Resource and Energy Economics...Resource and Energy Economics 32 (2010) 500–518 ARTICLE INFO Article history: Received 7 May 2008 Accepted 2 April 2010 Available online 18 April

Indirect management of invasive species throughbio-controls: A bioeconomic model of salmon andalewife in Lake Michigan

Eli P. Fenichel a,*, Richard D. Horan b, James R. Bence c

a Arizona State University, School of Life Sciences and ecoSERVICES Group, PO Box 874501, Tempe, AZ 85287-4501, USAb Michigan State University, Department of Agricultural, Food, and Resource Economics, 303A Agriculture Hall, East Lansing,MI 48824-1039, USAc Michigan State University, Department of Fisheries and Wildlife, Natural Resources Building, East Lansing, MI 48824, USA

Resource and Energy Economics 32 (2010) 500–518

A R T I C L E I N F O

Article history:

Received 7 May 2008

Accepted 2 April 2010

Available online 18 April 2010

JEL classification:

C61

Q22

Q27

Keywords:

Bioeconomics

Optimal control

Blocked intervals

Fisheries

Multiple-use species

Invasive species

A B S T R A C T

Invasive species are typically viewed as an economic bad because

they cause economic and ecological damages, and can be difficult to

control. When direct management is limited, another option is

indirect management via bio-controls. Here management is

directed at the bio-control species population (e.g., supplementing

this population through stocking) with the aim that, through

ecological interactions, the bio-control species will control the

invader. We focus on stocking salmon to control invasive alewives

in Lake Michigan. Salmon are valuable to recreational anglers, and

alewives are their primary food source in Lake Michigan. We

illustrate how stocking salmon can be used to control alewife, while

at the same time alewife can be turned from a net economic bad

(having a negative shadow value) into a net economic good (having

a positive shadow value) by providing valuable ecosystem services

that support the recreational fishery. Using optimal control theory,

we solve for a stocking program that maximizes social welfare.

Optimal stocking results in cyclical dynamics. We link concepts of

natural capital and indirect management, population dynamics,

non-convexities, and multiple-use species and demonstrate that

species interactions are critical to the values that humans derive

from ecosystems. This research also provides insight into the

management of salmon fisheries in the Great Lakes.

� 2010 Elsevier B.V. All rights reserved.

* Corresponding author. Tel.: +1 480 965 4027; fax: +1 480 965 6899.

E-mail addresses: [email protected] (E.P. Fenichel), [email protected] (R.D. Horan), [email protected] (J.R. Bence).

Contents lists available at ScienceDirect

Resource and Energy Economics

journal homepage: www.e lsev ier .com/ locate / ree

0928-7655/$ – see front matter � 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.reseneeco.2010.04.002

mailto:[email protected]



http://www.sciencedirect.com/science/journal/09287655

http://dx.doi.org/10.1016/j.reseneeco.2010.04.002

1. Introduction

Invasive species interact with other species in the ecosystem, thereby affecting the value of servicesthat humans derive from the ecosystem. Knowler (2005) emphasizes the need to consider interactionsamong ecosystem components when planning management and valuing the impact of invaders.While invaders often generate economic costs, some invaders may also produce some economicbenefits. Examples of positive impacts include service as a new prey species for prey-limited valuednative predators (Caldow et al., 2007), conservation of highly endangered species outside their nativerange (Bradshaw and Brook, 2007), values associated with introductions of charismatic species(Barbier and Shogren, 2004), and mitigating the impacts of previous invaders (Barton et al., 2005;Gozlan, 2008). In particular, non-native species may be intentionally introduced to mitigate theimpacts of previous invaders as part of bio-control programs (Hoddle, 2004). Such bio-control agentsmay also provide other benefits or damages, such that the net effect of such invasion could be positiveor negative.

In this paper, we examine a case in which the introduction of a bio-control agent turns the preyspecies (i.e., the previous invader) from a nuisance into a source of value, as measured by the shadowvalue of the original invader. Zivin et al. (2000) define multiple-use species as species that may causenet benefits or net damages to society, depending on ecological conditions and the managementregime. Multiple-use species have the potential to result in non-convexities that lead to multipleequilibria, each being a potential optima, in which case management history may affect whichequilibrium should be pursued (Zivin et al., 2000; Rondeau, 2001; Horan and Bulte, 2004). Previousstudies of multiple-use species considered cases where damages are a function of species density,while benefits may accrue through commodity-based harvests or existence values. These values,particularly benefits, arise as a result of direct feedbacks between humans and the species, and directpopulation management of the multiple-use species (Zivin et al., 2000; Rondeau, 2001; Horan andBulte, 2004). We examine a case where management of the invader is indirect, stemming frommanagement of a bio-control agent. Moreover, the source of value is indirect, stemming from theinvader supporting the bio-control agent that is valuable for recreational angling. Hoddle (2004)advocates greater consideration of bio-control to indirectly manage invasive species. Management ofnative species may also indirectly influence the impacts of an invader (Drury and Lodge, 2009).Indirect management tends to have (positive or negative) spillover effects on other ecosystemservices.

Spillover effects from non-selective management actions that only partially target the species ofconcern have been shown to lead to complex nonlinear feedback rules for efficient management(Mesterton-Gibbons, 1987; Horan and Wolf, 2005; Fenichel and Horan, 2007). Such imperfectly-targeted management can lead to cyclical dynamics in an optimally-managed system (Horan andWolf, 2005; Fenichel and Horan, 2007).

Bio-control represents a different form of indirect management. Management is selective, but it isdirected at a different species—the bio-control agent. The expectation is that management of the bio-control agent will influence predator–prey interactions, resulting in indirect management of theinvading species. But we still find that indirect management in this case can lead to non-convexitiesand complex feedback rules involving cyclical management.

We consider the case of Chinook salmon (Oncorhynchus tschawytscha) and alewife (Alosa

pseudoharengus) management in Lake Michigan. Salmon and alewife management is a dominant issueon Lakes Ontario, Huron, and Michigan. Alewives invaded Lake Michigan in 1949 and began directlygenerating ecosystem disservices and imposing costs on society by fouling beaches and drainpipes(O’Gorman and Stewart, 1999). Alewives also began indirectly generating ecosystem disservicesthrough negative impacts on native fish species (O’Gorman and Stewart, 1999). For example, there isevidence that alewife predation on lake trout (Salvelinus namaycush) fry impedes the restoration ofnative lake trout (Krueger et al., 1995; Madenjian et al., 2002), and that alewife predation on larval fishcontributed to the decline of yellow perch (Perca flavescens) (Shroyer and McComish, 2000), perhapsthe most widely targeted sport fish in Lake Michigan (Wilberg et al., 2005). Chinook salmon wereintroduced to Lake Michigan from the Pacific Northwest, both as a bio-control for alewives and togenerate a sport fishery (Madenjian et al., 2002). Chinook salmon sport fishing has become a valuable

E.P. Fenichel et al. / Resource and Energy Economics 32 (2010) 500–518 501

recreational activity (Hoehn et al., 1996), and it relies on alewife as a prey base. Alewives comprise themajority of the Chinook salmon diet in Lake Michigan (Madenjian et al., 2002), and it is believed thatthe Chinook salmon population – and hence the recreational Chinook salmon fishery – would collapsewithout a large alewife forage base (Stewart and Ibarra, 1991; Holey et al., 1998). Thus, from theanglers’ perspective, alewives provide an important in situ benefit in the production of Chinooksalmon. Management of the system is conducted by stocking Chinook salmon, as alewives are notharvested, and harvest from the recreational salmon fishery is largely unregulated.

