optimal rebuilding ofa metapopulation 2… · sanchirico,wilen,and coleman optimal rebuilding of a...

OPTIMAL REBUILDING OF A METAPOPULATION

JAMES N. SANCHIRICO, JAMES E. WILEN, AND CONRAD COLEMAN

A contentious issue in recent fisheries management debates has been how to rebuild overfished marinefish stocks. We investigate the economic and ecological nature of rebuilding plans in a metapopulationsystem. We find “most rapid approach paths” for system-wide recovery that differ from prototypicalpaths that might be chosen when populations are assumed independent. For example, moratoria insome patches may be extended in order to speed the recovery of other patches in the system. We alsoinvestigate the costs of second-best policies that depend on the degree of spatial heterogeneity, thenature of the linkages, and the dispersal rates.

Key words: bioeconomic, fisheries, optimal control, renewable resources, spatial ecology.

JEL codes: C61,Q2.

Introduction

Despite the nearly thirty years that haveelapsed since the Law of the Sea negotiationsenclosed much of the world’s most valuablefish resources, many fisheries are still economi-cally and ecologically depressed. In the UnitedStates’ exclusive economic zone, for example,one out every four major fish stocks was belowits target level in 2006. One of the more con-tentious issues during the reauthorization ofthe U.S. Magnuson-Stevens Fishery Conser-vation and Management Act (MSFCMA) in2006 was how fast to rebuild overfished fishecosystems.

In general, industry representatives lob-bied for rules that would stretch out rebuild-ing periods and avoid large upfront costs,whereas environmental groups argued thatit was important to rebuild depressed stocksquickly (Freeman 2006). The 2007 MSFCMAthat ultimately passed requires that most over-fished stocks be rebuilt over a period “as short

James N. Sanchirico is a professor, Department of Environmen-tal Science and Policy, University of California, Davis, a memberof the Giannini Foundation of Agricultural Economics, and anonresident fellow, Resources for the Future, Washington DC.James E. Wilen is a professor, Department of Agricultural andResource Economics, University of California, Davis and a mem-ber of the Giannini Foundation of Agricultural Economics. Con-rad Coleman is a research associate, Resources for the Future,Washington DC. The authors thank Erik Lichtenberg and ananonymous reviewer for insightful and helpful comments andsuggestions.

as possible and not to exceed 10 years.” Inaddition,the 2007 law contains admonishments(or caveats) that recovery plans should takeinto account “the status and biology of anyoverfished stocks of fish, the needs of fish-ing communities, and the interaction of theoverfished stock of fish within the marineecosystem.”

The political debate continues over howbest to rebuild overfished systems. Some leg-islators are looking to provide the RegionalFishery Management Councils with additional“flexibility” over how to address rebuild-ing (Winter 2007). A common suggestion bylegislators with fishing industry constituentsinvolves dropping the ten-year requirement,essentially permitting higher levels of fishingduring the rebuilding phase than the Councilswould have otherwise been able to authorize.The question that arises during the debateon rebuilding is: What are the economic andecological trade-offs associated with differ-ent levels of fishing during the rebuildingperiod?

Basic fisheries economics provides us withsome insights into the economic issues involvedin stock rebuilding. As A. D. Scott (1955)pointed out over fifty years ago, decisionsabout optimal use rates, whether they involverebuilding, draw down, or sustainable steady-state use, are all fundamentally capital theorydecisions. As such, optimal rebuilding deci-sions should involve trading off the currentrent losses associated with refraining from

Amer. J. Agr. Econ. 1–16; doi: 10.1093/ajae/aaq045Received June 2009; accepted March 2010

© The Author (2010). Published by Oxford University Press on behalf of the Agricultural and Applied EconomicsAssociation. All rights reserved. For permissions, please e-mail: [email protected]

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2 Amer. J. Agr. Econ.

harvest today against long-term yield gainsassociated with investing at higher biomasslevels.

Virtually all of our intuition about thepractical implications of this theory is basedon a single species, selective fishing gear,and an aspatial framework. For example,Clark (1990) shows that in a linear-in-fishing-effort optimization framework, it is optimalto rebuild a depressed population using a so-called most rapid approach path (MRAP).If there are no required minimal boundson effort, an MRAP rebuilding plan fora depressed population calls for a com-plete moratorium until the optimal desiredsteady state is attained. This economicallyoptimal policy coincides precisely with thepolicy language in the 2007 MSFCMA thatcalls for rebuilding over periods “as short aspossible.”

Related extensions to the classical bioeco-nomic model include papers by Wilen andBrown (1986),who focus on rebuilding popula-tions in a highly stylized two-species predator–prey system; Kaitala and Pohjole (1988), whoinvestigate cooperative and noncooperativesolutions for rebuilding a shared stock; andAgar and Sutinen (2004),who focus on rebuild-ing with time-invariant policies in a two-speciessystem with nonselective gear.

We contribute to the bioeconomic litera-ture by investigating how to optimally rebuilda population that consists of a system oflinked subpopulations. In particular, we modela three-patch metapopulation system wherethe regulator has control over the fishingeffort level in each patch in each period.Three patches are sufficient to characterizerebuilding plans for a rich variety of spatiallyexplicit structures, such as those illustrated

Figure 1. Ecological structures in a three-patch metapopulation model

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Sanchirico, Wilen, and Coleman Optimal rebuilding of a metapopulation 3

in figure 1. An important characteristic of ametapopulation is that there are two spatialscales operating simultaneously: the local orpatch population dynamics and the systemwide ecological dispersal process (Sale et al.2006).

An analytical solution for a three-control,three-state linear optimal control problem isnot possible in general, hence we solve forthe rebuilding plans using numerical meth-ods. As part of our analysis, we comparethe cost of inefficient, second-best policiesproposed during the recent political debates,such as time-invariant fishing effort policiesand dynamic policies with nonzero lowerbounds on fishing effort. The costs of second-best policies are measured as the reduc-tion in the net present value from theoptimal rebuilding plan. These comparisonsare useful because while allowing some fish-ing during the rebuilding to protect fishing-dependent communities is consistent withMSFMCA legislation, the main questionsrevolve around the ecological implicationsand efficiency benefits forgone with slowerrebuilding.

