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4Â RelationshipÂ betweenÂ PropaguleÂ PressureÂ Â andÂ EstablishmentÂ RiskÂ 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 Â 72Â

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 73Â Â 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

74Â Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â 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 FIGUREÂ 4â1Â Â CommonÂ shapesÂ forÂ relationshipsÂ betweenÂ twoÂ variables:Â (a)Â hyperbolic,Â (b)Â sigmoid,Â (c)Â linear,Â (d)Â exponentialÂ (bothÂ axesÂ linear).Â Â AÂ priori,Â anyÂ ofÂ theseÂ couldÂ representÂ theÂ relationshipÂ betweenÂ invasionÂ riskÂ (probabilityÂ ofÂ aÂ speciesÂ establishing)Â andÂ propaguleÂ pressureÂ (e.g.,Â numberÂ ofÂ individualsÂ released).Â Â SOURCE:Â Â Adapted,Â withÂ permission,Â fromÂ RuizÂ andÂ CarltonÂ (2003).Â Â Â©2003Â byÂ IslandÂ Press.Â

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 75Â Â Â Â Â A B Â FIGUREÂ 4â2Â Â ConceptualÂ applicationÂ ofÂ aÂ riskâreleaseÂ relationshipÂ toÂ informÂ ballastÂ waterÂ organismÂ dischargeÂ standards.Â Â SÂ isÂ theÂ observedÂ riskÂ ofÂ ballastâmediatedÂ speciesÂ invaâ sion;Â NÂ isÂ theÂ observedÂ numberÂ ofÂ organismsÂ released.Â Â (AxisÂ unitsÂ dependÂ onÂ whetherÂ theÂ modelÂ representsÂ aÂ singleÂ speciesÂ orÂ multipleÂ species.)Â Â S*Â isÂ theÂ targetÂ invasionÂ risk;Â N*Â isÂ theÂ correspondingÂ targetÂ releaseÂ value.Â Â AssumingÂ aÂ robustÂ riskâreleaseÂ relationâ ship,Â reducingÂ theÂ ballastÂ waterÂ releaseÂ byÂ theÂ proportionÂ RNÂ isÂ predictedÂ toÂ reduceÂ theÂ invasionÂ riskÂ byÂ theÂ proportionÂ RS.Â Â (A)Â UnderÂ theÂ assumptionÂ ofÂ aÂ linearÂ riskâreleaseÂ relationship,Â aÂ givenÂ reductionÂ inÂ theÂ releaseÂ rateÂ isÂ predictedÂ toÂ giveÂ theÂ sameÂ proporâ tionalÂ reductionÂ inÂ invasionÂ rateÂ (i.e.,Â RN=RS).Â Â (B)Â UnderÂ theÂ assumptionÂ ofÂ aÂ sigmoidÂ relationship,Â theÂ sameÂ reductionÂ inÂ releaseÂ (RN1=1â[N*1/N1]Â =Â RN2=1â[N*2/N2])Â isÂ preâ dictedÂ toÂ giveÂ aÂ muchÂ lesserÂ (RS1Â =1â[S*1/S1])Â orÂ aÂ muchÂ greaterÂ (RS2Â =1â[S*2/S2])Â reducâ tionÂ inÂ invasionÂ risk,Â dependingÂ onÂ theÂ rangeÂ overÂ whichÂ RNÂ occurs.Â Â SolidÂ dotÂ indicatesÂ theÂ inflectionÂ pointÂ aroundÂ whichÂ theÂ greatestÂ reductionÂ inÂ riskÂ isÂ obtainedÂ forÂ theÂ leastÂ reductionÂ inÂ release.Â Â OpenÂ dotsÂ indicateÂ theÂ pointsÂ atÂ whichÂ theÂ slopeÂ passesÂ throughÂ aÂ o 45 Â angle:Â outsideÂ theseÂ bounds,Â increasinglyÂ lessÂ riskÂ reductionÂ isÂ obtainedÂ forÂ theÂ sameÂ releaseÂ reduction.Â Â PanelÂ (A)Â providesÂ aÂ graphicalÂ illustrationÂ ofÂ theÂ multiâspeciesÂ linearÂ modelÂ proposedÂ byÂ CohenÂ (2005,Â 2010)Â andÂ ReusserÂ (2010),Â whereÂ theÂ yâaxisÂ isÂ theÂ inâ vasionÂ riskÂ (characterizedÂ asÂ invasionÂ rate,Â theÂ numberÂ ofÂ speciesÂ establishedÂ perÂ unitÂ time)Â andÂ theÂ xâaxisÂ isÂ organismÂ releaseÂ (characterizedÂ asÂ releaseÂ rate,Â theÂ totalÂ numberÂ organismsÂ perÂ unitÂ time).Â Â ReusserÂ (2010)Â definedÂ asÂ theÂ perÂ capitaÂ invasionÂ riskÂ theÂ numberÂ ofÂ introducedÂ speciesÂ thatÂ establishÂ perÂ organismÂ released,Â i.e.,Â theÂ slopeÂ ofÂ theÂ straightâlineÂ relationship.Â Â CohenÂ (2005,Â 2010)Â definedÂ asÂ theÂ reductionÂ factorÂ theÂ equalÂ proportionalÂ reductionsÂ inÂ releaseÂ andÂ riskÂ fromÂ currentÂ toÂ targetÂ levels.Â Â Â

76Â Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â 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.