We use the Lake Michigan system as a case study and develop a model that integrates the complexfeedbacks within the ecosystem, the multiple-use species problem, and indirect controls. Specifically,we focus on interactions between the two most important fish species in the system—alewives andChinook salmon. We then solve for an optimal stocking program from the agency’s perspective—onethat maximizes social welfare, defined as the sum of discounted net benefits from the open-access,unregulated, salmon sport fishery minus alewife-induced damages (which implicitly account foradverse impacts to aquatic species not explicitly modeled here) and stocking costs. In this case theagency is not a true social planner because the agency takes angler behavior as given. This can bethought of as an institutional constraint (Dasgupta and Maler, 2003). The solution, while efficient fromthe agency’s perspective, is ‘‘second best.’’ A ‘‘first best’’ solution would require that managers controlangler behavior, and therefore could optimally manage salmon and alewife harvests in addition tostocking.

We examine the tradeoffs associated with the stocking program in an analytical fashion, anddevelop general rules that can help guide stocking policy. We contribute to the bioeconomic literatureby linking non-convexities (Tahvonen and Salo, 1996; Rondeau, 2001; Dasgupta and Maler, 2003)with indirect management and expand understanding of biological capital. Indirect management iscompared and contrasted with imperfectly targeted management (Mesterton-Gibbons, 1987; Clark,2005; Horan and Wolf, 2005; Fenichel and Horan, 2007; Horan et al., 2008). We also provide insightsinto management of salmon fisheries on the Great Lakes by highlighting the tradeoffs implicit in theChinook salmon stocking program.

2. A bioeconomic model of salmon stocking

2.1. The managers’ problem

Consider a fishery management agency that aims to choose a level of stocking, w (mass per unittime) at each point in time that maximizes the discounted social net benefits (SNB) from a fisheryresource. SNB are the sum of individual salmon angler (consumer) surplus (B) minus the amountsociety invests in the fishery (stocking in kilograms of salmon) and damages (D) caused by the alewifestock (a, measured in biomass):

SNB ¼Z 1

0ðBðtÞ � DðaðtÞÞ � vwðtÞÞ e�rt dt (1)

where r is the discount rate and v is the constant marginal cost of stocking a unit of salmon biomass.Alewife-induced damages (e.g., due to fouled beaches and drainpipes, and adverse impacts to otheraquatic species) are denoted D(a) and are increasing in alewife biomass, D0(a)>0, at an increasing rate,D0 0(a)>0.

In order to choose a stocking program that maximizes expression (1), managers must account forthe constraints imposed by angler behavior, ecological dynamics, hatchery capacity, and the initialconditions. Models of angler behavior and ecological dynamics are constructed in the next two sub-sections.

2.2. An angler behavioral model

Including explicit models of angler behavior is important in fishery management (Wilen et al.,2002). A model of recreational angler behavior is necessarily different than the standard models ofcommercial fisher behavior (e.g., Clark, 1980, 2005; Knowler, 2005). Anglers in a recreational fishery

E.P. Fenichel et al. / Resource and Energy Economics 32 (2010) 500–518502

are not coordinated by the market, and each individual’s demand must be accounted for. The quantityof fishing trips demanded by each individual is a function of the angler’s individual preferences, skills,and costs. Knowler (2005) argues that that welfare losses from an invader in the Black Sea anchovyfishery are small because the fishery was open-access and all rents had already been dissipated beforethe invasion—there was nothing to lose (similarly, see McConnell and Strand (1989) for an applicationinvolving water quality impairments). This is not likely to be the case in a recreational fishery becauseindividuals’ diminishing marginal willingness to pay for an increased quantity of recreational daysimplies a positive angler surplus even under open-access (Anderson, 1983; McConnell and Strand,1989).

Assume all anglers have the same individual angling preferences and skills, but that fishing costsvary across individuals. Angler utility is U ¼ uðm; zðsÞÞ þ x. Following Anderson (1983), u is a quasi-concave, increasing function of days fished, m, and the quality of the fishery, z, which itself isincreasing in the salmon stock, s. In our simulation model below, we make the common assumptionthat quality is measured by the catch rate. Hence, um(m, z(s))>0, uz(m, z(s))>0, and z0(s)>0, wheresubscripts denote partial derivatives. We assume marginal utility is downward sloping with increasesin fishing days, umm(m, z(s))<0, and that marginal utility is increasing in fishing quality, umz(m,z(s))>0. Finally, the variable x is a composite numeraire good. Each individual has a budget constraintgiven by I=x+cm, where I is income and c is a unit cost of fishing that differs across individuals. Usingthe budget constraint, we can focus on the following affine transformation of utility, which is ameasure of individual angler surplus due to the assumption of quasi-linear utility

V ¼ uðm; zðsÞÞ � cm (2)

Quasi-linear utility is a common assumption in the empirical literature that may have small effectson welfare estimates (Herriges and Kling, 1999).

In a recreational fishery, the individual angler has two choices (i) whether or not to fish in a givenseason, and (ii) how many days to fish given that he chooses to participate (McConnell and Sutinen,1979; Anderson, 1983).1 An angler enters the fishery if V�0. Given that an angler participates, hechooses the number of fishing days, m, to maximize utility. The optimal value of m solvesumðm; zðsÞÞ � c ¼ 0, and is written m� ¼ m½zðsÞ; c�. The resulting surplus is V�ðs; cÞ ¼ uðm�; zðsÞÞ � cm�.

To determine the total level of effort in the fishery, we recognize that each angler has a unique costto fishing and think of c as a cost type. Each cost type is treated as a ‘‘micro-unit’’ (Hochman andZilberman, 1978).2 Cost types are ordered in increasing order, such that the last cost type to enter thefishery is c. That is, c is the cost at which the marginal angler is indifferent about entry and receiveszero surplus

V�ðs; cÞ ¼ uðm�; zðsÞÞ � cm� ¼ 0 (3)

This condition implicitly defines c as a function of s, cðsÞ, with c 0ðsÞ>0. A larger stock yields fishingquality improvements that induce entry by individuals with higher costs, increasing their willingnessto pay.

The assumption of heterogeneous anglers is important to derive a reasonable angler surplus. Ifanglers were homogeneous in preference and costs, then Eq. (3) would not define a threshold for entry.Either there would be no angler surplus or the total number of anglers participating would have to beimposed exogenously as a constant (McConnell and Sutinen, 1979) or as a function of the stock(Swallow, 1994).

Cost types, c, are continuously distributed over the interval [0,1] with the probability densityfunction c(c). If N is the total number of potential anglers, then the actual number of anglers in thefishery, n(s), depends on salmon biomass and is calculated as

nðsÞ ¼ N

Z cðsÞ

0cðcÞdc (4)

1 We assume all fish caught are landed as this generally depicts the Lake Michigan salmon fishery.2 This approach could also be extended to skills and preferences, but this would greatly complicate the model. Assuming

heterogeneous costs captures the general nature of the unregulated, open-access recreational fishery.


Total angler surplus, B, is the sum of angler surplus received by all individual anglers at time t, andis also a function of salmon biomass

BðtÞ ¼ BðsÞ ¼ N

Z cðsÞ

0Vðs; cÞcðcÞdc (5)

Total catch per unit time, h(s), is derived similarly and also depends on salmon biomass

hðsÞ ¼ N

Z cðsÞ

0m�ðzðsÞ; cÞzðsÞcðcÞdc: (6)

2.3. Fish population models

The fishery manager accounts for the dynamics of the fish stocks and their interactions with otherspecies. Define the dynamics of the harvestable salmon stock in terms of biomass as

s ¼ uðaÞðwþ bÞ � dðs; aÞ � hðsÞ: (7)

The first term in Eq. (7) is the total recruitment to the fishery, where b is the reproductive contributionfrom the stock and is independent of the stock size, i.e., b fish produced per unit time in nature rather thanby stocking. This is motivated by the assumption that the limited spawning habitat will be saturated(implying strong density-dependent mechanisms) (Kocik and Jones, 1999) and follows other models ofnatural salmon reproduction in Lake Michigan (Jones and Bence, 2009).3 The rate at which salmonbecome part of the fishery is given by the function u(a), which is increasing in the salmon’s prey, alewife.

