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4
Relationship between Propagule Pressure
and Establishment Risk
Chapter 3 highlighted the many factors that influence invasion risk, namely
(in broad terms) propagule pressure, species traits, abiotic environmental charac-
teristics, and biotic interactions. Managing invasion risk by setting a discharge
standard assumes that, despite these powerful modifying factors, organism den-
sity alone is a reasonable predictor of establishment probability. Consequently,
this chapter examines the relationship between organism density and invasion
risk, and considers how this relationship might help inform an organism-based
discharge standard. Subsequent chapters examine other, non-modeling-based
approaches to setting a discharge standard (Chapter 5) and evaluate the data re-
quirements and limitations in estimating the relationship between invasion risk
and propagule pressure, including uncertainty, variability over space and time,
the relative merits of historical, survey, and experimental data, and the use of
proxy variables (Chapter 6).
THE RISK–RELEASE RELATIONSHIP
Concepts and Terms
There are many definitions of invasion risk (e.g., Drake and Jerde, 2009).
This report uses the term invasion risk interchangeably with establishment prob-
ability to refer to the chance that an introduced group of individuals establishes a
self-maintaining population. In formal risk assessment, risk is defined a func-
tion of both exposure (the probability of a harmful event) and hazard (the effect
of the harmful event). In this framework, invasion risk could be defined as a
function of the probability of a species establishing (exposure) and its expected
impact (hazard); however, such terminology is not used here. Rather, the term
invasion risk is defined simply as the probability of establishment. For repeated
introductions of invasive species, it is important to consider the time scale for
establishment probability because, over infinite time, any invasion with a non-
zero probability will eventually occur. The term invasion rate refers to the
72

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Relationship Between Propagule Pressure and Establishment Risk 73
number of nonindigenous species that establish in a given region per unit time.
It is straightforward to convert invasion rate to establishment probability.
A propagule is any biological material (such as particles, cells, spores, eggs,
larvae, and mature organisms) that is or may become a mature organism. Prop-
agule pressure is a general term expressing the quantity, quality, and frequency
with which propagules are introduced to a given location. As discussed in
Chapter 3, propagule pressure is a function of a suite of variables reflecting the
nature of the species and the transport vector. The remainder of this chapter,
however, focuses on the quantity of propagules alone.
In the context of ballast water, it is useful to distinguish two measures of
propagule quantity. Following Minton et al. (2005), inoculum density is defined
as the density of organisms in released ballast water. Inoculum density (denoted
as DI in equations) is given simply as the total number of organisms in the in-
oculum (NI) divided by the inoculum, or ballast water, volume (VI):
DI = NI / VI.
The initial population size is the initial number of organisms released into
the environment in a given location at a given time, i.e., the inoculum number
(NI). As these organisms will tend to spread out into their new habitat, their
density in the environment (DE) is, in the simplest formulation, given as the
number of organisms released (NI) divided by the volume of water in the envi-
ronment (VE):
DE = NI / VE.
It is important to recognize that it is the inoculum density (DI) that is subject
to a ballast water discharge standard. However, both the initial number of or-
ganisms NI (conventionally denoted N0 in population modeling) and their densi-
ty in the new environment DE are expected to affect establishment probability.
In other words, the variable to be managed to reduce invasion risk is clearly dis-
tinct from the variable that is typically used in predicting invasion risk. This
disconnect is the central conceptual challenge in converting empirical and theo-
retical results in population establishment to an operational discharge standard
(see section below).
The relationship between invasion risk and propagule pressure is the risk–
release relationship. Understanding the risk–release relationship is essential to
predicting and comparing the invasion risk associated with different discharge
standards. However, understanding this relationship is not a straightforward
proposition. It is easiest to define, model, and estimate this relationship for sin-
gle species, focusing on the relationship between the number of individuals re-
leased at a given time in a given location, and the probability of that population
establishing. For larger-scale analyses of invasions by multiple species, the rela-
tionship becomes less easy to define, model, and measure. In general, however,
it can be thought of as the number of established species as a function of the

