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Potential Applications of Population Viability Analysis to Risk Assessment for Invasive Species

Posted on: Wednesday, 28 December 2005, 03:02 CST

By Andersen, Mark C

ABSTRACT

Population viability analysis, the use of ecological models to assess a population's risk of extinction, plays an important role in contemporary conservation biology. The premise of this review is that models, concepts, and data analyses that yield results on extinction risk of threatened and endangered species can also tell us about establishment risks of potentially invasive species. I briefly review important results for simple unstructured models, demographic models, and spatial models, giving examples of the application of each type of model to invasive species, and general conclusions about the applicability of each type of model to risk analysis for invasive species. The examples illustrate a portion of the range of potential applications of such models to invasive species, and some of the types of predictions that they can provide. They also highlight some of the limitations of such models. Finally, I present several conjectures and open research questions concerning the application of population viability analyses to risk analysis and control of invasive species.

KeyWords: exotic species, pest risk assessment, ecological model, demography, establishment, review.

INTRODUCTION

In managing threatened and endangered species, and in assessing threats to their persistence, it is frequently useful to integrate information from multiple sources in a quantitative modeling framework. This need has led to the development of the set of techniques collectively known as population viability analysis (PVA). PVA, broadly construed, is the use of quantitative methods of data analysis and modeling to assess the extinction risk of populations of threatened and endangered species. Viability analysis, and general principles derived from applications of viability analysis, plays a pivotal role in contemporary conservation biology, and is one of the most active areas of application of theoretical ecology and ecological modeling.

This review's premise is that models, concepts, and data analyses that yield results on extinction risk can also tell us something about risk of establishment. This implies that, both biologically and mathematically, there is an inverse relationship between extinction and establishment, that is, that establishment and extinction are two sides of the same coin, because the probability of establishment of a small local population is one minus its extinction probability (assuming, as seems biologically reasonable, that these are the only two possible outcomes). It is this equivalency that justifies the application of models and concepts developed for threatened and endangered species to invasive species.

The potential uses of PVA in risk analysis for invasive species are analogous to their uses for threatened and endangered species as summarized in Morris and Doak (2002), but with a few important differences, because not all possibilities apply to invasives. In the context of risk analysis for invasive species, PVA-based methods may be used for

1. Assessing the risk of establishment of a population of a particular species at a particular site.

2. Comparing risks of establishment at a particular site across several potential invasive species.

3. Analyzing monitoring data from established invasive species as a decision-support tool for management intervention.

4. Identifying key life cycle stages and/or demographic processes as targets of focused management interventions for established invasive species.

5. Determining a tolerable range of numbers of arrivals of potential invasives.

A crucial point to remember for all of these applications is that, in the case of invasives, extinction is a desirable outcome rather than an outcome to be avoided, as it is for threatened and endangered species.

Below I briefly review basic concepts and methods of PVA, assuming or citing most of the required background from theoretical ecology; I also present examples to show how the application of these methods may extend from threatened and endangered species to include risk assessments and evaluations of management options for potential or established invasive species. The presentation here closely follows that of Morris and Doak (2002) and others (Andersen 1994; case 2000; Caswell 2001). The intent is to introduce practitioners of ecological risk assessment and pest risk assessment to a range of under-utilized tools, and to encourage practitioners of PVA to apply their customary analytical tools to problems of invasive species.

BASIC POPULATION CONCEPTS

The simplest population models used in PVA consider changes in total population size without regard for such details as the age, size, or life-cycle stage of the individuals comprising the population, or for their movements through or position within the landscape. One factor that is considered by all such PVA models is stochasticity. One might think that it would be acceptable to ignore stochasticity in modeling extinction or establishment of populations because one might expect stochastic variation to even out in the end. However, populations with stochastic variation in the vital rates tend to grow more slowly than we might expect based on their mean vital rates. In other words, a deterministic model will consistently overestimate the growth rate of a population with stochastically varying vital rates. This is because, in general, the appropriate measure of population growth in a stochastic environment is the geometric mean growth rate rather than the arithmetic mean growth rate (Lewontin and Cohen 1969: Caswell 2001; Morris and Doak 2002).

