defining viable recovery paths toward sustainable fisheries

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ANALYSIS

Defining viable recovery paths toward sustainable fisheries

Vincent Martineta,⁎, Olivier Thébauda,b, Luc Doyenc

aMarine Economics Department, IFREMER, ZI Pointe du Diable BP 70, 29280 Plouzané, FrancebUBO-CEDEM, FrancecCNRS, CERSP, Muséum National d'Histoire Naturelle, France

A R T I C L E I N F O

⁎ Corresponding author. EconomiX-UniversitéE-mail addresses: vincent.martinet@u-paris

0921-8009/$ - see front matter © 2007 Elsevidoi:10.1016/j.ecolecon.2007.02.036

A B S T R A C T

Article history:Received 18 October 2006Received in revised form23 February 2007Accepted 23 February 2007Available online 16 April 2007

This paper develops a formal analysis of the recovery process for a fishery, from crisissituations to desired levels of sustainable exploitation, using the theoretical framework ofviable control. We define sustainability as a combination of biological, economic and socialconstraints which need to bemet for a viable fishery to exist. Biological constraints are basedon the definition of a minimum resource stock to be preserved. Economic constraints relateto the existence of a guaranteed profit per vessel. Social constraints refer to themaintenanceof aminimumsize of the fleet, and to themaximumspeedatwhich fleet adjustment can takeplace. Using fleet size adjustment and fishing effort per vessel as control variables, we firstidentify the states of this bioeconomic system for which sustainable exploitation is possible,i.e. for which all constraints can be dynamically met. Such favorable states are called viablestates. We then examine possible transition phases, from non-viable to viable states. Wecharacterize recovery paths with respect to the time of crisis of the trajectory, which is thenumber of periods duringwhich the constraints are not respected. The approach is applied tothe single stock of the bay of Biscay Nephrops fishery. The transition path identified throughthe viability approach is compared to the historical recovery process, and to both open-access and optimal harvesting scenarios.

© 2007 Elsevier B.V. All rights reserved.

Keywords:Sustainable fishingRecoveryFishery policiesBio-economic modeling

JEL classification:Q22; C61

1. Introduction

According to recent studies, the maximum production poten-tial of marine fisheries worldwide was reached at least twodecades ago; since then, due to the widespread developmentof excess harvesting capacity, there has been an increase inthe proportion of marine fish stocks which are exploitedbeyond levels at which they can produce their maximumsustainable yields (FAO, 2004; Garcia and Grainger, 2005).Hence, the problem of managing fisheries is increasingly castin terms of restoring them to higher sustainable levels of fishstocks, catches, and revenues from fishing.

Paris X., France. Fax: +3310.fr (V. Martinet), olivier

er B.V. All rights reserved

The problems posed by fisheries restoration are dynamic innature: beyond the issue of choosing adequate objective levelsfor restored fisheries, a key question is the identification andthe selection of the possible paths towards these objectivelevels. In practical situations, this question is crucial as itrelates to the feasibility (technical, economic, biological) and tothe social and political acceptability of the adjustmentsrequired for fisheries to be restored, hence to the actualpossibility of driving fisheries back towards decided sustain-ability objectives.

The definition of preferred strategies for the harvesting ofmarine fish stocks has been widely studied in the literature onrenewable resource management. While most of the initial

2 98 22 47 [email protected] (O. Thébaud), [email protected] (L. Doyen).

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http://dx.doi.org/10.1016/j.ecolecon.2007.02.036

1 Information about this software of scientific calculus and freedownload are available at http://www.scilab.org/.

412 E C O L O G I C A L E C O N O M I C S 6 4 ( 2 0 0 7 ) 4 1 1 – 4 2 2

work focused on the comparative statics of the problem,analysis of the dynamics of bio-economic systems has de-veloped as a substantial body of literature. In the domain offisheries, early studies focused on the dynamics of open accessand competitive harvesting of common pool resources (Smith,1968; Wilen, 1976), and on the means to optimally drive suchdynamic bioeconomic systems towards a given stationarystate (Clark and Munro, 1975; Clark, 1976, 1985), taking intoaccount issues of capital malleability (Clark et al., 1979;McKelvey, 1985), and uncertainty (Beddington and May, 1977;Hilborn and Walters, 1992). Developments have been largelytheoretical, and are based on a single command variable suchas fishing effort, and a single optimization criterion such asthe net present value of the expected benefits derived fromharvesting. In some cases, data concerning the evolution ofspecific fisheries has been used to estimate the parametersof discrete-time simulation models of the dynamics of thesefisheries (Bjorndal and Conrad, 1987; Ward and Sutinen, 1994;Homans andWilen, 1997). Suchmodels have been used for thesimulation of specific adjustment trajectories for given bioe-conomic systems, according to predetermined scenarios, andon their a posteriori evaluation with respect to a given set ofcriteria, e.g. economic, social and/or biological (Smith, 1969;Holland, 2000; Mardle and Pascoe, 2002).