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 77Â Â TABLEÂ 4â1Â Â ApproachesÂ TakenÂ toÂ ModelingÂ theÂ RelationshipÂ BetweenÂ InvasionÂ RiskÂ andÂ PropaguleÂ PressureÂ Â Â Scale SampleÂ structure1Â TypeÂ SingleÂ speciesÂ Â MultipleÂ speciesÂ examplesÂ Â examplesÂ DescriptiveÂ StatisticalÂ Â MemmottÂ etÂ al.Â LonsdaleÂ (1999)Â Â LogisticÂ regression:Â (2005)Â LevineÂ andÂ DâAntonioÂ ï¦yï¶ JongejansÂ etÂ al.Â (2003)Â ï½ b0 ï« b1 x Â (2007)Â ln ï§ ï¨1ï yï· RicciardiÂ (2006)Â ï¸ BertolinoÂ (2009)Â CohenÂ (2005,Â 2010)*Â FunctionalÂ formÂ Â DrakeÂ andÂ JerdeÂ ReusserÂ (2010)*Â SpeciesâareaÂ curve:Â (2009)Â z y=cx Â MechanisticÂ Â ProbabilisticÂ SheaÂ andÂ Possingâ DrakeÂ etÂ al.Â (2005)*Â ïï¡ N pE ï½ 1 ï e Â hamÂ (2000)Â CostelloÂ etÂ al.Â (2007)*Â LeungÂ etÂ al.Â (2004)Â USCGÂ (2008)*Â DynamicÂ demographicÂ MemmottÂ etÂ al.Â SeeÂ textÂ dN (2005)Â ï½ rN Â dt DrakeÂ etÂ al.Â (2006)Â USCGÂ (2008)*Â DrakeÂ andÂ JerdeÂ (2009)Â JerdeÂ etÂ al.Â (2009)Â BaileyÂ etÂ al.Â (2009)Â KramerÂ andÂ DrakeÂ (2010)Â Notes:Â Â ModelsÂ areÂ categorizedÂ asÂ beingÂ onÂ theÂ descriptiveÂ orÂ theÂ mechanisticÂ endÂ ofÂ aÂ spectrum,Â andÂ asÂ representingÂ singleÂ orÂ multipleÂ invadingÂ species.Â Â SampleÂ structuresÂ showÂ simple,Â genericÂ formsÂ ofÂ theseÂ modelÂ types.Â Â MostÂ ofÂ theseÂ approachesÂ haveÂ beenÂ widelyÂ developedÂ andÂ impleâ mentedÂ throughoutÂ theÂ biologicalÂ literature;Â onlyÂ aÂ fewÂ recentÂ examples,Â furtherÂ discussedÂ inÂ theÂ text,Â areÂ listedÂ here.Â Â SomeÂ studiesÂ illustrateÂ moreÂ thanÂ oneÂ modelingÂ approach.Â 1 Â Parameters:Â y,Â dependentÂ variableÂ (invasionÂ risk);Â x,Â independentÂ variableÂ (organismÂ dischargeÂ orÂ proxyÂ variable);Â b0,Â b1,Â c,Â z,Â shapeÂ parameters;Â pE,Â populationÂ establishmentÂ probability;Â Î±,Â ln(individualÂ establishmentÂ probability);Â N,Â numberÂ ofÂ individuals;Â r,Â perÂ capitaÂ populationÂ growthÂ rate.Â Â Â *Â ProposedÂ applicationÂ ofÂ modelingÂ approachÂ toÂ ballastÂ waterÂ management.Â Â

78Â Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â 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

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 79Â Â 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)

80Â Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â 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

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 81Â Â A B C Â FIGUREÂ 4â3Â Â SingleâspeciesÂ riskâreleaseÂ relationshipsÂ obtainedÂ fromÂ descriptiveÂ models.Â Â (A)Â SplineÂ fitÂ toÂ shortâtermÂ establishmentÂ probabilityÂ ofÂ scentlessÂ chamomileÂ (MatricariaÂ perforata)Â (DrakeÂ andÂ Jerde,Â 2009).Â Â (B)Â GeneralizedÂ linearÂ modelÂ fitÂ toÂ establishmentÂ outcomeÂ (successÂ orÂ failure)Â forÂ EuropeanÂ thistleÂ (CarduusÂ acanthoides)Â acrossÂ threeÂ levelsÂ ofÂ nativeÂ plantÂ biomassÂ reducedÂ byÂ clippingÂ (JongejansÂ etÂ al.,Â 2007).Â Â (C)Â LogisticÂ regressionÂ fitÂ toÂ establishmentÂ probabilityÂ ofÂ squirrelÂ Sciurus.Â Â ReplottedÂ fromÂ BertolinoÂ (2009)Â withÂ dataÂ generouslyÂ provideÂ byÂ S.Â Bertolino.Â Â SOURCES:Â (A)Â Reprinted,Â withÂ perâ missionÂ from,Â DrakeÂ andÂ JerdeÂ (2009).Â Â©Â 2009Â byÂ OxfordÂ UniversityÂ Press.Â Â (B)Â Reâ printed,Â withÂ permission,Â fromÂ JongejansÂ etÂ al.Â (2009).Â Â©Â byÂ Springer.Â Â (C)Â Reprinted,Â withÂ permission,Â fromÂ BertolinoÂ (2009).Â Â©Â 2009Â byÂ JohnÂ WileyÂ andÂ Sons.Â Â Â

82Â Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â 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

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 83Â Â 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).