Stochastic effects can arise in two ways in natural populations. Environmental stochasticity is stochastic variation in vital rates due to environmental variability. All populations to some degree experience a natural sequence of good years and bad years; this variation drives environmental stochasticity. Demographic stochasticity, on the other hand, is stochastic variation due to inherent variability in demographic processes. This leads to stochastic effects analogous to genetic drift in population genetics (Gillespie 1998).

Because the effects of demographic stochasticity are particularly strong for the smallest populations, it will be especially important for potential invasive species. This contrasts with the case for threatened and endangered species, where demographic stochasticity is often neglected in the belief that, if a population is small enough to experience a strong influence from demographic stochasticity, it is already in imminent danger of extinction. In addition, stochasticity in general (especially environmental stochasticity, because its effects are constant across all population sizes) decreases population growth below what one expects in a constant environment (Lewontin and Cohen 1969; Tuljapurkar and Orzack 1980; Tuljapurkar 1986). Like the effect of demographic stochasticity, this makes establishment of a potentially invasive species less likely.

Allee effects (positive density-dependence, especially at low population densities) are also potentially very important in the establishment of invasive species. These effects have been shown capable of producing latent periods in the early phases of establishment as well as minimum threshold population sizes for establishment (Lewis and Kareiva 1993). Thus, although negative density-dependence (mostly a large-population phenomenon) may be safely ignored in applications of PVA-based models to invasive species, it may not be safe to ignore positive density-dependence. In addition, it may prove that the Allee effect, the geometric mean effect, and demographic stochasticity together may be responsible for much of the strong filtering of species (Williamson 1996) between the entry and establishment phases of species introductions and invasions.

EXAMPLES

An article by Drake (2004) specifically examines the role of Allee effects in establishment of invasives. In general, Allee effects have been shown to generate thresholds in establishment probability, depending on the size or density of the population (Dennis 1989,2002). Drake models the population dynamics of the invasive freshwater cladoceran Bythotrephes longimanus, including in the model both the Allee effect and the seasonal parthenogenesis characteristic of most cladocera. He finds that the Allee effect lowers the risk of establishment for sexually reproducing populations, producing a threshold population size for establishment. He further finds that seasonal or facultative parthenogenesis can reduce this threshold to zero. Drake also suggests that Allee effects may influence the spatial spread of invasive species because dispersal acts essentially like mortality at the level of the local population, potentially reducing the population below a size that can persist. This article is an example of the use of PVA-based models to compare risks of establishment across several species. It also provides a useful general principle for invasive species risk assessments: ecological or life-history traits (such as facultative parthenogenesis) that allow a species to circumvent the Allee effect increase that species' risk of establishment and spread.

Several authors have derived results on the establishmen\t of mutant alleles (Keiding 1975; Ludwig 1975; Chesson and Ellner 1989). Haccou and Iwasa (1996) provide one of the few studies to directly address the question of establishment of invaders. Their model is an inhomogeneous branching process; thus it incorporates both demographic and environmental stochasticity, but does not include age or stage structure in its description of the population. To my knowledge, inhomogeneous branching process models have not been used for any actual population viability analyses, presumably because of their mathematical complexity. Still, the model of Haccou and Iwasa (1996) provides some useful insights. They show that the probability of success of a single invader at any given time will depend on whether the invader arrives during a favorable or unfavorable period (because of environmental stochasticity), and thus that the probability of invasion success differs for simultaneous or sequential arrivals at a single site and for invasions at different sites. Their most important finding is that the probability of success of sequential arrivals at a single point of entry exceeds that of simultaneous arrivals at multiple points of entry. Although not strictly speaking an application of PVA to invasive species, this article nevertheless provides potentially useful general principles concerning establishment risk.