In this article, we develop a formal analysis of the recoverypaths for a fishery, based on viable control theory. The viablecontrol framework (Aubin, 1991) focuses on intertemporalfeasible paths. This consists in the definition of a set of con-straints that represent the “good health” of the system at anymoment, and in the analysis of conditions which allow theseconstraints to be satisfied along time. More specifically, in thecontext of renewable resource harvesting, viability may implythe satisfactionof both economicandecological constraints. Inthis sense, it is amulti-criteria approach sometimes referred toas a “co-viability” approach. Moreover, an intergenerationalequity feature is naturally integrated within this framework(Martinet and Doyen, 2007). From the ecological viewpoint, theso-called population viability analysis (PVA) (Morris and Doak,2003) and conservation biology have concerns close to viablecontrol by focusing on extinction processes generally within astochastic framework. Cury et al. (2005) advocate the use ofthe viability framework as part of the ecosystem approach tofisheries, and it has been proposed as a useful basis for theanalysis of renewable resources management (Béné andDoyen, 2000; Béné et al., 2001; Doyen and Béné, 2003; Eisenacket al., 2006; De Lara et al., in press).

In particular, the viable control approach allows character-ization of the dynamics of a fishery in terms of its capacity toremain within predefined constraints, beyond which its con-tinued long-term existence would be jeopardized. We adoptthis approach, and define constraints on a fishery related tomicro-economic, biological and social factors. Any path thatdoes not respect the ecological, economic or social constraints(or any combinationof these)will be associated to a crisis of thefishery. Following Béné et al. (2001), we first use the mathe-matical concept of viability kernel to identify the set of states ofthe fishery fromwhich it is possible to satisfy these constraintsdynamically and thus to avoid crisis. This kernel representsthe “target” states for a perennial fishery. In a second step, ouranalysis focuses on crisis situations, outside the viability

kernel. In particular, attention is paid to the ways by whichthe fishery can recover from such crisis, and on the ‘recoverypaths’ leading the system back to sustainable states lying inthe viability kernel. We use the concept of time of crisisintroduced in Doyen and Saint-Pierre (1997) to define thehorizon over which such targets can be reached, and examinetransition paths with respect to that time of crisis. Morespecifically, the minimum time of crisis with respect topossible decision paths is identified, revealing the shortestpath(s) to a viable state. Suchoptimal control problemprovidesjoint information on viability, restoration and irreversibilityissues. Consequently it sheds an interesting light on thesustainability issues of the bio-economic model. However,this minimal time of crisis is not a usual optimal controlproblem. It turns out to bedifficult to solvemathematically andnumerical approximations are often required. The analysis isthen applied to a simplified representation of the Bay of Biscaynephrops fishery (ICES area VIII), and focuses on the historicalchanges as observed via this simple model, estimated usingavailable data.We discuss the viability and recovery of varioustrajectories, including the estimated historical trajectory, andsimulated alternative trajectories. Numerical analysis hasbeen computed in Scilab code.1

The article is structured as follows. The fishery is describedin Section 2. The simplified model of the bay of Biscay ne-phrops fishery used for the analysis is presented (2.1). Theeconomic, biological and social constraints determining theviability of the fishery are defined (2.2), and the case study isdescribed (2.3). In Section 3, we develop the theoretical frame-work that allows us to analyse the conditions under which theviability constraints can be satisfied throughout time (3.1), andto study recovery processes from crisis situations (3.2). We usethis framework in Section 4 to study possible recovery pathsfrom a historical crisis situation with respect to the estimatedhistorical trajectory. Section 5 concludes.

2. Defining a sustainable fishery

2.1. A bio-economic model of the fishery

In this paper, we consider a single stock fishery, characterizedfor each year t by the biomass Bt of the exploited resourcestock and the size Kt of the fleet. The dynamics of the bio-economic system are controlled by the effort et correspondingto the days at sea per period and per vessel, and by the changein fleet size ξt, namely the number of boats entering or leavingthe fleet.