84Â Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â Â BOXÂ 4â1Â ProbabilisticÂ ModelÂ FrameworkÂ Â ThisÂ boxÂ illustratesÂ theÂ developmentÂ ofÂ theÂ simpleÂ probabilisticÂ modelÂ inÂ LeungÂ etÂ al.Â (2004)Â andÂ showsÂ howÂ itÂ providesÂ insightÂ intoÂ theÂ overallÂ shapeÂ ofÂ theÂ riskâreleaseÂ relationship.Â Â ItÂ beginsÂ withÂ aÂ simpleÂ probabilityÂ statementÂ inÂ whichÂ NÂ isÂ theÂ numberÂ ofÂ propagulesÂ releasedÂ andÂ pÂ isÂ theÂ individualÂ establishâ mentÂ probabilityÂ ofÂ eachÂ propagule.Â Â InÂ thisÂ case,Â 1âpÂ isÂ theÂ probabilityÂ ofÂ aÂ sinâ gleÂ propaguleÂ failingÂ toÂ establish,Â andÂ (1âp)NÂ isÂ theÂ cumulativeÂ probabilityÂ ofÂ allÂ NÂ individualsÂ failingÂ toÂ establish.Â Â TheÂ probabilityÂ ofÂ theÂ populationÂ establishing,Â pE,Â isÂ thereforeÂ Â Â pEÂ =Â 1â(1âp)NÂ (4â1)Â Â ItÂ isÂ mathematicallyÂ convenientÂ toÂ defineÂ anÂ additionalÂ parameter,Â ï¡Â =Â âln(1âp),Â whichÂ allowsÂ usÂ toÂ rewriteÂ (4â1)Â synonymouslyÂ asÂ Â pE ï½ 1 ï eïï¡ N Â (4â2)Â Â ForÂ thisÂ equationÂ (4â1,Â 4â2),Â theÂ shapeÂ ofÂ theÂ riskâreleaseÂ relationshipÂ betweenÂ pEÂ andÂ NÂ isÂ hyperbolic,Â asymptotingÂ towardsÂ pEÂ =Â 1Â (FigureÂ 4â4a).Â Â ThisÂ mustÂ beÂ theÂ caseÂ becauseÂ evenÂ ifÂ theÂ individualÂ establishmentÂ probabilityÂ (p)Â isÂ low,Â theÂ totalÂ probabilityÂ pEÂ increasesÂ inexorablyÂ toÂ oneÂ asÂ moreÂ andÂ moreÂ individualsÂ areÂ releasedÂ (LeungÂ etÂ al.,Â 2004).Â ThisÂ modelÂ canÂ beÂ extendedÂ toÂ incorporateÂ theÂ negativeÂ densityÂ depenâ dence,Â orÂ AlleeÂ effects,Â thatÂ mayÂ beÂ expectedÂ toÂ reduceÂ pEÂ atÂ lowÂ valuesÂ ofÂ NÂ (LeungÂ etÂ al.,Â 2004;Â JerdeÂ etÂ al.,Â 2009).Â Â TheÂ resultingÂ sigmoidÂ shapeÂ (FigureÂ 4â 4b)Â canÂ beÂ producedÂ byÂ addingÂ theÂ shapeÂ coefficientÂ cÂ >Â 1Â toÂ equationÂ 4â2Â (whichÂ followsÂ theÂ cumulativeÂ WeibullÂ distribution),Â givingÂ Â c pE ï½ 1 ï e ï ï¡ N Â Â Â (4â3)Â Â EquationÂ (4â2)Â isÂ theÂ specialÂ caseÂ ofÂ (4â3)Â whereÂ cÂ =Â 1Â andÂ thereÂ isÂ noÂ strongÂ AlleeÂ effectÂ (LeungÂ etÂ al.,Â 2004).Â Â Thus,Â theÂ biologicalÂ meaningÂ ofÂ cÂ canÂ beÂ interâ pretedÂ asÂ anÂ AlleeÂ parameter.Â Â However,Â itÂ shouldÂ beÂ notedÂ thatÂ ifÂ AlleeÂ effectsÂ areÂ toÂ beÂ considered,Â bothÂ ï¡Â andÂ cÂ needÂ toÂ beÂ fitÂ simultaneouslyÂ toÂ describeÂ aÂ biologicalÂ system.Â Â

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 85Â Â Â Â Â TheseÂ equationsÂ illustrateÂ thatÂ simpleÂ probabilityÂ statementsÂ combinedÂ withÂ basicÂ principlesÂ ofÂ populationÂ growthÂ revealÂ twoÂ candidateÂ shapesÂ forÂ theÂ overallÂ pEÂ vs.Â NÂ curve:Â hyperbolicÂ orÂ sigmoid.Â Â However,Â theÂ shapeÂ ofÂ theÂ curveÂ forÂ aÂ givenÂ empiricalÂ datasetÂ overÂ aÂ limitedÂ parameterÂ spaceÂ mayÂ appearÂ linear,Â particularlyÂ forÂ highÂ (bothÂ curves)Â andÂ lowÂ (sigmoidÂ curve)Â valuesÂ ofÂ N.Â Â ShortÂ sectionsÂ ofÂ theÂ sigmoidÂ curveÂ couldÂ alsoÂ appearÂ exponentialÂ (leftâhandÂ sideÂ ofÂ curve)Â orÂ hyperbolicÂ (rightâhandÂ side).Â Â Â (A)Â Â Â (B)Â FIGUREÂ 4â4Â Â PredictedÂ relationshipÂ betweenÂ pEÂ andÂ NÂ givenÂ byÂ equationÂ 4â3.Â Â (A)Â HyperbolicÂ shapeÂ withÂ noÂ AlleeÂ effectsÂ (cÂ =Â 1).Â Â (B)Â SigmoidÂ shapeÂ characteristicÂ ofÂ anÂ AlleeÂ effectÂ (cÂ >Â 1).Â Â InÂ bothÂ panels,Â theÂ upper,Â middle,Â andÂ lowerÂ curvesÂ areÂ forÂ pÂ âÂ ï¡Â =Â 0.01,Â 0.005,Â andÂ 0.001.

86Â Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â AÂ BÂ

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 87Â Â 1.0 0.8 Establishment probability 0.6 0.4 0.2 0.0 1 3 5 7 9 11 Initial number CÂ DÂ FIGUREÂ 4â5Â Â ContinuedÂ Â SingleâspeciesÂ riskâreleaseÂ relationshipsÂ predictedÂ byÂ probabilisticÂ modelsÂ forÂ theÂ establishmentÂ probabilityÂ ofÂ (A)Â theÂ psyllidÂ Arytaiâ nillaÂ spartiophilaÂ (MemmottÂ etÂ al.,Â 2005),Â (B)Â ChineseÂ mittenÂ crabÂ (EriocheirÂ sinensis)Â (JerdeÂ etÂ al.,Â 2009),Â (C)Â spinyÂ waterfleaÂ (BythotrephesÂ longimanus)Â withÂ demographicÂ (openÂ dots)Â andÂ environmentalÂ (solidÂ dots)Â stochasticityÂ (reâ drawnÂ fromÂ DrakeÂ etÂ al.,Â 2006),Â asÂ aÂ functionÂ ofÂ theÂ initialÂ numberÂ ofÂ organâ isms,Â andÂ (D)Â threeÂ cladoceransÂ (BosminaÂ spp.,Â circles;Â BosminaÂ coregoni,Â trianâ gles;Â DaphniaÂ retrocurva,Â squares)Â asÂ aÂ functionÂ ofÂ theÂ initialÂ organismÂ densityÂ (BaileyÂ etÂ al.,Â 2009).Â Â SOURCES:Â Â (A)Â Reprinted,Â withÂ permission,Â fromÂ MemmottÂ etÂ al.Â (2005).Â Â Â©Â 2005Â byÂ JohnÂ WileyÂ andÂ Sons.Â Â (B)Â Reprinted,Â withÂ permission,Â fromÂ JerdeÂ etÂ al.Â (2009).Â Â Â©Â 2009Â byÂ TheÂ UniversityÂ ofÂ ChicagoÂ Press.Â (C)Â Reâ printed,Â withÂ permission,Â fromÂ DrakeÂ etÂ al.Â (2006).Â Â Â©Â 2006Â byÂ Springer.Â Â (D)Â Reprinted,Â withÂ permission,Â fromÂ BaileyÂ etÂ al.Â (2009).Â Â Â©Â 2009Â byÂ NRCÂ ResearchÂ Press.Â

