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Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water (2011)

Chapter: 4 Relationship between Propagule Pressureand Establishment Risk

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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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Suggested Citation:"4 Relationship between Propagule Pressureand Establishment Risk." National Research Council. 2011. Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water. Washington, DC: The National Academies Press. doi: 10.17226/13184.
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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 

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

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. 

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.   

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.

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.   

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

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)

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

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.   

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

Relationship Between Propagule Pressure and Establishment Risk   83    success or failure. In contrast, dynamic demographic models directly represent population dynamics in order to predict invasion outcomes. Dynamic Demographic Models Dynamic models of population growth are written as a system of one or more discrete or differential equations whose solution gives the population size at a given time. Familiar examples include exponential (geometric) and logistic growth, and Leslie and Lefkovitch matrix models. The parameters in these models represent the demographic rates or probabilities of birth, death, growth, and reproduction. Dynamic demographic models do not predict establishment probability directly. Instead, a number of stochastic simulations must be gener- ated for each initial population size. A subsequent model, either descriptive or probabilistic, must then be fit to the outcome of the simulations. Parameterizing and validating this kind of stochastic dynamic model requires estimating the distribution of each demographic parameter (e.g., birth rate, death rate). In their simpler forms, demographic models require more data to parameterize and vali- date than do probabilistic models. Dynamic demographic models serve as the basis for the population viability analysis (PVA) of declining species (Morris and Doak, 2002), an approach that has been applied more recently to the analysis of establishing invaders (Bartell and Nair, 2004; Neubert and Parker, 2004; Andersen, 2005), and that has been proposed for use in comparing ballast water discharge standards (USCG, 2008). It is useful to recall that the goals and outcome of population viability analysis are not the exclusive domain of demographic models. Although the term PVA typically refers to the analysis of these models, any modeled population may be subjected to an analysis of its viability; indeed, such analysis is inherent in the construction of a probabilistic establishment model. Traditionally, PVA in- volves dynamic demographic models that are matrix-based and use age- or stage-specific dynamic rates to estimate population growth and hence viability. These models use dynamic information for each stage including growth, surviv- al, and reproduction to estimate population growth rate (Caswell, 1989). The most basic dynamic demographic models are count-based and assume no variation among individuals in the population. These count-based models make several assumptions including that the mean and variance of population growth remain constant, no density dependence, dynamic stochasticity, etc. (Morris and Doak, 2002). However, more complex count-based models can incorporate positive and negative density dependence, Allee effects, stochastici- ty, spatial structure, etc. The more realistic dynamic demographic models expli- citly use different dynamic parameters for each age or stage class in the popula- tion (see example from Kramer and Drake, 2010, below). Although various kinds of stochasticity and autocorrelation in vital rates can be includedthrough simulation methods, these models have their own restrictive assumptions of time invariance, stable age or stage distribution, etc. (Morris and Doak, 2002).

84    Propagule Pressure and Invasion Risk in Ballast Water      BOX 4‐1  Probabilistic Model Framework    This  box  illustrates  the  development  of  the  simple  probabilistic  model  in  Leung  et  al.  (2004)  and  shows  how  it  provides  insight  into  the  overall  shape  of  the  risk–release  relationship.   It  begins  with  a  simple  probability  statement  in  which  N  is  the  number  of  propagules  released  and  p  is  the  individual  establish‐ ment  probability  of  each  propagule.   In  this  case,  1‐p  is  the  probability  of  a  sin‐ gle  propagule  failing  to  establish,  and  (1‐p)N  is  the  cumulative  probability  of  all  N individuals failing to establish.  The probability of the population establishing,  pE, is therefore     pE = 1‐(1‐p)N  (4‐1)    It  is  mathematically  convenient  to  define  an  additional  parameter,    = ‐ln(1‐p),  which allows us to rewrite (4‐1) synonymously as    pE  1  e N   (4‐2)    For  this  equation  (4‐1,  4‐2),  the  shape  of  the  risk–release  relationship  between  pE  and  N  is  hyperbolic,  asymptoting  towards  pE  =  1  (Figure  4‐4a).   This  must  be  the  case  because  even  if  the  individual  establishment  probability  (p)  is  low,  the  total  probability  pE  increases  inexorably  to  one  as  more  and  more  individuals  are released (Leung et al., 2004).  This  model  can  be  extended  to  incorporate  the  negative  density  depen‐ dence,  or  Allee  effects,  that  may  be  expected  to  reduce  pE  at  low  values  of  N  (Leung  et  al.,  2004;  Jerde  et  al.,  2009).   The  resulting  sigmoid  shape  (Figure  4‐ 4b)  can  be  produced  by  adding  the  shape  coefficient  c  >  1  to  equation  4‐2  (which follows the cumulative Weibull distribution), giving    c pE  1  e   N    (4‐3)    Equation  (4‐2)  is  the  special  case  of  (4‐3)  where  c  =  1  and  there  is  no  strong  Allee effect (Leung et al., 2004).  Thus, the biological meaning of c can be inter‐ preted  as  an Allee  parameter.   However,  it  should  be  noted  that  if  Allee effects  are  to  be  considered,  both    and  c  need  to  be  fit  simultaneously  to  describe  a  biological system.   