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74 Propagule Pressure and Invasion Risk in Ballast Water
number of released species, organisms, or combination thereof, over a given
time period. In the case of ballast water, the central but somewhat indirect risk–
release relationship is the number of species that establish as a function of the
large-scale release of a varying number of varying species at varying densities.
The Hypothetical Risk–Release Relationship
In general, the relationship between invasion risk and propagule pressure is
expected to be positive, although its shape is unknown. A priori, it might take
any of a number of standard shapes including linear, exponential, hyperbolic,
and sigmoid (Ruiz and Carlton, 2003) as shown in Figure 4-1. The shape of the
risk–release relationship has important implications for managing invasion risk.
If the relationship were linear, then a given reduction in release density
would always lead to a proportional reduction in invasion risk (Figure 4-2A). If
the relationship were nonlinear, with one or more inflection or slope-balance
points, then interesting management thresholds would emerge (Figure 4-2B). If
the relationship were exponential, a reduction from high to moderate release
density leads to a much greater reduction in invasion risk than a similar-sized
reduction from moderate to low release density. The opposite would hold for
FIGURE 4‐1 Common shapes for relationships between two variables: (a) hyperbolic, (b)
sigmoid, (c) linear, (d) exponential (both axes linear). A priori, any of these could
represent the relationship between invasion risk (probability of a species establishing)
and propagule pressure (e.g., number of individuals released). SOURCE: Adapted, with
permission, from Ruiz and Carlton (2003). ©2003 by Island Press.

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Relationship Between Propagule Pressure and Establishment Risk 75
A B
FIGURE 4‐2 Conceptual application of a risk–release relationship to inform ballast water
organism discharge standards. S is the observed risk of ballast‐mediated species inva‐
sion; N is the observed number of organisms released. (Axis units depend on whether
the model represents a single species or multiple species.) S* is the target invasion risk;
N* is the corresponding target release value. Assuming a robust risk–release relation‐
ship, reducing the ballast water release by the proportion RN is predicted to reduce the
invasion risk by the proportion RS. (A) Under the assumption of a linear risk‐release
relationship, a given reduction in the release rate is predicted to give the same propor‐
tional reduction in invasion rate (i.e., RN=RS). (B) Under the assumption of a sigmoid
relationship, the same reduction in release (RN1=1‐[N*1/N1] = RN2=1‐[N*2/N2]) is pre‐
dicted to give a much lesser (RS1 =1‐[S*1/S1]) or a much greater (RS2 =1‐[S*2/S2]) reduc‐
tion in invasion risk, depending on the range over which RN occurs. Solid dot indicates
the inflection point around which the greatest reduction in risk is obtained for the least
reduction in release. Open dots indicate the points at which the slope passes through a
o
45 angle: outside these bounds, increasingly less risk reduction is obtained for the same
release reduction. Panel (A) provides a graphical illustration of the multi‐species linear
model proposed by Cohen (2005, 2010) and Reusser (2010), where the y‐axis is the in‐
vasion risk (characterized as invasion rate, the number of species established per unit
time) and the x‐axis is organism release (characterized as release rate, the total number
organisms per unit time). Reusser (2010) defined as the per capita invasion risk the
number of introduced species that establish per organism released, i.e., the slope of the
straight‐line relationship. Cohen (2005, 2010) defined as the reduction factor the equal
proportional reductions in release and risk from current to target levels.