88Â Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â 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

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 89Â Â AÂ BÂ FIGUREÂ 4â6Â Â SingleâspeciesÂ riskâreleaseÂ relationshipsÂ obtainedÂ fromÂ dynamicÂ demoâ graphicÂ simulationÂ models.Â Â (A)Â PredictedÂ proportionÂ ofÂ populationsÂ exceedingÂ 20Â indiâ vidualsÂ afterÂ 100Â yearsÂ fromÂ aÂ populationÂ growthÂ modelÂ ofÂ treeÂ squirrelsÂ (WoodÂ etÂ al.,Â 2007).Â Â (B)Â PredictedÂ establishmentÂ probabilityÂ fromÂ aÂ populationÂ growthÂ modelÂ ofÂ theÂ copepodÂ PseudodiaptomusÂ marinusÂ (plottedÂ fromÂ dataÂ inÂ BarryÂ andÂ Levings,Â 2002).Â Â (A)Â Reprinted,Â withÂ permission,Â fromÂ WoodÂ etÂ al.Â (2007).Â Â Â©Â 2007Â byÂ AmericanÂ SocietyÂ ofÂ Mammalogists.Â Â Â

90Â Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â BOXÂ 4â2Â DynamicÂ DemographicÂ ModelÂ FrameworkÂ Â ThisÂ boxÂ illustratesÂ theÂ developmentÂ ofÂ aÂ dynamicÂ demographicÂ populationÂ model,Â followingÂ thatÂ formulatedÂ byÂ KramerÂ andÂ DrakeÂ (2010).Â Â TheÂ modelÂ frameworkÂ beginsÂ withÂ theÂ standardÂ continuousÂ timeÂ equationÂ forÂ aÂ homogeâ neousÂ populationÂ ofÂ sizeÂ N growingÂ asÂ aÂ functionÂ ofÂ theÂ differenceÂ betweenÂ theÂ birthÂ rate (ï¢) andÂ theÂ deathÂ rate (ï). TheÂ populationÂ growthÂ rateÂ isÂ givenÂ asÂ Â dN ï½ ï¢N ï ïN (4-4) dt To examineÂ theÂ effectsÂ ofÂ predation,Â anÂ additionalÂ mortalityÂ function, g(N), wasÂ addedÂ toÂ representÂ aÂ standardÂ predationÂ typeÂ IIÂ functionalÂ response,Â suchÂ that:Â Pï¡N g( N ) ï½ (4-5) (1 ï« ï¡Th N ) where P isÂ theÂ numberÂ ofÂ predators, ï¡ isÂ theÂ attackÂ rate,Â and Th isÂ theÂ handlingÂ time,Â givingÂ theÂ populationÂ growthÂ rate:Â Â dN ï½ ï¢N ï ïN ï g ( N ) (4-6) dt Â TheÂ modelÂ wasÂ extendedÂ toÂ representÂ twoÂ sizeÂ classes,Â juvenilesÂ (J)Â andÂ adultsÂ (A),Â whereÂ theÂ juvenilesÂ areÂ producedÂ byÂ adultsÂ atÂ rate ï¢ andÂ matureÂ toÂ adultsÂ atÂ rate ï¤, andÂ predationÂ isÂ aÂ functionÂ ofÂ totalÂ populationÂ size:Â Â Â 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 ThisÂ modelÂ wasÂ parameterizedÂ fromÂ laboratoryÂ experimentsÂ withÂ theÂ claâ doceranÂ DaphniaÂ magnaÂ andÂ aÂ nonâvisualÂ ambushÂ predator,Â larvaeÂ ofÂ theÂ midgeÂ ChaoborusÂ trivittatus.Â Â StochasticÂ modelÂ simulationsÂ showedÂ thatÂ predaâ tionÂ inducedÂ aÂ sigmoidÂ riskâreleaseÂ relationship,Â comparedÂ toÂ theÂ hyperbolicÂ curveÂ predictedÂ inÂ theÂ absenceÂ ofÂ predation.Â Â InÂ otherÂ words,Â predationÂ inâ ducedÂ anÂ AlleeÂ effectÂ inÂ thisÂ systemÂ (FigureÂ 4â7).Â Â Â Â Â Â Â Â Â Â boxÂ continuesÂ nextÂ pageÂ Â

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 91Â Â BOXÂ 4â2Â ContinuedÂ Â FIGUREÂ 4â7Â Â PredictedÂ singleâspeciesÂ riskâreleaseÂ relationshipÂ obtainedÂ fromÂ simulationsÂ ofÂ aÂ dynamicÂ demographicÂ model.Â Â TheoreticalÂ (dashedÂ line)Â andÂ simulatedÂ (pointsÂ withÂ fittedÂ solidâlineÂ spline)Â predictionsÂ ofÂ establishmentÂ probabilityÂ forÂ aÂ DaphniaÂ magnaÂ populationÂ asÂ aÂ functionÂ ofÂ initialÂ populationsÂ sizeÂ underÂ differentÂ predationÂ levels.Â Â SOURCE:Â AdaptedÂ fromÂ KramerÂ andÂ DrakeÂ (2010);Â courtesyÂ J.Â Drake.Â Â Reprinted,Â withÂ permission,Â fromÂ DrakeÂ (2010).Â Â Â©Â 2010Â byÂ JohnÂ WileyÂ andÂ Sons.Â Â 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

92Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â 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

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 93Â Â 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

94Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â TABLEÂ 4â2Â Â SpatialÂ andÂ TemporalÂ ScaleÂ ofÂ HistoricalÂ InvasionÂ Records,Â andÂ Spatial,Â Temâ poral,Â andÂ SamplingÂ ScaleÂ ofÂ BallastedÂ OrganismÂ SurveysÂ inÂ InlandÂ andÂ CoastalÂ WatersÂ ofÂ theÂ U.S.Â andÂ CanadaÂ Â InvasionÂ RecordsÂ BallastÂ SurveysÂ SampleÂ type LocationÂ DecadesÂ SourcesÂ YearsÂ (N)Â SourcesÂ (meshÂ size)Â LaurentianÂ 1840sâ2000s RicciardiÂ 1990â91Â waterÂ (41Â LockeÂ etÂ al.Â GreatÂ LakesÂ (2006)Â (86)Â Âµm,Â 110Â Âµm)Â (1993),Â SubbaÂ RaoÂ andÂ St.Â Lawâ Â Â etÂ al.Â (1994);Â Â renceÂ SeawayÂ Â Â Â 2000â02Â waterÂ Â BaileyÂ etÂ al.Â Â (39) Â (unfiltered)Â (2005),Â DugganÂ etÂ Â sediment al.Â (2005)Â ChesapeakeÂ 1600â2000sÂ FofonoffÂ etÂ 1993â94Â waterÂ SmithÂ etÂ al.Â (1999)Â Â Bay,Â MDÂ al.Â (2009)Â (60) Â (80Â Âµm)Â Â Â Â Â Â 1996â97Â (7) waterÂ Â LavoieÂ etÂ al.Â (80Â Âµm)Â (1999)Â SanÂ FranciscoÂ 1850sâ1990s CohenÂ andÂ Bay,Â CAÂ CarltonÂ âÂ âÂ âÂ (1998)Â HumboldtÂ Bay,Â 1920sâ1990s WonhamÂ CAÂ andÂ CarltonÂ âÂ âÂ âÂ (2005)Â CoosÂ Bay,Â OR 1940sâ1990s RuizÂ etÂ al.Â 1986â91Â waterÂ CarltonÂ andÂ GellerÂ Â (2000),Â (159) (80Â Âµm)Â (1993)Â WonhamÂ andÂ CarltonÂ (2005)Â WillapaÂ Bay,Â 1930sâ1990s WonhamÂ WAÂ andÂ CarltonÂ âÂ âÂ âÂ (2005)Â PugetÂ Sound,Â 1900sâ1990s RuizÂ etÂ al.Â 2001â07Â waterÂ CordellÂ etÂ al.Â Â WAÂ (2000),Â (372)Â (73Â Âµm) (2009),Â LawrenceÂ WonhamÂ andÂ CordellÂ andÂ CarltonÂ (2010)Â (2005)Â Vancouver,Â BCÂ Â 1900sâ1990s WonhamÂ 2000Â (15) waterÂ LevingsÂ etÂ al.Â (andÂ Â regionalÂ andÂ CarltonÂ Â (80Â Âµm)Â (2004)Â waters)Â (2005)Â Â Â Â 2007â08Â waterÂ Â KleinÂ etÂ al.Â (2010)Â Â Â (23) (unfiltered) PrinceÂ Â Â Â Â Â Â Â Â Â 1800â1990sÂ HinesÂ andÂ 1998â1999 waterÂ HinesÂ andÂ RuizÂ WilliamÂ Sound,Â RuizÂ (2000)Â (80Â Âµm)Â (2000)Â AKÂ Note:Â AK,Â Alaska;Â BC,Â BritishÂ Columbia;Â CA,Â California;Â MD,Â Maryland;Â OR,Â Oregon;Â WA,Â Washingâ ton.Â BallastÂ surveysÂ includeÂ studiesÂ ofÂ N>5Â vesselsÂ orÂ voyages.Â

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 95Â Â 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

96Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â AâCÂ DâE Â FIGUREÂ 4â8Â Â DescriptiveÂ modelsÂ ofÂ multiâspeciesÂ riskâreleaseÂ relationships.Â Â ComparisonÂ ofÂ logâlogÂ (dotted),Â logâlinearÂ (dashed),Â andÂ MichaelisâMentenÂ (solâ id)Â equationsÂ fitÂ toÂ numberÂ ofÂ (A)Â mollusks,Â (B)Â plantÂ pathogens,Â andÂ (C)Â insects,Â vs.Â cumulativeÂ importsÂ overÂ timeÂ inÂ theÂ U.S.Â (LevineÂ andÂ DâAntonio,Â 2003).Â Â (D)Â LogâlogÂ plotÂ ofÂ numberÂ ofÂ visitorsÂ vs.Â numberÂ ofÂ nativeÂ plantÂ speciesÂ inÂ natureÂ reservesÂ worldwide;Â (E)Â logâlinearÂ plotÂ ofÂ numberÂ ofÂ nonindigenousÂ plantÂ speâ ciesÂ vs.Â visitorÂ residualsÂ (nonânativeÂ plantÂ speciesÂ asÂ aÂ functionÂ ofÂ theÂ residualsÂ fromÂ theÂ relationshipÂ inÂ D)Â (p<0.001)Â (Lonsdale,Â 1999).Â Â SOURCE:Â Â (E)Â Reprinted,Â withÂ permission,Â fromÂ LonsdaleÂ (1999).Â Â Â©Â 1999Â byÂ EcologicalÂ SocietyÂ ofÂ Ameriâ ca.Â Â