We use a discrete time version of the “logistic model”(Schaefer, 1954) to represent the fish stock's renewal function.The growth of the resource stock is given by

RðBtÞ ¼ rBt 1−Bt

Bsup

� �; ð1Þ

where Bsup is the carrying capacity of the ecosystem for theresource stock.

http://www.scilab.org/

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The fleet is assumed homogeneous. Each vessel has thesame access to the resource and the same technical char-acteristics. Global catches are defined by

Ct ¼ qBtetKt; ð2Þ

where q represents the catchability of the resource. The dy-namics of the resource, combining Eqs. (1) and (2), can thus bedescribed, following Gordon (1954), by

Btþ1 ¼ Bt þ RðBtÞ−Ct ¼ Bt þ rBt 1−Bt

Bsup

� �−qBtetKt: ð3Þ

The economic status of the fleet is characterized by pervessel profit. This profit dependson landings Lt of the resource,defined with respect to the per vessel catches ct=Ct /Kt=qBtetand a discard rate τd

Lt ¼ ð1−sdÞqBtet: ð4Þ

Based on these landings, gross return associated to thetargeted species is defined as a part λ of the vessel's total grossreturn.2 Vessel profit thus reads

pt ¼ ðpð1−sdÞqBtetÞ 1k −ðxf þ xvetÞ; ð5Þ

where p is an exogenous resource price that is consideredconstant.ωf represents fixed costs andωv a per effort unit cost.Thus defined, fleet profit represents the remuneration ofproduction factors (capital and labor) at vessel level.

The production structure is assumed to be slowly flexible,in terms of both capital and labor. The size of the fleet evolvesaccording to a decision control ξt,

Ktþ1 ¼ Kt þ nt: ð6Þ

A degree of capital inertia is assumed to exist in the fishery.Due to technical and regulatory constraints, a limited numberξsup of vessels can enter the fishery in any time period. Also,the number of vessels exiting the fleet in any time periodcannot exceed ξinf, due mainly to social and political con-straints (see below). Such rigidities are captured by condition

−ninf VntVnsup: ð7Þ

On the other hand, fleet activity (effort per period et) canchange, and even be set to nil. Moreover, the effort is boundedby a maximum number of days at sea per period3 esup. Hencethe technical constraint on vessel effort:

0VetVesup: ð8Þ

2.2. Defining viability constraints

We define the viability of the fishery by a set of biological,economic and social constraints that have to be satisfied overtime.

2 Taking λ=1 means that the species studied is the only oneexploited by the fleet.3 Which obviously cannot exceed 365 days per year (366 for leap

years), but will usually be lower.

2.2.1. Biological constraintIn order to preserve the renewable resource, a minimum re-source stock Bmin is considered.

BtzBmin: ð9Þ

2.2.2. Economic constraintAn economic constraint on vessel performance is also con-sidered: profit per vessel is required to be greater than athreshold πmin for economic units to be viable.

ptzpmin: ð10Þ

This minimum profit is defined such as to ensure remu-neration of both capital and labor at their opportunity costs.

This minimum profit constraint actually leads to inducedconstraints on a minimal effort e(B) and a minimal stock sizeB, for any given fleet size, as proven in Appendix A.2.

2.2.3. Social constraintTo take into account social concerns, the viability of the fisheryinduces a constraint on fleet size. In particular, the number ofvessels is required to be greater than a threshold Kmin:

KtzKmin; ð11Þ

ensuring a minimum employment and activity in the fishery.In addition to this minimum fleet size, we assume that the

speed at which fleet size can be reduced is also limited. Theconstraint on the adjustment possibilities regarding the fleetsize (Eq. (7)) can be interpreted as a social and political con-straint limiting the number of vessels (and employment)leaving the fleet during any time period. This interpretationis somewhat different from that encountered in the literatureregarding capital inertia, which is assumed to result mainlyfrom the lack of possibilities to quickly reallocate specificfishing assets to alternative uses, a technical, rather than asocial constraint.

2.3. A simplified model for the Bay of Biscay Nephropsfishery

To illustrate the method and type of results produced, theanalysis is applied to a simplified model of the Bay of BiscayNephrops fishery (ICES area VIII). The empirical modelingapproach, based on the estimation of a surplus productionmodel, was chosen with the aim to limit analytical and com-putational difficulties. This provides a reasonable represen-tation of the qualitative dynamics of the fishery over the pastten years, which is then used to apply the viability frame-work. However, it ignores certain important characteristicsof the fishery, in particular the age structure of the pop-ulation and the uncertainty in recruitment, which limits theusefulness of the model for policy recommendations. In-cluding such factors in the analysis would require thedevelopment of a more realistic model of the biologicalcomponent of this fishery.