Natural mortality of salmon in the fishery, d(s, a), is a function of salmon and alewife biomass. Weassume salmon mortality is a decreasing convex function of alewife biomass, @d(s, a)/@a<0, @2d(s, a)/(@a)2�0, so that as alewife biomass increases, ultimately salmon reach a minimum mortality rate. Wealso assume that @uðaÞ=@a>0 and @2uðaÞ=ð@aÞ2 <0. Specific functional forms are specified in thesimulation section.

The alewife population is defined in terms of biomass and follows logistic growth,

a ¼ ar 1� a

K

� �� sP að Þ (8)

where r is the net recruitment rate of biomass in the limit as stock size approaches zero (recruitmentminus non-predation mortality) and K is the alewife carrying capacity. The term sP(a) is salmonpredation on alewife, where P(a) is the biomass of alewife consumed by a biomass unit of salmon.Salmon consume more alewife as alewife biomass increases, such that P0(a)>0. However, the rate atwhich salmon consume more alewife biomass as alewife biomass increases may decline withincreasing alewife biomass, P0 0(a)�0. That is, the response of salmon to increases in alewife biomassmay be saturating. In this general framework d(s, a) and P(a) are separate functions. In the numericalanalysis below these functions are explicitly linked reflecting the biological connection between foodconsumption and mortality that stems from food limitation.

3. Optimizing social welfare through stocking

The agency’s problem is defined as choosing w to maximize (1) subject to Eqs. (7) and (8). Thisrequires that the agency explicitly consider Eqs. (5) and (6). The agency’s problem can be solved usingthe maximum principle (Clark, 2005). Write the Hamiltonian as

H ¼ BðsÞ � DðaÞ � vwþ lsþma; (9)

where l and m are the co-state variables associated with the salmon and alewife stocks, respectively.This problem is linear in the control variable, w.

3 Biological rates are often written as a proportional change, i.e., s=s. A standard logistic growth function can be written

s=s ¼ b f ðsÞ:::; where f(s) is the density dependence component. In our model strong density dependence is included as

s=s ¼ b=s:::.


The marginal impact of stocking salmon is given by

@H

@w¼ �vþ luðaÞ (10)

The right-hand-side (RHS) of expression (10) is the coefficient of stocking from the Hamiltonian. If@H/@w>0, then stocking always increases the value of the Hamiltonian and so stocking should be setat the maximum limit (an exogenously imposed limit, denoted wmax, that represents fixed hatcherycapacity).4 On the other hand, if @H/@w<0, then stocking always decreases the value of theHamiltonian and so stocking should be set to zero. These are constrained solutions. Another possibilityis that @H/@w=0. When this occurs, then w should be set at its singular value w*. In this case, l equalsthe marginal cost of stocking scaled for survival to the fishery, l ¼ v=uðaÞ. The complete solution canbe written as a feedback rule dependent upon the alewife stock.

w ¼w ¼ 0; if luðaÞ � v< 0w ¼ w� if luðaÞ � v ¼ 0w ¼ wmax if luðaÞ � v> 0

8<: (11)

Conrad and Clark (1987, p. 76) state that linear control problems guarantee the optimality ofconstrained solutions, which they refer to as ‘‘bang-bang’’ controls. In the linear control problem theydescribe, a constrained solution is pursued to a steady-state equilibrium, and then the singular solutionis adopted to maintain the system at equilibrium.5 This illustrates a case of rapid adjustment and thenmaintenance at equilibrium. This result relies on the existence of control variables for each state variable,where each control is perfectly targeted and directly affects only a single state variable at a single point intime. More complex feedback rules may emerge when these conditions do not hold (Mesterton-Gibbons,1987; Horan and Wolf, 2005), such as in the present case where only a single control variable (stocking) isused to affect two states—salmon are controlled directly but imperfectly (because stocking onlyincreases the salmon stock), while alewife are controlled indirectly via ecological interactions. Underthese conditions, the singular solution is specified as a nonlinear feedback rule that is optimally pursuedwhen the system is out of equilibrium. This means that adjustment is not as rapid as it is when thecontrols perfectly target the states of the system. The slow adjustment under imperfect targeting is akinto the slow adjustment in capital that arises in investment models with convex investment costs (e.g.,Liski et al., 2001). Imperfect targeting in the current application effectively generates convex adjustmentcosts that make the nonlinear feedback rule optimal.

Regardless of the type of solution, an optimal program requires that two adjoint equationsassociated with the co-state variables be satisfied at each point in time (Conrad and Clark, 1987)

l ¼ rl� @H

@s¼ rl� Bs þ l½dsðs; aÞ þ h0ðsÞ� þmPðaÞ (12)

m ¼ rm� @H

@a¼ rmþ D0ðaÞ � lððbþwÞu0ðaÞ � daðs; aÞÞ �mr 1� 2a

K

� �þmsP0ðaÞ (13)

These conditions prevent intertemporal arbitrage opportunities. If these conditions are notsatisfied, then welfare gains can be made by reallocating stocking across time. The arbitrageconditions may be manipulated into two ‘‘golden rule’’ equations that must hold at each point in time(Conrad and Clark, 1987):

r ¼ llþ Bs

l�m

lPðaÞ � dsðs; aÞ � h0ðsÞ (14)

4 We assume fixed hatchery capacity, which is consistent with our understanding of the current Michigan and Wisconsin

hatchery plans. Though it is possible to also model investment in hatchery capacity, doing so would greatly complicate the

analysis without adding significant insight into the bio-control problem. For instance, Clark et al.’s (1979) results suggest that

investment in non-malleable hatchery capital would likely lead to undershooting and overshooting of the optimal number of

hatcheries.5 This is the case for autonomous problems. For non-autonomous problems, the singular solution will be a path and the

optimal solution generally involves moving to this path as quickly as possible (Conrad and Clark, 1987).


r ¼ mm� D0ðaÞ

mþ l

m½ðbþwÞu0ðaÞ � daðs; aÞ� þ r 1� 2a

K

� �� sP0ðaÞ (15)

These golden rule equations highlight tradeoffs associated with increases or decreases in salmonand alewife biomass.

Consider Eq. (14). The discount rate, r, is society’s rate of time preference and can be thought of asthe opportunity cost of investing in the salmon stock. The RHS of Eq. (14) collectively represents therate of return to investing in the salmon stock. The first term on the RHS is a capital gains term that willbe zero at a steady-state equilibrium. The second RHS term is the normalized angler surplus gainedfrom a larger salmon stock at the margin, and is always positive. This term indicates that, ceteris

paribus, anglers are better off with a larger salmon stock because a larger salmon stock increasesfishing quality, thereby increasing angler welfare. The third RHS term is the marginal impact of anincrease in the salmon population on the alewife resource, on which the salmon population depends.This term is negative, as increasing the salmon population decreases the alewife population, and areduced alewife stock supports a smaller salmon stock. The fourth and fifth RHS terms together are theeffect of the salmon stock on its own marginal growth. The fourth term, dsðs; aÞ, is the direct effect ofthe salmon stock on its own natural mortality: in our specified model below this term equals thenatural mortality rate. The final RHS term represents the change in salmon mortality with an increasein the salmon stock due to changes in angler behavior. This term is negative: an increase in the salmonstock increases the total effort and thus total catch in the fishery, reducing future fishing opportunitiesceteris paribus. This is a fishing mortality cost.