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 97Â Â BOXÂ 4â3Â LinearÂ StatisticalÂ MultispeciesÂ ModelsÂ Â LinearÂ modelsÂ haveÂ beenÂ usedÂ toÂ estimateÂ theÂ multiâspeciesÂ riskâreleaseÂ relationshipÂ forÂ ballastÂ water.Â Â ToÂ date,Â theÂ mostÂ widelyÂ analyzedÂ dataÂ atÂ thisÂ scaleÂ areÂ invasionÂ trendsÂ inÂ theÂ GreatÂ Lakes.Â Â AÂ varietyÂ ofÂ analysesÂ haveÂ beenÂ conductedÂ forÂ thisÂ systemÂ usingÂ differentÂ dataÂ subsets,Â andÂ differentÂ depenâ dentÂ andÂ independentÂ variables.Â Â InÂ allÂ theÂ analyses,Â theÂ dataÂ haveÂ beenÂ parsedÂ intoÂ temporalÂ intervalsÂ toÂ provideÂ multipleÂ dataÂ pointsÂ forÂ modelÂ fitting.Â Â TheÂ resultsÂ areÂ notÂ consistentÂ amongÂ analyses.Â DependingÂ onÂ theÂ dataÂ subset,Â thereÂ mayÂ orÂ mayÂ notÂ appearÂ toÂ beÂ aÂ signifiâ cantÂ riskâreleaseÂ relationship.Â Â RicciardiÂ (2001)Â usedÂ aÂ linearÂ regressionÂ toÂ estiâ mateÂ theÂ rateÂ ofÂ allÂ speciesÂ invasionÂ vs.Â shippingÂ tonnageÂ inÂ netÂ tons,Â byÂ decadeÂ fromÂ 1900Â toÂ 1999Â (yÂ =Â 0.062x,Â r2Â adjÂ =Â 0.62,Â p<0.004).Â Â ThisÂ analysisÂ wasÂ upâ datedÂ byÂ RicciardiÂ (2006)Â forÂ onlyÂ thoseÂ freeâlivingÂ invadersÂ assumedÂ toÂ haveÂ beenÂ introducedÂ byÂ shippingÂ (yÂ =Â 0.05x;Â r2Â =Â 0.516,Â p<0.019;Â FigureÂ 4â9A).Â Â InÂ contrast,Â GrigorovichÂ etÂ al.Â (2003)Â analyzedÂ GreatÂ LakesÂ invasionÂ dataÂ inÂ 5âyearÂ intervalsÂ fromÂ 1959â1999;Â theirÂ dataÂ showÂ noÂ clearÂ trendÂ inÂ newÂ invadersÂ asÂ aÂ functionÂ ofÂ theÂ netÂ tonnageÂ ofÂ overseasÂ ballastedÂ traffic,Â andÂ ifÂ anythingÂ aÂ negâ ativeÂ relationshipÂ withÂ theÂ numberÂ ofÂ overseasÂ ballastedÂ vesselsÂ (FigureÂ 4â9Bâ C).Â LinearÂ relationshipsÂ haveÂ beenÂ usedÂ toÂ estimateÂ aÂ perâshipÂ invasionÂ rate.Â Â DrakeÂ andÂ LodgeÂ (2004)Â reanalyzedÂ theÂ dataÂ inÂ RicciardiÂ (2001)Â againstÂ shippingÂ tonnageÂ inÂ metricÂ tons,Â usingÂ aÂ linearÂ regressionÂ withÂ aÂ PoissonÂ errorÂ distribuâ tionÂ (yÂ =Â 8.47Â xÂ 10â8x;Â p<0.0001).Â Â RescalingÂ byÂ theÂ averageÂ shipÂ tonnage,Â theyÂ estimatedÂ aÂ perâshipÂ probabilityÂ ofÂ causingÂ anÂ invasionÂ asÂ 0.00044Â (95%Â CIÂ =Â 0.00008),Â equivalentÂ toÂ 1Â speciesÂ perÂ 2275Â shipsÂ orÂ 0.44Â invasionsÂ perÂ 1000Â shipsÂ (95%Â CLÂ 0.36,Â 0.52).Â Â ThisÂ estimateÂ wasÂ basedÂ onÂ allÂ nonindigenousÂ speâ ciesÂ inÂ theÂ GreatÂ Lakes,Â regardlessÂ ofÂ theirÂ presumedÂ vector.Â Â InÂ contrast,Â Cosâ telloÂ etÂ al.Â (2007)Â usedÂ annualÂ dataÂ onÂ shipâmediatedÂ animalÂ introductionsÂ aloneÂ fromÂ 1959â2000,Â asÂ aÂ functionÂ ofÂ numberÂ ofÂ shipsÂ (FigureÂ 4â9D),Â andÂ obâ tainedÂ aÂ maximumÂ likelihoodÂ estimateÂ ofÂ 0.14Â animalÂ invasionsÂ perÂ 1000Â shipsÂ (95%Â CLÂ 0.02,Â 5.2).Â LinearÂ relationshipsÂ haveÂ alsoÂ beenÂ usedÂ toÂ estimateÂ aÂ perâorganismÂ invaâ sionÂ rateÂ forÂ 17Â NorthÂ AmericanÂ portsÂ (Reusser,Â 2010).Â Â TheseÂ dataÂ wereÂ notÂ separatedÂ intoÂ timeÂ intervals,Â soÂ theÂ relationshipÂ forÂ eachÂ portÂ wasÂ basedÂ onÂ aÂ singleÂ dataÂ point.Â Â TheÂ dependentÂ variableÂ wasÂ theÂ totalÂ numberÂ ofÂ establishedÂ invadersÂ (invertebratesÂ andÂ macroalgae)Â fromÂ 1981â2006Â consideredÂ toÂ likelyÂ toÂ haveÂ beenÂ introducedÂ byÂ ballastÂ water.Â Â TheÂ independentÂ variableÂ wasÂ theÂ totalÂ volumeÂ ofÂ foreignÂ ballastÂ waterÂ dischargedÂ fromÂ 2005â2007,Â multipliedÂ onÂ aÂ perâshipÂ basisÂ byÂ aÂ randomÂ selectionÂ fromÂ anÂ empiricallyÂ determinedÂ zoopâ lanktonÂ densityÂ distributionÂ thatÂ spannedÂ eightÂ ordersÂ ofÂ magnitudeÂ (basedÂ onÂ boxÂ continuesÂ nextÂ pageÂ

