Received 4 July 2002 Accepted 29 August 2002 Published online 6 November 2002 An ounce of prevention or a pound of cure: bioeconomic risk analysis of invasive species Brian Leung1* , David M. Lodge1, David Finnoff 2, Jason F. Shogren3, Mark A. Lewis4 and Gary Lamberti1 1 Department Department 3 Department 4 Department 2 of of of of Biological Sciences, University of Notre Dame, Notre Dame, IN 46556, USA Economics, University of Central Florida, Orlando, FL 32816-1400, USA Economics and Finance, University of Wyoming, Laramie, WY 82071-3985, USA Mathematical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada Numbers of non-indigenous species—species introduced from elsewhere—are increasing rapidly worldwide, causing both environmental and economic damage. Rigorous quantitative risk-analysis frameworks, however, for invasive species are lacking. We need to evaluate the risks posed by invasive species and quantify the relative merits of different management strategies (e.g. allocation of resources between prevention and control). We present a quantitative bioeconomic modelling framework to analyse risks from nonindigenous species to economic activity and the environment. The model identiŽes the optimal allocation of resources to prevention versus control, acceptable invasion risks and consequences of invasion to optimal investments (e.g. labour and capital). We apply the model to zebra mussels (Dreissena polymorpha), and show that society could beneŽt by spending up to US$324 000 year21 to prevent invasions into a single lake with a power plant. By contrast, the US Fish and Wildlife Service spent US$825 000 in 2001 to manage all aquatic invaders in all US lakes. Thus, greater investment in prevention is warranted. Keywords: stochastic dynamic programming; non-indigenous; exotic species; risk assessment 1. INTRODUCTION Non-indigenous species are increasing worldwide (Sala et al. 2000), are one of the top causes of global biodiversity loss and environmental change (Mack et al. 2000; Lodge 2001) and are economically expensive (e.g. estimated to cost the US$137 billion year2 1 ; Pimentel et al. 1999). Recent efforts across many countries have highlighted the urgent need for more rigorous and comprehensive riskanalysis frameworks for non-indigenous species so that prevention and control strategies can be targeted appropriately (McNeely et al. 2001; National Invasive Species Council 2001). To develop an appropriate framework, we need to recognize that risk analysis of species invasions is inherently an interdisciplinary problem, involving ecology, economics and mathematics. Ecosystem conditions and species’ characteristics determine whether a nonindigenous species will establish itself in a new location and whether it will cause damage (although these conditions may be difŽcult to quantify; Mack et al. 2000). Economic conditions inuence the transport of nonindigenous species, and inuence the resources that are spent on preventing an invasion versus control after an invasion. Reasonable resource expenditure is, in turn, inuenced by the expected consequences of the invader. Thus, ecological and economic parameters together deŽne non-indigenous species risks, with mathematics providing the techniques that allow the most rigorous analysis possible. Accounting for the ecological and economic links and feedbacks is now critical in invasion biology (Perrings et * Author for correspondence (bleung@nd.edu). Proc. R. Soc. Lond. B (2002) 269, 2407–2413 DOI 10.1098/rspb.2002.2179 al. 2002; Simberloff 2002) and requires an interdisciplinary effort to unite risk assessment with risk management (Committee on Environment and Natural Resources 1999). We use stochastic dynamic programming (SDP; Bellman 1961) as the mathematical basis underlying our riskanalysis framework. SDP is efŽcient (Žnds global solutions to exponentially complex problems in linear time), permits uncertainty to be included in the analysis and has the exibility to incorporate the entire invasion process, including biology and economic components. Furthermore, management responses change with environmental and economic conditions, and the responses, in turn, moderate the environment and economy. In the SDP framework we can explicitly incorporate such forecasted interactions based on the best available data. We can identify the combination of prevention and control efforts that maximizes social welfare given uncertain invasion events. SDP has been useful in other management contexts (Shea & Possingham 2000), but this application to merge the ecology and economics of species invasions is novel. The conceptual underpinnings of the framework