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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 RISKâRELEASE 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  72Â
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 riskâ release relationship. Understanding the riskârelease 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
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 riskâ 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 RiskâRelease 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 riskârelease 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 4â1  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.Â
Relationship Between Propagule Pressure and Establishment Risk  75     A B  FIGURE 4â2  Conceptual application of a riskârelease relationship to inform ballast water organism discharge standards.  S is the observed risk of ballastâmediated species invaâ 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 riskârelease relationâ 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 riskârelease relationship, a given reduction in the release rate is predicted to give the same proporâ tional reduction in invasion rate (i.e., RN=RS).  (B) Under the assumption of a sigmoid relationship, the same reduction in release (RN1=1â[N*1/N1] = RN2=1â[N*2/N2]) is preâ dicted to give a much lesser (RS1 =1â[S*1/S1]) or a much greater (RS2 =1â[S*2/S2]) reducâ 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 multiâspecies linear model proposed by Cohen (2005, 2010) and Reusser (2010), where the yâaxis is the inâ vasion risk (characterized as invasion rate, the number of species established per unit time) and the xâaxis 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 straightâline relationship.  Cohen (2005, 2010) defined as the reduction factor the equal proportional reductions in release and risk from current to target levels.  Â
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 riskârelease 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 shapesâ the hyperbolic and sigmoid curvesâcan represent the overall riskârelease 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 riskârelease 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 riskârelease 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 riskârelease 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 riskârelease relationship. The following section reviews a range of approaches that have been taken to fitting riskârelease curves to empirical data. Modeling the RiskâRelease Relationship An ideal analysis of the riskârelease 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 riskârelease relationship and (2) how strong this relationship is relative to other potential explanatory variables. Several approaches have been taken to modeling the riskârelease 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 categoriesâa given model may include both mechanistic and descriptive componentsâbut 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.
Relationship Between Propagule Pressure and Establishment Risk  77  TABLE 4â1  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 DâAntonio ï¦yï¶ Jongejans et al. (2003)Â ï½ b0 ï« b1 x  (2007) ln ï§ ï¨1ï yï· Ricciardi (2006)Â ï¸ Bertolino (2009) Cohen (2005, 2010)* Functional form  Drake and Jerde Reusser (2010)* Speciesâarea curve: (2009) z y=cx  Mechanistic  Probabilistic Shea and Possingâ Drake et al. (2005)* ïï¡ N pE ï½ 1 ï e  ham (2000) Costello et al. (2007)* Leung et al. (2004) USCG (2008)* Dynamic demographic Memmott et al. See text dN (2005)Â ï½ 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 impleâ 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. Â
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 riskârelease relationship for ballast water. SINGLE-SPECIES MODELS It is informative to examine the riskârelease 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 dynamicsâdemographic stochasticity, positive density dependence, and the spatial environmentâand must be considered in developing an effective riskârelease 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 riskârelease 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 riskârelease
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 riskârelease 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, oneâs 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)
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 invadersâ 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 riskârelease relationship, the probabilistic models con- sidered here are composed of probability statements beginning with the proba- bility of an individualâs 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 modelâs constituent probabilities are represented as frequency distributions rather than as
Relationship Between Propagule Pressure and Establishment Risk  81  A B C  FIGURE 4â3  Singleâspecies riskârelease relationships obtained from descriptive models.  (A) Spline fit to shortâterm 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 perâ mission from, Drake and Jerde (2009). © 2009 by Oxford University Press.  (B) Reâ printed, with permission, from Jongejans et al. (2009). © by Springer.  (C) Reprinted, with permission, from Bertolino (2009). © 2009 by John Wiley and Sons.  Â
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 parameterâthe 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
Relationship Between Propagule Pressure and Establishment Risk  83  success or failure. In contrast, dynamic demographic models directly represent population dynamics in order to predict invasion outcomes. Dynamic Demographic Models Dynamic models of population growth are written as a system of one or more discrete or differential equations whose solution gives the population size at a given time. Familiar examples include exponential (geometric) and logistic growth, and Leslie and Lefkovitch matrix models. The parameters in these models represent the demographic rates or probabilities of birth, death, growth, and reproduction. Dynamic demographic models do not predict establishment probability directly. Instead, a number of stochastic simulations must be gener- ated for each initial population size. A subsequent model, either descriptive or probabilistic, must then be fit to the outcome of the simulations. Parameterizing and validating this kind of stochastic dynamic model requires estimating the distribution of each demographic parameter (e.g., birth rate, death rate). In their simpler forms, demographic models require more data to parameterize and vali- date than do probabilistic models. Dynamic demographic models serve as the basis for the population viability analysis (PVA) of declining species (Morris and Doak, 2002), an approach that has been applied more recently to the analysis of establishing invaders (Bartell and Nair, 2004; Neubert and Parker, 2004; Andersen, 2005), and that has been proposed for use in comparing ballast water discharge standards (USCG, 2008). It is useful to recall that the goals and outcome of population viability analysis are not the exclusive domain of demographic models. Although the term PVA typically refers to the analysis of these models, any modeled population may be subjected to an analysis of its viability; indeed, such analysis is inherent in the construction of a probabilistic establishment model. Traditionally, PVA in- volves dynamic demographic models that are matrix-based and use age- or stage-specific dynamic rates to estimate population growth and hence viability. These models use dynamic information for each stage including growth, surviv- al, and reproduction to estimate population growth rate (Caswell, 1989). The most basic dynamic demographic models are count-based and assume no variation among individuals in the population. These count-based models make several assumptions including that the mean and variance of population growth remain constant, no density dependence, dynamic stochasticity, etc. (Morris and Doak, 2002). However, more complex count-based models can incorporate positive and negative density dependence, Allee effects, stochastici- ty, spatial structure, etc. The more realistic dynamic demographic models expli- citly use different dynamic parameters for each age or stage class in the popula- tion (see example from Kramer and Drake, 2010, below). Although various kinds of stochasticity and autocorrelation in vital rates can be includedthrough simulation methods, these models have their own restrictive assumptions of time invariance, stable age or stage distribution, etc. (Morris and Doak, 2002).