98Â CÂ AÂ Â BOXÂ 4â3Â ContinuedÂ BÂ DÂ FIGUREÂ 4â9Â Â InvasionÂ patternsÂ inÂ theÂ GreatÂ Lakes.Â Â (A)Â TheÂ numberÂ ofÂ invadersÂ scalesÂ positivelyÂ withÂ netÂ shippingÂ tonnageÂ byÂ decadeÂ 1900â1999Â (Ricciardi,Â 2006).Â Â TheÂ sameÂ trendÂ isÂ notÂ evidentÂ fromÂ scatterÂ plotsÂ ofÂ (B)Â numberÂ ofÂ invadersÂ atÂ 5âyearÂ intervalsÂ fromÂ 1959â1999Â vs.Â netÂ tonnageÂ orÂ (C)Â vs.Â numberÂ ofÂ ballastedÂ shipsÂ (replottedÂ fromÂ GrigorovichÂ etÂ al.,Â 2003)Â orÂ ofÂ (D)Â annualÂ numberÂ ofÂ invadersÂ vs.Â numberÂ ofÂ shipsÂ (plottedÂ fromÂ dataÂ inÂ CostelloÂ etÂ al.Â 2007,Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ appendicesÂ AâB).Â Â SOURCE:Â (A)Â Reprinted,Â withÂ permission,Â fromÂ RicciardiÂ (2006).Â Â Â©Â 2006Â byÂ JohnÂ WileyÂ andÂ Sons.Â Â (B,Â C)Â Reprinted,Â withÂ permission,Â fromÂ GrigorovichÂ etÂ al.Â (2003).Â Â Â©Â 2003Â byÂ NRCÂ ResearchÂ Press.Â Â (D)Â Reprinted,Â withÂ permission,Â fromÂ CostelloÂ etÂ al.Â 2007.Â Â Â©Â 2007Â byÂ EcologicalÂ SocietyÂ ofÂ Amerâ

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 99Â Â Â Â Â 354Â shipsÂ sampledÂ inÂ fourÂ U.S.Â ports,Â ofÂ whichÂ threeÂ areÂ includedÂ inÂ theÂ 17Â anaâ lyzed;Â MintonÂ etÂ al.,Â 2005).Â Â RepeatedÂ randomÂ drawsÂ generatedÂ aÂ bootstrappedÂ estimateÂ ofÂ theÂ medianÂ andÂ theÂ firstÂ andÂ thirdÂ quantileÂ invasionÂ ratesÂ forÂ eachÂ port.Â Â Together,Â theseÂ perâcapitaÂ invasionÂ ratesÂ spannedÂ fourÂ ordersÂ ofÂ magniâ tudeÂ fromÂ 10â11Â toÂ 10â8,Â orÂ oneÂ invasionÂ forÂ everyÂ 10Â millionÂ toÂ 10Â billionÂ organâ ismsÂ (Reusser,Â 2010).Â Â Interestingly,Â theÂ dataÂ providedÂ noÂ evidenceÂ ofÂ aÂ strongÂ riskâreleaseÂ relationshipÂ acrossÂ ports,Â basedÂ onÂ eitherÂ numberÂ ofÂ vesselsÂ orÂ ballastÂ waterÂ volumeÂ (FigureÂ 4â10AâB).Â Â AÂ BÂ Â FIGUREÂ 4â10Â Â AcrossÂ 17Â U.S.Â coastalÂ ports,Â theÂ numberÂ ofÂ invadersÂ reportedÂ fromÂ 1981â 2006Â showsÂ noÂ strongÂ relationshipÂ withÂ (A)Â numberÂ ofÂ shipsÂ withÂ foreignÂ ballastÂ 2005â 2007Â orÂ (B)Â volumeÂ ofÂ ballastÂ waterÂ dischargedÂ 2005â2007Â (plottedÂ fromÂ dataÂ inÂ Reussâ er,Â 2010,Â TableÂ 3â2).Â Â Â Â boxÂ continues nextÂ pageÂ Â Â Â

100Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â Â BOXÂ 4â3Â ContinuedÂ Â TheÂ resultsÂ fromÂ theseÂ attemptsÂ toÂ characterizeÂ aÂ multispeciesÂ riskâreleaseÂ relationshipÂ areÂ ambiguousÂ andÂ highlightÂ theÂ challengesÂ inÂ quantifyingÂ propaâ guleÂ pressure.Â Â TheÂ GreatÂ LakesÂ analysesÂ (FigureÂ 4â9)Â useÂ shippingÂ trafficÂ vaâ riablesÂ thatÂ inÂ andÂ ofÂ themselvesÂ doÂ notÂ directlyÂ causeÂ invasions,Â andÂ thatÂ haveÂ notÂ beenÂ testedÂ forÂ theirÂ correspondenceÂ toÂ organismÂ density.Â Â InÂ otherÂ words,Â theyÂ areÂ servingÂ asÂ proxiesÂ forÂ propaguleÂ pressureÂ underÂ theÂ untestedÂ assumpâ tionÂ thatÂ theyÂ scaleÂ linearlyÂ withÂ propaguleÂ pressure.Â Â TheÂ coastalÂ analysisÂ (Reusser,Â 2010)Â isÂ anÂ attemptÂ toÂ useÂ aÂ moreÂ directÂ measureÂ ofÂ propaguleÂ presâ sure.Â Â However,Â ballastÂ waterÂ volumeÂ isÂ scaledÂ upÂ assumingÂ theÂ sameÂ organismÂ densityÂ distributionÂ forÂ allÂ ships,Â andÂ theÂ relationshipÂ isÂ basedÂ onÂ aÂ mismatchedÂ datasetÂ ofÂ invasionÂ data,Â shippingÂ data,Â andÂ organismÂ dataÂ fromÂ differentÂ yearsÂ andÂ ports.Â Â BothÂ CohenÂ (2005,Â 2010)Â andÂ ReusserÂ (2010)Â haveÂ proposedÂ usingÂ aÂ linearÂ riskâreleaseÂ relationshipÂ toÂ informÂ ballastÂ waterÂ dischargeÂ standardsÂ [FigureÂ 4â 2A;Â reviewedÂ inÂ LeeÂ etÂ al.Â (2010)];Â DrakeÂ andÂ LodgeÂ (2004)Â usedÂ aÂ linearÂ riskâ releaseÂ relationshipÂ embeddedÂ withinÂ aÂ gravityÂ modelÂ toÂ investigateÂ riskâ reductionÂ strategies.Â Â TheÂ primaryÂ theoreticalÂ challengeÂ inÂ developingÂ theseÂ approachesÂ isÂ identifyingÂ theÂ expectedÂ shapeÂ ofÂ theÂ relationship,Â particularlyÂ givenÂ thatÂ evenÂ totalÂ organismÂ numberÂ cannotÂ beÂ expectedÂ toÂ directlyÂ predictÂ speciesÂ establishment.Â Â TheÂ primaryÂ practicalÂ challengeÂ isÂ theÂ currentÂ absenceÂ ofÂ theÂ appropriateÂ data,Â i.e.,Â spatiallyÂ andÂ temporallyÂ matchedÂ variables,Â andÂ untestedÂ orÂ unrepresentativeÂ proxyÂ variables.Â Â 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.