focus on processes common to all species invasions in all ecosystems (the ecology column of Žgure 1a): a species is transported in a pathway and is released into a new environment, where it may establish, spread and become abundant, with environmental and economic impact. Success at each step in this invasion process is probabilistic, and only a small proportion (but a large absolute number) of species survive the entire process and cause an impact. Ecological forecasts of success or failure at each transition interact with the economic circumstances associated with each transition (Žgure 1a). Social welfare (beneŽts minus 2407 Ó 2002 The Royal Society 2408 B. Leung and others Bioeconomic risk analysis of invasive species a) economics ecology source location of species objective transport and introductions prevention transport and survival in pathway costs control establishment value added investment abundance«spread production maximize societal welfare benefits non-market values impact implementation: model structure b) SDP memorize states, optimize strategies future social welfare cost_benefit and strategy current population info and strategy growth models, age structure seasonality control strategies convert to discrete states probability matrix of future population info calculate optimal labour, capital, species impact, invasion probability production, and non-market valuation, strategy cost–benefit analysis of life -history traits, propagule pressure, Allee effects prevention strategies Figure 1. Bioeconomic framework for invasions. (a) The conceptual approach to the ecological and economic components of a generalized invasion process. Both economic input and ecological states change over time and inuence one another. Our goal is to determine the optimal set of strategies that maximize welfare, where welfare can be a function of both market and nonmarket values. (b) Implementation of the conceptual approach through an operational model structure. The boxes and bold text represent modules, within which details (italic text) may be hidden (encapsulated) and modiŽed without affecting the entire model. Plain text represents the interfaces (information passed between modules). costs) is determined jointly by ecological and economic processes (Žgure 1a). By contrast, traditional risk analysis considers risk assessment (determining environmental goals, such as reducing invader abundance to a speciŽed level) and risk management (determining methods to reach those goals) separately (Simberloff 2002). Such separation implicitly assumes that the beneŽts of prevention and control are negligible below the level speciŽed in the risk assessment. Our perspective, however, is that it is important to determine quantitatively the beneŽts of incremental improvements in the environment relative to the costs of achieving them, thereby recognizing explicitly the interaction between ecology and economics. In our conceptual frameProc. R. Soc. Lond. B (2002) work, the level of risk society should accept occurs when the projected damages are less than the costs associated with prevention and control. This framework may be applied at different scales: for a nation, a region or a local area. We implement the conceptual model (Žgure 1a) into the computational SDP model, breaking the conceptual model into four operational modules (Žgure 1b). The ‘abundance and spread’ and ‘transport and establishment’ modules contain the biological aspects of an invasion and their interaction with control and prevention strategies. The functions encapsulated within these modules include processes such as recruitment and survival. Importantly, these processes represent details that may differ depending Bioeconomic risk analysis of invasive species B. Leung and others 2409 on the biological system, but do not affect the overall structure of the model. Only the movement between states (e.g. uninvaded, invaded, population density of invader) needs to be passed to the economics module. The economics module determines the costs and beneŽts of each state, and analytically determines the optimal labour and capital investment. The SDP module keeps track of the future states that have been calculated and determines the optimal strategies based on the economic beneŽt–cost analysis of the current state and the accumulation of future states based on expected trajectories. We Žrst present a hypothetical example to demonstrate the general application of our framework, so that its properties are not obscured by the idiosyncrasies that are inherent in speciŽc biological systems. We then test the utility of our framework on the real-world example of the invasions by zebra mussels (Dreissena polymorpha) of uninvaded lakes. We examined zebra mussels for the following reasons: they currently cost US industries an estimated US$100 million year2 1 (Pimentel et al. 1999); power plants and water Žrms continue to experiment with new control measures and schedules in an effort to maximize the beneŽts of zebra mussel control; zebra mussels have enormous environmental impacts (Ricciardi & Rasmussen 1998; Lodge 2001); and prevention of new infestations remains timely because zebra mussels are still expanding their range within North America (Bossenbroek et al. 2001). 