and δ ) 
govern the meta-stable lifetime through their impact on the nucleation rate per unit volume I and 
the radial growth velocity v [Eq. (A.6)], leaving the functional form of the of the time-dependent 
density unchanged in Eq. (A.7). 
 

 
17
Figures 
 
 
 
 
 
Fig. 1. Time-dependent global densities obtained by numerically integrating the mean-field equations for 
4
10−
=
β
, 
6
10−, and 
8
10− (from left to right, respectively). 
70
.0
=
r
α
, 
80
.0
=
i
α
, and 
10
.0
=
µ
 
throughout this paper. Matching pairs of ρr(t) and ρi(t) intersect near a density of  0.425. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
18
 
 
 
 
Fig. 2. Single-cluster invasion mode in lattice Monte Carlo simulations. Times are given in units of Monte 
Carlo steps per site (MCSS); L = 128 and β = 10-6. White represents empty sites, blue and red correspond to 
sites occupied by resident and invasive species, respectively.  The waiting time for successful introduction 
exceeds the time between initiation of invasion and the resident becoming numerically superior. 
 

 
19
 
 
 
Fig. 3. Multi-cluster invasion mode in lattice Monte Carlo simulations. Times are given in units of MCSS; 
L = 128 and β = 10-4. Sites are color coded as in Fig. 2. The first successful introduction occurs rapidly; 
additional clusters nucleate during invasion. Note that the system size L is the same as in Fig. 1. The 
temporal sequence of configurations in Fig. 2 and Fig. 3 demonstrates the importance of the interplay of the 
characteristic length scales: the typical cluster separation and the system size.  In our simulations, critical 
cluster size is clearly smaller than either underlying length scale. 
 

 
20
 
 
 
 
Fig. 4. Five independent realizations of the time series of the two species’ global densities during (a) 
single-cluster invasion mode. Note that matching pairs of ρr(t) and ρi(t) intersect near a density of  0.425. 
(b) During multi-cluster (self-averaging) invasion mode.  The parameters are the same as those in Fig. 2 
and Fig. 3 for (a) and (b) respectively, i.e., L = 128 for both, β = 10-6 in (a) and β = 10-4 in (b). 
 

 
21
 
 
 
 
Fig. 5. Cumulative probability distribution for the lifetime of the resident species (the probability that the 
resident’s density has not decayed to rms/2 by time t ) in the single-cluster invasion regime for (a) β = 10-6 
and L=32, 64, 128 (from top to bottom, respectively); (b) for L=128 and β = 10-8, 10-7, 10-6 (from top to 
bottom, respectively). The log-linear scales indicate an exponential distribution for the nucleation process, 
equation (9). The lifetime distributions were constructed using 104 independent Monte Carlo runs for β =10-
6; L=32, 64, 128 and 103 independent runs for L=128; β = 10-8, 10-7. 

 
22
 
 
 
 
Fig. 6. (a) Cumulative probability distribution for the lifetime of the resident species (the probability that 
the resident’s density has not decayed to rms/2 by time t ) in the multi-cluster (self-averaging) invasion with 
β = 10-4 for system sizes indicated. Error functions, with the recorded averages and standard deviations as 
parameters, are plotted over the corresponding simulation data. The lifetime distributions were constructed 
using 104 independent Monte Carlo runs. (b) Time-series of the density of the resident species ρr(t) for a 
256×256 system (using an ensemble average of 100 independent runs) for the same value of β as in (a).  
The solid line represents Avrami’s law, Eq. (4), not distinguishable from simulation data on these scales. 
The inset shows ρr(t) vs t3 on log-linear scales indicating where the deviation from Avrami’s law becomes 
noticeable. 

 
23
 
 
 
 
Fig. 7. Comparison of the mean-field approximation with lattice Monte Carlo (MC) simulation results. (a) 
Time-dependent global densities for 
4
10−
=
β
. Here the system size for MC simulations was 
256
=
L
, 
yielding self-averaging multi-cluster invasion. (b) The resident’s global density vs 
3t  shown on log-normal 
scales illustrate the validity of nucleation theory and Avrami’s law, and the weakness of the mean-field 
approximation. (c) Lifetime of the resident in the mean-field approximation and by Monte Carlo (MC) 
simulations for L=128 as a function of the introduction rate β.  Log-log scales are used to capture the range 
of β spanning several orders of magnitude and the resulting disparate timescales for the lifetime 〈τ〉 in the 
MC simulations. Standard error for the MC lifetime data is less than 4% for all data points (and would 
correspond to error bars less than the symbol size for the average lifetime on the graph). The two straight-
line segments correspond to the β-dependence of the average lifetime predicted by nucleation theory in the 
multi-cluster regime (slope –1/3 on log-log plot) and in the single-cluster regime (slope –1 on the log-log 
plot). The inset shows the same data on linear-log scales to illustrate the linear dependence of the lifetime 
on log(β) for the mean-field approximation. The straight line is the best linear fit, 
)
log(β
b
a −
, to the 
mean-field model. 

 
24
Table 1 
List of model symbols, definitions (numerical value or range used in the simulations, 
where appropriate) 
 
Symbol 
Definition 
 
L 
 
Lattice length/width (32 ≤ L ≤ 512) 
σ 
 
Set of lattice site’s elementary states (empty, invader, resident) 
βi 
 
Invader’s introduction rate (10-8 ≤ βi ≤ 10-2) 
βr 
 
Resident’s introduction rate (βr = βi = β) 
δ 
 
Neighborhood size for clonal growth (4) 
αi 
 
Invader’s clonal propagation rate (0.8) 
αr 
 
Resident’s clonal propagation rate (0.7) 
ηi(x, t)  
Number of invader neighbors around site x at time t 
ηr(x, t)  
Number of resident-species neighbors around site x at time t 
µi 
 
Invader’s mortality rate (0.1) 
µr 
 
Resident’s mortality rate (µi = µr = µ = 0.1) 
ρi 
 
Invader’s global density 
ρr 
 
Resident’s global density 
rms 
 
Resident’s “metastable” global density 
<τ> 
 
Resident’s metastable lifetime 
ti 
 
Waiting time for invader’s nucleation 
tg 
 
Time for successful invader to grow to competitive dominance 

 
25
I(β) 
 
Nucleation rate per unit area 
v 
 
Velocity at which cluster radius grows 
S(t, t’)  
Volume of cluster at time t formed at time t’ < t 
rc 
 
Initial radius of nucleating cluster 
Cd 
 
Dimension-dependent multiplier 
d 
 
Dimension of volume within which nucleation occurs 
 