Eq. (15) is also a golden rule equation, but its interpretation depends on whether alewives are anuisance (m<0; a net economic bad) or an asset (m>0; a net economic good). When alewives are anuisance, then the discount rate represents the opportunity cost of diverting resources fromelsewhere in the economy to manage the invasive alewives and their associated damages. The RHSthen represents the rate of return from nuisance control. When alewives are an asset, then theinterpretation is the same as it was for salmon: the RHS is the rate of return to investing in alewives,while r is the opportunity cost of that investment. In our numerical analysis we find that alewives areprimarily an asset (m>0) under optimal management. In that case, the first RHS term is a capital gain/loss term that will be zero at equilibrium. The second RHS term represents the marginal alewife-induced damages. The third RHS term is the marginal benefits of alewife as prey for the recreationalsalmon fishery. The fourth and fifth terms together represent the marginal impact of alewife onreproduction.

3.1. Solving for the singular feedback rule

The approach for finding the nonlinear feedback rule for the singular solution is similar to Fenicheland Horan’s (2007) procedure for finding a partial singular solution (their model has two controlvariables). First, set Eq. (10) equal to zero and solve for l ¼ lðaÞ. Next, take the derivative of l withrespect to time, yielding dlðaÞ=dt ¼ l0ðaÞa� zðs; aÞ, and substitute l(a) for l and z(s, a) for l incondition (12). Solve the resulting equation for m ¼ mðs; aÞ and take the time derivativedmðs; aÞ=dt ¼ mssþmaa�Cðs; a;wÞ. Finally, substitute Cðs; a;wÞ for m, l(a) for l, and m(s,a) for min condition (13), and solve for w. The solution is a nonlinear feedback rule for the singular solutionw* =w*(s, a).

As indicated in Eq. (11), the singular solution is only part of the optimal stocking program.Constrained controls are also possible. Combinations of state variables such that w�ðs; aÞ � 0 orw�ðs; aÞ�wmax provide a necessary, though not sufficient, condition for a constrained solution (i.e.,w ¼ 0 or w ¼ wmax).6 The arcs where these equalities hold outline the boundaries, in state space, ofblocked intervals—intervals over which the control is constrained or blocked from taking on itssingular value (Arrow, 1968; Clark, 2005 pp. 56–57, 96).

6 One may be tempted to write out the Kuhn-Tucker conditions in attempt to uniquely identify the switching points; however

Conrad and Clark (1987 p. 87) point out that this is not a helpful approach and can actually lead to the wrong conclusions.


The problem of when to pursue the singular solution and when to pursue a constrained solution(i.e., the delineation of blocked intervals) is inherently numerical, even in relatively simple problems(Arrow, 1968). This is particularly true for the current problem, where the multiple-use species aspectof the problem and the complex human–ecological relationships resulting from indirect managementof the two species can lead to non-convexities and multiple equilibria (Zivin et al., 2000; Rondeau,2001; Crepin, 2003; Horan and Bulte, 2004). For problems with non-convexities, ‘‘what is required isthe sheer brute force of computing welfare along candidate programs and comparing them’’ (Dasguptaand Maler, 2003). Therefore, we now turn to a numerical analysis of the problem.

4. Specifying the numerical model

We begin by specifying explicit functional forms for the remaining implicit functions in the model.To maintain tractability and to make use of available data, we have adopted parsimonious functionalforms that are consistent with economic and biological knowledge and theory. The best available dataon the salmon-alewife-angler system were used to calibrate the model (all parameter values arespecified in Appendix A). Though we have strived for realism, our specific numerical results should beviewed as an illustrative example that provides insights into managing the complex ecological andeconomic interactions associated with the salmon stocking program in the presence of non-convexities and imperfectly targeted controls.

4.1. Economic functions

The exact nature of the alewife damage function is unknown, but given the assumptions thatD0(a)>0 and D00(a)>0, we use a second order approximation to a convex function D(a)=Da2, where D

is a damage parameter.The angler ‘‘inverse demand’’ function takes the form um ¼ &1 � &2mþ &3zðsÞ, where z(s)=qs, q is

the catchability coefficient, and &i is a parameter. This implies that marginal willingness to pay fordays fishing is declining in days fishing but increasing in fishing quality. The linearity of thisspecification results in a choke price, &1 þ &3zðsÞ, at which an individual will cease to fish when his orher unit costs exceed this value. The choke price is increasing in s, and so changes in the salmon stockaffect ongoing fishing effort by existing participants, as well as entry and exit behavior. Thedistribution of cost types is assumed to be log-normally distributed with a mean of h and standarddeviation s. Just and Antle (1990) recommend this distribution for micro-parameter models.

4.2. Ecological functions

We define dðs; aÞ ¼ as� gsPðaÞ so that salmon mortality declines linearly with increases in thebiomass of alewife consumed per biomass unit of salmon. The instantaneous (per unit salmonbiomass) annual mortality rate is a� gPðaÞ so that the parameter a is the rate with zero alewife.Following Jones and Bence (2009) we assume that the per unit salmon biomass rate of consumption ofalewife biomass follows the function PðaÞ ¼ va=ð1þ baÞ, with parameters v (the limit of P0(a) as a

approaches zero) and b. This function follows our general assumption P0(a)>0, and is saturating withP00(a)�0; predation on alewife asymptotically goes to v/b as alewife biomass goes to infinity. In theecological literature P(a) is called a functional response. Our ‘‘Holling type II’’ saturating responsefunction is widely used in modeling predation to reflect that handling prey takes time away fromsearching (Holling, 1959).7 Note that in addition to specifying a model that agrees with the general

7 An alternative form of the function is parameterized in terms of handling time and catchability. The Holling type II response

is based on the assumption that predators spend some time searching for prey and some time handling or processing their prey

after catching them. Our functional form of the type II response is a parameterization of what is known as Holling’s disk

equation. Denote search time, Ts, and handling time, Th and let T be the proportion of total time spent feeding, so that in one unit

of time T=Ts +Th. Denote ts to be the catchability rate of the prey, so that the amount of prey caught in one period equals Tstsa.

Given this catch and defining th as the per unit handling time, the total handling time is Th =thTstsa, Using the time constraint,

T=Ts[1+thtsa], thus Ts =T/[1+thtsa], and P(a)=Ttsa/[1+thtsa]. Hence, the parameter v=ts, and the parameter b=thts. Both

forms are widely used in the ecological literature (see Gotelli, 2008 p. 137).


conditions we gave above we have also made the reasonable direct connection between predatorconsumption rates and their mortality. The alewife population also affects the stocking survival andrecruitment to the fishery. We model this with a scaling function, uðaÞ ¼ yaf, where 0<f<1 and y areparameters.

5. Results

We begin by analyzing the results of a base model using the parameters from Appendix A. Later, weexamine the sensitivity of these results to alternative parameter values. Mathematica 6.0 (WolframResearch) was used to implement the model.

Given an initial state of the world, s0 and a0, the agency planner must choose a stocking program.This program may include constrained or singular values of w (i.e., w ¼ 0, w ¼ wmax, or w ¼ w�ðs; aÞ) atdifferent points in time. The choice of when to apply which type of solution is a common problemwhen multiple populations are managed with a smaller number of controls than states, or withimperfectly-targeted controls (Mesterton-Gibbons, 1987; Horan and Wolf, 2005; Fenichel and Horan,2007; Horan et al., 2008).