Relationship Between Propagule Pressure and Establishment Risk   85          These  equations  illustrate  that  simple  probability  statements  combined  with  basic  principles  of  population  growth  reveal  two  candidate  shapes  for  the  overall  pE  vs.  N  curve:  hyperbolic  or  sigmoid.   However,  the  shape  of  the  curve  for a given empirical dataset over a limited parameter space may appear linear,  particularly  for  high  (both  curves)  and  low  (sigmoid  curve)  values  of  N.   Short  sections  of  the  sigmoid  curve  could  also  appear  exponential  (left‐hand  side  of  curve) or hyperbolic (right‐hand side).      (A)      (B)  FIGURE 4‐4  Predicted relationship between pE and N given by equation 4‐3.  (A)  Hyperbolic  shape  with  no  Allee  effects  (c  =  1).   (B)  Sigmoid  shape  characteristic  of  an  Allee  effect  (c  >  1).   In  both  panels,  the  upper,  middle,  and  lower  curves  are for p ≈  = 0.01, 0.005, and 0.001.

86    Propagule Pressure and Invasion Risk in Ballast Water    A  B 

Relationship Between Propagule Pressure and Establishment Risk   87    1.0 0.8 Establishment probability 0.6 0.4 0.2 0.0 1 3 5 7 9 11 Initial number C  D  FIGURE  4‐5    Continued    Single‐species  risk–release  relationships  predicted  by  probabilistic  models  for  the  establishment  probability  of  (A)  the  psyllid  Arytai‐ nilla  spartiophila  (Memmott  et  al.,  2005),  (B)  Chinese  mitten  crab  (Eriocheir  sinensis)  (Jerde  et  al.,  2009),  (C)  spiny  waterflea  (Bythotrephes  longimanus)  with  demographic  (open  dots)  and  environmental  (solid  dots)  stochasticity  (re‐ drawn  from  Drake  et  al.,  2006),  as  a  function  of  the  initial  number  of  organ‐ isms, and (D) three cladocerans (Bosmina spp., circles; Bosmina coregoni, trian‐ gles;  Daphnia  retrocurva,  squares)  as  a  function  of  the  initial  organism  density  (Bailey et al., 2009).  SOURCES:  (A) Reprinted, with permission, from Memmott  et  al.  (2005).   ©  2005  by  John  Wiley  and  Sons.   (B)  Reprinted,  with  permission,  from  Jerde  et  al.  (2009).    ©  2009  by  The  University  of  Chicago  Press.  (C)  Re‐ printed,  with  permission,  from  Drake  et  al.  (2006).    ©  2006  by  Springer.    (D)  Reprinted, with permission, from Bailey et al. (2009).  © 2009 by NRC Research  Press. 

88    Propagule Pressure and Invasion Risk in Ballast Water    Dynamic demographic models have been developed for a tremendous varie- ty of plant and animal species. The following examples illustrate their applica- tion to predicting the establishment probability of introduced or re-introduced species. Wood et al. (2007) used life history data to parameterize an individual- based simulation model of tree squirrel re-introductions, and predicted the pro- portion of populations above a threshold abundance (Figure 4-6A). Other mod- els represent the establishment of age- or stage-structured populations (Parker, 2000; Barry and Levings, 2002; Kramer and Drake, 2010) (Figure 4-6B). Kra- mer and Drake (2010) used experimental laboratory results to parameterize a demographic model of the cladoceran Daphnia magna, and found that increas- ing predation shifted the risk–release 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 reaction–diffusion 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 risk–release rela- tionships. For the application of a reaction–diffusion 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 risk–release relationship could provide in- sight into discharge standards in two main ways: to illustrate a best-case scena- rio, and to serve as a building block for multi-species models. To illustrate a best-case scenario, a model could be constructed and parameterized for fast- growing, high-impact, or commonly released species. Invasion risk could then be predicted under the assumption that all ballasted organisms belonged to this species, and were released under optimal conditions. This approach would lead to a conservative discharge standard. The greatest difficulty in developing a discharge standard from a single- species model is that these models are generally constructed to represent a one- time introduction of a known initial number of individuals. However, ballast water discharge is a repeated event, which will tend to increase invasion risk, and the organisms may rapidly be redistributed in the physical environment, which may immediately alter the effective initial number of individuals with the