2. MATERIAL AND METHODS (a) Hypothetical example We used a logistic growth model allowing uncertainty («) S D 12N dN = rN 1 «. dt k (2.1) SDP required discrete states (u), deŽned here as discrete levels of population abundance (u = N). We modelled probability of invasion (I9) per time interval as a function of base rate invasion probability (I) and prevention effort (E) I9 = Ie2 L E. (2.2) K¤ was analogously calculated. The welfare (w) of each state (u) was used in the SDP calculation MAX Wu ,t = wu ,X,E 1 X,E O Pu ,X,E,iWi,t 1 1, (2.8) where W was the cumulative welfare from the end time horizon (T ) to the current time (t), and P was the probability of moving from a state u to state i, given strategies X and E (chosen to maximize W ). W was calculated by moving backwards from t = T to t = 1. Thus, for each state at each time interval, we knew the optimal strategies and future trajectories. (b) Zebra mussel invasion (i) Biological sub-model Seasonally dependent recruitment rates were determined by counting settled postveligers on artiŽcial substrates from a power plant (electronic Appendix A, available on The Royal Society’s Publications Web site). We obtained age-dependent survival estimates from the literature (Akcakaya & Baker 1998) (electronic Appendix A). Estimates were skewed and non-negative, and were log(X 1 1) transformed (Zar 1984). Rates were converted to monthly bases. We used a point estimate of growth rate (DS) (Akcakaya & Baker 1998) as follows: DS = 16.3 2 0.343S, (2.9) where S is the size (in mm). We modiŽed growth to monthly rates occurring during May–October. We related volume to size by measuring water displacement and regressing shell length versus displacement (electronic Appendix A). We collapsed recruitment and survival rate distributions, age structure, seasonality, growth rates and size–volume relation (biological details) into a few variables directly causing industrial impacts to avoid the ‘curse of dimensionality’ (Bellman 1961) and keep SDP efŽcient. Here, we deŽned SDP states (u) as each possible control scenario and the associated total zebra mussel volume, which could then be related to damage estimates and control strategy. Based on nuclear power industry data, we used a point estimate of zebra mussel control cost (CX = US$1.6 million) and effectiveness (95% reduction in abundance of all age classes) (E. C. Mallen, personal communication). (ii) Economic sub-model Production (Q) followed an economic Cobb–Douglas functional form (Archer & Shogren 1996), relating labour, capital investment and damage due to the pest (G). Based on data from six power plants from 1994 to 2000 (electronic Appendix B, available on The Royal Society’s Publications Web site), we estimated the price per unit of production ( p), cost of labour (CL) and cost of capital (CK) using regression techniques (electronic Appendix C, available on The Royal Society’s Publications Web site). We assumed Žrms made choices to maximize their welfare (equation (2.3)). The production function (equation (2.4)) and optimal labour (equation (2.7)) were expressed in regression form as: Q = aLaKbG(N)c, (2.4) ln(Q) = ln(a) 1 aln(L) 1 bln(K ) 1 «, G(N) = 1 2 e2 l/N, (2.5) Nt 1 1 = Nte2 n X. (2.6) Welfare (w) for any given state is a function of production (Q), price per unit of production ( p) and the cost (C) per unit of labour (L), capital (K ), prevention (E) and control (X): w = pQ 2 CLL 2 C KK 2 CEE 2 CXX, (2.3) We extended the usual economic models by integrating damage with an explicit biological model. We determined optimal labour (L¤) analytically as follows: L¤ = S D apaKbGc CL 1/(1 2 a) = apQ¤ . w Proc. R. Soc. Lond. B (2002) (2.7) (2.10) and 1 1 =a 1 «. pQ ¤ CLL¤ (2.11) Optimal capital was analogously calculated. We estimated a, b and a using SURE (seemingly unrelated regression equations) (Zellner 1962). Zebra mussels reduce production (i.e. cause damage) by clogging pipes and reducing water ow. We modelled ow as being linearly related to production and pipe cross-sectional area (A), 