Our approach to developing an understanding of system dynamics and the optimal managementstrategy is to first consider dynamics with no stocking, then dynamics with stocking at the maximumrate, and third dynamics when w ¼ w�ðs; aÞ is a possibility for some combinations of s and a. Finally,we explore an optimal solution in which the singular stocking level and the two constrained stockinglevels are all feasible options.

5.1. Dynamics when w ¼ 0

Consider the dynamics when there is no stocking (Fig. 1), but with salmon established in thesystem. There are two equilibria; a high alewife state (A) and a low alewife state (B). The phasedynamics indicate that equilibrium point A is locally stable. This is confirmed by examining theeigenvalues of the linearized system at equilibrium A (both eigenvalues are negative). Equilibrium B isonly stable if approached along a unique saddle path (SP). This is confirmed by examining theeigenvalues of the linearized system at equilibrium B (one is positive, the other is negative). For initialpoints above SP, the system will tend towards equilibrium A. For initial points below SP, alewives willbe eradicated and salmon will disappear.8 Path SP therefore bifurcates the system, dividing the phaseplane into alternate basins of attraction.

5.2. Dynamics when w ¼ wmax

Now consider the dynamics when stocking always occurs at the maximal value w ¼ wmax (Fig. 2).In this case the salmon stock builds up, which then reduces the alewife stock via predation. Indeedfrom any initial condition, pursuit of a maximum stocking strategy leads to alewife eradication. Recallthat alewives are necessary for salmon recruitment and survival, for both wild and stocked salmon.This means that alewife eradication implies collapse of the salmon stock—even in the presence ofcontinued stocking.

5.3. Dynamics when w ¼ w�ðs; aÞ

Now consider the non-linear feedback rule associated with the singular solution, w ¼ w�ðs; aÞ. Thisfeedback rule can only be valid in the portion of the state space where 0 � w�ðs; aÞ � wmax, R3 (Fig. 3).We therefore begin by defining the boundaries of this interior region, where the boundaries are givenby w�ðs; aÞ ¼ 0 and w�ðs; aÞ ¼ wmax (see Fenichel and Horan, 2007 and Horan et al., 2008 for a similarapproach). Boundaries are plotted in Fig. 3 (and later in Figs. 4 and 5) as dashed lines. Beyond eachboundary (shaded regions, R1 and R2), stocking necessarily becomes constrained (though, as we

8 In our model the use of a type-II predator response function and that alewives and salmon are the only species in the model

leads to the eradication result. This should be kept in mind when reading the results.


describe below, stocking at w ¼ 0 is not necessarily optimal in the region R2), and the dynamics underthese constraints follow the corresponding dynamics from Figs. 1–3 is a ‘‘feedback control diagram’’and not a standard phase plane (Clark et al., 1979; Conrad and Clark, 1987). The feedback controldiagram is constructed by splicing together the phase planes in Figs. 1 and 2 and a phase planerepresenting the singular solution dynamics.

Fig. 2. Phase plane showing the dynamics of salmon and alewife when stocking is set to its maximum level (w ¼ wmax).

Fig. 1. Phase plane showing the dynamics of salmon and alewife when stocking is set to zero (w ¼ 0).

Fig. 3. Feedback control diagram showing the optimal management of salmon and alewife for the baseline parameters.


The feedback control diagram divides the state space into three regions. The first region, R1, liesabove the w ¼ wmax boundary, where the singular solution would imply stocking at greater than themaximum rate (this appears as two regions on the phase plane). Paths that move into this region arenecessarily constrained along a blocked interval. The second region, R2, lies below the w ¼ 0boundary, where the singular solution would imply negative stocking. Paths that move into this regionare also necessarily constrained along a blocked interval. The third region, R3, lies between theboundaries, and represents the region where the singular solution w ¼ w�ðs; aÞ is feasible. Theoptimality of pursuing the singular solution at any point within the interior region is not guaranteed;hence, this region does not necessarily represent a free interval.

In Fig. 3, the s ¼ 0 isocline bends sharply when it enters and leaves region R3. The a ¼ 0 isocline isunaffected by the stocking level because stocking only has indirect effects on alewife. Within region R3the s ¼ 0 and a ¼ 0 isoclines cross at points Y and Z. The eigenvalues of the linearized system at point Y

are imaginary with positive real parts, indicating that equilibrium Y is an unstable focus. The localdynamics are indicated by the phase arrows. There is one positive and one negative eigenvalue for thelinearized system at point Z, resulting in a conditionally stable or saddle point equilibrium. Theseparatrices that lead to point Z within the interior region vanish at the boundaries of the constrainedregions. In each of the constrained regions, there is also one unique path that, if followed, can move thesystem from the constrained region to the separatrix in the interior region (dotted line in Fig. 3).Splicing these paths together yields a piece-wise continuous ‘‘saddle path’’ to equilibrium Z.

Fig. 4. Feedback control diagram showing the optimal management of salmon and alewife for increased alewife related

damages (D=1.79�10�11).

Fig. 5. Feedback control diagram showing the optimal management of salmon and alewife with no wild salmon reproduction

(b=0).


5.4. Optimal stocking strategies

Economically optimal solutions to dynamic resource problems very often involve moving to asaddle path and then following that path to a saddle point equilibrium (e.g., Clark, 2005). We will showthis is not the solution here: it is impossible to move to the ‘‘saddle path’’ when starting to the right ofit, and an alternative path dominates the ‘‘saddle path’’ when starting to the left of it. The ‘‘saddle path’’divides the feedback control diagram into two basins of attraction, with the optimal stocking strategydependent upon whether the system initially lies to the left or the right of the ‘‘saddle path’’. Thathistory matters for the optimal solution is not uncommon in problems with non-convexities(Tahvonen and Salo, 1996; Rondeau, 2001; Horan and Bulte, 2004).

First, consider the solution for initial points lying to the right of the ‘‘saddle path’’. We consider aninitial point in the w ¼ wmax region, though the results would be unaffected by starting at any otherpoint to the right of the ‘‘saddle path.’’ Starting in the w ¼ wmax region and setting w ¼ wmax, thesystem will stay to the right of the saddle path and eventually move into the interior region. Once inthe interior region, we could switch to the feedback rule w ¼ w�ðs; aÞ, which would continue to steerthe system away from the saddle path until the w ¼ 0 boundary is crossed. Then we could adopt w ¼ 0,which again would continue to steer the system away from the saddle path until the systemeventually collapses, as occurs below SP in Fig. 1. However, given that the system eventually collapses,maintaining w ¼ wmax throughout is optimal. This can be thought of as a most rapid approach path(MRAP) to the eventual equilibrium. The MRAP generates larger benefits than the alternative pathbecause stocking generates large angler benefits while the system moves towards collapse. That theMRAP generates larger net benefits is verified numerically.

Setting w ¼ 0 in the w ¼ 0 region does not lead to a ‘‘jump’’ to the saddle path. This is because theonly way for salmon to be removed from the system is through harvests and through naturalmortality, neither of which is controlled by the managers. The salmon population, therefore, respondsslowly to management. In contrast, adopting a policy of no stocking while in the w ¼ wmax regionwould move the system back to the ‘‘saddle path’’ in this region. However, this does not reduce thesalmon stock and move the system back into the interior fast enough, and in the interim the benefitsfrom salmon harvests are reduced relative to the case of maximum stocking. Recognizing this, it isoptimal for managers to stock the maximum amount of salmon and capture the short-term anglingbenefits. The same holds true for initial values in the interior region to the right of the ‘‘saddle path’’(numerical results support these conclusions).