Appendix A.1 describes how parameters have been esti-mated and provides the values and constraint levels used inthe analysis. In particular, based on 2003 data, it appearsthat the minimum profit per vessel πmin in our stylized model

Fig. 1 –The viability kernel (in white), stationary states and the historical dynamics as estimated via the model.


of the fishery is 130,000 euros per year.4 The minimum stocksize Bmin was fixed at 5000 tons based on the availablebiological reference point defined in the literature for the re-source stock. The minimal fleet size Kmin was arbitrarily fixedat 100 vessels, and themaximumspeed of entry/exit of vesselsin the fleet (respectively ξsup and ξinf) at 10 per year.

In2003, the fleetwas composedof 235vesselswithanaverageprofit of 165,000 euros. The resource stock was estimatedat about 18,600 tons. Viability constraints, as defined in theAppendix,were thusmet for that particular year.However, in theearly 90's, the stock was lower, and the associated per vesselprofit as estimated through themodel appeared to be lower thanthe economic viability constraint. The estimated resource stockreached its lowest level in 1994, at about 14,000 tons, with anestimatedper vessel profit at 78,000 euros. The fishery thus faceda crisis period (at least from an economic point of view), fromwhich it appears to have recovered since then.

3. Characterizing viable situations and crisissituations

3.1. The viability kernel and viable harvesting strategies

The aim of this section is to define state configurations for thefishery, including resource stock and fleet size, which arecompatible with the viability constraints as defined in Section

4 This level corresponds to the actual (mean) profit required toensure the remuneration of production factors (labor and capital)at their opportunity costs in the Bay of Biscay Nephrops fishery,as measured via the economic surveys carried out by Ifremer in2003.

2.2. The question is to determine whether the dynamics arecompatiblewith the set of constraints. For this purpose,weusetheviable control approachand study the consistence betweendynamics (3) and (6) and the constraints (7) (9)–(11). The set ofbioeconomic states from which there exist intertemporalpaths respecting the entire set of constraints is called the via-bility kernel of the problem. We associate it with the sus-tainable exploitation configurations.

3.1.1. Viable statesFormally, for our problem, the viability kernel is defined by

Viab ¼ ðB0;K0Þjaðeð:Þ; nð:ÞÞ and ðBð:Þ;Kð:ÞÞ; starting from ðB0;K0Þsatisfying dynamics ð3Þ and ð6Þand constraints ð7Þ; ð9Þ; ð10Þ and ð11Þ for any taNþ

8<:

9=;: ð12Þ

The viability kernel Viab for our simplified model of thenephrops fishery (using parameter values presented inAppendix A.1) is represented in Fig. 1. The estimated historicaltrajectory of the fishery is also displayed. Note that thesituation in 1994, as defined via the model, is not viable inthe sense that it does not belong to the viability kernel.

Fig. 1 also presents the estimated position of specific‘reference points’ for the fishery, including the states compat-iblewith aMaximumSustainableYield (MSY), theOpenAccessEquilibrium (OAE), and the Maximum Economic Yield (MEY).These states belong to the broader set of viable stationarystates of the bioeconomic system, which is also represented inthe figure: this set is composed of all the states between thedecreasing straight line and the curved line (see Appendix A.3for a formal description of stationary states).

For any given initial state (B0, K0) in the viability kernel,there exists at least one intertemporal set of decisions (e(.), ξ(.))for which the associated trajectory starting from (B0, K0)

Fig. 2 –Minimum time of crisis C(B0,K0) and scale of viability.


respects all the constraints forever. From some viable statesthere are several viable decisions leading to several viablepaths. However, some admissible decisions are not viable inthe sense that they do not respect the viability constraints.For example, the estimated historical dynamics of the fisheryshow that it entered the viability kernel in 1996. This meansthat, from that year, it would have been possible to follow aviable exploitation trajectory, in the sense that there wereadmissible decisions such that all of the constraints could havebeen met in the following years. Nevertheless, harvestingdecisions from 1996 onwards were not viable, since the profitconstraint was not met, according to the model.5

3.2. Outside the kernel: crisis situations

The viability kernel represents the “goal” for recovery pathsstarting from initial states outside the kernel, i.e. the set ofstates the system must reach to make a viable exploitationpath possible. From its very definition, from any initial stateoutside the kernel, there are no decisions thatmake it possibleto satisfy the constraints in the long run. At least one of theconstraints will be violated in a finite time, whatever thedecisions taken. The system thus faces a crisis situation ifeither the bioeconomic state is located outside the kernel, orthe intertemporal path is bound to leave the kernel in thesubsequent periods.