2410 B. Leung and others Bioeconomic risk analysis of invasive species and therefore damage (Gc) was proportional to reduction in pipe area. Vo = pl S D 1D 2 2 (2.12) and Vzm = surface area O divi = plD O divi, (2.13) where Vo was the pipe volume without zebra mussels (m3), l was the pipe length (m), D was diameter (m), d was the density (individuals m2 2) and v was the volume of an individual zebra mussel (m3). We summed over all the age classes i (simpliŽed to homogeneous sizes within an age class). The proportional reduction in ow was Gc = Ao 2 Azm Vo 2 Vzm = Ao Vo Gc = 1 2 O divi 4 [MIN = 0]. D (2.14) Annual production, labour and capital costs were converted to monthly values, whereas zebra mussel control was expressed as the cost per event. Welfare was determined using equations (2.3) and (2.8). The coefŽcient c was not estimated explicitly, because our estimate of damage was externally derived rather than Žtted from industry data. (iii) Probability of invasion To estimate invasion probability, we considered 95 at-risk lakes in Michigan monitored over 7 years from before invasion (http://www.msue.msu.edu/seagrant/zmŽles/). A total of 56% of the lakes remained uninvaded (Ut = 0.56). If the invasion probability was constant, Ut = e2 pt, where p was the probability of invasion per time interval, and t was time: p = 2ln(Ut)/t. Given monthly time-intervals, p = 2ln(0.56)/(7 ´ 12) = 0.007, i.e. a 0.7% probability of being invaded each month. (iv) Simulations To determine the optimal control strategy after invasion, we ran the simulation for 120 monthly time intervals (10 years) with the two possible control strategies at each time-step (do nothing or perform a control event). To determine the acceptable prevention expenditure for a given initial invasion probability and a given proportional reduction in invasion probability, simulations were conducted at a range of prevention expenditures, in increments of US$1000. These results were for quasi-stationarity, i.e. the time-frame where the welfare of each state plateaued, and, therefore, the strategies became constant for each state. 3. RESULTS In our hypothetical example, we compared a lake that was initially uninvaded with an invaded lake, simulated over 25 years. For the uninvaded lake, we also examined a shorter (e.g. politically driven) time-frame (5 years). In each of the scenarios (Žgure 2), we considered the consequences for cumulative welfare of alternative strategies (Žgure 2a–c) and examined the average optimal welfare and the associated optimal expenditures on control (Žgure 2d2f ), labour, capital and prevention (Žgure 2g2i ) over time. We also derived the invasion probabilities associated with optimal management strategies for 5 and 25 year durations (Žgure 2j–l). Proc. R. Soc. Lond. B (2002) In our example, the cost of optimal control in the invaded lake reduced welfare by one-half relative to welfare in a lake in which optimal prevention measures were adopted before an invasion (Žgure 2a versus Žgure 2b); thus, under these model assumptions, prevention would be a good investment. Additionally, ‘subopt’, ‘random’ and ‘do nothing’ represented null strategies, which society could also apply; the optimal strategies resulted in the highest cumulative welfares, demonstrating internal model consistency and that the SDP works within the constraints of the model (Žgure 2a–c). In the invaded lake, the optimal investment in control declined over time because control was effective; low welfare was acceptable in the Žrst few years to maximize welfare over the long term (Žgure 2e). Time-frame was a major determinant of the optimal strategy. With a 5 year time horizon and an uninvaded lake, the optimal strategy was to spend nothing on either prevention (Žgure 2i ) or control (Žgure 2f ). Our modelling framework thus permitted an explicit demonstration of the consequences of long-term environmental perspectives versus short-term political perspectives for optimal control. Investments in labour and capital were inuenced by the initial environmental state, representing another level of interaction between ecological and economic parameters (Žgure 2g versus Žgure 2h). The level of risk of invasion that society should accept changed over time, but was never reduced to zero because the cost of achieving the last increment of risk reduction was not offset by the extra beneŽts (Žgure 2j–l). In all scenarios, the acceptable risk increased toward the end of the time horizon because the costs beyond the time horizon were excluded from consideration (Žgure 2j–l). While these results reect one hypothetical parameter set, they demonstrate that the modelling