The results of our analysis indicate thatthe nature of ecosystem interconnectionsbetween subpopulations has significant qual-itative and quantitative implications for opti-mal harvest strategies. We find interesting,counterintuitive examples in which ecologi-cal interconnections can be optimally man-aged at each point in time and space toachieve system-wide objectives, using strate-gies quite different than would be used ifpopulations were thought to be indepen-dent. For example, we find cases where eventhough a patch’s population is above itsoptimal steady state and the classical sin-gle stock result is to fish down the popula-tion (Clark 1990), the optimal solution in themetapopulation context is to initially closethe patch to allow the stock to achieve evenhigher levels before beginning the fishing-down phase.

We also find that the cost of second-best policies depends on the degree ofspatial heterogeneity, the nature of the link-ages, and the dispersal rate. Higher disper-sal rates in highly connected systems tend toreduce the costs, everything else being equal.Differences between first- and second-bestrebuilding times are often pronounced, withsecond-best rebuilding times as little as 30%,or as much as five times those of first-besttimes.

Bioeconomic Metapopulation Model

Building a realistic and analytically tractablemodel of a fish population is difficult. Con-sequently, models and results that generategeneral qualitative principles are very useful,since we simply do not have the data to makemore detailed predictions or to justify the useof particular models. While there are certainlycomplexities in marine systems that are overlysimplified in our metapopulation model orfall outside its scope, we start with the sim-plest model that still captures the fundamentalprocesses of interest.

We assume that the objective is to maximizethe present discounted value of system-widefishery profits or rent by choosing the effortlevels in each patch. The objective function ofthe regulator is:

(1) maxEi(t)

∫∞

0e−δt

(N∑

i=1

πi(xi(t))Ei(t)

)dt

where Ei(t) is the effort level in patch i in periodt, πi(xi(t)) is the net rent per unit of effort,assumed to be a quasi-concave function of thelevel of the population in patch i, denoted xi(t).The rent in each patch is assumed to be linearin effort and implicitly a function of parame-ters such as the price received at the dock fromfish in patch i, and fishing costs.1 The discountrate is δ.

The regulator maximizes the present dis-counted value from the system subject to themetapopulation dynamics, effort limits, andstock density initial conditions.We assume thatthe metapopulation dynamics are given by:

(2)dxi

dt= fi(xi) +

N∑j=1

dijxj − hi(xi)Ei

where fi(xi) is the biological growth functionin patch i, hi(xi) is harvest per unit effort,and dij is the dispersal rate between patches

1 The linear-in-effort model utilizes a convenient structure firstintroduced into fisheries economics by Colin Clark (see, e.g., dis-cussion in Clark 1990). Clark showed that linear models may bethought of as approximations to more general concave/convexproblems and may be useful for distilling qualitative results fromanalytically tractable systems before generalizing. Linear prob-lems involve control trajectories that are either at extremes orsingular values, and hence the solution is characterized by findingswitch points where controls change qualitatively. Despite beingrestrictive, qualitative patterns in optimal controls and state vari-ables synthesized from linear models generally carry over to moregeneral problems.

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i and j (dii � 0, dij � 0 i, j). This formulationof the metapopulation problem follows Clark(1976) in assuming that the complete maxi-mization problem is linear in its controls fortractability. The initial conditions and controllimits are xi(0) = X0

i and Emini � Ei(t) � Emax

ifor each patch i. Connectivity structures suchas those in figure 1 can be depicted with var-ious restrictions on the dispersal parameters.We also impose an adding up restriction suchthat there is no mortality or entrants in thedispersal process.

Metapopulation models with similar struc-tures have been used to address a numberof issues. For example, Huffaker, Bhat,and Lenhart (1992) investigate optimalmanagement of a nuisance beaver populationin a two-patch model. Tuck and Possingham(1994) investigate optimal steady-state man-agement in a two-patch sink-source fisherysystem with no economic heterogeneity, andBrown and Roughgarden (1997) use a modelwith sedentary adults and larval dispersal toillustrate the value of larval pools to system-wide fishery profits. Drawing on a two-patchformulation in Clark (1990), Sanchirico andWilen (2005) derive necessary conditions forthe N-patch linear-in-effort model and thencharacterize the nature of the steady-stateconditions of a two-patch model with eco-nomic heterogeneity. They compare first-bestspatially explicit policy instruments withsecond-best uniform instruments and showthat optimal management may involve closinga patch to fishing, a finding also in Jannmatt(2005). Sanchirico et al. (2006) explore theoptimality of these “corner solutions” morefully, showing bioeconomic conditions underwhich complete closures of patches may beoptimal as elements of a system-wide optimalprogram. Costello and Polasky (2008) coversimilar ground in a metapopulation model withstochasticity that is an extension of Reed’s(1979) single-patch model, finding similarconditions for which closures are optimal.Sanchirico and Wilen (2008) elaborate onthe concept of economically optimal marinereserve closures by showing how nonfishingbenefits of marine populations enhance thedecision to optimally close individual patches.Some recent papers by Fenichel and Horan(2007) and Horan and Wolf (2005) examinethe optimal management of wildlife diseasesin a two-patch setting over time.

The analysis of optimal metapopulationmanagement over the past decade or so hasgenerated useful qualitative insights about

how connectivity between patches altersconclusions about optimal management ofthe simpler single-patch model. Most, if notall, of the analysis of optimal metapopula-tion management has focused on steady-stateconditions (in linear-in-effort models), leavingdiscussion of the complete optimal approachpaths to the steady state unexplored. To ourknowledge, we are the first to explore in detailoptimal approach paths, including both interiorand boundary portions of the paths to the fullspatial-dynamic solution for a metapopulationmodel. Our analysis, therefore, is of generalinterest to complete the understanding of howto manage metapopulations, given any set ofinitial conditions, including initial conditionswith pristine or underexploited populationsthat are subject to fishing down. But it is alsoof particular interest for the problem of stockrebuilding and recovery. In that (real world)case, the more contentious issues in public pol-icy debates center around both the ultimatelevels to which stocks should be rebuilt andhow fast rebuilding and recovery should pro-ceed. These are the issues we explore with ourdiscussion of economically optimal rebuildingstrategies.