Now consider starting points to the left of the ‘‘saddle path.’’ Without loss, we take equilibrium Afrom Fig. 1 as the initial condition. Starting from point A, there are potentially three candidateprograms: (i) always stock at the maximum rate and move past the ‘‘saddle path’’ and eradicatealewives, (ii) move to the ‘‘saddle path’’ and pursue it to equilibrium Z, and (iii) move to a stable limitcycle around equilibrium Y, provided a stable limit cycle exists. Each solution must be evaluatednumerically and compared.

First, consider stocking at the maximum rate, moving past the ‘‘saddle path,’’ and eradicatingalewife. The value of expression (1) is first evaluated numerically along a path from equilibrium A bystocking at the maximum rate until a<1 (i.e., a MRAP). This yields $54 million in net benefits.9

Now consider moving quickly to the ‘‘saddle path’’ and then pursuing it to the equilibrium Z. Eq. (1)is evaluated along a path originating at equilibrium A and stocking at the maximum rate to the ‘‘saddlepath’’, and then following the stocking program implied by the ‘‘saddle path’’ until the system arrivesat equilibrium Z. This program yields $44 million in net benefits.

Finally, consider the possibility of a piece-wise continuous stable limit cycle around equilibrium Y

(see Fig. 3). Starting at equilibrium A the feedback rule implied by the singular solution is applied.Doing so leads to the w ¼ wmax boundary, and the start of a blocked interval with w ¼ wmax. Stocking

9 Given the continuous nature of the system neither alewife nor salmon can be truly eradicated, but as Hastings (2001) points

out such low values can only be interpreted as extinction. An alternative to the MRAP is to stock at the maximum rate until just

after the saddle path is crossed, then stocking at the interior value until w� ¼ 0, and then cease stocking until the system

collapses. This yields $45 million in net benefits. Comparison of this result with that of the MRAP supports our earlier assertion

that always stocking at the maximum rate to the right of the saddle path is optimal.


at wmax eventually leads back to the interior region, where stocking at w ¼ w�ðs; aÞ is again feasible.Continuation along the path defined by the singular solution eventually leads to another blockedinterval where w ¼ 0. Pursuit of the singular solution to the constraining boundary is myopic, and it isoptimal to set stocking to zero prior to encountering the constraining boundary (i.e., the prematureswitching principle; see Arrow, 1968; Clark et al., 1979; Clark, 2005).10 Stocking is forgone until thesystem re-enters the interior region at the w ¼ 0 boundary (to left of equilibrium Y in Fig. 3). In theinterior region the singular solution is pursued until the w ¼ wmax boundary is again approached,where stocking is constrained to the maximum. Stocking continues at the maximum rate until thesystem re-enters the interior region. The singular solution is then pursued until the arc that resultedfrom the previous pre-mature switch is intersected. At that point the far-sighted planner switches tow ¼ 0, creating a piece-wise continuous stable limit cycle. Expression (1) evaluated along such aprogram (as drawn in Fig. 3) results in net benefits of $210 million.

5.5. Intuition and explanation

Stable limit cycles have not been a common outcome of bioeconomic models of renewableresources, but Wirl (1992, 1995) hypothesized that they should be—especially in systems withpredator–prey interactions. Wirl (1992) examined a class of optimal control problems involving twostates and one control and, in the context of our model and notation, he showed a necessary conditionfor a stable limit cycle is satisfied when saas <0.11 This inequality holds in our model as sa >0 andas <0. These relations imply a particular type of interdependency between the two stocks: more of a

generates positive spillovers on population s, while more of s generates negative spillovers onpopulation a. Increasing salmon, via stocking, reduces the rate of increase in alewife biomass. But,reduced growth of alewife biomass negatively impacts the salmon population, which in turn has apositive effect on the alewife population. It is difficult for the system to stabilize under thesecircumstances when managers possess a single control.

Intuition can be obtained by recalling that biological stocks are capital resources. Individualbiological stocks exist in complex ecosystems and interact with other biological stocks, therebynecessarily linking the growth path of biological capital resources within a given ecosystem. Yet,controls, or the abilities to reallocate biological capital, are often limited and therefore imperfectlytargeted. For instance, a manager may utilize a control that simultaneously impacts multiple stocks(e.g., Fenichel and Horan, 2007). In the present case, the manager may only be able to direct controlsat one stock, meaning the other stock is managed indirectly through ecological interactions. A first-best outcome in our model would involve controlling harvests of both species in addition tostocking. But, in our model managers only have the power to increase the salmon stock, as angling isunregulated. This is essentially a problem of irreversible investment in salmon capital, which issomewhat akin to Clark et al.’s (1979) problem of irreversible investment in fishing capacity. In thepresent case, managers have no direct mechanisms for managing the alewife capital stock.However, investments in salmon can be seen as an investment in the capacity for alewife predation,akin to investments in fishing capital in Clark et al.’s model. Like Clark et al., managers do not havethe ability to dis-invest in predation capacity (i.e., the salmon stock) as quickly as they may like.Rather, they rely on depreciation via natural mortality and unregulated angling mortality to depletethe salmon stock. Irreversible investment in Clark et al.’s (1979) model resulted in convexadjustment costs (Liski et al., 2001) and sluggish capital investment, leading to overshooting andundershooting the equilibrium. Managers face additional restrictions in the current problem incomparison with that addressed by Clark et al’s model. Hence, our model also results in sluggishadjustment because direct setting of predation mortality (analogous to fishing effort in Clark et al.’smodel) is not a choice.

10 Whenever a blocked interval is encountered, it is optimal to impose the extreme value of the control prior to intersecting the

constraint. In case of the w ¼ wmax constraint this occurs instantaneously before the constraint is intersected, but the switch is

significantly earlier for the w ¼ 0 constraint.11 This is one of four possible necessary conditions; only one must be satisfied. Wirl (1992) emphasizes that determining

sufficiency is often a numerical exercise.


This overshooting and undershooting, resulting from the single control and the condition saas <0,contributes to the stable-limit cycle in our model. But, in contrast to Wirl’s (1992) results, additionalelements are necessary for the current problem. Wirl’s problem was nonlinear in the control, and thisfeature is what allowed him to derive a stable-limit cycle that is everywhere continuous and tocharacterize the necessary conditions for this outcome. Our problem is linear in the control, whichmeans the premise of Wirl’s necessary conditions for limit cycles are not satisfied in our model. Still,the condition saas <0, along with a single control, is important for the reasons described above. Apiecewise continuous stable-limit cycle can emerge in our model if the following conditions are alsosatisfied: (i) an unstable equilibrium arises in the singular region of our feedback control diagram, and(ii) the dynamics in the relevant constrained regions (state space where blocked intervals must belocated) move the system back into the interior. For our model, this second requirement means thatthe separatrix associated with w ¼ 0 must lie beyond the w ¼ 0 and w ¼ wmax boundaries within theneighborhood of the singular portion of the cycle. We do not prove these statements formally since thesolution is inherently numerical. However, we do illustrate the point by examining a case in thesensitivity analysis (below) in which these conditions are not satisfied and there is no interior cycle.

Condition (ii) is associated with hysteresis, as the separatrix associated with w ¼ 0 divides thesystem into two basins of attraction. Once the system crosses the separatrix from above, there is nomanagement action that can be taken to return the system to the region above the separatrix. Theseparatrix incorporates the anglers’ decisions, but is exogenous with respect to the managers’decision. Fenichel and Horan (2007) and Horan et al. (2008) also find piecewise continuous stable-limit cycles to be optimal in a linear control setting, and their solutions also satisfy conditions (i) and(ii) above. But their models involved an additional control. As a result, the separatrices in their modelswere endogenous and hysteresis was not required to satisfy condition (ii). When one control variablewas removed from the Fenichel and Horan (2007) model, then condition (i) was not satisfied and theequilibrium was conditionally stable.