Relationship Between Propagule Pressure and Establishment Risk   89    A  B  FIGURE  4‐6    Single‐species  risk–release  relationships  obtained  from  dynamic  demo‐ graphic  simulation  models.   (A)  Predicted  proportion  of  populations  exceeding  20  indi‐ viduals  after  100  years  from  a  population  growth  model  of  tree  squirrels  (Wood  et  al.,  2007).   (B)  Predicted  establishment  probability  from  a  population  growth  model  of  the  copepod  Pseudodiaptomus  marinus  (plotted  from  data  in  Barry  and  Levings,  2002).   (A)  Reprinted,  with  permission,  from  Wood  et  al.  (2007).    ©  2007  by  American  Society  of  Mammalogists.   

90    Propagule Pressure and Invasion Risk in Ballast Water    BOX 4‐2  Dynamic Demographic Model Framework    This box illustrates the development of a dynamic demographic population  model,  following  that  formulated  by  Kramer  and  Drake  (2010).    The  model  framework  begins  with  the  standard  continuous  time  equation  for  a  homoge‐ neous population of size N growing as a function of the difference between the  birth rate () and the death rate (). The population growth rate is given as    dN  N  N (4-4) dt To examine  the  effects  of  predation,  an  additional  mortality  function, g(N), was  added  to  represent  a  standard  predation  type  II  functional  response,  such that:  PN g( N )  (4-5) (1  Th N ) where P is the number of predators,  is the attack rate, and Th is the handling  time, giving the population growth rate:    dN  N  N  g ( N ) (4-6) dt   The  model  was  extended  to  represent  two  size  classes,  juveniles  (J)  and  adults  (A),  where  the  juveniles  are  produced  by  adults  at  rate  and  mature  to  adults  at rate , and predation is a function of total population size:     dJ  A  J  g ( J  A)  J dt (4-7) dA  J  A  g ( J  A) dt This  model  was  parameterized  from  laboratory  experiments  with  the  cla‐ doceran  Daphnia  magna  and  a  non‐visual  ambush  predator,  larvae  of  the  midge  Chaoborus  trivittatus.   Stochastic  model  simulations  showed  that  preda‐ tion  induced  a  sigmoid  risk–release  relationship,  compared  to  the  hyperbolic  curve  predicted  in  the  absence  of  predation.    In  other  words,  predation  in‐ duced an Allee effect in this system (Figure 4‐7).            box continues next page   

Relationship Between Propagule Pressure and Establishment Risk   91    BOX 4‐2 Continued    FIGURE 4‐7  Predicted single‐species risk–release relationship obtained from simulations  of  a  dynamic  demographic  model.   Theoretical  (dashed  line)  and  simulated  (points  with  fitted  solid‐line  spline)  predictions  of  establishment  probability  for  a  Daphnia  magna  population  as  a  function  of  initial  populations  size  under  different  predation  levels.   SOURCE:  Adapted  from  Kramer  and  Drake  (2010);  courtesy  J.  Drake.    Reprinted,  with  permission, from Drake (2010).  © 2010 by John Wiley and Sons.   potential to establish. The closer together small releases occur in space and time, the more they will approximate a single large release with a correspon- dingly higher establishment probability (analogous to the rescue effect in meta- population dynamics, Gotelli, 1991). Theoretical studies have demonstrated that, due to environmental stochasticity, the likelihood of success of multiple arrivals at a single entry point is higher than that for simultaneous arrivals at multiple sites (Haccou and Iwasa, 1996). In a homogeneous environment, or- ganisms will disperse and the effective initial population size will rapidly de- crease; an advective environment and intraspecific behavior may either enhance