Optimal Solution

Because the general solution techniques for alinear optimal control problem are well doc-umented in Clark (1990) and Kamien andSchwartz (1991) and the general techniquesdo not change with the number of controlvariables (Bryson 1999;Fraser-Andrews 1989a;Kamien and Schwartz 1991; Volker 1996), wepresent an abridged discussion here. The opti-mal fishing effort levels are determined bymaximizing the current value Hamiltonian

H ={

N∑i=1

πi(xi(t))Ei(t)

}+

N∑i=1

λi(t)(3)

×⎛⎝fi(xi(t)) − hi(xi(t))Ei(t)

+N∑

j=1

dijxj(t)

⎞⎠

where λi(t) is the biomass shadow value inpatch i (costate variables). Since the Hamil-tonian in Equation (3) is linear in the effort

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control variables, it can be rewritten toisolate the control variables by using switchingfunctions σi = πi(xi(t)) − λihi(xi(t)) that serveas terms multiplying the controls with i ∈{1, 2, 3, . . ., N}. Maximization of Equation (3)requires that the optimal controls be eitherextreme values or singular values, dependingupon the signs of the switching functions:

Ei(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Emaxi when πi(xi(t))

−λihi(xi(t)) > 0Es

i when πi(xi(t))−λihi(xi(t)) = 0

Emini when πi(xi(t))

−λihi(xi(t)) < 0

(4)

i ∈ {1, 2, 3 . . . N}.

Maximizing the Hamiltonian at each instantrequires that the optimal control for that patchbe set at its maximum when the switching func-tion is positive and that the control be set at itsminimum allowable value when the switchingfunction is negative. When the switching func-tion is zero, further analysis needs to be carriedout to determine the optimal controls along thesingular arc. On the singular arc, we can derivea feedback rule for the optimal level of effortas a function of the stock size and the economicand ecological conditions.

In addition to Equation (4), the first-orderconditions include the state Equations (2) andthe costate equations, which are derived fromdλi/dt = δλi − dH/dxi. Specifically, the costateequations for each patch i are:

dλi

dt= δλi − ∂H

∂xi= δλi − ∂πi

∂xiEi(5)

− λi

(∂fi(·)∂xi

− ∂hi

∂xiEi

)−

N∑j=1

λjdji.

This costate equation shows clearly that theown shadow value is partially determinedby the contribution that own biomass makesto other patches (via dispersal), weighted bymarginal profits or cross-system shadow values.

Solving a single-state, single-control linearcontrol problem explicitly requires a numberof steps. First,one characterizes conditions thathold when the switching function is identicallyzero for some finite period. This involves set-ting the switching function and its higher-orderderivatives equal to zero, inserting Pontryaginconditions where appropriate, and solving for

the singular value of the biomass, or a singularpath of the biomass. The second consists of cal-culating the singular value of the control thatsustains the corresponding singular biomasslevel. The final step in characterizing the solu-tion is to “synthesize” the solution, i.e., deter-mine when over the horizon extreme controlshold and when the singular solution holds. Thisstep involves piecing the various parts of thesolution together, and it depends upon initialconditions. If, for example, the initial biomassis low, the synthesized solution for a single-state variable problem will typically involve aperiod with the lowest effort possible so thatbuildup occurs rapidly, then a switch to a sin-gular solution holding biomass at its singularvalue. With an infinite time horizon, the solu-tion to an autonomous problem will involve a“most rapid approach” to the singular value,and then sustained harvesting at that value forthe remainder of the time horizon. The key isto determine the switching time at which pointcontrols switch from extreme to singular con-trols. With a single-state autonomous problem,this generally involves finding one initial switchtime, at which point either maximum or mini-mum controls are switched to singular controls.

In a multiple-state problem, there arenumerous combinations of the order for whichcontrols switch from extreme to singular. Forlarge dimension problems, it becomes diffi-cult almost immediately to derive any quali-tative insights about the nature of the optimalapproach path. In what follows, we analyze athree-patch model that permits us to derive andcharacterize the solution characteristics. Weassume that the growth function is quadraticwith intrinsic growth rates and carrying capac-ities ri and ki, respectively. We also assumethat the net rent function per unit effort isπi ≡ piqixi − ci, where pi is the patch-specificprice, qi is the catchability coefficient, and ci isthe cost coefficient.

Following Bryson (1999), Fraser-Andrews(1989), and Clark (1990), the analogous firststep in our system is to find the treble singu-lar solution,where all three switching functionsare simultaneously zero. The treble singularsolution can be expressed as three equationsthat implicitly define the optimal equilibriumbiomass densities, which depend on biologi-cal growth and dispersal parameters as wellas the economic parameters. The implicitequations are generalizations of the “goldenrule” of resource economics and are alsoderived for a two-patch system in Sanchiricoand Wilen (2005). Specifically, the spatially

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modified “golden rule” for patch i is:

(pi − ci

qixi

) (δ − ∂fi

∂xi

)︸︷︷︸

I

− ci

qix2i

fi(xi)︸︷︷︸II

(6)

−N∑

j=1

((pj − cj

qjxj

)dji + dij

cixj

qix2i

)=

︸︷︷︸III

0.

The spatial part of the problem is embed-ded in III. With no spatial connectivity, thecross-patch diffusion coefficients in III will bezero, and the treble singular solution reducesto three independent biomass levels deter-mined by terms I + II. With spatial connectiv-ity, the dispersal coefficients in Equation (2)will appear in the partial derivatives of III,and the solutions for each patch are inter-twined in ways that depend upon the spatialinterconnections.