84  Propagule Pressure and Invasion Risk in Ballast Water   BOX 4â1 Probabilistic Model Framework  This box illustrates the development of the simple probabilistic model in Leung et al. (2004) and shows how it provides insight into the overall shape of the riskârelease relationship.  It begins with a simple probability statement in which N is the number of propagules released and p is the individual establishâ ment probability of each propagule.  In this case, 1âp is the probability of a sinâ gle propagule failing to establish, and (1âp)N is the cumulative probability of all N individuals failing to establish.  The probability of the population establishing, pE, is therefore   pE = 1â(1âp)N (4â1)  It is mathematically convenient to define an additional parameter, ï¡Â = âln(1âp), which allows us to rewrite (4â1) synonymously as  pE ï½ 1 ï eïï¡ N  (4â2)  For this equation (4â1, 4â2), the shape of the riskârelease relationship between pE and N is hyperbolic, asymptoting towards pE = 1 (Figure 4â4a).  This must be the case because even if the individual establishment probability (p) is low, the total probability pE increases inexorably to one as more and more individuals are released (Leung et al., 2004). This model can be extended to incorporate the negative density depenâ dence, or Allee effects, that may be expected to reduce pE at low values of N (Leung et al., 2004; Jerde et al., 2009).  The resulting sigmoid shape (Figure 4â 4b) can be produced by adding the shape coefficient c > 1 to equation 4â2 (which follows the cumulative Weibull distribution), giving  c pE ï½ 1 ï e ï ï¡ N    (4â3)  Equation (4â2) is the special case of (4â3) where c = 1 and there is no strong Allee effect (Leung et al., 2004).  Thus, the biological meaning of c can be interâ preted as an Allee parameter.  However, it should be noted that if Allee effects are to be considered, both ï¡Â and c need to be fit simultaneously to describe a biological system. Â
Relationship Between Propagule Pressure and Establishment Risk  85     These equations illustrate that simple probability statements combined with basic principles of population growth reveal two candidate shapes for the overall pE vs. N curve: hyperbolic or sigmoid.  However, the shape of the curve for a given empirical dataset over a limited parameter space may appear linear, particularly for high (both curves) and low (sigmoid curve) values of N.  Short sections of the sigmoid curve could also appear exponential (leftâhand side of curve) or hyperbolic (rightâhand side).   (A)   (B) FIGURE 4â4  Predicted relationship between pE and N given by equation 4â3.  (A) Hyperbolic shape with no Allee effects (c = 1).  (B) Sigmoid shape characteristic of an Allee effect (c > 1).  In both panels, the upper, middle, and lower curves are for p â ï¡Â = 0.01, 0.005, and 0.001.