92  Propagule Pressure and Invasion Risk in Ballast Water    or counter this effect. The mathematical framework of a single-species model could readily be modified to analyze the risk associated with multiple repeated inocula that are dispersed or concentrated in the local environment and to deter- mine an adjusted discharge target. In summary, both descriptive and mechanistic models have been developed to examine the risk–release 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 risk–release relationship at this scale re- quires both risk and release data spanning the same large spatial, temporal, and taxonomic scales. At present, however, there are only loosely corresponding empirical estimates of risk and release (see Table 4-2). Before proceeding with multi-species examples, this section considers the nature of the available data for both invasion risk and organism release, and the resulting constraints on model construction and interpretation. For invasion risk, there are historical records of the invaders that have ac- cumulated in various ports over the past decades (see Table 4-2). These inva- sion records are characterized by considerable uncertainty stemming from in- complete collections, the cryptogenic nature of many species, the taxonomic bias of field samples, and the uncertainty associated with ascribing a given inva- sion to ballast transport over other candidate vectors (Chapman and Carlton, 1991; Ruiz et al., 2000; Costello et al., 2007; Fitzpatrick et al., 2009; Jerde and Bossenbroek, 2009). These are standard sampling difficulties that plague any assessment of nonindigenous species, and are not unique to the problem of bal- last water management. The consequences, however, are that the best empirical estimates of invasion rate and risk are nevertheless incomplete and uncertain. Furthermore and more crucially, there are no good estimates of the scale of that uncertainty. For organism release from ballast water, there are snapshot surveys of particular size classes of organisms collected from a subset of tanks on a subset of ships arriving in selected locations over brief and recent time periods, identi- fied to the lowest taxonomic level possible which nevertheless is often well

Relationship Between Propagule Pressure and Establishment Risk   93    above that of species (see Table 4-2). As discussed in Chapter 1, organism iden- tities and densities vary within ballast tanks, vessels, and routes (Lavoie et al., 1999; Smith et al., 1999; Wonham et al., 2001; Verling et al., 2005; Lawrence and Cordell, 2010). As a result, the best empirical estimates of organism release from ballast water are recent, local, taxonomically variable subsamples of the process. Again, the degree of uncertainty is not well characterized. Thus, two difficulties emerge in parameterizing the multi-species risk– 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 risk–release 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 risk–release 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), local–regional 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 risk– 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 risk– release relationship offer a phenomenological description of a pattern without requiring that the underlying mechanisms be specified. The majority of these studies, recognizing the difficulty of measuring propagule pressure directly, have measured a proxy variable of human activity ranging from population to transport to economic indices. Some statistical analyses of large-scale invasion vectors have focused on a single transport or economic variable as a substitute for propagule pressure (e.g., Levine and D'Antonio, 2003; Taylor and Irwin, 2004; Ricciardi, 2006; Costello et al., 2007; see examples in Figure 4-8). Others have used multivariable

94  Propagule Pressure and Invasion Risk in Ballast Water    TABLE  4‐2   Spatial  and  Temporal  Scale  of  Historical  Invasion  Records,  and  Spatial,  Tem‐ poral, and Sampling Scale of Ballasted Organism Surveys in Inland and Coastal Waters of  the U.S. and Canada    Invasion Records  Ballast Surveys  Sample type Location  Decades  Sources  Years (N)  Sources  (mesh size)  Laurentian  1840s–2000s Ricciardi  1990–91  water (41  Locke et al.  Great Lakes  (2006)  (86)  µm, 110 µm)  (1993), Subba Rao  and St. Law‐     et al. (1994);   rence Seaway        2000–02  water   Bailey et al.    (39)   (unfiltered)  (2005), Duggan et    sediment al. (2005)  Chesapeake  1600–2000s  Fofonoff et  1993–94  water  Smith et al. (1999)    Bay, MD  al. (2009)  (60)   (80 µm)            1996–97 (7) water   Lavoie et al.  (80 µm)  (1999)  San Francisco  1850s–1990s Cohen and  Bay, CA  Carlton  —  —  —  (1998)  Humboldt Bay,  1920s–1990s Wonham  CA  and Carlton  —  —  —  (2005)  Coos Bay, OR 1940s–1990s Ruiz et al.  1986–91  water  Carlton and Geller    (2000),  (159) (80 µm)  (1993)  Wonham  and Carlton  (2005)  Willapa Bay,  1930s–1990s Wonham  WA  and Carlton  —  —  —  (2005)  Puget Sound,  1900s–1990s Ruiz et al.  2001–07  water  Cordell et al.    WA  (2000),  (372)  (73 µm) (2009), Lawrence  Wonham  and Cordell  and Carlton  (2010)  (2005)  Vancouver, BC   1900s–1990s Wonham  2000 (15) water  Levings et al.  (and  regional  and Carlton    (80 µm)  (2004)  waters)  (2005)        2007–08  water   Klein et al. (2010)      (23) (unfiltered) Prince           1800–1990s  Hines and  1998–1999 water  Hines and Ruiz  William Sound,  Ruiz (2000)  (80 µm)  (2000)  AK  Note:  AK,  Alaska;  BC,  British  Columbia;  CA,  California;  MD,  Maryland;  OR,  Oregon;  WA,  Washing‐ ton. Ballast surveys include studies of N>5 vessels or voyages. 

