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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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