In the single patch case, Equation (6) can beinterpreted as the net value of refraining froma marginal unit of harvest and adding it to thestanding biomass. The marginal gain is givenby the terms in I + II, which equal zero at thegolden rule biomass level. In a multiple-patchsetting, refraining from harvest also alters dis-persal among and between the patches. Hencea unit of forgone harvest ends up influenc-ing profits in other destinations after dispersal(captured in term III in Equation (6)).Whethera unit of biomass is more “valuable” in termsof profit in one or the other patch depends onthe relative prices and costs in the two patches.

We are interested in exploring optimal tran-sition paths that ultimately converge to theoptimal system-wide steady state described byEquation (6). This transition path problem hasnot been solved for a multipatch system to ourknowledge. Analogous to a single-patch case,the full dynamic transition path to this con-trol problem can consist of periods where one(or more) controls are singular and the oth-ers are at extremes. The complication is thatoptimal controls can be extreme or singular inmany different combinations. For example, forarbitrary initial conditions, the initial optimalcontrols can be configured twenty-seven differ-ent ways for a three-patch problem (max, max,max; max, max, min; max, min, min; etc.).

To determine the values of the singularcontrol Es

i when one or two σi = 0 for afinite time period, we need a singular feed-back rule (Fraser-Andrews 1989;Volker 1996).

We illustrate this process for the case of onesingular control. Without loss of generality,we assume that Es

1 needs to be determined,while E2 = Emin

2 or E2 = Emax2 and E3 = Emin

3 orE3 = Emax

3 . Since σ1 = 0 for a measurable inter-val, we know that d(n)σ1/dt(n) = 0 on that inter-val. In order to obtain our feedback rule,we dif-ferentiate the switching function with respectto time as many times as needed (Bryson 1999;Fraser-Andrews 1989; Volker 1996), substitut-ing equations where appropriate, until appearsin the expression. For our model, turns outto be two. Therefore, using d(2)σ1/dt(2) = 0,d(1)σ1/dt(1) = 0, and σ1 = 0 along with dx1/dt,dλ1/dt, d2λ1/dt2, we obtain a complicatedclosed-form expression for the singular feed-back law

Es1(t) = g1(E2, E3, x1(t), x2(t), x3(t),(7)

λ2(t), λ3(t); �1)

where �i is the set of economic and ecologicalparameters in patch i. The closed form solutionof Equation (7) is illustrated in the supplemen-tary material. A similar process is used for thecase of a double singular arc. When the thirdpatch switches onto its singular arc, we canshow that the system has reached the treblesingular steady-state solution.

By inspection of Equation (7), we see thatwhen E2 or E3 jump off of either extreme andonto their singular paths, then the correspond-ing time-varying singular path Es

1(t) will alsojump or be discontinuous at that point in time.For example, if patch 3 switches on next, wehave the following system of equations:

Es1(t) = g1(E2, x1(t), x2(t), x3(t), λ2(t); �1)(8)

Es1(t) = g1(E2, x1(t), x2(t), x3(t), λ2(t); �1).

The discontinuity on the time-varying por-tion of the singular arc is a result of havingmore than two controls that are interdepen-dent along with the extreme/singular nature ofthe control paths. This is a new result that doesnot arise in single- or two-control problemsthat have previously been utilized in metapop-ulation modeling. With the case of two controlsin our setup (e.g., two-patch model), when thesecond control switches to the singular pathit must be the case that the singular pathsfor both patches correspond to the steady-state control levels. In general, interdependentmultiple-control problems such as this oneare a fundamental feature of spatial-dynamiceconomic problems.

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0 t1* t2

* t3*

E1=E1sing

E2=0E3=0

E1=E1max

E2=0E3=0

E1=E1ss

E2=E2ss

E3=E3ss

E1=E1sing

E2=E2sing

E3=0

Figure 2. Illustration of the full dynamic solu-tionNote: The illustration corresponds to the casewhere patch 1 switches onto its singular patchfirst at t∗1 , patch 2 switches on next at t∗2 , andpatch 3 switches on last at t∗3 . Patch switchtimes and order are endogenous. For exam-ple,depending on the initial stock densities andecological and economic conditions, we couldfind that patch 3 switches on before patch 1(t∗3 < t∗1 ).

Given the initial values of the fish stocks,xi(0), and the upper and lower bounds onfishing effort, the full solution to the optimalcontrol problem is found by solving for thethree switch times (t1, t2, t3), the dates at whicheach of the controls switches to its singulararc. Figure 2 illustrates the role of the switchtimes and a possible solution.We search for theswitch times using a shooting algorithm that isdescribed in the online supplementary mate-rial. We also describe in the supplementarymaterial the necessary but not sufficient gener-alized Legendre-Clebsch (GLC) condition foroptimality that we employ (Fraser-Andrews1989b). It is important to note that our solu-tion utilizes the numerical closed-form singularfeedback law along the singular arcs to describethe singular dynamics, which we accomplishby building the symbolic solver capabilitiesof Matlab (release 2008a) into our numericalalgorithm.

Numerical Analysis

In designing optimal rebuilding plans, the reg-ulator needs to trade off not just the economicvalues associated with standing biomass in thelocal patch, but also the value associated withthe local patch’s contribution to productivityof other patches via various dispersal pro-cesses (see, e.g., Equation (6)). Because theregulator is setting the optimal effort in eachpatch in each period, she will need to trade offcatching more fish in patch 1, which implieslower population levels and therefore feweradults dispersing to patch 2 and/or patch 3,against catching less fish in patch 1 and shiftingeffort to patch 2 and/or patch 3. This trade-offexplicitly accounts for the relative profitabilityassociated with harvesting in the patch, itselfa function of bioeconomic parameters associ-ated with each patch, as well as with the natureof connectivity between patches.

Our analysis focuses first on the homoge-neous case, where all patch-specific bioeco-nomic parameters are identical, except forinitial biomass levels. We then investigate aspatially heterogeneous case with cost hetero-geneity. In each case, we assume that patch1’s initial biomass density is above its steadystate and that those of patches 2 and 3 arebelow. Specifically, the initial conditions are{X1(0), X2(0), X3(0)} = {.775, .3, .1}. Assumingthat all three patch populations need to berebuilt is another interesting case, but lessinformative than our current case.