86  Propagule Pressure and Invasion Risk in Ballast Water  A BÂ
Relationship Between Propagule Pressure and Establishment Risk  87  1.0 0.8 Establishment probability 0.6 0.4 0.2 0.0 1 3 5 7 9 11 Initial number C D FIGURE 4â5  Continued  Singleâspecies riskârelease relationships predicted by probabilistic models for the establishment probability of (A) the psyllid Arytaiâ nilla spartiophila (Memmott et al., 2005), (B) Chinese mitten crab (Eriocheir sinensis) (Jerde et al., 2009), (C) spiny waterflea (Bythotrephes longimanus) with demographic (open dots) and environmental (solid dots) stochasticity (reâ drawn from Drake et al., 2006), as a function of the initial number of organâ isms, and (D) three cladocerans (Bosmina spp., circles; Bosmina coregoni, trianâ gles; Daphnia retrocurva, squares) as a function of the initial organism density (Bailey et al., 2009).  SOURCES:  (A) Reprinted, with permission, from Memmott et al. (2005).  © 2005 by John Wiley and Sons.  (B) Reprinted, with permission, from Jerde et al. (2009).  © 2009 by The University of Chicago Press. (C) Reâ printed, with permission, from Drake et al. (2006).  © 2006 by Springer.  (D) Reprinted, with permission, from Bailey et al. (2009).  © 2009 by NRC Research Press.Â
88  Propagule Pressure and Invasion Risk in Ballast Water  Dynamic demographic models have been developed for a tremendous varie- ty of plant and animal species. The following examples illustrate their applica- tion to predicting the establishment probability of introduced or re-introduced species. Wood et al. (2007) used life history data to parameterize an individual- based simulation model of tree squirrel re-introductions, and predicted the pro- portion of populations above a threshold abundance (Figure 4-6A). Other mod- els represent the establishment of age- or stage-structured populations (Parker, 2000; Barry and Levings, 2002; Kramer and Drake, 2010) (Figure 4-6B). Kra- mer and Drake (2010) used experimental laboratory results to parameterize a demographic model of the cladoceran Daphnia magna, and found that increas- ing predation shifted the riskârelease relationship from a hyperbolic to the sig- moid shape characteristic of Allee effects (Box 4-2). Once a group of organisms is released, they will disperse through advection and locomotion. These may lead to a net aggregation or dispersal. The effects of dispersal on population establishment have been explored in considerable detail by extending demographic models to a reactionâdiffusion framework and its extensions (Skellam, 1951; Shigesada and Kawasaki, 1997; Lubina and Le- vin, 1988; Neubert and Parker, 2004; Lewis et al., 2005; Hastings et al., 2005). These models have been used to explore the persistence and spread of aquatic and marine species (Drake et al., 2005; Pachepsky et al., 2005; Lutscher et al., 2007, 2010; Dunstan and Bax, 2007; reviewed for marine invasions by Wonham and Lewis, 2009), but in general have not been used to predict riskârelease rela- tionships. For the application of a reactionâdiffusion model to a multi-species scenario, see Drake et al. (2005). Obtaining a Discharge Standard from Single Species Models A single-species model of the riskârelease relationship could provide in- sight into discharge standards in two main ways: to illustrate a best-case scena- rio, and to serve as a building block for multi-species models. To illustrate a best-case scenario, a model could be constructed and parameterized for fast- growing, high-impact, or commonly released species. Invasion risk could then be predicted under the assumption that all ballasted organisms belonged to this species, and were released under optimal conditions. This approach would lead to a conservative discharge standard. The greatest difficulty in developing a discharge standard from a single- species model is that these models are generally constructed to represent a one- time introduction of a known initial number of individuals. However, ballast water discharge is a repeated event, which will tend to increase invasion risk, and the organisms may rapidly be redistributed in the physical environment, which may immediately alter the effective initial number of individuals with the
Relationship Between Propagule Pressure and Establishment Risk  89  A B FIGURE 4â6  Singleâspecies riskârelease relationships obtained from dynamic demoâ graphic simulation models.  (A) Predicted proportion of populations exceeding 20 indiâ viduals after 100 years from a population growth model of tree squirrels (Wood et al., 2007).  (B) Predicted establishment probability from a population growth model of the copepod Pseudodiaptomus marinus (plotted from data in Barry and Levings, 2002).  (A) Reprinted, with permission, from Wood et al. (2007).  © 2007 by American Society of Mammalogists.  Â
90  Propagule Pressure and Invasion Risk in Ballast Water  BOX 4â2 Dynamic Demographic Model Framework  This box illustrates the development of a dynamic demographic population model, following that formulated by Kramer and Drake (2010).  The model framework begins with the standard continuous time equation for a homogeâ neous population of size N growing as a function of the difference between the birth rate (ï¢) and the death rate (ï). The population growth rate is given as  dN ï½ ï¢N ï ïN (4-4) dt To examine the effects of predation, an additional mortality function, g(N), was added to represent a standard predation type II functional response, such that: Pï¡N g( N ) ï½ (4-5) (1 ï« ï¡Th N ) where P is the number of predators, ï¡ is the attack rate, and Th is the handling time, giving the population growth rate:  dN ï½ ï¢N ï ïN ï g ( N ) (4-6) dt  The model was extended to represent two size classes, juveniles (J) and adults (A), where the juveniles are produced by adults at rate ï¢ and mature to adults at rate ï¤, and predation is a function of total population size:   dJ ï½ ï¢A ï ïJ ï g ( J ï« A) ï ï¤J dt (4-7) dA ï½ ï¤J ï ïA ï g ( J ï« A) dt This model was parameterized from laboratory experiments with the claâ doceran Daphnia magna and a nonâvisual ambush predator, larvae of the midge Chaoborus trivittatus.  Stochastic model simulations showed that predaâ tion induced a sigmoid riskârelease relationship, compared to the hyperbolic curve predicted in the absence of predation.  In other words, predation inâ duced an Allee effect in this system (Figure 4â7).          box continues next page Â