We now explain the clockwise rotation of the stable-limit cycle in Fig. 3. The dynamics illustratechanges in stocking levels in addition to changes in the states. Note that stocking increases as thesystem moves from the w ¼ 0 boundary to the w ¼ wmax boundary.

Stocking and alewives are complements in the production of fishing services to salmon anglers.When the alewife population is small, stocking has little impact on salmon productivity at the margin.But as the alewife stock increases, the marginal impact of stocking on salmon production increases. Atthe same time, stocking increases the salmon population, which reduces alewives. The reduction inalewives reduces damages, but it also reduces the productivity of salmon. So the choice of when andhow much to stock is a balancing act.

Stocking is initiated in the stable-limit cycle in Fig. 3 when the alewife population is sufficientlylarge and increasing (at the point where the path crosses the w ¼ 0 boundary). Here, marginaldamages due to alewives are small in comparison to the marginal benefits that alewives provide to thesalmon fishery (indicated by m>0). Stocking levels continue to increase as the alewife populationcontinues to grow. When the path crosses the a ¼ 0 isocline, the alewife population has become toolarge and the marginal damages it generates exceed its marginal benefits to the salmon fishery.Stocking then grows to its maximum level, increasing the salmon population and benefitting anglerswhile simultaneously controlling the alewife population to reduce damages. As the alewife populationis diminished, the incentives to stock decline and stocking levels fall and eventually cease. The reducedalewife stock means that the salmon stock will also decline (as the path crosses the s ¼ 0 isocline), andboth populations decline for a time. Eventually, the reduction in salmon numbers allows the alewifepopulation to recover as the system intersects the a ¼ 0 isocline. This increase in alewives is initiallybeneficial as the marginal benefits to the salmon population exceed the marginal damages generated.Eventually stocking is re-implemented and the cycle begins again.

The marginal value of additional alewives, m, is generally positive within the optimal program. Inthe absence of the salmon fishery, however, m would necessarily be negative and alewives would be anuisance. This result extends prior work on multiple-use species to a broader ecosystem servicecontext; prior work focused only on using existing or newly-created markets to tap into the harvest-related demand for a nuisance species and turn the species into a source of net value (Zivin et al., 2000;Horan and Bulte, 2004).


5.6. Sensitivity analysis

In the interest of space we focus on three parameters of interest: the alewife-induced damageparameter, D, the amount of natural recruitment, b, and the discount rate, r. This model has a largenumber of parameters and we could, in principle, evaluate how our model results change in responseto changes in each of them. Doing so would require drawing and analyzing a new feedback controldiagram for each parameter change. Parameters D and b were chosen due to the high degree ofuncertainty associated with these parameters. The parameter r was chosen because the discount ratecan have a strong affect on intertemporal tradeoffs.

5.6.1. Alewife induced damages

The relationship between alewife damages and the stock size is unknown. Varying this parameterhelps understand the multiple-use species nature of the problem. We investigate the effect ofincreasing D by up to two orders of magnitude. First consider an order of magnitude increase in D. Inthis case the feedback control diagram (not illustrated) looks almost identical to Fig. 3 (Figs. 1 and 2 areunaffected). We follow the same process of evaluating expression (1) given various managementprograms, and find that a path involving a piece-wise continuous stable limit cycle similar to the onein Fig. 3 is optimal, yielding net benefits of $120 million. Net benefits decline relative to the base casedue to the increased damages.

If the damage parameter is increased by another order of magnitude, then there is a qualitativechange. Fig. 4 shows the feedback control diagram for this case. There are no interior equilibria.Equilibria A and B (from Figs. 1 and 2) would both lie region R1. No stable limit cycle is possible in thiscase. Second, the w ¼ 0 saddle path lies in the interior region. Any interior path moving towards theright must cross this saddle path before turning back around. However, once the saddle path is crossednothing can be done to prevent the system from collapsing. A cycle is therefore not possible. In thiscase stocking at wmax until alewife are eradicated yields�$38 million in net benefits. For comparison,stocking at the maximum rate until SP is intersected and then ceasing stocking to arrive at equilibriumB yields �$111 million in net benefits. It is optimal to stock at the maximum rate and eradicatealewives. Furthermore, in contrast to the baseline scenario, we find that alewives generally have anegative marginal value along the optimal path (i.e., m<0).

5.6.2. Wild reproduction

Though there have been reports of Chinook salmon naturally reproducing in the Great Lakes sincethe mid-1970s (Kocik and Jones, 1999), there were no wild spawned salmon in the Great Lakes whenthe Chinook salmon stocking program began in the early 1960s. We therefore consider the case inwhich there is no wild Chinook salmon reproduction, i.e., b=0 (and all other parameters are held attheir baseline levels). This is the problem managers may have believed they faced when Chinooksalmon were first introduced.

The dynamics of the constrained solutions are similar to those displayed in Figs. 1 and 2. In theinterest of space we do not redraw those phase planes. Note, however, that in the case of no stockingthere are no interior equilibria. Equilibrium A shifts to the point (0, K), and equilibrium B shifts to theorigin. The local dynamics near the equilibria remain the same, with a saddle path leading to theorigin. When stocking is set at the maximum rate, the s ¼ 0 isocline, conditional on w ¼ wmax, shifts tothe left (relative to Fig. 2) but there are still no interior equilibria. Otherwise the dynamics are thesame.

The feedback control diagram is illustrated in Fig. 5. We continue to evaluate the optimal strategyfrom the point of equilibrium A (Fig. 1) in order to facilitate comparison with the case of naturalreproduction. As with the baseline case, we find optimal management is achieved by a piece-wisecontinuous stable limit cycle, though there are differences from the baseline case.

The no-wild-reproduction (b=0) scenario yields net benefits of $237 million, which exceeds the netbenefits in our baseline scenario with wild reproduction. Wild reproduction has two effects. First, wildreproduction represents a stocking subsidy during those times when it is optimal to increase thesalmon stock. This effect should increase net benefits. Second, wild reproduction makes it moredifficult to reduce the salmon stock during those times when it is optimal to decrease the salmon


stock. This effect decreases net benefits. The second effect dominates because it is more difficult andhence costly to control the system during periods of salmon depletion.

Managers have greater control over salmon depletion in the b=0 case (Fig. 5); the optimal cyclenever comes close (relative to Fig. 3) to the ‘‘saddle path’’ (dotted line) or w ¼ 0 boundary. Rather, thelimit cycle becomes ‘‘tighter’’ around the focal point Y, as there is less overshooting and undershootingin the no-wild-reproduction case.

5.6.3. The discount rate

To examine changes in time preference, we again adopt the baseline parameters but increase thediscount rate from 5% to 10%. The feedback control diagram is not illustrated for this case, but theeffect is to shift equilibrium Y down and left slightly relative to Fig. 3. A stable-limit cycle is stilloptimal, though it will also shift down and to the left so that there will be fewer salmon and alewife onaverage. These shifts come about because the higher discount rate reduces the value of future salmonangling, reducing the incentives to conserve salmon or its prey. Discounted net benefits are alsoreduced in this scenario to $90 million.