DEMOGRAPHIC MODELS

Demographic approaches such as the classic Leslie and Lefkovitch matrix models (Caswell 2001) are also widely used in PVA. These models account for differences between individuals in the population due to such factors as age or size. Whether one is considering extinction or establishment, it is essential to consider stochastic extensions to the basic deterministic population projection matrix models (Tuljapurkar 1989; Tuljapurkar 1990, 1994; Caswell 2001). It makes sense to consider such models as representing random draws from some set of possible population projection matrices with a given joint distribution of matrix elements. In practice (e.g., for simulations on a computer) we can either draw an entire matrix at once from some given set, or draw individual matrix elements from some probability distribution and assemble them into a population projection matrix.

The expected population growth rate for such a population, that is, the most likely long-term logarithmic population growth rate (Tuljapurkar and Orzack 1980; Caswell 2001 ) is referred to as the stochastic logarithmic growth rate λ^sub s^. Analysis of an asymptotic approximation for λ^sub s^ (Tuljapurkar 1990) shows that larger environmental fluctuations lead to lower growth rates. In addition, positive covariances among vital rates increase the variation in population growth, thus decreasing growth rates, whereas negative covariances among vital rates raise the overall growth rate (because negatively covarying fluctuations tend to cancel out). Thus life history theory, which deals explicitly with covariances among vital rates, can potentially tell us a great deal about the establishment of invasive species (Fitzgerald 1994; Mangel 1994; Heppell and Crowder 1998; Heppell et al. 2000; Rieman and Dunham 2000).

If the two outcomes of establishment and extinction are mutually exclusive and collectively exhaustive, as seems reasonable, then it can be assumed that the probability of establishment is simply one minus the probability of extinction. If this is the case, then some well-known results for stochastic matrix models can be applied to give approximate establishment probabilities as a function of λ^sub s^ and the variance of environmental fluctuations (Tuljapurkar and Orzack 1980; Lande and Orzack 1988). However, these formulae are not likely to provide more than a rough guide for comparing the establishment probabilities of several species.

It is also common for PVAs to examine the sensitivity of λ^sub s^ to changes in the vital rates, in other words, the partial derivatives of λ^sub s^ with respect to each projection matrix element. Stochastic sensitivities are influenced, not just by mean vital rates, but also by their variances and covariances. For ease of comparison, sensitivity values are typically rescaled; these rescaled sensitivities are called elasticities. They are simply the proportional change in λ^sub ^s given a proportional change in the vital rate or matrix element of interest.

Rather than using available approximation formulas (Tuljapurkar 1982, 1986, 1989, 1990), it is often easier to compute sensitivities and elasticities by simulation. To implement this, one may simply vary each population projection matrix element (or variance or covariance) one at a time over a predetermined range of values, and then compute the sensitivity as the change in the stochastic growth rate divided by the change in the vital rate or matrix element.

These sensitivity values may be put to a number of uses. For example, they enable us to evaluate the effects of errors in estimation of individual vital rates on errors in estimation of λ^sub ^s thus allowing more focused data-collection for monitoring efforts. They also allow us to evaluate the effects of management strategies, because these strategies are almost always targeted at particular life-cycle stages. It is for this purpose that sensitivity analysis is most frequently used in PVA (Crooks et al 1998; Heppell 1998; Heppell and Crowder 1998), and for which it is most likely to be of use for interdiction and control of invasive species. Still, there are a number of caveats in the application of sensitivity analysis in conservation biology (Mills et al 1999); presumably these caveats also apply to the use of such methods for invasive species.