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76 Propagule Pressure and Invasion Risk in Ballast Water
a hyperbolic relationship. For an S-shaped curve, a reduction over the middle
range of release densities would to the greatest reduction in invasion risk (Figure
4-2B). Because of qualitative change in the risk–release relationship at the in-
flection point and at the slope-balance points (where the tangent to the curve
passes through 45o), non-linear relationships present influential management
thresholds.
Theory tells us that for a single population, only two of these four shapes—
the hyperbolic and sigmoid curves—can represent the overall risk–release rela-
tionship (see Box 4-1). The combined curves of multiple populations would
also be expected to be nonlinear. Thus, it is expected that there should be at
least one threshold in the risk–release relationship that could in principle prove
useful in informing discharge standards.
Despite understanding that the overall theoretical shape is hyperbolic or
sigmoid for a single species, it is possible for a given set of risk–release data to
be better characterized by a linear or even an altogether different model. This
apparent discrepancy could emerge for two main reasons. First, there may be
insufficient data points to support a curved line over a straight line. This diffi-
culty will be exacerbated as the true slope decreases (for example, the lower-left
or upper-right ends of the hyperbolic or sigmoid curves). Second, any underly-
ing theoretical risk–release relationship may be swamped out by other more im-
portant sources of variation that affect establishment probability (see Chapter 3),
such that it cannot be recovered from the data.
To quantitatively predict the effects of a discharge standard on invasion
risk, and to compare the risk associated with different discharge standards, it is
essential to understand the shape and strength of the risk–release relationship.
The following section reviews a range of approaches that have been taken to
fitting risk–release curves to empirical data.
Modeling the Risk–Release Relationship
An ideal analysis of the risk–release relationship would involve developing
and testing a suite of candidate theoretical models, collecting multiple rigorous
empirical datasets, and comparing the fit of the models to the data to determine
(1) which model best captures the risk–release relationship and (2) how strong
this relationship is relative to other potential explanatory variables.
Several approaches have been taken to modeling the risk–release relation-
ship, categorized in Table 4-1 along two axes. First, models can range from
descriptive models that simply represent the shape of the relationship to mecha-
nistic models that define the processes generating the relationship (e.g., Drake
and Jerde, 2009). These are not mutually exclusive categories—a given model
may include both mechanistic and descriptive components—but at their ex-
tremes they represent very different modeling philosophies and goals, and they
define a useful spectrum for organizing modeling approaches. Second, Table 4-
1 distinguishes models that focus on the establishment of a single species vs.

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Relationship Between Propagule Pressure and Establishment Risk 77
TABLE 4‐1 Approaches Taken to Modeling the Relationship Between Invasion Risk and
Propagule Pressure
Scale
Sample structure1
Type Single species Multiple species
examples examples
Descriptive Statistical Memmott et al. Lonsdale (1999)
Logistic regression: (2005) Levine and D’Antonio
y Jongejans et al. (2003)
b0 b1 x (2007)
ln
1 y
Ricciardi (2006)
Bertolino (2009) Cohen (2005, 2010)*
Functional form Drake and Jerde Reusser (2010)*
Species‐area curve: (2009)
z
y=cx
Mechanistic Probabilistic Shea and Possing‐ Drake et al. (2005)*
N
pE 1 e ham (2000) Costello et al. (2007)*
Leung et al. (2004) USCG (2008)*
Dynamic demographic Memmott et al. See text
dN (2005)
rN
dt Drake et al. (2006)
USCG (2008)*
Drake and Jerde
(2009)
Jerde et al. (2009)
Bailey et al. (2009)
Kramer and Drake
(2010)
Notes: Models are categorized as being on the descriptive or the mechanistic end of a spectrum,
and as representing single or multiple invading species. Sample structures show simple, generic
forms of these model types. Most of these approaches have been widely developed and imple‐
mented throughout the biological literature; only a few recent examples, further discussed in the
text, are listed here. Some studies illustrate more than one modeling approach.
1
Parameters: y, dependent variable (invasion risk); x, independent variable (organism discharge or
proxy variable); b0, b1, c, z, shape parameters; pE, population establishment probability; α,
ln(individual establishment probability); N, number of individuals; r, per capita population growth
rate.
* Proposed application of modeling approach to ballast water management.