Relationship Between Propagule Pressure and Establishment Risk   95    analyses to tease out the relative importance of propagule pressure, again usually by proxy, among other factors contributing to invasion success (ballast water, Drake and Lodge, 2004; plants, Lonsdale, 1999, Dehnen-Schmutz et al., 2007, Castro and Jaksic, 2008, Dawson et al., 2009; earthworms, Cameron and Bayne, 2009; vertebrates, Jeschke and Strayer, 2006; birds, Chiron et al., 2009; fish, Copp et al., 2010) (see Figure 4-8A-C). A variety of linear and non-linear rela- tionships have emerged from these analyses. However, even in the case of a strong statistical relationship, the question of causation must be examined care- fully to minimize spurious significant effects caused by confounding variables (Lonsdale, 1999; Figure 4-8D-E) and to distinguish observed patterns from null expectations (Lockwood et al., 2009). For the case of ballast water, linear risk–release relationships have been es- timated in a number of systems (Box 4-3). There are both theoretical assump- tions and logistical challenges in developing these models. The first assumption is that total organism number, regardless of the number or abundance of the con- stituent species, is a reasonable predictor of the number of successfully estab- lishing species. Although these two variables do not have an explicit causal connection, it is intuitively clear that increasing total abundance requires in- creasing species number, abundance, or both, any of which would be predicted to increase invasion risk. However, the causation is indirect and the precise na- ture of this relationship is unclear. The second assumption is that this relationship is linear. Although theory predicts that this relationship should be non-linear, and would be expected to be sigmoidal (as in the single-species case) if Allee effects were operative for spe- cies with the highest likelihood of establishment, the trend in a limited dataset may be indistinguishable from linear. Therefore, statistical model fitting should compare multiple candidate models before selecting a linear (or any other) shape. In fitting such a model to data, the operational challenges quickly become clear. We have estimates of organism density and number of invaders for only a handful of locations (Table 4-2). The density measures have been made with different methods and taxonomic foci, are recent and short-term relative to the accumulation of invaders over decades of ballast water release (Table 4-2), and are patchy and possessed of considerable uncertainty (Verling et al., 2005; Min- ton et al., 2005; Lawrence and Cordell, 2010). Even if density estimates were entirely accurate and precise, we would not necessarily expect current estimates to predict historical invasion success. In the face of these difficulties, some authors have used shipping metrics such as vessel number, vessel tonnage, and ballast volume as proxies for propa- gule pressure (Box 4-3). At first glance, proxy variables appear to offer an ap- pealing way to proceed, since unlike organism density data, vessel traffic data are relatively easy to collect, can be collected retroactively, and might seem to be plausible stand-ins for organism density. However, their use relies on the

96  Propagule Pressure and Invasion Risk in Ballast Water    A–C  D–E   FIGURE  4‐8    Descriptive  models  of  multi‐species  risk–release  relationships.   Comparison of log‐log (dotted), log‐linear (dashed), and Michaelis‐Menten (sol‐ id) equations fit to number of (A) mollusks, (B) plant pathogens, and (C) insects,  vs.  cumulative  imports  over  time  in  the  U.S.  (Levine  and  D’Antonio,  2003).   (D)  Log‐log  plot  of  number  of  visitors  vs.  number  of  native  plant  species  in  nature  reserves  worldwide;  (E)  log‐linear  plot  of  number  of  nonindigenous  plant  spe‐ cies  vs.  visitor  residuals  (non‐native  plant  species  as  a  function  of  the  residuals  from the relationship in D) (p<0.001) (Lonsdale, 1999).  SOURCE:  (E) Reprinted,  with permission, from Lonsdale (1999).  © 1999 by Ecological Society of Ameri‐ ca.  