Table 1 lists the set of economic and ecolog-ical parameters used to derive the results intables 2 and 3 and figures 3 to 5. The over-all strategy was to choose parameter values

Table 1. Parameter definitions, symbols, and levels used in the numerical analysis

OptimalHomogenous Cost Hetero. Moratorium

Parameter Symbol Base Case (figures 3 & 4) (figure 5)

Intrinsic growth rate ri ri = .1875 i = 1, 2, 3 unchanged unchangedDispersal rate dij d = .12∗max(ri) (see note) unchanged unchangedCatchability coefficient qi qi = 1.25, i = 1, 2, 3 unchanged unchangedEx-vessel price pi pi = 1.25, i = 1, 2, 3 unchanged p1 = 1.0 = p2,

p3 = 3.85Cost per unit of fishing effort ci ci = .3, i = 1, 2, 3 c1 = .3, c2 = .2, c1 = .3, c2 = .2,

c3 = .125 c3 = .125Discount rate δ .05 unchanged .03Percent of optimal effort w .4 unchanged unchanged

Note: We impose the adding up restriction in the dispersal matrix by multiplying the rate of emigration by the number of patches that are connected tothe patch.

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Table 2. Percent differences in switch times, steady-state stock levels, and net present valuebetween the closed and other metapopulation structures

Structure t∗1 t∗2 t∗3 X∗1 X∗

2 X∗3 NPV

Homogenous Closed 3.47 4.79 11.99 0.51 0.51 0.51 1.92Fully −10.1% −3.1% −24.2% 0.0% 0.0% 0.0% 0.4%Nearest −4.1% −2.2% −12.0% 0.0% 0.0% 0.0% 0.16%Step −5.3% −9.0% −21.1% −4.0% −0.01% 3.72% 0.2%Circle −5.6% −8.8% −12.4% 0.0% 0.0% 0.0% 0.1%Source −10.4% −16.5% −24.6% −7.7% 3.5% 3.5% 0.6%

Cost Closed 3.47 3.82 10.25 0.51 0.47 0.43 2.20Heterogeneity Fully −18.3% −3.5% −29.6% 1.7% −0.1% −2.6% 1.8%

Nearest −7.7% −1.6% −13.6% 0.7% −0.1% 1.0% 0.7%Step −8.7% −5.7% −22.0% −2.8% 1.2% 2.5% 2.8%Circle −8.9% −5.6% −15.2% 0.4% 1.3% −2.1% 0.9%Source −19.1% −17.2% −27.1% −4.7% 3.2% 2.6% 4.6%

Note: We illustrate the percentage deviation from the closed case. For the case of t1, a negative percentage corresponds to less time fishing down patch 1 than inthe closed case. For the case of t2, a positive (negative) number means a longer (shorter) moratorium than in the closed case. Finally, for t3, a negative numberimplies a shorter moratorium in patch 3, which in these runs also corresponds to the time at which the system is fully rebuilt.

Table 3. Percent differences in switch times and net present value between second-best andfirst-best policies

Time-invariantTime-varying Second Best Second Best

Structure T1 T2 T3 NPV NPV 2

Homogenous Closed 0.0% 49.7% 39.0% −3.01% −15.2%Fully −3.7% 54.6% 32.9% −2.01% −9.4%Nearest −0.4% 57.0% 35.0% −2.5% −12.3%Step −0.3% 49.9% 36.7% −2.46% −13.0%Circle −0.2% 50.0% 35.0% −2.48% −11.6%Source −4.7% 48.1% 36.2% −2.47% −12.5%

Cost Closed 0.0% 53.3% 44.2% −1.70% −10.0%Heterogeneity Fully −2.7% 57.2% 37.3% −0.91% −5.27%

Nearest 0.5% 60.2% 40.0% −1.30% −7.60%Step −.05% 52.6% 41.8% −1.37% −8.41%Circle 0.1% 53.0% 39.9% −1.26% −7.08%Source −3.8% 51.4% 40.3% −1.19% −7.13%

Note: These results demonstrate the percent deviation from the optimal results, as presented in table 1. In each case the steady-state stock and fishing effortare identical. The time-invariant policy does not have switch times, because effort is held constant at the optimal steady-state levels.

that supported interior steady-state solutions.2With respect to the dispersal rate, we utilizea common dispersal rate (d) and measure thedispersal rate as a fraction of the maximumintrinsic growth rate to ensure a nonzero unex-ploited equilibrium for all the connectivitystructures.3

The maximum effort levels (Emax) are arbi-trary but set to ensure that it is possible totransition from open-access steady states to

2 The population levels were rescaled to be between 0 and 1 forthe numerical analysis.

3 We impose the adding-up restriction by multiplying the rate ofemigration (dii) by the number of patches the patch is connectedwith.

optimal steady-state levels. In practice, manyfisheries in severely overfished open-accessequilibria have gotten there after a “bloom”of investment by overly optimistic entrants.After the open-access equilibrium is achieved,there are often redundant vessels left idle fora considerable period. We set Emax to 125%of the open-access steady-state effort levels inthe independent system. The maximum effortlevels do not change across the connectivitystructures, although they differ under simula-tions with parameter heterogeneity.4 Lowering

4 The maximum effort levels in the base case are: Emaxi = .1515

with i = 1, 2, and 3. The maximum effort levels corresponding to

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Figure 3. Optimal Effort Levels in the case of cost heterogeneityNote: Under these specific parameter conditions (see table 1), the time (t∗3 ) at which patch threeswitches from Emin to Es corresponds to the time at which the system is rebuilt. t∗3 also correspondsto the time at which the system reaches the trebly singular path, which itself corresponds to thesteady state.

(raising) the upper bound on effort reduces(increases) the speed at which the fishery canbe fished down. The lower levels of effort(Emin) are set to zero,which represents a fishingmoratorium.