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78 Propagule Pressure and Invasion Risk in Ballast Water
that of multiple species. The following sections highlight illustrative examples
of each approach and outline their key advantages and disadvantages. Since
population establishment theory applies across all species, habitats, and vectors,
many of the given examples fall outside the immediate realm of ballast-
mediated invasions. Nevertheless, the approaches illustrated are readily appli-
cable to the risk–release relationship for ballast water.
SINGLE-SPECIES MODELS
It is informative to examine the risk–release relationship at the scale of a
single species, for two main reasons. First, this approach allows examination of
invasion scenarios for certain model species, such as fast growing, high impact,
or commonly released invaders, which could be used to obtain upper bounds for
discharge standards under best-case (for invasion) scenarios. Second, it allows
for clarification of model structures and assumptions before scaling up to the
more realistic scenario of multi-species releases. The primary disadvantage of
the single-species approach, in the context of managing ballast water, is that it
does not represent the reality of the simultaneous and continuous release of
many species from ballast water. The greatest challenge in this approach is in
converting experimental and theoretical results premised on N0 to a discharge
standard applicable to DI.
A ballast-mediated invasion may be expected to begin from the introduction
of relatively few individuals. Three factors are particularly relevant to small
population dynamics—demographic stochasticity, positive density dependence,
and the spatial environment—and must be considered in developing an effective
risk–release model. Their net effect can be captured by a descriptive model;
their individual effects can be tested by incorporating them explicitly into a me-
chanistic model.
Demographic stochasticity is the natural variability in individual survival
and reproduction that occurs in populations of any size, but that in small popula-
tions can lead to large fluctuations in population growth rate. On average, de-
mographic stochasticity makes extinction more likely than the equivalent deter-
ministic model would predict; on the other hand, in a given realization, stochas-
ticity can lead to establishment when a deterministic model would otherwise
predict extinction (Morris and Doak, 2002; Drake, 2004; Andersen, 2005; Drake
et al., 2006; Jerde et al., 2009). Demographic stochasticity is expected to lower
the risk–release curve at low density.
Positive density dependence, or Allee effects, is the intuitively logical notion
that as organism density decreases, individuals may suffer increasing difficulty
finding mates or foraging. Population growth rate would thus be expected de-
cline at low density, rather than to increase as per an exponential growth model
(Allee, 1931; Courchamp et al., 2009). Thus, across the range of low initial
densities, it takes a higher density to achieve the same invasion risk when Allee
effects are at work than when they are not. Allee effects lower the risk–release

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Relationship Between Propagule Pressure and Establishment Risk 79
curve at low density, leading to a characteristic sigmoid curve (see Curve B in
Figure 4-1). Allee effects are logically appealing and have been applied to mod-
els of both sexually reproducing and parthenogenetic organisms (Drake, 2004).
However, they have proved elusive to document empirically (see discussion and
examples in Dennis, 2002; Morris and Doak, 2002; Leung et al., 2004; Drake et
al., 2006; Courchamp et al., 2009; Jerde et al., 2009; Kramer and Drake, 2010).
The third feature is the effect of the spatial environment on population den-
sity. One of the greatest challenges in population modeling is that organisms
released into an environment will tend to spread out, through both passive and
active dispersal. Depending on the relative scales of dispersal and population
growth, the effective initial population size may be very much lower than the
original number of organisms released. This reduction in density will presuma-
bly exacerbate the impacts of demographic stochasticity and Allee effects.
Thus, in general it is expected that both individual and population establishment
probabilities to be much lower in the wild than in contained laboratory experi-
ments. (It is possible, of course, that hydrodynamic features or aggregative be-
havior will have the opposite effect, tending to concentrate organisms in a locale
and enhancing their chances of establishment; see Chapter 3).
Since establishment probability seems generally likely to be dominated by
the dynamics of small populations, the most rigorous modeling approach is to
construct models that incorporate (or can phenomenologically reflect the effects
of) demographic stochasticity, Allee effects, and their modification by dispersal,
and to allow the empirical data to indicate on a case-by-case basis the impor-
tance of these additional features. The following examples consider none, some,
or all of these features.
Descriptive Models
Descriptive models, such as regression and similar statistical techniques, of-
fer a phenomenological characterization of the risk–release relationship. That is,
they can be formulated and parameterized without having to understand or spe-
cify the underlying mechanisms by which the independent variables explain the
dependent variable (Drake and Jerde, 2009). As a result, one’s confidence in
their predictive ability is limited. Familiar descriptive models include statistical
models such as regression and functional forms such as species-area curves and
behavioral responses.
These models have been applied to the results of both experimental and his-
torical survey data. In a simple example, Drake and Jerde (2009) fit a spline, or
a series of local regressions, to establishment probability as a function of propa-
gule pressure in the scentless chamomile (Matricaria perforata) (Figure 4-3A).
In this case, establishment was defined as survival simply from seed to flower-
ing, but the same method could be applied to a longer-term study of population
establishment. The same data were also fit with a probabilistic model (see later
section). In a slightly more complex field experiment, Jongejans et al. (2007)