Relationship Between Propagule Pressure and Establishment Risk  91  BOX 4â2 Continued  FIGURE 4â7  Predicted singleâspecies riskârelease relationship obtained from simulations of a dynamic demographic model.  Theoretical (dashed line) and simulated (points with fitted solidâline spline) predictions of establishment probability for a Daphnia magna population as a function of initial populations size under different predation levels.  SOURCE: Adapted from Kramer and Drake (2010); courtesy J. Drake.  Reprinted, with permission, from Drake (2010).  © 2010 by John Wiley and Sons.  potential to establish. The closer together small releases occur in space and time, the more they will approximate a single large release with a correspon- dingly higher establishment probability (analogous to the rescue effect in meta- population dynamics, Gotelli, 1991). Theoretical studies have demonstrated that, due to environmental stochasticity, the likelihood of success of multiple arrivals at a single entry point is higher than that for simultaneous arrivals at multiple sites (Haccou and Iwasa, 1996). In a homogeneous environment, or- ganisms will disperse and the effective initial population size will rapidly de- crease; an advective environment and intraspecific behavior may either enhance
92 Propagule Pressure and Invasion Risk in Ballast Water  or counter this effect. The mathematical framework of a single-species model could readily be modified to analyze the risk associated with multiple repeated inocula that are dispersed or concentrated in the local environment and to deter- mine an adjusted discharge target. In summary, both descriptive and mechanistic models have been developed to examine the riskârelease relationship for single species. This relationship can reasonably easily be defined and parameterized for a one-time release under controlled laboratory or field conditions. It is somewhat more difficult to define and parameterize a model that would represent repeated releases in an advective environment, making it challenging to scale up to a discharge standard. The models could be useful either for setting a discharge standard based on a best- case species, or for developing modeling frameworks that would help inform multi-species scenarios. MULTI-SPECIES APPROACHES Broad-scale vectors like ballast water (Carlton and Geller, 1993; Smith et al., 1999), shipping containers (Suarez et al., 2005), or commercial imports (Copp et al., 2007; Dehnen-Schmutz et al., 2007) repeatedly release assemblages of tens to hundreds of species into the environment, of which only a small subset establish successfully. To model the riskârelease relationship at this scale re- quires both risk and release data spanning the same large spatial, temporal, and taxonomic scales. At present, however, there are only loosely corresponding empirical estimates of risk and release (see Table 4-2). Before proceeding with multi-species examples, this section considers the nature of the available data for both invasion risk and organism release, and the resulting constraints on model construction and interpretation. For invasion risk, there are historical records of the invaders that have ac- cumulated in various ports over the past decades (see Table 4-2). These inva- sion records are characterized by considerable uncertainty stemming from in- complete collections, the cryptogenic nature of many species, the taxonomic bias of field samples, and the uncertainty associated with ascribing a given inva- sion to ballast transport over other candidate vectors (Chapman and Carlton, 1991; Ruiz et al., 2000; Costello et al., 2007; Fitzpatrick et al., 2009; Jerde and Bossenbroek, 2009). These are standard sampling difficulties that plague any assessment of nonindigenous species, and are not unique to the problem of bal- last water management. The consequences, however, are that the best empirical estimates of invasion rate and risk are nevertheless incomplete and uncertain. Furthermore and more crucially, there are no good estimates of the scale of that uncertainty. For organism release from ballast water, there are snapshot surveys of particular size classes of organisms collected from a subset of tanks on a subset of ships arriving in selected locations over brief and recent time periods, identi- fied to the lowest taxonomic level possible which nevertheless is often well
Relationship Between Propagule Pressure and Establishment Risk  93  above that of species (see Table 4-2). As discussed in Chapter 1, organism iden- tities and densities vary within ballast tanks, vessels, and routes (Lavoie et al., 1999; Smith et al., 1999; Wonham et al., 2001; Verling et al., 2005; Lawrence and Cordell, 2010). As a result, the best empirical estimates of organism release from ballast water are recent, local, taxonomically variable subsamples of the process. Again, the degree of uncertainty is not well characterized. Thus, two difficulties emerge in parameterizing the multi-species riskâ release relationship from empirical data. First, neither the dependent nor the independent variable is well resolved. Second, there is a spatial and temporal mismatch between the dependent and independent variables, in that invasion risk is estimated from the outcome of a cumulative century-long historic process, whereas organism release is estimated over a very short time period of months to years (as is evident in Table 4-2). As a result, our ability to rigorously explore the riskârelease relationship at the multiple species scale with existing data is