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 101Â Â 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.

102Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â 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.

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 103Â Â BOXÂ 4â4Â OutlineÂ ofÂ aÂ SimpleÂ MultispeciesÂ ProbabilisticÂ ModelÂ Â TheÂ singleâspeciesÂ probabilisticÂ modelÂ developedÂ inÂ BoxÂ 4â1Â isÂ readilyÂ exâ tendedÂ toÂ aÂ multispeciesÂ probabilisticÂ model.Â Â EquationÂ 4â3Â inÂ BoxÂ 4â1Â definesÂ theÂ establishmentÂ probabilityÂ forÂ aÂ singleÂ speciesÂ as pE = 1-e-ï¡N. ThisÂ equationÂ canÂ beÂ modifiedÂ toÂ accommodateÂ SÂ species,Â eachÂ withÂ itsÂ ownÂ establishmentÂ probabilty ps. FollowingÂ theÂ sameÂ generalÂ approachesÂ describedÂ inÂ SheaÂ andÂ PossinghamÂ (2000)Â andÂ USCGÂ (2008),Â theÂ modelÂ thenÂ describesÂ SE,Â theÂ exâ pectedÂ numberÂ ofÂ speciesÂ thatÂ establish,Â asÂ Â S SE ï½ ï¥1 ï e ïï¡i Ni ci s ï½1 (4-8) and ps, theÂ probabilityÂ thatÂ atÂ leastÂ oneÂ speciesÂ establishes,Â asÂ Â s pS ï½ 1 ï ï eïï¡i Ni (4-9) ci i ï½1 TheÂ sameÂ principlesÂ canÂ beÂ usedÂ toÂ extendÂ theÂ modelÂ toÂ considerÂ variationÂ inÂ establishmentÂ probabilityÂ acrossÂ multipleÂ locationsÂ (L),Â multipleÂ vesselsÂ arrivâ ingÂ inÂ thoseÂ locationsÂ (VL),Â andÂ soÂ on.Â Â InÂ thisÂ case,Â theÂ expectedÂ numberÂ ofÂ speâ ciesÂ thatÂ establishÂ canÂ beÂ writtenÂ asÂ 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) andÂ theÂ probabilityÂ thatÂ atÂ leastÂ oneÂ speciesÂ establishesÂ asÂ 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) boxÂ continuesÂ nextÂ pageÂ Â

104Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â BOXÂ 4â4Â ContinuedÂ Â TheseÂ modelsÂ followÂ theÂ sameÂ logicÂ asÂ theÂ basicÂ oneÂ (4â3),Â whereÂ theÂ shapeÂ parametersÂ ï¡Â andÂ cÂ describeÂ theÂ speciesÂ establishmentÂ probabilityÂ asÂ aÂ functionÂ ofÂ theÂ initialÂ numberÂ ofÂ individualsÂ N;Â theÂ subscriptsÂ s,Â l,Â andÂ vÂ allowÂ variationÂ amongÂ species,Â locations,Â andÂ vessels;Â andÂ theÂ complementÂ ofÂ allÂ theÂ propagulesÂ failingÂ toÂ establishÂ givesÂ theÂ finalÂ probabilityÂ ofÂ establishment.Â Â ThisÂ modelÂ doesÂ notÂ accountÂ forÂ variationÂ inÂ parameterÂ valuesÂ overÂ timeÂ orÂ forÂ poâ tentialÂ interactionsÂ amongÂ species.Â Â Nevertheless,Â parameterizingÂ suchÂ aÂ modâ el,Â particularlyÂ inÂ aÂ hierarchicalÂ structureÂ whereÂ eachÂ parameterÂ isÂ characteâ rizedÂ asÂ aÂ frequencyÂ distribution,Â wouldÂ requireÂ aÂ tremendousÂ amountÂ ofÂ data.Â Â Qualitatively,Â theÂ overallÂ establishmentÂ probabilityÂ psÂ obtainedÂ fromÂ aÂ mulâ tiâspeciesÂ modelÂ canÂ onlyÂ beÂ theÂ sameÂ as,Â orÂ greaterÂ than,Â theÂ largestÂ estabâ lishmentÂ probabilityÂ pEÂ ofÂ theÂ constituentÂ species.Â Â 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-

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 105Â Â 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

106Â PropaguleÂ PressureÂ andÂ InvasionÂ RiskÂ inÂ BallastÂ WaterÂ Â 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

RelationshipÂ BetweenÂ PropaguleÂ PressureÂ andÂ EstablishmentÂ RiskÂ Â 107Â Â 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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