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 identies 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 benet 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 difcult to quantify; Mack et al. 2000).
Economic conditions inuence the transport of nonindigenous species, and inuence the resources that are
spent on preventing an invasion versus control after an
invasion. Reasonable resource expenditure is, in turn,
inuenced by the expected consequences of the invader.
Thus, ecological and economic parameters together dene
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 efcient (nds 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 gure 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 (gure 1a). Social welfare (benets 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 inuence 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 modied without affecting the
entire model. Plain text represents the interfaces (information passed between modules).
costs) is determined jointly by ecological and economic
processes (gure 1a).
By contrast, traditional risk analysis considers risk
assessment (determining environmental goals, such as
reducing invader abundance to a specied level) and risk
management (determining methods to reach those goals)
separately (Simberloff 2002). Such separation implicitly
assumes that the benets of prevention and control are
negligible below the level specied in the risk assessment.
Our perspective, however, is that it is important to determine quantitatively the benets 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 (gure 1a) into
the computational SDP model, breaking the conceptual
model into four operational modules (gure 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 benets 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 benet–cost
analysis of the current state and the accumulation of
future states based on expected trajectories.
We rst 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 specic 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 rms continue to experiment with new
control measures and schedules in an effort to maximize
the benets 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), dened 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 articial 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 modied 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
efcient. Here, we dened 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 rms 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 (simplied
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 coefcient c was not estimated explicitly,
because our estimate of damage was externally derived rather
than tted 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/zmles/). 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 (gure 2), we considered the consequences for cumulative welfare of alternative strategies
(gure 2a–c) and examined the average optimal welfare
and the associated optimal expenditures on control (gure
2d2f ), labour, capital and prevention (gure 2g2i ) over
time. We also derived the invasion probabilities associated
with optimal management strategies for 5 and 25 year durations (gure 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 (gure 2a versus gure 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 (gure 2a–c). In the invaded lake, the optimal investment in control declined over time because control was effective; low welfare was acceptable in the rst
few years to maximize welfare over the long term (gure
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 (gure 2i ) or control (gure 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 inuenced by
the initial environmental state, representing another level
of interaction between ecological and economic parameters (gure 2g versus gure 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 benets (gure 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 (gure 2j–l). While these results reect 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 efciency 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 trafc ( 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 benets. We
considered spread by boat trafc, 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 benets of reducing invasions by a given
probability (i.e. society would derive a net benet 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 benecial
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 (gure 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 benecial 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 benets,
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 benet
from more information about probable values that
are attached to ecological services. Invaders have the
potential to inuence 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 (Efer & 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. Quantication 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-specic
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 ramications 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
efciently. Efcient 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–benet
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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