In each of the analyses, the GLC conditionis satisfied and the treble singular steady-statesolutions are all interior and stable. That is,we are investigating only parameter sets whereonce the control is on the singular path, itremains within the upper and lower bounds.Investigating corner solution cases where thesingular path could be blocked by the upperand lower bounds is beyond the scope of thecurrent paper.

We start with the spatially unconnectedsystem where the solution for each patch

figures 3 and 4 are Emax1 = .1515, Emax

2 = .1635, Emax3 = .1725 and

the levels corresponding to figure 5 are Emax1 = .171, Emax

2 = .189,and Emax

3 = .2192.

independently mimics the prototypical MRAP(bang-bang control) single-patch results foundin Spence and Starrett (1975). Table 2 presentsthe switch times.5 The initially underexploitedpatch 1 is fished at the maximum rate possi-ble (Emax

1 ),driving down the population densityuntil the steady-state level is reached, at whichpoint (t∗1 = 3.47) the effort level switches tothe singular path, which is time invariant. Sim-ilarly, overexploited patches 2 and 3 employeffort moratoria—patch 2 switches from Emin

2to Es

2 at t∗2 = 4.79 and patch 3 switches fromEmin

3 to Es3 at t∗3 = 11.99. The differences in

switch times between patch 2 and patch 3 aredue to different initial biomass levels. With thecurrent set of parameters, the system is fullyrebuilt (all patches are equal to their optimal

5 The switch times are reported in simulation-scale time and donot necessarily translate into calendar scales.

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Figure 4. Optimal biomass density levels in the case of cost heterogeneity

steady-state levels) when patch 3 switchesto Es

3.What are the dynamic implications of dif-

ferent kinds of ecological connectivity in asystem that is otherwise parametrically homo-geneous? To answer this question, table 2presents the percentage changes between theswitch times, steady-state stock levels, and netpresent value for the different structures infigure 1. Note first that when the system isparametrically homogeneous, the steady statesolutions for the fully integrated and nearestneighbor system will be the same as for theindependent system.6 But the switch times willvary depending upon differences in ecologi-cal connectivity, which affect the time for thewhole system to recover. In the fully integratedcase,patch 1 switches off of Emax

1 10.1% sooner,patch 2 maintains a moratorium a little longer

6 In the steady state, the biomass density levels are identi-cal across the system (all at an interior solution), and hencedispersal based on differences in relative densities cancels out.Costello and Polasky (2008) identify similar conditions underwhich optimal escapement is independent of dispersal and spatialinterconnections are irrelevant.

(about 3% longer),and patch 3 recovers in 24%less time than in the closed case. By switchingoff of Emax

1 sooner and onto its singular path,for example, the planner maintains a higherpopulation density in patch 1, which then dis-perses to the other patches. Because of theecological connectivity, the fully integrated sys-tem recovers faster than the independent case,since the planner is able to utilize the flow ofdispersal to help enhance patches with lowerpopulation densities. Biomass disperses frompatch 1 to patches 2 and 3 and from patch2 to patch 3 during the transition. Althoughthe difference is small, the ability to “manage”dispersal results in an increase in net presentvalue.

Comparing the fully integrated case withthe nearest neighbor dispersal case, we findthat qualitatively the results are similar, exceptthat without a direct linkage between patch1 and patch 3 the regulator maintains alonger moratorium patch 2. As a result, thetime to recovery of the system is greaterthan in the fully integrated system but isstill about 12% shorter than in the closedsystem.

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Figure 5. Optimal effort levels in the presence of cost and price heterogeneityNote: Under these specific parameter conditions (see table 1), in all cases but the independentand sink-source system, the time (t∗2 ) at which patch two switches from Emin to Es correspondsto the time at which the system is rebuilt. In the independent and sink-source system, the lastpatch to turn on is patch 3. Note that in the fully integrated and sink-source systems, the initiallevel of effort is set at zero (Emin) rather than at Emax to allow the system to rebuild faster.

The asymmetrical connectivity cases (step-ping stone, circle, and sink-source) exhibit dif-ferent dynamics depending on the nature oflinkages with patch 3. The sink-source case,for example, with patch 1 a source, rebuildspatch 3 slightly faster than the fully integratedsystem. The circle structure that has flows outof patch 3 into patch 1 independently of rela-tive densities in the respective patches has theslowest rebuilding time among the connectedcases, even though it is still 12% faster thanthe closed case. Furthermore, because the cir-cle and fully integrated systems have the samesteady-state levels as the independent systemwith homogeneous bioeconomic parameters,the differences in rebuilding time and thedynamics illustrate rather clearly the role of theconnectivity structure in the spatial-dynamicsolutions.

Next we turn to the case of parametric het-erogeneity, where the cost per unit of effortvaries across space. In particular, we assumethat patch 1 is the highest cost patch, patch2 has costs 33% lower, and patch 3 has costs60% lower than patch 2; with all other param-eters equal.This mimics circumstances in whichthe degrees of initial overexploitation in eachpatch are due to differential unit profits, acase likely to reflect reality in many fisheries.In this case, steady-state biomass levels dif-fer from one another, and approach timesreflect both biological connectivity and differ-ences between initial conditions and optimalsteady-state levels. Figures 3 and 4 illustratethe effort and stock dynamics, and the sec-ond half of table 2 reports the differencesbetween the optimal switching times underthese assumptions.

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Rather than going through each connectivitystructure separately, we focus on the qualita-tive differences between them. Figure 4 showsthat because of the cost (profit) differences, theoptimal steady-state biomass is lowest in high-profit patch 3 and highest in low-profit patch 1.The opposite is true for the steady-state effortlevels. Note the bang-bang nature of the con-trol paths in the independent system in figure 3.In the other panels in figure 3, singular pathsare time varying, due to the integration of thepopulation dynamics across the patches. Wealso observe jumps in these singular paths thatoccur when a patch switches onto its singularpath. Overall, the three patches are on a “jointMRAP,” even though each individual patch isdeviating from the MRAP it would follow inthe independent system.