Relationship Between Propagule Pressure and Establishment Risk   97    BOX 4‐3  Linear Statistical Multispecies Models    Linear  models  have  been  used  to  estimate  the  multi‐species  risk–release  relationship  for  ballast  water.   To  date,  the  most  widely  analyzed  data  at  this  scale  are  invasion  trends  in  the  Great  Lakes.   A  variety  of  analyses  have  been  conducted  for  this  system  using  different  data  subsets,  and  different  depen‐ dent and independent variables.  In all the analyses, the data have been parsed  into  temporal  intervals  to  provide  multiple  data  points  for  model  fitting.   The  results are not consistent among analyses.  Depending on the data subset, there may or may not appear to be a signifi‐ cant  risk–release  relationship.   Ricciardi  (2001)  used  a  linear  regression  to  esti‐ mate the rate of all species invasion vs. shipping tonnage in net tons, by decade  from  1900  to  1999  (y  =  0.062x,  r2  adj  =  0.62,  p<0.004).   This  analysis  was  up‐ dated  by  Ricciardi  (2006)  for  only  those  free‐living  invaders  assumed  to  have  been  introduced  by  shipping  (y  =  0.05x;  r2  =  0.516,  p<0.019;  Figure  4‐9A).    In  contrast,  Grigorovich  et  al.  (2003)  analyzed  Great  Lakes  invasion  data  in  5‐year  intervals  from  1959‐1999;  their  data  show  no  clear  trend  in  new  invaders  as  a  function  of  the  net  tonnage  of  overseas  ballasted  traffic,  and  if  anything  a  neg‐ ative  relationship  with  the  number  of  overseas  ballasted  vessels  (Figure  4‐9B‐ C).  Linear  relationships  have  been  used  to  estimate  a  per‐ship  invasion  rate.   Drake and Lodge (2004) reanalyzed the data in Ricciardi (2001) against shipping  tonnage  in  metric  tons,  using  a  linear  regression  with  a  Poisson  error  distribu‐ tion  (y  =  8.47  x  10‐8x;  p<0.0001).   Rescaling  by  the  average  ship  tonnage,  they  estimated  a  per‐ship  probability  of  causing  an  invasion  as  0.00044  (95%  CI  =  0.00008),  equivalent  to  1  species  per  2275  ships  or  0.44  invasions  per  1000  ships  (95%  CL  0.36,  0.52).   This  estimate  was  based  on  all  nonindigenous  spe‐ cies  in  the  Great  Lakes,  regardless  of  their  presumed  vector.   In  contrast,  Cos‐ tello  et  al.  (2007)  used  annual  data  on  ship‐mediated  animal  introductions  alone  from  1959–2000,  as  a  function  of  number  of  ships  (Figure  4‐9D),  and  ob‐ tained  a  maximum  likelihood  estimate  of  0.14  animal  invasions  per  1000  ships  (95% CL 0.02, 5.2).  Linear  relationships  have  also  been  used  to  estimate  a  per‐organism  inva‐ sion  rate  for  17  North  American  ports  (Reusser,  2010).    These  data  were  not  separated  into  time  intervals,  so  the  relationship  for  each  port  was  based  on  a  single  data  point.   The  dependent  variable was  the  total  number  of  established  invaders  (invertebrates  and  macroalgae)  from  1981–2006  considered  to  likely  to  have  been  introduced  by  ballast  water.   The  independent  variable  was  the  total volume of foreign ballast water discharged from 2005–2007, multiplied on  a  per‐ship  basis  by  a  random  selection  from  an  empirically  determined  zoop‐ lankton density distribution that spanned eight orders of magnitude (based on  box continues next page 

98  C  A    BOX 4‐3 Continued  B  D  FIGURE  4‐9  Invasion patterns in the Great Lakes.  (A) The number of invaders scales positively with net shipping tonnage by decade 1900–1999  (Ricciardi,  2006).  The same trend is not evident from scatter plots of (B) number of invaders at 5‐year intervals from 1959–1999 vs. net tonnage or (C) vs. number of  ballasted ships (replotted from Grigorovich et al., 2003) or of (D) annual number of invaders vs. number of ships (plotted from data in Costello et al. 2007,  Propagule Pressure and Invasion Risk in Ballast Water  appendices A‐B).  SOURCE: (A) Reprinted, with permission, from Ricciardi (2006).  © 2006 by John Wiley and Sons.  (B, C) Reprinted, with permission, from  Grigorovich et al. (2003).  © 2003 by NRC Research Press.  (D) Reprinted, with permission, from Costello et al. 2007.  © 2007 by Ecological Society of Amer‐