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80 Propagule Pressure and Invasion Risk in Ballast Water
estimated the establishment probability (defined as persistence over six years) of
the European thistle (Carduus acanthoides) as a function of propagule pressure
and native plant biomass. Using a generalized linear model, they found that
establishment probability increased significantly with higher initial seed num-
ber, and tended to increase with reduced native biomass; together these variables
accounted for 37 percent of the variation (Figure 4-3B). Bertolino (2009) mod-
eled the success of global historical squirrel introductions (defined as persistence
to the present day of populations introduced over a >100-year period) as a func-
tion of propagule pressure, environmental matching, native diversity, and the
invaders’ biogeographical origin. For the genus Sciurus (squirrels), a logistic
regression fit to the initial number of individuals alone explained 55 percent of
the variation in establishment probability (Figure 4-3C).
Statistical models can be made increasingly complex by adding ever more
independent variables, and have been used to describe invasion risk over a spa-
tial domain using species distribution (environmental niche) modeling (Peterson
and Vieglais, 2001; Herborg et al., 2007, 2009; Dullinger et al., 2009).
Mechanistic Models
In contrast to descriptive statistical models, mechanistic models represent
invasion establishment as a function of parameters that have a readily defined
biological meaning. Whereas statistical models describe a relationship only
over the range of data to which they are fit, mechanistic models are presumed to
extrapolate well over the entire biologically realistic parameter space. Further-
more, descriptive models allow one to investigate the shape of a relationship,
while mechanistic models force the user to specify the processes driving the
relationship and to link causative variables explicitly.
It is useful to distinguish two general classes of mechanistic models: proba-
bilistic statements and dynamic, demographic models. In their simpler forms,
these two model classes possess different mathematical structures and require
different data to parameterize and validate. In more complex models of popula-
tion establishment, this distinction blurs and a given model may incorporate
elements of both classes (e.g., Jerde and Lewis, 2007; Leung and Mandrak,
2007; Jerde et al., 2009).
Probabilistic Models
In the context of the risk–release relationship, the probabilistic models con-
sidered here are composed of probability statements beginning with the proba-
bility of an individual’s establishment probability and scaling up to a population
level. A probabilistic model is written immediately in terms of its solution,
namely, in terms of population establishment probability. When the model’s
constituent probabilities are represented as frequency distributions rather than as

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Relationship Between Propagule Pressure and Establishment Risk 81
A
B
C
FIGURE 4‐3 Single‐species risk–release relationships obtained from descriptive models.
(A) Spline fit to short‐term establishment probability of scentless chamomile (Matricaria
perforata) (Drake and Jerde, 2009). (B) Generalized linear model fit to establishment
outcome (success or failure) for European thistle (Carduus acanthoides) across three
levels of native plant biomass reduced by clipping (Jongejans et al., 2007). (C) Logistic
regression fit to establishment probability of squirrel Sciurus. Replotted from Bertolino
(2009) with data generously provide by S. Bertolino. SOURCES: (A) Reprinted, with per‐
mission from, Drake and Jerde (2009). © 2009 by Oxford University Press. (B) Re‐
printed, with permission, from Jongejans et al. (2009). © by Springer. (C) Reprinted,
with permission, from Bertolino (2009). © 2009 by John Wiley and Sons.