greatly limited. Despite the empirical difficulties, both descriptive and mechanistic model- ing frameworks have been developed for the multi-species riskârelease relation- ship, and to some extent parameterized. One response to the absence of robust release data has been to use proxy variables in place of direct measures of prop- agule pressure. The merits of this strategy are discussed in some detail below, using the examples that follow. It should be noted that additional theoretical probability and demographic models of species assemblages have developed in the context of island biogeo- graphy (MacArthur and Wilson, 1967), localâregional species richness patterns (e.g., Shurin et al., 2000), community assembly (Case, 1990, 1995), and meta- communities (Holyoak et al., 2005). All of these approaches explore the riskâ release relationship in its broadest sense. However, since they do not directly address the question of invasion risk vs. organism density, they are unlikely to provide major insight into the question of ballast water standard setting and thus are not reviewed further. Descriptive Models As for the single-species scale, statistical models of the multi-species riskâ release relationship offer a phenomenological description of a pattern without requiring that the underlying mechanisms be specified. The majority of these studies, recognizing the difficulty of measuring propagule pressure directly, have measured a proxy variable of human activity ranging from population to transport to economic indices. Some statistical analyses of large-scale invasion vectors have focused on a single transport or economic variable as a substitute for propagule pressure (e.g., Levine and D'Antonio, 2003; Taylor and Irwin, 2004; Ricciardi, 2006; Costello et al., 2007; see examples in Figure 4-8). Others have used multivariable
94 Propagule Pressure and Invasion Risk in Ballast Water  TABLE 4â2  Spatial and Temporal Scale of Historical Invasion Records, and Spatial, Temâ poral, and Sampling Scale of Ballasted Organism Surveys in Inland and Coastal Waters of the U.S. and Canada  Invasion Records Ballast Surveys Sample type Location Decades Sources Years (N) Sources (mesh size) Laurentian 1840sâ2000s Ricciardi 1990â91 water (41 Locke et al. Great Lakes (2006) (86) µm, 110 µm) (1993), Subba Rao and St. Lawâ   et al. (1994);  rence Seaway    2000â02 water  Bailey et al.  (39)  (unfiltered) (2005), Duggan et  sediment al. (2005) Chesapeake 1600â2000s Fofonoff et 1993â94 water Smith et al. (1999)  Bay, MD al. (2009) (60)  (80 µm)      1996â97 (7) water  Lavoie et al. (80 µm) (1999) San Francisco 1850sâ1990s Cohen and Bay, CA Carlton â â â (1998) Humboldt Bay, 1920sâ1990s Wonham CA and Carlton â â â (2005) Coos Bay, OR 1940sâ1990s Ruiz et al. 1986â91 water Carlton and Geller  (2000), (159) (80 µm) (1993) Wonham and Carlton (2005) Willapa Bay, 1930sâ1990s Wonham WA and Carlton â â â (2005) Puget Sound, 1900sâ1990s Ruiz et al. 2001â07 water Cordell et al.  WA (2000), (372) (73 µm) (2009), Lawrence Wonham and Cordell and Carlton (2010) (2005) Vancouver, BC  1900sâ1990s Wonham 2000 (15) water Levings et al. (and  regional and Carlton  (80 µm) (2004) waters) (2005)    2007â08 water  Klein et al. (2010)   (23) (unfiltered) Prince          1800â1990s Hines and 1998â1999 water Hines and Ruiz William Sound, Ruiz (2000) (80 µm) (2000) AK Note: AK, Alaska; BC, British Columbia; CA, California; MD, Maryland; OR, Oregon; WA, Washingâ ton. Ballast surveys include studies of N>5 vessels or voyages.Â
Relationship Between Propagule Pressure and Establishment Risk  95  analyses to tease out the relative importance of propagule pressure, again usually by proxy, among other factors contributing to invasion success (ballast water, Drake and Lodge, 2004; plants, Lonsdale, 1999, Dehnen-Schmutz et al., 2007, Castro and Jaksic, 2008, Dawson et al., 2009; earthworms, Cameron and Bayne, 2009; vertebrates, Jeschke and Strayer, 2006; birds, Chiron et al., 2009; fish, Copp et al., 2010) (see Figure 4-8A-C). A variety of linear and non-linear rela- tionships have emerged from these analyses. However, even in the case of a strong statistical relationship, the question of causation must be examined care- fully to minimize spurious significant effects caused by confounding variables (Lonsdale, 1999; Figure 4-8D-E) and to distinguish observed patterns from null expectations (Lockwood et al., 2009). For the case of ballast water, linear riskârelease relationships have been es- timated in a number of systems (Box 4-3). There are both theoretical assump- tions and logistical challenges in developing these models. The first assumption is that total organism number, regardless of the number or abundance of the con- stituent species, is a reasonable predictor of the number of successfully estab- lishing species. Although these two variables do not have an explicit causal connection, it is intuitively clear that increasing total abundance requires in- creasing species number, abundance, or both, any of which would be predicted to increase invasion risk. However, the causation is indirect and the precise na- ture of this relationship is unclear. The second assumption is that this relationship is linear. Although theory predicts that this relationship should be non-linear, and would be expected to be sigmoidal (as in the single-species case) if Allee effects were operative for spe- cies with the highest likelihood of establishment, the trend in a limited dataset may be indistinguishable from linear. Therefore, statistical model fitting should compare multiple candidate models before selecting a linear (or any other) shape. In fitting such a model to data, the operational challenges quickly become clear. We have estimates of organism density and number of invaders for only a handful of locations (Table 4-2). The density measures have been made with different methods and taxonomic foci, are recent and short-term relative to the accumulation of invaders over decades of ballast water release (Table 4-2), and are patchy and possessed of considerable uncertainty (Verling et al., 2005; Min- ton et al., 2005; Lawrence and Cordell, 2010). Even if density estimates were entirely accurate and precise, we would not necessarily expect current estimates to predict historical invasion success. In the face of these difficulties, some authors have used shipping metrics such as vessel number, vessel tonnage, and ballast volume as proxies for propa- gule pressure (Box 4-3). At first glance, proxy variables appear to offer an ap- pealing way to proceed, since unlike organism density data, vessel traffic data are relatively easy to collect, can be collected retroactively, and might seem to be plausible stand-ins for organism density. However, their use relies on the