Relationship Between Propagule Pressure and Establishment Risk   99          354 ships sampled in four U.S. ports, of which three are included in the 17 ana‐ lyzed; Minton et al., 2005).  Repeated random draws generated a bootstrapped  estimate  of  the  median  and  the  first  and  third  quantile  invasion  rates  for  each  port.   Together,  these  per‐capita  invasion  rates  spanned  four  orders  of  magni‐ tude  from  10‐11  to  10‐8,  or  one  invasion  for  every  10  million  to  10  billion  organ‐ isms  (Reusser,  2010).   Interestingly,  the  data  provided  no  evidence  of  a  strong  risk–release  relationship  across  ports,  based  on  either  number  of  vessels  or  ballast water volume (Figure 4‐10A‐B).    A  B    FIGURE  4‐10   Across  17  U.S.  coastal  ports,  the  number  of  invaders  reported  from  1981– 2006  shows  no  strong  relationship  with  (A)  number  of  ships  with  foreign  ballast  2005– 2007  or  (B)  volume  of  ballast  water  discharged  2005–2007  (plotted  from  data  in  Reuss‐ er, 2010, Table 3‐2).       box continues next page      

100  Propagule Pressure and Invasion Risk in Ballast Water      BOX 4‐3 Continued    The results from these attempts to characterize a multispecies risk–release  relationship  are  ambiguous  and  highlight  the  challenges  in  quantifying  propa‐ gule  pressure.    The  Great  Lakes  analyses  (Figure  4‐9)  use  shipping  traffic  va‐ riables  that  in and  of  themselves  do  not directly cause  invasions,  and  that  have  not  been tested  for  their  correspondence  to  organism density.   In  other words,  they  are  serving  as  proxies  for  propagule  pressure  under  the  untested  assump‐ tion  that  they  scale  linearly  with  propagule  pressure.    The  coastal  analysis  (Reusser,  2010)  is  an  attempt  to  use  a  more  direct  measure  of  propagule  pres‐ sure.   However,  ballast  water  volume  is  scaled  up  assuming  the  same  organism  density distribution for all ships, and the relationship is based on a mismatched  dataset of invasion data, shipping data, and organism data from different years  and ports.   Both  Cohen  (2005,  2010)  and  Reusser  (2010)  have  proposed  using  a  linear  risk–release  relationship  to  inform  ballast  water  discharge  standards  [Figure  4‐ 2A;  reviewed  in  Lee  et  al.  (2010)];  Drake  and  Lodge  (2004)  used  a  linear  risk– release  relationship  embedded  within  a  gravity  model  to  investigate  risk‐ reduction  strategies.    The  primary  theoretical  challenge  in  developing  these  approaches  is  identifying  the  expected  shape  of  the  relationship,  particularly  given  that  even  total  organism  number  cannot  be  expected  to  directly  predict  species  establishment.   The  primary  practical  challenge  is  the  current  absence  of  the  appropriate  data,  i.e.,  spatially  and  temporally  matched  variables,  and  untested or unrepresentative proxy variables.    critical assumption that organism density is homogeneous across tanks and ves- sels–an 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 risk–release relationship. Any statistically significant relationship that emerges may represent a spurious correlation. Any non-significant relationship could be the result of a non-representative proxy, or from the absence of a fun- damental underlying relationship between risk and release density resulting from the myriad other factors that influence success (see Chapter 3). The results from analyses to date are ambiguous and highlight that proxy variables may not al- ways be reliable predictors of invasion risk, particularly across regions (see Box 4-3). The principle of using proxy variables is not without merit, but it is essen- tial to select and test candidate variables with care before assigning any meaning to their relationships, or lack thereof.

Relationship Between Propagule Pressure and Establishment Risk   101    Mechanistic Models Mechanistic modeling frameworks can be scaled up from single species models to represent the release of multiple species from multiple ships at mul- tiple locations and multiple times. As the mathematical framework of such a model expands, so too do the data requirements for model parameterization and validation. It is crucial to recognize that it is not possible, mechanistically, to predict the invasion risk associated with the release of an unknown number of unidenti- fied species at unknown abundance, density, and frequency. Any mechanistic multi-species model is necessarily parameterized for a specific group of taxa, and its output is therefore as case-specific as that of a single-species model. To parameterize such a model requires knowing the identities and numbers of all released species, and knowing which of those incipient introductions established and failed. These data can be obtained from controlled experimental studies (e.g., Tilman, 1997; Shurin, 2000; Lee and Bruno, 2009), but not at the full scale of ballast water discharge. At present, therefore, there are not sufficient taxonomic information or em- pirical data to parameterize either a probabilistic or a demographic model for all the species in a ballast assemblage. Even if such data and information were available, the time scale mismatch between ballast water discharge and invasion record datasets would still prevent the validation (testing) of a mechanistic mod- el against the empirical data. Nevertheless, one can examine what the frame- work of such a model might look like with an eye to parameterizing it in the future. Probabilistic Models Probabilistic models of the single-species invasion process can be scaled up to create a framework representing the introduction of multiple species, as well as multiple vessels, locations, and releases. This model expresses invasion risk as either the expected number of established species, or the probability that at least one species will establish, in a given time frame. Organism release is spe- cified as the number of individuals of each species released, and may also in- clude separate releases from multiple vessels at multiple locations on multiple occasions (Costello et al., 2007; USCG, 2008). As for single-species models, when the constituent probabilities of the multispecies model are drawn from distributions, the approach is described as a hierarchical probability model (HPM). HPM has the advantage of explicitly representing the known uncertain- ty in the inherently stochastic invasion process, in the same way that a dynamic demographic model can be made stochastic by drawing parameter values from a distribution.