6. Conclusion

The terms ‘‘invasive’’ or ‘‘invader’’ are officially reserved for non-native species that cause somedegree of harm (Clinton, 1999). In practice, the terms are applied to almost any non-native species,which may be justified by the fact that non-native species generally alter the functions and services oftheir new ecosystem—with at least some services being adversely affected. The definition of‘‘invasive’’ is not a strict economic one, however, as it focuses on harm while ignoring benefits. It ispossible that an ‘‘invasive’’ species, as traditionally-defined (and the definition that we adopt), cancause some harm but still provide a net benefit. For instance, Pimentel et al. (2000) lists dogs and catsas invasive because they are non-native and have had some adverse ecosystem effects, though wedoubt many would call these animals an economic nuisance. Whether or not an invasive species is aneconomic nuisance, i.e., has a negative shadow value due to it being a net source of damages, is aneconomic question. This question is at the heart of the invasive species problem.

The net value of an invasive species depends on ecological and economic relations, includinginteractions with other species and how the species and its new ecosystem directly or indirectlyprovide services or disservices for people. Invasive species can be multiple-use species, providing amix of services and disservices. In this case the net value of the invader depends on how it is managed.More efficient management enhances the benefits and reduces the damages. Prior work has shownthat efficient, direct management of an invader can turn it from a nuisance into a valuable asset thatprovides positive net benefits. We have shown that this can also occur indirectly through themanagement of other species that interact through ecological relationships with the introducedspecies. Specifically, our analysis focused on the use of a bio-control species to manage the invader, acommon indirect approach to managing invasive species. Moreover, our management of the bio-control species, salmon, is also imperfect. Anglers of the bio-control species are left unregulated; theonly control for salmon is the government’s stocking program.

Indirect management of an invader, though beneficial in turning the invader into an asset, canresult in a host of complexities that may not arise under direct management approaches. Indirectmanagement actions represent a useful but imperfectly-targeted method of controlling the invader.Imperfectly-targeted controls often lead to spillovers affecting other parts of the ecosystem. Inparticular, adverse spillovers imply adjustment costs that impede management. Wirl (1992, 1995)analyzed a class of problems where such spillovers arise and found that optimal management in suchinstances can sometimes involve limit cycles as opposed to steady states. Our results, for a related butslightly different class of models, lend further support to Wirl’s findings. The complex cycles thatemerge from this relatively simple model illustrate the complexities and difficulties real managersface in managing ecological capital efficiently.

Our efforts to incorporate ecological and behavioral realism into our management model can beviewed as part of broader efforts among environmental scientists to capture the complexity of coupledhuman–ecological systems. This enhanced realism is costly in terms of computation and also yields


complicated emergent management recommendations. Nevertheless, there are real benefits. Forinstance, if the analysis was simplified and interactions between the invader and other speciesignored, then there would be no way to turn the invader into an asset. Moreover, the available set ofmanagement options would be more restricted and would not include a bio-control that yields thedual benefits of nuisance control and sport-fishing values.

Finally, our analysis can be used to inform Great Lakes stocking policy in concert with morecomplex simulation models (e.g., Szalai, 2003; Jones and Bence, 2009). Other authors have usedsimulation modeling to test the hypothesis that state-dependent stocking rules outperform constantstocking strategies (e.g., Szalai, 2003; Jones and Bence, 2009). Those analyses focused on biologicalindicators such as the salmon and alewife stock size and did not directly consider ecosystem servicesprovide by the system or the human response to changing ecological conditions. We showed thatcomplex state-dependent stocking rules are often optimal on economic grounds and can be developedconsidering net benefits from both the recreational fishery for the control agent (salmon) and bio-control of alewives.

Acknowledgements

This work was supported by the Great Lakes Fishery Commission, Michigan Department of NaturalResources Fisheries Division, MSU through the Quantitative Fisheries Center (QFC), the MichiganAgricultural Experiment Station, and USFWS Sportfish Rehabilitation Project F-80-R. This is QFCpublication number 2010-07. The views expressed are the authors’ alone. We are grateful for thecomments of Mike Jones, Jean Tsao, Frank Lupi, Jim Wilen, Jim Sanchirico, two anonymous reviewers,and Quantitative Fisheries Center staff during the development of this manuscript.

Appendix A

This appendix explains the calibration of the model. Most biological parameters for fish are basedon size, we use age and weight data to convert these for parameters relevant to a ‘‘representative fish’’.The instantaneous net recruitment rate for alewife, r, was calculated based on per biomassrecruitment of alewife minus the natural mortality rate for alewife as 5.7 (Szalai, 2003). The carryingcapacity for alewife, K, was based on simulations of the Lake Michigan Decision Analysis Model withno stocking, and was estimated at 2.89�109 kilograms of alewife (Jones and Bence, 2009). Thepredation parameters b and v were calculated based on bioenergetics, weight, and age data reportedin Szalai (2003) as b=1.67�10�4 and v=3.62�10�2. The salmon mortality parameters a=5.82 andg=0.02 were chosen such that salmon had a high instantaneous mortality rate in the absence ofalewives and were comparable to the imputed representative instantaneous mortality rates reportedby Benjamin and Bence (2003) associated with comparable alewife biomasses. Salmon recruitment tofishery parameters f=0.1 and y=58 where based on bioenergetics parameters in Szalai (2003) andthen manipulated to fit historic salmon levels. Salmon natural recruitment, b, was assumed to be10,000kg. This value is approximately 1/4 the maximum feasible stocking rate, conditional onhatchery capacity, wmax. The value of wmax was chosen to be slightly greater than historic high stockinglevels at 40,500kg (Michael Jones, Michigan State University, personal communication).

The recreational angling behavior parameters included 1 =19.95, 2 =6.57, 3 =29.94, h=12.31, ands=12.37. An additional parameter was added to the cost function so that the lognormal distributionwas shifted to the right so that it was defined from $35.05 to infinity. Therefore the average cost of atrip was $47.36. Applying these parameters in the angler behavior model, and using the 1994 Chinooksalmon stock size reported in Benjamin and Bence (2003), our model predicts an average per trip netbenefit of approximately $15 (nominal). This value is reasonable as compared to an average per tripvalue of $22 (nominal) for all 1994 Great Lakes recreation fishing by Michigan residents reported byLupi and Hoehn (1998). These parameters were estimated by assuming 13% of angling licenses sold inMichigan resulted in Lake Michigan salmon fishing trips, and fitting a time series of the number ofanglers, effort, catch, and salmon biomass (Benjamin and Bence, 2003 and Michael Jones, MichiganState University, personal communication) to the angler response model by minimizing the sum of the


squared error between that model projection and observations, weighted by the inverse of thestandard deviation of the individual data sets. The daily catchablity, q=6.43�10�8 is based onBenjamin and Bence (2003) converted for different time units. The marginal cost of stocking akilogram of salmon, v ¼ $19:55 was based on personal communication with Gary Whelan (MichiganDepartment of Natural Resources) and weight-at-age data (Szalai, 2003). The maximum number ofanglers was assumed to be the largest value in the number of angler time series. The discount rate, r,was assumed to be 5%.

The specific relationship between alewife stock size and damages is unknown. The damagesrepresent the sum of lost beach days and forgone ecosystems services associated with native fishes.For the base case we assume that if alewives were at carrying capacity, then there would be $1.5million in annual damages, all else equal. This seems feasible even if highly uncertain adverseecological effects are not large, given our experience with the Lake Michigan system and its history.This value for damages results in D=1.79�10�13. Given the rather arbitrary nature of this choice, weconduct sensitivity analysis to investigate the implications of alewife-induced damages being one ortwo orders of magnitude higher (D=1.79�10�12 and D=1.79�10�11).

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Resource and Energy Economics...Resource and Energy Economics 32 (2010) 500–518 ARTICLE INFO Article history: Received 7 May 2008 Accepted 2 April 2010 Available online 18 April