96 Propagule Pressure and Invasion Risk in Ballast Water  AâC DâE  FIGURE 4â8  Descriptive models of multiâspecies riskârelease relationships.  Comparison of logâlog (dotted), logâlinear (dashed), and MichaelisâMenten (solâ id) equations fit to number of (A) mollusks, (B) plant pathogens, and (C) insects, vs. cumulative imports over time in the U.S. (Levine and DâAntonio, 2003).  (D) Logâlog plot of number of visitors vs. number of native plant species in nature reserves worldwide; (E) logâlinear plot of number of nonindigenous plant speâ cies vs. visitor residuals (nonânative plant species as a function of the residuals from the relationship in D) (p<0.001) (Lonsdale, 1999).  SOURCE:  (E) Reprinted, with permission, from Lonsdale (1999).  © 1999 by Ecological Society of Ameriâ ca. Â
Relationship Between Propagule Pressure and Establishment Risk  97  BOX 4â3 Linear Statistical Multispecies Models  Linear models have been used to estimate the multiâspecies riskârelease relationship for ballast water.  To date, the most widely analyzed data at this scale are invasion trends in the Great Lakes.  A variety of analyses have been conducted for this system using different data subsets, and different depenâ dent and independent variables.  In all the analyses, the data have been parsed into temporal intervals to provide multiple data points for model fitting.  The results are not consistent among analyses. Depending on the data subset, there may or may not appear to be a signifiâ cant riskârelease relationship.  Ricciardi (2001) used a linear regression to estiâ mate the rate of all species invasion vs. shipping tonnage in net tons, by decade from 1900 to 1999 (y = 0.062x, r2 adj = 0.62, p<0.004).  This analysis was upâ dated by Ricciardi (2006) for only those freeâliving invaders assumed to have been introduced by shipping (y = 0.05x; r2 = 0.516, p<0.019; Figure 4â9A).  In contrast, Grigorovich et al. (2003) analyzed Great Lakes invasion data in 5âyear intervals from 1959â1999; their data show no clear trend in new invaders as a function of the net tonnage of overseas ballasted traffic, and if anything a negâ ative relationship with the number of overseas ballasted vessels (Figure 4â9Bâ C). Linear relationships have been used to estimate a perâship invasion rate.  Drake and Lodge (2004) reanalyzed the data in Ricciardi (2001) against shipping tonnage in metric tons, using a linear regression with a Poisson error distribuâ tion (y = 8.47 x 10â8x; p<0.0001).  Rescaling by the average ship tonnage, they estimated a perâship probability of causing an invasion as 0.00044 (95% CI = 0.00008), equivalent to 1 species per 2275 ships or 0.44 invasions per 1000 ships (95% CL 0.36, 0.52).  This estimate was based on all nonindigenous speâ cies in the Great Lakes, regardless of their presumed vector.  In contrast, Cosâ tello et al. (2007) used annual data on shipâmediated animal introductions alone from 1959â2000, as a function of number of ships (Figure 4â9D), and obâ tained a maximum likelihood estimate of 0.14 animal invasions per 1000 ships (95% CL 0.02, 5.2). Linear relationships have also been used to estimate a perâorganism invaâ sion rate for 17 North American ports (Reusser, 2010).  These data were not separated into time intervals, so the relationship for each port was based on a single data point.  The dependent variable was the total number of established invaders (invertebrates and macroalgae) from 1981â2006 considered to likely to have been introduced by ballast water.  The independent variable was the total volume of foreign ballast water discharged from 2005â2007, multiplied on a perâship basis by a random selection from an empirically determined zoopâ lankton density distribution that spanned eight orders of magnitude (based on box continues next pageÂ
98 C A  BOX 4â3 Continued B D FIGURE 4â9  Invasion patterns in the Great Lakes.  (A) The number of invaders scales positively with net shipping tonnage by decade 1900â1999 (Ricciardi, 2006).  The same trend is not evident from scatter plots of (B) number of invaders at 5âyear intervals from 1959â1999 vs. net tonnage or (C) vs. number of ballasted ships (replotted from Grigorovich et al., 2003) or of (D) annual number of invaders vs. number of ships (plotted from data in Costello et al. 2007, Propagule Pressure and Invasion Risk in Ballast Water appendices AâB).  SOURCE: (A) Reprinted, with permission, from Ricciardi (2006).  © 2006 by John Wiley and Sons.  (B, C) Reprinted, with permission, from Grigorovich et al. (2003).  © 2003 by NRC Research Press.  (D) Reprinted, with permission, from Costello et al. 2007.  © 2007 by Ecological Society of Amerâ
Relationship Between Propagule Pressure and Establishment Risk  99     354 ships sampled in four U.S. ports, of which three are included in the 17 anaâ lyzed; Minton et al., 2005).  Repeated random draws generated a bootstrapped estimate of the median and the first and third quantile invasion rates for each port.  Together, these perâcapita invasion rates spanned four orders of magniâ tude from 10â11 to 10â8, or one invasion for every 10 million to 10 billion organâ isms (Reusser, 2010).  Interestingly, the data provided no evidence of a strong riskârelease relationship across ports, based on either number of vessels or ballast water volume (Figure 4â10AâB).  A B  FIGURE 4â10  Across 17 U.S. coastal ports, the number of invaders reported from 1981â 2006 shows no strong relationship with (A) number of ships with foreign ballast 2005â 2007 or (B) volume of ballast water discharged 2005â2007 (plotted from data in Reussâ er, 2010, Table 3â2).    box continues next page   Â