EXAMPLE

Bartell and Nair (2004) used a stage-based stochastic matrix population model to assess the risk of establishment of populations of Asian longhorned beetle (Anoplophora glaimpennis) in North America. In particular, they used the model to examine the relationship between propagule pressure (i.e., exposure in the terminology of risk assessment) and risk of establishment of a beetle population (i.e., one possible response to the stressor represented by the beetle itself). Their choice of a stage- structured matrix model was driven both by the level of detail in available data on the beetle, and by the need to provide predictions specific to different stages in the beetle's life cycle. The model recognizes four life-cycle stages (eggs, larvae, pupae, and adults), with parameters estimated from published and unpublished data. Risk characterization is accomplished through stochastic simulation. The authors thoroughly discussed formulation of the model, estimation of parameter values, and use of the model in risk assessment. They also presented a quantitative uncertainty analysis that could be used to guide collection of additional data. Their study demonstrates the high data requirements for applying such models to quantitative risk assessment. This article is an example of the use of PVA-like models to assess the risk of establishment of a particular potentially invasive species.

MODELS OF SPATIAL AND SPATIOTEMPORAL DYNAMICS

Spatial processes and phenomena can be crucial in understanding the ecology of populations (Pulliam 1988; Holmes et al. 1994; Wennergren et al. 1995; Moilanen and Hanski 2001). Populations occupying multiple sites are influenced by movement rates between sites (Gilliam and Fraser 2001; Campbell et al. 2002), habitat quality variation across sites and its effects on demographic processes (Doak 1995; Pulliam 2000; Amarasekare and Nisbet 2001; Donahue et al. 2003), and correlations between sites (Engen et al 2002; Lindenmayer et al. 2002; Urban et al. 2002). Populations spreading across a more-or-less continuous habitat are influenced by spatial variations in habitat permeability and quality (Murray et al. 1986; Okubo et al. 1989; Andow et al. 1993; Higgins and Richardson 1996; Suarez 2000), as well as by other sources of variation in movement rates and by the probability structure of dispersal distances (Kot 1992; Lewis and Kareiva 1993; Lewis et al. 1996; Neubert et al. 2000).

The complex questions associated with spatiotemporal population dynamics have led to a flourishing of theory addressing these questions. This body of theory includes stochastic models (Mollison 1977, 1978; Durrett and Levin 1994a, b) and deterministic diffusion models (Skellam 1951; Okubo and Levin 2001), as well as integrodifference models (Andersen 1991; Kot 1992; Neubert et al. 1995; Alien et al. 1996) and multiregional models (Lebreton 1996; Brooks and Lebreton 2001). In terms of applications to invasive species, spatial spread and geographic range expansion are important factors in the ecology of invasions (Williamson 1996).

In addition, propagule pressure, one of the main determinants of successful establishment (Haccou and Iwasa 1996; Hanski et al. 1996; Williamson 1996; Keeling 2002), has an important spatial component. Many imported products that may harbor potential invasive species arrive at multiple ports of entry. The number of ports of entry is certainly a component of the overall propagule pressure for potential pest species on that commodity (Manchester and Bullock 2000; Richardson et al. 2000). In the entry and establishment phases of biological invasions, multiple ports of entry may or may not function as classical metapopulations, however (Gutierrez and Harrison 1996). In a classical metapopulation, local populations are linked by dispersal, or at least by potential dispersal links between the multiple sites (Hanski 1998; Hill et al. 2002). For the case of potential pests arriving at multiple ports of entry, this would only be true if the ports were linked by, for example, ground transport of the host commodity or some other product that could harbor the species (Bartell and Nair 2004).

If this were the case, simple pa\tch-occupancy models (Keymer et al. 1998; vanRensburg et al. 2000; Hokit et al. 2001; Ovaskainen and Hanski 2002; Hanski and Ovaskainen 2003) might reveal conditions under which a species could persist, based on the chance of establishment of the species at one or more ports of entry, and the rate of extinction of incipient local populations. Alternatively, one could consider the case of multiple ports as a type of mainland- island system, in which there is a single source of colonists, the mainland (the country of origin of the commodity), and a series of islands (the ports of entry) receiving colonists. However, a simple patch-occupancy model for this situation shows that some fraction of the sites will always be occupied regardless of the relative magnitudes of the extinction and colonization rates, because of the constant propagule pressure from the mainland (Gotelli2001).