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82 Propagule Pressure and Invasion Risk in Ballast Water
fixed points, it is known as a hierarchical probability model (HPM). The HPM
approach to representing stochastic events has been extensively applied to medi-
cal, engineering, and ecological problems, and it allows the explicit incorpora-
tion and analysis of uncertainty (Dennis and Lele, 2009; Ponciano et al., 2009).
Parameterizing and validating probabilistic models requires comparatively sim-
ple data: the outcome, as success or failure, of a series of introductions inocu-
lated over a range of initial organism numbers. As will be discussed later, prob-
abilistic models can readily be expanded to represent multiple species and envi-
ronmental conditions that have different associated probabilities of establish-
ment.
Probabilistic models of population establishment have been developed to
serve as the basis for a metapopulation model of biocontrol release (Shea and
Possingham, 2000) and for a gravity model of zebra mussel spread (Leung et al.,
2004). In their simplest, non-spatial form, they contain a sole parameter—the
probability of a single individual producing an established population. This
value is then scaled up to obtain the probability of a group of individuals leading
to an established population (Shea and Possingham, 2000; Leung et al., 2004,
Leung and Mandrak, 2007; Jerde and Lewis, 2007; Jerde et al., 2009). The dif-
fusion approximation to exponential growth shares the same core probability
structure, and has likewise been used to model population establishment (Drake
et al., 2006; Bailey et al., 2009).
The basic construction of a simple probabilistic model is outlined in Box 4-
1, and its application is illustrated in several examples below.
To implement these probabilistic models, studies have examined population
establishment over a range of initial population sizes either from directed expe-
riments or from descriptive population data. Memmott et al. (2005) fit both a
logistic regression and a probabilistic model to the success after six years of
biocontrol insect releases (Arytainilla spartiophila; Figure 4-5A). Drake and
Jerde (2009) used short-term data for the success of the prairie weed scentless
chamomile from seed to flowering (Matricaria perforata; Jerde and Lewis,
2007) to scale up to a population-level establishment model. This model was
subsequently extended by Jerde et al. (2009) to incorporate mate-finding limita-
tions that led to a biologically driven Allee effect, and it was used to predict
invasion risk for Chinese mitten crabs (Eriocheir sinensis) and apple snails
(Pomacea canaliculata) (Jerde et al., 2009; Figure 4-5B). Drake et al. (2006)
used descriptive population growth data of the spiny waterflea Bythotrephes
longimanus in three lakes over four years, and Bailey et al. (2009) conducted
100-day mesocosm studies of a variety of cladocerans, to parameterize diffusion
approximations to exponential growth models (Figure 4-5C,D).
Probabilistic models have also been constructed to investigate the accumu-
lation of nonindigenous species over time (Solow and Costello, 2004; Wonham
and Pachepsky, 2005); this type of model has been extended to examine the rela-
tionship between propagule pressure and invasion risk (Costello et al., 2007; see
later section). Basic probabilistic models can represent the invasion outcome
alone, without necessarily representing the population dynamics leading to

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Relationship Between Propagule Pressure and Establishment Risk 103
BOX 4‐4
Outline of a Simple Multispecies Probabilistic Model
The single‐species probabilistic model developed in Box 4‐1 is readily ex‐
tended to a multispecies probabilistic model. Equation 4‐3 in Box 4‐1 defines
the establishment probability for a single species as pE = 1-e-N. This equation
can be modified to accommodate S species, each with its own establishment
probabilty ps. Following the same general approaches described in Shea and
Possingham (2000) and USCG (2008), the model then describes SE, the ex‐
pected number of species that establish, as
S
SE 1 e i Ni
ci
s 1 (4-8)
and ps, the probability that at least one species establishes, as
s
pS 1 ei Ni (4-9)
ci
i 1
The same principles can be used to extend the model to consider variation
in establishment probability across multiple locations (L), multiple vessels arriv‐
ing in those locations (VL), and so on. In this case, the expected number of spe‐
cies that establish can be written as
VL
S L
SE 1 e s,l ,v Ns,l ,v
cs , l , v
l 1 v 1
s 1 (4-10)
and the probability that at least one species establishes as
VL
S L
pS 1 e s,l ,v Ns,l ,v
cs ,l , v
s 1 l 1 v 1 (4-11)
box continues next page