100 Propagule Pressure and Invasion Risk in Ballast Water   BOX 4â3 Continued  The results from these attempts to characterize a multispecies riskârelease relationship are ambiguous and highlight the challenges in quantifying propaâ gule pressure.  The Great Lakes analyses (Figure 4â9) use shipping traffic vaâ riables that in and of themselves do not directly cause invasions, and that have not been tested for their correspondence to organism density.  In other words, they are serving as proxies for propagule pressure under the untested assumpâ tion that they scale linearly with propagule pressure.  The coastal analysis (Reusser, 2010) is an attempt to use a more direct measure of propagule presâ sure.  However, ballast water volume is scaled up assuming the same organism density distribution for all ships, and the relationship is based on a mismatched dataset of invasion data, shipping data, and organism data from different years and ports.  Both Cohen (2005, 2010) and Reusser (2010) have proposed using a linear riskârelease relationship to inform ballast water discharge standards [Figure 4â 2A; reviewed in Lee et al. (2010)]; Drake and Lodge (2004) used a linear riskâ release relationship embedded within a gravity model to investigate riskâ reduction strategies.  The primary theoretical challenge in developing these approaches is identifying the expected shape of the relationship, particularly given that even total organism number cannot be expected to directly predict species establishment.  The primary practical challenge is the current absence of the appropriate data, i.e., spatially and temporally matched variables, and untested or unrepresentative proxy variables.  critical assumption that organism density is homogeneous across tanks and ves- selsâan assumption that ballast surveys tell us categorically does not hold (Verl- ing et al., 2005; Minton et al., 2005; Lawrence and Cordell, 2010; see also refer- ences in Table 4-2). As a result, these variables cannot mechanistically explain the riskârelease relationship. Any statistically significant relationship that emerges may represent a spurious correlation. Any non-significant relationship could be the result of a non-representative proxy, or from the absence of a fun- damental underlying relationship between risk and release density resulting from the myriad other factors that influence success (see Chapter 3). The results from analyses to date are ambiguous and highlight that proxy variables may not al- ways be reliable predictors of invasion risk, particularly across regions (see Box 4-3). The principle of using proxy variables is not without merit, but it is essen- tial to select and test candidate variables with care before assigning any meaning to their relationships, or lack thereof.
Relationship Between Propagule Pressure and Establishment Risk  101  Mechanistic Models Mechanistic modeling frameworks can be scaled up from single species models to represent the release of multiple species from multiple ships at mul- tiple locations and multiple times. As the mathematical framework of such a model expands, so too do the data requirements for model parameterization and validation. It is crucial to recognize that it is not possible, mechanistically, to predict the invasion risk associated with the release of an unknown number of unidenti- fied species at unknown abundance, density, and frequency. Any mechanistic multi-species model is necessarily parameterized for a specific group of taxa, and its output is therefore as case-specific as that of a single-species model. To parameterize such a model requires knowing the identities and numbers of all released species, and knowing which of those incipient introductions established and failed. These data can be obtained from controlled experimental studies (e.g., Tilman, 1997; Shurin, 2000; Lee and Bruno, 2009), but not at the full scale of ballast water discharge. At present, therefore, there are not sufficient taxonomic information or em- pirical data to parameterize either a probabilistic or a demographic model for all the species in a ballast assemblage. Even if such data and information were available, the time scale mismatch between ballast water discharge and invasion record datasets would still prevent the validation (testing) of a mechanistic mod- el against the empirical data. Nevertheless, one can examine what the frame- work of such a model might look like with an eye to parameterizing it in the future. Probabilistic Models Probabilistic models of the single-species invasion process can be scaled up to create a framework representing the introduction of multiple species, as well as multiple vessels, locations, and releases. This model expresses invasion risk as either the expected number of established species, or the probability that at least one species will establish, in a given time frame. Organism release is spe- cified as the number of individuals of each species released, and may also in- clude separate releases from multiple vessels at multiple locations on multiple occasions (Costello et al., 2007; USCG, 2008). As for single-species models, when the constituent probabilities of the multispecies model are drawn from distributions, the approach is described as a hierarchical probability model (HPM). HPM has the advantage of explicitly representing the known uncertain- ty in the inherently stochastic invasion process, in the same way that a dynamic demographic model can be made stochastic by drawing parameter values from a distribution.