Diffusion Models

Many sources contain maps of the progress of species invasions (Elton 1958; Hengeveld 1989; Williamson 1996). The expansion of the geographic range of an invading species is one of the most spectacular aspects of biological invasions, and one of the most readily modeled. Diffusion models have been the tool of choice to describe this process for some time (Fisher 1937; Skellam 1951).

If individuals tend to move in relatively small increments, and if their movements are uncorrelated, then the individuals will be moving in simple random walks (Turchin 1998) and the aggregate or collective result of these individual random walks will be a diffusion (case 2000). For both exponential (Skellam 1951 ) and logistic (Fisher 1937) population growth, the asymptotic rate of spread of the population front is 2[the square root of]rD where r is the population's intrinsic rate of increase and LAs its diffusion coefficient (Okubo and Levin 2001); thus the rate of geographic range expansion of an invasive species will depend on both reproduction (through r) and dispersal (through D). Adding stochasticity to these models can result in slightly higher asymptotic rates of spread, up to 30% or so higher than 2[the square root of]rD (Mollison 1977; Andow et al 1990, 1993). Age-structured models show up to threefold increases in rate of spread over the Skellam and Fisher models (van den Bosch et al. 1990, 1992). Allee effects, on the other hand, can slow down the rate of spread to as low as one-third to one-half of 2[the square root of]rD; this slower rate of spread due to Allee effects may account at least in part for the long latent periods observed in many biological invasions (Lewis and Kareiva 1993; Shigesada and Kawasaki 1997).

All these variations involve changing the basic assumptions of the Fisher and Skellam models concerning reproduction. There are also two variations of dispersal behavior that have been examined. If the movements of individual organisms are correlated rather than independent as assumed by the diffusion equation, dispersal is described by a telegraph equation (Turchin 1998; Okubo and Levin 2001). The rates of spread predicted by the telegraph equation are very close to those predicted by the Fisher and Skellam equations (Holmes 1993). If we change the focus from individual movement paths to the probability distribution of dispersal distances or redistribution kernel, we may ask what happens if the redistribution kernel is a fattailed distribution such as the Cauchy distribution (Shaw 1994, 1995). Under this assumption no actual wave front exists, and new populations may appear at practically any distance from the current edge of the population's geographic distribution. This resembles the pattern of spread seen in many plant pathogens (Williamson 1996).

Examples

One interesting application of metapopulation models comes from the work of Hanski (1999). The motivation for this application is the increasing emphasis in conservation biology on habitat restoration and species reintroductions. Reintroductions into networks of habitat patches may have greater success than reintroductions at single sites. Interestingly, the same concerns apply to eradication of established invasive species, and initial establishment of invasives. Multiple sites of entry are a major component of propagule pressure, itself a major determinant of establishment risk.

To assess strategies for allocating available individuals for reintroduction across available habitat patches, Hanski examines a model with Ricker local dynamics, stochastic extinctions, and local dispersal and colonization. Phrasing his findings in terms of invasive species, the implications of his results are that (a) large areas of high-quality habitat with other habitat areas nearby are particularly vulnerable sites for entry and establishment, (b) the larger the number of arriving organisms, the greater the probability of establishment, (c) risk of establishment is higher in less variable environments, and (d) high growth rates and dispersal rates make a species more likely to establish. Thus results obtained specifically for species restoration efforts also have direct applicability to species invasions.

Neubert and Parker (2004) present an application of integrodifference equations to predict rates of spread of Scotch broom Cytisus scoparius at local scales. Scotch broom is an aggressive invasive plant with seeds that are both ballistically dispersed and ant-dispersed. Integrodifference equations are discrete-time equivalents of the diffusion models discussed earlier. Their chief advantage is that they model dispersal through a redistribution kernel; this lends itself to empirical applications because histograms of dispersal distances are often the most readily available form of dispersal data, at least for plants. Neubert and Parker discuss deriving redistribution kernels from both mechanistic models of propagule dispersal and from measured distributions of dispersal distances. They also present extensions to periodic environments and stage-structured populations, and illustrate the use of sensitivity analysis in the context of integrodifference equations. They argue that an understanding of the dynamics and mechanisms of spread can inform decisions about invasive species control. This article is an example of the use of PVA-based models to identify key demographic processes as targets of management intervention for an established invasive species.