framework is useful for examining the interaction of ecological and economic factors, and for providing quantitative guidance for public policy. Next, we tested the application of our framework to a real-world problem—zebra mussel invasions in lakes. We modelled changes in zebra mussel population over time, using rates of recruitment, growth and survival structured by age or size and seasonality. We focused on the impact on industry of reduced water intake efŽciency caused by fouling of pipes by zebra mussels. Although lake-wide control of established zebra mussels is currently impossible, industry applies toxins to pipes to reduce fouling (see Deng (1996) for these and other methods). We used the model to choose between alternative control strategies in terms of when and how often power plants should perform control efforts using toxins. As prevention of zebra mussel spread to uninfected lakes is possible through public education and the management of boat trafŽc ( Johnson et al. 2001), we also considered how much society should be willing to pay to reduce the probability of invasion of currently uninvaded lakes, to maximize the net beneŽts. We considered spread by boat trafŽc, as this is the primary vector of zebra mussel spread to inland lakes ( Johnson et al. 2001). Results illustrated that after invasion the optimal strategy for our modelled power plant is to perform one control event per year in September (different plants may have different optimal strategies). Control is now conducted twice per year in the power plant that provided us with Bioeconomic risk analysis of invasive species B. Leung and others 2411 a) b) c) cost do nothing random subopt optimum do nothing random f) e) 40 10 30 0 _10 20 _20 10 _30 0 _40 g) h) i) k) l) welfare d) subopt optimum _140 subopt _80 do nothing 40 _20 random 100 optimum cumulative welfare 160 2.5 cost 1.5 0.5 _0.5 invasion probability j) 1 0.5 0 0 5 10 15 20 25 0 5 10 15 20 25 0 2 4 6 time years) Figure 2. An analysis of a hypothetical invasion. Each datum was a projected value at each time interval, weighted by the probability of being in a state, summed across all states. The panels show the projected cumulative welfares (a–c), optimal welfare (squares) and control expenditure (circles (d2f )), labour (triangles), capital (diamonds) and prevention expenditure (dashed line ( g–i )), and the invasion probability that society should accept for optimal welfare ( j–l). ‘Optimal’ used optimal strategies, ‘subopt’ used random strategies during one time interval, ‘random’ used random strategies at all intervals and ‘do nothing’ spent nothing. The error bars represent one standard deviation. Shown for 25 year uninvaded (a, d, g and j ), 25 year invaded (b, e, h and k) and 5 year uninvaded (c, f, i and l ) time horizons. data, but moving to a single control event in September is currently under consideration. On the question of preventing new lakes from infestation, society should implement prevention policies when CE , E(W0 ,t 2 W1 ,t), where CE is the cost of prevention, E is the absolute reduction in probability of invasion and W0 ,t and W1 ,t are the cumulative welfares given optimal control and prevention strategies summed from time t to the time horizon of an uninvaded lake and newly invaded lake, respectively. Using data from 95 lakes, we derived an estimate for probability of invasion of 0.7% per month (see § 2b(iii)). As empirical estimates of the costs or effectiveness of particular prevention strategies do not yet exist, we modelled how much society should be willing to pay to reduce the probability of invasion by a given amount, considering only damages to industry. We determined the Proc. R. Soc. Lond. B (2002) level of expenditure at which the costs of prevention equalled the beneŽts of reducing invasions by a given probability (i.e. society would derive a net beneŽt if expenditures below this level achieved a given effectiveness of prevention). For example, assuming a constant probability of invasion of 0.7% per month, our analyses indicated that to reduce the probability by 10% it would be beneŽcial for society to pay up to US$27 000 month2 1 (US$324 000 year2 1 ) on prevention for a single lake containing the modelled power plant (Žgure 3). 