102 Propagule Pressure and Invasion Risk in Ballast Water  The multiple-species, multiple-invasion framework readily allows new in- formation about the characteristics of release and establishment (e.g., vessel type and source location; habitat and seasonal differences) to be incorporated into the hierarchical framework. This is particularly useful for evaluating which of many possible scenarios resulting in the observed number of species invasions is best supported by the data. Box 4-4 outlines a general framework for a multi- species HPM of invasion establishment. While this mechanistic hierarchical probability model poses an interesting framework for thinking formally about multi-species and multi-variable risk-release relationships, it has not yet been parameterized or validated with an empirical dataset, and the prospects of doing so are currently remote. Nonetheless, HPM for both single and multi-species scenarios holds advantages over other models because (1) it offers a mechanistic representation of the invasion process, (2) in the absence of detailed distribution data for all the parameters, it can be used in a simplified (i.e., non-hierarchical, point-estimate) version, and (3) as more data become available, it can be easily expanded to incorporate different species, locations, seasons, vessels, etc. See the conclusions of this chapter for a summary recommendation about this ap- proach. In a somewhat different approach, Costello et al. (2007) adapted a probabil- istic model of species introduction and detection over time (Solow and Costello, 2004) to test the relationship between invasion rate and number of vessels arriv- ing annually in the Great Lakes. This analysis highlights the influence of detec- tion lag in confounding our ability to assess the effectiveness of changes in bal- last management. Although this relationship is formulated as a probabilistic model, it is based on a proxy variable (number of vessels) so its mechanistic interpretation is unclear. It should be noted that at present, most available data are surrogates of propagule pressure (e.g., number of ships, ballast volume, or ballast discharge) and the number of invaders observed within a given time frame. The existing multispecies models (and most single species models) as- sume that a reduction in ballast reduces the number of invaders linearly in a sys- tem or probabilistically per ship (e.g., Drake and Lodge 2004; Costello, 2007). This may be reasonable given the limited data currently available to construct these models. Dynamic Demographic Models Like probabilistic models, dynamic demographic models could in principle be scaled up from the single-species scenario to model the combined risk of many species establishing. Again, such an exercise would require constructing and parameterizing a model with the identity, initial number, and invasion suc- cess of each population, and again, the resulting relationship would apply only to that suite of species.
Relationship Between Propagule Pressure and Establishment Risk  103  BOX 4â4 Outline of a Simple Multispecies Probabilistic Model  The singleâspecies probabilistic model developed in Box 4â1 is readily exâ tended to a multispecies probabilistic model.  Equation 4â3 in Box 4â1 defines the establishment probability for a single species as pE = 1-e-ï¡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 exâ pected number of species that establish, as  S SE ï½ ï¥1 ï e ïï¡i Ni ci s ï½1 (4-8) and ps, the probability that at least one species establishes, as  s pS ï½ 1 ï ï eïï¡i Ni (4-9) ci i ï½1 The same principles can be used to extend the model to consider variation in establishment probability across multiple locations (L), multiple vessels arrivâ ing in those locations (VL), and so on.  In this case, the expected number of speâ cies that establish can be written as VL S L SE ï½ ï¥1 ï ï ï e ïï¡ s,l ,v Ns,l ,v cs , l , v l ï½1 v ï½1 s ï½1 (4-10) and the probability that at least one species establishes as VL S L pS ï½ 1 ï ï ï ï eïï¡ s,l ,v Ns,l ,v cs ,l , v s ï½1 l ï½1 v ï½1 (4-11) box continues next page Â
104 Propagule Pressure and Invasion Risk in Ballast Water  BOX 4â4 Continued  These models follow the same logic as the basic one (4â3), where the shape parameters ï¡Â 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 poâ tential interactions among species.  Nevertheless, parameterizing such a modâ el, particularly in a hierarchical structure where each parameter is characteâ rized as a frequency distribution, would require a tremendous amount of data.  Qualitatively, the overall establishment probability ps obtained from a mulâ tiâspecies model can only be the same as, or greater than, the largest estabâ lishment probability pE of the constituent species.  To the Committeeâs 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-
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 riskârelease 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 riskârelease 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
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 riskârelease 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 riskârelease 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 riskârelease relationships thus far have tested multiple models and quantified uncertainty. The predicted shape of the riskârelease 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 riskârelease 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 riskârelease curve. Life histo- ry traits promoting such sensitivity to small inoculum density possibly include
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 riskârelease 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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