Across the different cases, the qualitativenature of the singular paths in patch 2 isinteresting and surprising. For example, the ini-tial period on the singular path in the fullyintegrated, nearest neighbor, and sink-sourcecase is characterized by increasing effort levelsalong the time-varying portion of the singulararc,whereas in the stepping stone and circle,theinitial effort singular levels are decreasing overtime. The time-varying nature of effort allowsthe biomass to either increase but at a slowerrate than with no fishing or decrease at a slowerrate than in the case of maximum fishing.

We also investigated cases where thereexisted both price and cost heterogeneity andthe case of ecological heterogeneity with dif-ferent intrinsic growth rates. Because in eachof the cases the qualitative nature of the switchtimes is similar to the case of cost heterogene-ity, we report the difference in the switch timesin the supplementary material. We also inves-tigated many other parameter configurations,including varying discount rates, initial con-ditions, maximum and minimum effort levels,dispersal rates, and catchability coefficients.

In what follows, we present two of the moreinteresting and policy-relevant scenarios thatemerged from our extensive analysis. First, weinvestigate conditions in which understandingthe nature of ecological connectivity is abso-lutely critical to making qualitatively correctinitial policy decisions. In particular, we con-sider whether it is ever optimal to implementa moratorium in a linked system when singlepatch optimization would indicate a most rapidapproach reduction of biomass. Second, weinvestigate what happens when moratoria areoptimal but infeasible due to political consid-erations. In particular, we consider cases where

effort in each patch is constrained not to fallbelow a positive minimum level,correspondingto cases where fishing is allowed to continue inorder to keep the industry operating during thetransition to a new steady state.

We address the question of when under-standing the nature of ecological connectivityis critical for policy by considering cases wherethe planner finds it optimal to initiate initialMRAP reductions in patch 1 under some eco-logical configurations but adopts patch 1 mora-toria in the fully integrated and sink-sourcesystems, illustrated in figure 5. To generate thisresult, we increased the price in patch 3 from1 to 3.85 and reduced the discount rate from5% to 3%; all other parameters are identicalto the case with cost heterogeneity. By raisingthe patch 3 price, the return from reducing therebuilding time in patch 3 increases. Patch 3can be rebuilt faster by forgoing profits fromfishing patch 1 and by using patch 1 to feed viadispersal the growth in the overharvested patch3. The precise nature of connectivity is criticalto whether this trade-off is optimal; with theseparameters, dispersal benefits in the nearestneighbor, stepping stone, and circle ecosystemsare too low for the patch 1 moratorium to payoff.

Circumstances that lead to an optimal mora-torium in patches where the initial fish popula-tion is above its steady state depend on manyfactors, including the discount and dispersalrates. For example, when we increase the dis-persal rate by 10%, the optimal moratorium inthe fully integrated case is three times longer,and in the sink-source case,it is 1.3 times longer.This occurs because the ecosystem-wide impactof a patch 1 moratorium is stronger duringthe early period of the policy. On the otherhand, even at the 10% higher dispersal rate,an increase in the discount rate from 3% to7% results in the dynamics returning to thosefound in figure 3.7 We also find, for example,at the dispersal rate used in figure 5, a smallincrease in the discount rate from 3% to 4%yields an optimal patch 1 moratorium in thesink-source setting but not in the fully inte-grated system. A similar result holds when wehold the discount rate at 3% but lower the dis-persal rate by 20%. As the costs of waitingdecrease (as the discount rate decreases), thelength of the moratorium increases for a given

7 At a 7% discount rate, if we increase the dispersal rate by 25%off of the base case, a moratorium in the fully integrated and sink-source case is optimal.

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rate of dispersal.8 As we increase the dispersalrate, however, the benefits due to connectivitycan offset the costs, and the moratorium will beoptimal for a wider range of the discount rates.

Up to this point, we have assumed that thesocial planner is able to implement a mora-torium on fishing by reducing effort to zero.In our model, effort reductions cause harvestreductions that then rebuild stocks, and thecosts of such policies are forgone harvest ben-efits net of savings in effort. But reducingfishing effort may generate indirect communityspillovers if crew, processing equipment, andcommunity infrastructure cannot be adaptedto other productive activities.Thus,not surpris-ingly,a controversial issue in designing rebuild-ing plans is whether to permit some fishing dur-ing the recovery period as a means to reducealleged short-term impact on local communi-ties and businesses (Safina et al. 2005). Unfor-tunately, there is no empirical evidence thatmeasures whether these spillover or adjust-ment costs are significant in practice or not.In the absence of empirical information aboutadjustment costs, we address this issue by com-paring the gains from rapid rebuilding (usingmoratoria) with those from options that allowsome fishing and thus feature slower recoverytimes. In particular, we measure the presentvalues of second-best policies that either main-tain a time-invariant level of fishing effort (onethat allows the stock to rebuild) or imposethe constraint that the minimum effort levelis positive.9

The time-invariant policy assumes that theplanner holds effort levels in each patch fixedfor all time at the optimal steady-state levels(Ei(t) = Esteady-state

i for i = 1, 2, 3 and for all t).In this case,the dynamics of the recovery followthe population dynamics with fixed (possiblydifferent) effort levels in each patch. The sys-tem converges to the steady state in a logisticfashion—there are no switching times, sinceeffort is constant. The other policy we consideris time varying, where instead of assuming thatthe minimum effort level is zero, which permitsthe moratoria, we set the minimum bound at

8 Note that the discount rate plays two roles. First, lower dis-count rates imply higher steady-state biomass levels,ceteris paribus.Second, discount rates scale the role of impatience relative to thespeeds of dispersal and growth. For given dispersal rates and max-imum recovery times, lower discount rates make the immediatelosses associated with moratoria less costly.

9 Clark,Clarke,and Munro (1979) consider the case of malleablevs. nonmalleable fishing capital and conclude that fishing moratoriaare more likely to be optimal in the case of malleable capital butare possible in both settings, especially when the fish stocks areseverely overfished.

some (arbitrary) positive level. In this case, weare solving for the full path of effort to max-imize the net present value in the system asbefore, except now Emin

i = w∗Esteady-statei rather

than Emini = 0. By reducing w to zero, our time-

varying second-best policy converges to thefirst best; and as we increase w toward one, thetwo second-best policies converge. Both poli-cies are consistent with various positions in cur-rent debates, and our analysis provides someinsights into the likely magnitude of the costsof following one over another. In both second-best policies, the costs are incurred during thetransition to the same long-run equilibrium asthe first-best policies. A sample set of transi-tion paths are provided in the supplementarymaterial.