102  Propagule Pressure and Invasion Risk in Ballast Water    The multiple-species, multiple-invasion framework readily allows new in- formation about the characteristics of release and establishment (e.g., vessel type and source location; habitat and seasonal differences) to be incorporated into the hierarchical framework. This is particularly useful for evaluating which of many possible scenarios resulting in the observed number of species invasions is best supported by the data. Box 4-4 outlines a general framework for a multi- species HPM of invasion establishment. While this mechanistic hierarchical probability model poses an interesting framework for thinking formally about multi-species and multi-variable risk-release relationships, it has not yet been parameterized or validated with an empirical dataset, and the prospects of doing so are currently remote. Nonetheless, HPM for both single and multi-species scenarios holds advantages over other models because (1) it offers a mechanistic representation of the invasion process, (2) in the absence of detailed distribution data for all the parameters, it can be used in a simplified (i.e., non-hierarchical, point-estimate) version, and (3) as more data become available, it can be easily expanded to incorporate different species, locations, seasons, vessels, etc. See the conclusions of this chapter for a summary recommendation about this ap- proach. In a somewhat different approach, Costello et al. (2007) adapted a probabil- istic model of species introduction and detection over time (Solow and Costello, 2004) to test the relationship between invasion rate and number of vessels arriv- ing annually in the Great Lakes. This analysis highlights the influence of detec- tion lag in confounding our ability to assess the effectiveness of changes in bal- last management. Although this relationship is formulated as a probabilistic model, it is based on a proxy variable (number of vessels) so its mechanistic interpretation is unclear. It should be noted that at present, most available data are surrogates of propagule pressure (e.g., number of ships, ballast volume, or ballast discharge) and the number of invaders observed within a given time frame. The existing multispecies models (and most single species models) as- sume that a reduction in ballast reduces the number of invaders linearly in a sys- tem or probabilistically per ship (e.g., Drake and Lodge 2004; Costello, 2007). This may be reasonable given the limited data currently available to construct these models. Dynamic Demographic Models Like probabilistic models, dynamic demographic models could in principle be scaled up from the single-species scenario to model the combined risk of many species establishing. Again, such an exercise would require constructing and parameterizing a model with the identity, initial number, and invasion suc- cess of each population, and again, the resulting relationship would apply only to that suite of species.

Relationship Between Propagule Pressure and Establishment Risk   103    BOX 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   

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-

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

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

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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The human-mediated introduction of species to regions of the world they could never reach by natural means has had great impacts on the environment, the economy, and society. In the ocean, these invasions have long been mediated by the uptake and subsequent release of ballast water in ocean-going vessels. Increasing world trade and a concomitantly growing global shipping fleet composed of larger and faster vessels, combined with a series of prominent ballast-mediated invasions over the past two decades, have prompted active national and international interest in ballast water management.

Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water informs the regulation of ballast water by helping the Environnmental Protection Agency (EPA) and the U.S. Coast Guard (USCG) better understand the relationship between the concentration of living organisms in ballast water discharges and the probability of nonindigenous organisms successfully establishing populations in U.S. waters. The report evaluates the risk-release relationship in the context of differing environmental and ecological conditions,including estuarine and freshwater systems as well as the waters of the three-mile territorial sea. It recommends how various approaches can be used by regulatory agencies to best inform risk management decisions on the allowable concentrations of living organisms in discharged ballast water in order to safeguard against the establishment of new aquatic nonindigenous species, and to protect and preserve existing indigenous populations of fish, shellfish, and wildlife and other beneficial uses of the nation's waters.

Assessing the Relationship Between Propagule Pressure and Invasion Risk in Ballast Water provides valuable information that can be used by federal agencies, such as the EPA, policy makers, environmental scientists, and researchers.

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