4. DISCUSSION Our hypothetical example and our real-world example with zebra mussels for a single generic power plant demonstrate the utility of our quantitative framework for ask- acceptable prevention expenditure $US month_1) 2412 B. Leung and others Bioeconomic risk analysis of invasive species 450000 300000 150000 0 20 40 60 80 100 % reduction in probability of invasion Figure 3. Acceptable expenditure for prevention. The graph shows the prevention expenditure (in US$) that society should accept to obtain a given reduction in the probability of invasion, given an initial probability of invasion. We simulated three initial probabilities of invasion per month: 0.7% (diamonds: result based on 95 lakes), 10% (squares), and 20% (triangles). For each, we simulated a reduction in invasion due to prevention of 10%–90%. Acceptable expenditure increased linearly with reduction in invasion probability. Heterogeneous slopes indicated an interaction between reduction due to prevention and the initial probability of invasion. ing key questions and guiding policy. For instance, our analysis for zebra mussels suggests that it would be beneŽcial to spend up to US$324 000 year2 1 to obtain a modest reduction in the probability of zebra mussel invasion into a single lake containing our modelled power plant. The actual cost of such a reduction would probably be much lower (e.g. a full-time manned inspection station would cost less). For comparison, in 2001 the US Fish and Wildlife Service distributed to all states combined a total of US$825 000 for prevention and control efforts for all aquatic non-indigenous species in all lakes. Thus, although separate analyses would need to be done for different lakes and different industries, and could also include alternative methods of prevention, these results suggest that prevention is currently underfunded. There are a number of challenges that remain to be addressed to improve policy decisions further. (i) For a comprehensive accounting of social beneŽts, it is important to consider non-market values. While we recognize the challenges of nonmarket valuation (Brown & Shogren 1998; Costanza 2000; Heal 2000; Ludwig et al. 2001), policymakers can beneŽt from more information about probable values that are attached to ecological services. Invaders have the potential to inuence biodiversity, aesthetics, recreation and property values. For instance, zebra mussels are decimating native mussels, 60% of which are already endangered (Ricciardi et al. 1998), which may increase the amount that society is willing to pay for prevention. Conversely, zebra mussels may improve water clarity, which may be positively valued (Efer & Siegfried 1998). The analysis of positive and negative impacts on ecosystems will allow a more accurate depiction of the true consequences of invasive species. (ii) We have only considered risk-neutral optimizations. Society may be risk averse towards some potential Proc. R. Soc. Lond. B (2002) invaders. QuantiŽcation and understanding of such preferences would allow better assessment of the amount society should spend on prevention. (iii) We should expand analyses to incorporate multiple species simultaneously. Although species-speciŽc assessments are the core of screening protocols for intentional pathways of introduction, other pathways, especially unintentional pathways, such as ballast water in ships, carry hundreds of species at a time (Aquatic Sciences Inc. 1996). Risk analyses on multiple species would be useful to understand better the risks created by the pathway. (iv) The framework should be extended to permit largescale heterogeneous landscapes, where the environment and optimal policy may differ between areas, and where policies in one local area may have ramiŽcations in others. Such analyses can use available information to facilitate the interaction between risk assessors and managers (Committee on Environment and Natural Resources 1999), provide quantitative rationale for policy decisions and help policymakers allocate society’s resources most efŽciently. EfŽcient resource allocation is crucial given that inadequate funding to achieve environmental objectives is the norm (Brown & Shogren 1998). Furthermore, even if limited to market values, bioeconomic cost–beneŽt analysis of non-indigenous species can strengthen the rationale for actions that achieve environmental goals (Van Wilgen et al. 2001). For example, our zebra mussel analysis—based only on market values of damage to industry—suggests that a much higher value should be placed on prevention than is currently spent. Greater prevention will protect the environment while also protecting industry. The authors thank E. M. Mallen for power plant data, and S. Stevens, J. Frentress, J. Drake, T. Robbins, R. Keller and R. Bechtel for help in collecting the data and for constructive comments. This project was funded by an NSF Biocomplexity Incubation grant to D.M.L. M.A.L. was supported by the Canada Research Chairs programme. 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