CONCLUSIONS

The basic premise of this review is that the theoretical problem of the establishment of an invasive species is essentially the inverse of the problem of extinction for an endangered species, and thus that models developed for one of these two situations may successfully be applied to the other. I have shown that simple unstructured models can yield useful general principles and specific results, and that demographic models and models of metapopulations and population spread can add substantial realism, data permitting. Thus I conclude that methods developed by conservation biologists to assess risks of extinction can provide considerable insight into risks of establishment as well.

The examples presented reinforce this conclusion. Drake (2004) allows us to compare risk of establishment across an entire taxonomic group (the Cladocera), Bartell and Nair (2004) estimates the risk of establishment of a single species, and Neubert and Parker (2004) provides a way of comparing and assessing management strategies for established invasive plants. Not all the potential uses of PVA-based models enumerated in the Introduction appear in the examples cited, however. Additional attempts to apply PVA-based methods to invasive species are needed to assess their usefulness for the range of potential uses listed in the Introduction to this review.

Theoretical results have contributed a great deal to the development of widely used general principles of extinction- proneness in conservation biology. Many of these results derive from PVA models in one form or another (Day and Possingham 1995; Gosselin 1996; Lindenmayer and Possingham 1996; Pagan et al. 1999; Hill and Caswell 2001 ; Brook et al 2002; Couvet 2002; Henle et al 2004). The same theoretical concepts and models can help clarify general principles of establishment-proneness in the study of invasive species. The work of Hanski (1999) and Haccou and Iwasa (1996) are examples of research leading to these types of general principles.

Still, many questions must be addressed before the potential of PVA-based models to contribute to risk analysis for invasive species can be fairly assessed or fully realized. I conjecture that it might be safe to ignore negative density-dependence in models of the risk of establishment, but that it is probably not safe to ignore positive density-dependence (i.e., the Allee effect). Further research is needed to clarify the relative importance of the two forms of density-dependence in determining risks of establishment. I have also predicted that demographic stochasticity may be more important in establishment than in extinction; research on the relative importance of these two factors is also needed. The results of Haccou and Iwasa (1996) suggest that rate of arrival may be more important than number of sites of arrival in determining invasion risk; it is not known whether this result also holds true for the mathematically simpler types of models usually employed in viability analysis.

I feel strongly that there is a role for PVA-based analyses in risk assessment and control of invasive species. PVA methods can provide numerous general principles for screening of potential species of concern, and for qualitative risk assessments, as well as a useful modeling approach for quantitative risk assessments. Although modeling is not likely to become a routine part of pest risk assessments, models should routinely be employed in assessing control options for established invasives. PVA-based models and the lessons learned fr\om their application in conservation biology are a good starting point, both for quantitative risk assessments and for developing invasive species control strategies.

ACKNOWLEDGMENTS

This research was supported by Cooperative Agreement 02-0101- 0047-CA between New Mexico State University and the USDA/APHIS. My thinking on this subject has been refined by conversations with Mark Powell, Richard Fite, Wendy Fineblum-Hall, Craig Chioino, and David Oryang. Additional useful comments were provided by two anonymous referees. Stephanie Caballero and Shelley Cowden provided clerical assistance, Jason Northcott provided library assistance, and Megan Ewald provided library and editorial assistance. This is a publication of the New Mexico State Agricultural Experiment Station.

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Mark C. Andersen

Department of Fishery and Wildlife Sciences, New Mexico State University, Las Graces, New Mexico, USA

Address correspondence to Professor Mark C. Andersen, Department of Fishery and Wildlife Sciences, New Mexico State University, Las Cruces, New Mexico 88003-0003, USA. E-mail: manderse@nmsu.edu

Copyright CRC Press Dec 2005

Source: Human and Ecological Risk Assessment

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