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104 Propagule Pressure and Invasion Risk in Ballast Water
BOX 4‐4 Continued
These models follow the same logic as the basic one (4‐3), where the
shape parameters and c describe the species establishment probability as a
function of the initial number of individuals N; the subscripts s, l, and v allow
variation among species, locations, and vessels; and the complement of all the
propagules failing to establish gives the final probability of establishment. This
model does not account for variation in parameter values over time or for po‐
tential interactions among species. Nevertheless, parameterizing such a mod‐
el, particularly in a hierarchical structure where each parameter is characte‐
rized as a frequency distribution, would require a tremendous amount of data.
Qualitatively, the overall establishment probability ps obtained from a mul‐
ti‐species model can only be the same as, or greater than, the largest estab‐
lishment probability pE of the constituent species.
To the Committee’s knowledge, this approach has not yet been applied to
predicting the success of multiple nonindigenous species. However, a related
approach using a reaction-diffusion model, which is a standard spatial extension
of a demographic model, offers interesting insights. Reaction-Diffusion (R-D)
models represent a class of models that were originally developed to model the
spread of organisms in continuous time and space (Skellam, 1951). These mod-
els were later developed to model the spread of invading organisms across a
one- or two-dimensional landscape with the goal of defining the rate of spread
as a travelling wave and so provide a description of the rate of spread and the
area occupied by the invasion (Okubo et al., 1989). The classical version of
these models typically involves several restrictive assumptions including spatial
homogeneity and random movement (at least at the population level). However,
they are comparatively easy to parameterize, requiring only estimates of per
capita rate of population increase and the mean squared displacement per unit
time of individuals in the population. Among the advantages of this approach
include the ability to approximate the spread of the invading population as a
travelling wave (Okubo et al., 1989). This permits the estimation of the rate of
spread as a linear function of time, so that the arrival of an invader at a new site
could be reasonably estimated under the model assumptions.
These models typically do not provide an estimate of the rate of establish-
ment, although recent applications have attempted this for multiple species using
some simplifying assumptions (Drake et al., 2005). In Drake et al. (2005), the
authors combined a generic exponential-growth reaction-diffusion model with
an allometric relationship between body size and population growth rate to ex-

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Relationship Between Propagule Pressure and Establishment Risk 105
amine the establishment probability of a variety of aquatic species. Their goal
was not to predict establishment probabilities of any particular species, but to
predict invasion rates over a range of species. Under a series of assumptions,
particularly concerning Allee dynamics, this model estimates the risk–release
relationship in terms of the proportion of species of a given body size that estab-
lish vs. the volume of water released. From this output, the chance of a single
invasion by a size class of organism can be predicted as a function of the num-
ber and volume of releases, independent of the number of individual organisms.
This proposed approach is specific to a size class, and has not yet been vali-
dated, but seems reasonable.
Obtaining a Discharge Standard from Multiple-Species Models
Multi-species models represent an attempt to capture the complexity of
wholesale ballast water release. Descriptive and mechanistic models can readily
be formulated in conceptual and mathematical terms at this scale. In the current
absence of data for parameterizing and validating mechanistic models, descrip-
tive statistical models can be developed. However, these must be interpreted
with caution given the uncertainty in the estimates, and disconnect between the
scales of the independent and dependent variables. The use of proxy variables
introduces a further challenge: although well-fitting models may be obtained,
proxy variables (such as ballast volume, shipping tonnage, vessel abundance)
must be evaluated for their relationship to the direct variable of interest (dis-
charge density) before ascribing any mechanistic meaning to their relationships.
To summarize, a multi-species approach focuses on the assemblage of spe-
cies released from ballast water. Because of the associated data requirements,
descriptive models are more likely than mechanistic models to yield estimates of
the risk–release relationship at this scale. Even so, given the uncertainty and
mismatch in both the independent and dependent variables, the applicability of
any apparent relationships is questionable. Relative to the single-species ap-
proach, a multi-species approach has the advantage of being conceptually more
realistic in the context of ballast water release, and the disadvantage of being
more complex and more difficult to ground in the relevant empirical data.
CONCLUSIONS AND RECOMMENDATIONS
Models are generally useful in environmental management because they
provide a transparent framework, force an explicit statement of assumptions,
allow us to predict and compare future projections under different management
scenarios, and can be updated in their structure and parameter estimates as new
information emerges. In principle, a well-supported model of the relationship
between invasion risk and organism release could be used to inform a ballast
water discharge standard. For a given discharge standard, the corresponding