Table 3 shows how our analysis with homo-geneous parameters and cost heterogeneity isaffected when w = .4. Table 3 illustrates switchtimes in the time-varying policy as measured asthe percent difference from the first-best time(found in table 2). A negative sign means thatthe switch time is shorter, and a positive levelimplies that the switch time is later. We alsopresent the percent difference between the netpresent value in the first-best to each second-best policy, where a negative level means thatthe net present value is lower than in theoptimal setting.

Focusing first on the time-varying policy, wefind no difference in the independent basecase (the closed system) between the first-and second-best switch times in patch 1. Thisis because patch 1 is fished at its maximumrate initially and then the patch immedi-ately switches to the singular path thereafter.Patches 2 and 3 are being rebuilt, however,and are directly impacted even in the inde-pendent base case setting, since the recoveryis delayed by the nonzero minimum bound onfishing effort.

Some general observations from table 3 areworth highlighting. First, the difference in netpresent values between first- and second-bestpolicies is not large. This could be signifi-cant; if these results hold in more realisticmodels with accurate parameter calibration,they could suggest that allowing some fish-ing during recovery is cost-effective. Second,the difference between the net present val-ues in the first- and second-best policies islargest for the independent system comparedwith cases where there are linkages. This ispotentially good news, since it implies that dis-persal can mitigate potential costs of deviatingfrom optimal policies. Third, not surprisingly,

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the costs from the time-invariant policy aresignificantly higher than the second-best time-varying policy. These costs would converge asw approaches one.10

While the present value differences betweenfirst- and second-best policies might not belarge, we do find that increases in the length ofrecovery time can be significant. In all cases, thepercent increase in the switch time, t∗3 , indicatesthe increase in the recovery time. These levelsrange from 30% to 44%. These are also poten-tially significant findings. If they hold in morerealistic settings, they might amplify disagree-ment between those focused on net economicvalues and those focused on getting the ecosys-tem back to a safe sustainable level as quicklyas possible. The recovery periods for the time-invariant policy (not shown) are even moredramatically different, on the order of three tofive times those of the first-best policies, withsome variation due to the different structuresand assumptions. In an ecosystem context, theecological and economic costs of these delayscould be much larger when the species playsan important role in ecosystem health. Thesebroader impacts are important topics for futureresearch.

Discussion

A frontier research area of natural resourceeconomics and policy concerns the implica-tions for resource management of spatial-dynamic processes (Sanchirico andWilen 1999;Wilen 2007;Smith,Sanchirico,andWilen 2009).This paper contributes to the literature inves-tigating these features by focusing on a real-world policy problem, namely, how spatial eco-logical connectivity affects optimal rebuildingpolicies for a metapopulation.

As we have shown, the specific nature of spa-tial connectivity has important impacts on thequalitative nature of optimal recovery pathsfor an integrated ecological system. States orregions that have carefully rebuilt or main-tained their particular subpopulations mayconclude that they should be allowed to resumeharvesting, which seems reasonable from asingle-patch perspective. But as we show, poli-cies that seem locally sensible may in fact be

10 Applying the parameters used to derive figure 5 with w = .4,we find that the optimal policy in the fully integrated and sink-source setting is to start fishing effort at the minimum rather thanthe maximum.

just the opposite of what is needed from theglobal and system-wide perspective. In viewof the potential for spatial externalities, theremay be a need for new institutional mecha-nisms that utilize side payments or other meansof inducing cooperation when local interestsdiverge from system-wide interests.

The relatively low costs of the time-varyingsecond-best policy imply that allowing somefishing (although greatly curtailed from the lev-els that led to overfishing) might not resultin large losses in the present value of rentsfrom slowing the recovery of the system. Thisis an interesting result but one driven partlyby our model structure. In principle, if the fish-ing industry receives the bulk of the rents fromfishery policy, it ought to prefer policies thatmaximize the present value of those rents, asassumed in our objective function. But the factthat there is political resistance to such rapidstock recovery suggests at least two potentialdiscrepancies between our model and the realworld. One possibility is that the industry facessome kind of adjustment costs (or nonconvex-ities) that are not captured by the simple eco-nomic trade-off between less fishing today andmore fishing in the future. Expectations thatthere are economic and social adjustment costsassociated with capital infrastructure (ports,fishing, and processing capacity) and jobs seemto be behind the continued calls for rolling backthe rebuilding requirements in the MSFCMA(Winter 2010). For example,“fishing groups saythat without more drastic relief from the reg-ulatory regime [MSFCMA rebuilding require-ments], they could lose thousands of jobs forfishermen and all of the industries that dependon them” (Winter 2010).

The other possibility is that political resis-tance will be yet another fallout from incom-plete or nonexistent property rights. If fisher-men do not have secure access privileges, thecosts of investing in stock rebuilding may fallon one group and the benefits may accrue toother groups. In this case, we would expectpolitical opposition even when present value-maximizing alternatives are identified (see,e.g.,Karpoff 1987 for a discussion of suboptimalcontrols in fishery management).

While our focus is on rebuilding metapop-ulations, we also found new complex controldynamics when more than two controls werespatially linked. Most important are the dis-continuities on the time-varying portions ofthe singular arcs—results that to our knowl-edge have not been shown previously in thenatural resource economics literature with

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smaller-dimensioned systems. These findingsare relevant in other settings where policies inany particular location affect the set and tim-ing of policy options in other locations and inother periods, such as the control of invasivespecies and animal and human diseases.

Funding

Sanchirico acknowledges support from the U.S.EPA Science to Achieve Results (R832223),Resources for the Future, and CaliforniaAgricultural Experimentation Station (AES)project CA-D-ESP-7084-H.

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