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106 Propagule Pressure and Invasion Risk in Ballast Water
invasion risk could be predicted, or, for a given target invasion risk, the corres-
ponding target release level could be obtained.
Candidate risk–release models developed to date include single- and mul-
tiple-species scales, and extend along the spectrum from descriptive to mecha-
nistic in their construction. Mechanistic single-species models require fewer
data to parameterize than do mechanistic multi-species models, but do not
represent the more realistic scenario of ballast discharge of an assemblage of
species. Descriptive single-species models are simpler, but offer none of the
predictive advantages of mechanistic ones. Descriptive multi-species models
are an appealing tool for investigating large correlative datasets, but are ham-
pered by a current lack of appropriate data.
The rigorous use of models requires that multiple candidate models be for-
mulated and compared in their ability to represent the data. This approach is
well established in the population dynamic literature at the single-species scale.
However, currently there are insufficient data to distinguish among risk–release
relationship models at the multi-species scale. The following conclusions and
recommendations identify how models might be put to use at present, and in the
future, to help inform a discharge standard.
Ballast water discharge standards should be based on models, and be
explicitly expressed in an adaptive framework to allow the models to be
updated in the future with new information. Before being applied, it is es-
sential that candidate models be tested and compared, and their compounded
uncertainty be explicitly analyzed. Only a handful of quantitative analyses of
invasion risk–release relationships thus far have tested multiple models and
quantified uncertainty.
The predicted shape of the risk–release relationship is non-linear. In-
flection points and slope-balance points could provide natural breakpoints for
informing a discharge standard. However, the apparent shape of the relation-
ship for a given system will depend on the quantity, error, and parameter range
of the empirical data, as well as the biology of the species and the nature of the
environment.
In the short term, mechanistic single-species models are recommended
to examine risk–release relationships for best-case (for invasion) scenario
species. This approach makes sense biologically because in general concerns
are only about the small subset of released species that establish as high-impact
invaders. Such an approach to setting a standard is conservative and would pro-
vide maximum safety against invader establishment.
Candidate best-case-scenario species should be those with life histories that
would favor establishment with the smallest inoculum density. Species with the
highest probability of establishment relative to inoculum density will have the
greatest influence in determining the shape of the risk–release curve. Life histo-
ry traits promoting such sensitivity to small inoculum density possibly include

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Relationship Between Propagule Pressure and Establishment Risk 107
fast-growth, parthenogenetic or other asexual reproductive abilities, lecitho-
trophic larvae, etc. Other considerations of best-case species might include
those that have a high ecological or economic impact, or are frequently intro-
duced. The greatest challenge in this approach will be converting the re-
sults of small-scale studies to an operational discharge standard.
Developing a mechanistic multi-species model of risk and release, parame-
terized for an assemblage of best-case scenario species, would only be recom-
mended over the longer term. This model would allow a detailed theoretical
investigation of the relationship between total organism number and invasion
risk, by permitting the analysis of the risk associated with different species rich-
ness and frequency distributions that sum to the same total organism number.
The challenges in this approach include the massive time and effort needed to
gather the necessary data as well as converting model results to a fleet-wide dis-
charge standard. The implications of these models would therefore be highly
specific, no more (and possibly less) informative than those of single-species
models, and will require more data and computational effort to construct, para-
meterize, and validate.
Developing a robust statistical model of the risk–release relationship is
recommended. It is unclear whether the current lack of a clear pattern across
ports reflects a true absence of pattern, or the absence of appropriate data to test
this model. Nevertheless, given spatial variation in shipping patterns and envi-
ronmental variables, it is anticipated that this approach will be more fruitful at a
local scale than a nation-wide scale.
Within a region, this relationship should be estimated across multiple time
intervals, rather than from a single point. The effect of temporal bin sizes on the
shape of the relationship must be examined. The choice of independent variable
must be carefully considered. Since long-term historical data on ballast-
organism density do not exist, the committee recommends an extremely careful
analysis and validation of any proxy variables. The greatest challenge in this
approach is the currently insufficient scope and scale of the data. There is no
evidence that any proxy variable used thus far is a reliable stand-in for or-
ganism density.
Finally, models of any kind are only as informative as their input data.
In the case of ballast water, both invasion risk and organism density dis-
charged from ballast water are characterized by considerable and largely
unquantified, uncertainty. At the multi-species scale in particular, the existing
data (historical invasion records vs. recent ballast surveys) are substantially
mismatched in time, and patchy in time, space, and taxonomy; current statistical
relationships with these or proxy variables are of dubious value. The judicious
use of an appropriate model combined with robust data may help inform stan-
dard setting in the future.

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108 Propagule Pressure and Invasion Risk in Ballast Water
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