To recover from such crisis situations, fleet size and effortper vessel must be adjusted so that the trajectories of the

5 See Appendix A.1 for the data describing the estimatedhistorical path followed by the fishery.

system reenter the viability kernel. This will only be possibleif at least one of the constraints is relaxed during a certainperiod of time, in order for the system to recover.

3.2.1. Minimum time of crisisOur study investigates the ways by which the fishery canrecover from crisis, and identifies the ‘recovery paths’leading the system back to sustainable states. FollowingDoyen and Saint-Pierre (1997) or Béné and Doyen (2000), wedefine the time of crisis as the time spent outside theviability constraints by a trajectory. A transition phase canthen be characterized by a time of transition, correspondingto this length of time. Starting from a given bioeconomicstate, various transition paths exist, that allow to reach theviability kernel more or less rapidly. Of particular interest isthe minimum time of crisis associated with the decisionstrategy that generates the shortest recovery path. Moreformally, the optimal control problem associated with theminimum time of crisis, i.e. the minimum time spent outsidethe viability constraints by trajectories starting from (B0, K0),is defined by

CðB0;K0Þ ¼ inffBð:Þ;Kð:Þ; eð:Þ; nð:ÞÞadmissible path

Xlt¼0

1IðBt;Kt; et; ntÞ ð13Þ

where• 1I, the characteristic function that counts the number

of period for which viability constraints do not hold true, isdefined by

1IðB;K; e; nÞ ¼ 0 if ðB;K; e; nÞ satisfy constraints ð9Þ; ð10Þ and ð11Þ1 otherwise

�ð14Þ


• path (B(.), K(.), e(.), ξ(.)) is said to be admissible wheneverit satisfies dynamics and control constraints (3) (6) (7) whilestarting from (B0, K0).

As demonstrated in Doyen and Saint-Pierre (1997), thisminimum time of crisis approach allows to simultaneouslydeal with viability and target6 problems. In this sense, thisoptimal control problem provides information on the poten-tial sustainability of the system. Hence, the set of states forwhich the minimum time of crisis is lower than or equal totime T is said to be viable at scale T. In particular, the viabilitykernel defined in the previous section is a specific case whichcorresponds to a 0 minimal time of crisis (viability at scale 0).Another very informative case related to irreversibility issuesoccurs when the minimal time of crisis is infinite (or verylarge). More globally, it turns out that the minimal time ofcrisis can be related to the time spent outside the viabilitykernel and to the shortest transition path towards theviability kernel. In this regard, the minimum time of crisisexpands the viability approach and inherits most of itsadvantages which includes the multi-criteria perspective.However, it is worth pointing out that such an optimalcontrol problem is not usual and turns out to be difficult tosolve mathematically. One major technical reason is that it isa “degenerated” (non-regular) optimal control problem in themathematical sense as the intertemporal criterion dependson a characteristic function 1I which displays non-smoothfeatures (0 or 1 Boolean values). Numerical methods, based forinstance on dynamic programming, are therefore needed toapproximate this value function and the associated optimalcontrols.

Theminimum time of crisis approach is applied here to theBay of Biscay Nephrops fishery. The set values of the func-tion Cð:; :Þ, computed through numerical simulations is repre-sented in Fig. 2. The numbers 1, 2, 3… represent the number ofperiods of the minimum time of crisis.

4. Recovering from a crisis situation

In this section, we characterize recovery processes from crisissituations to viable situations. To illustrate the analysis, we usethe estimated status of the fishery in 1994 as the initial statefrom which recovery paths can be discussed. As mentionedpreviously, the situation of the fishery as estimated via themodel for that year appears critical as it corresponds to thefarthest estimated state from the viability kernel (lowestestimated stock and per vessel profit). From this crisis situation,the fisheryhas recovered according to anestimatedpath,whichcan be compared to various paths defined a priori. We dis-tinguish three paths:

• open access fishery;• (economically) optimal intertemporal harvesting;• minimum time of crisis.

6 In fact, problems known as “minimal time” or “minimal timeto hit a target”. It is mainly used in robotics and engineeringscience.

4.1. The three scenarios

We first present the 3 scenarios before comparing their pathsto the historical trajectories.

4.1.1. Open accessThe open access case corresponds to a situation in whichvessels can freely enter and exit the fishery, subject to theinertia constraint (7) described above, and choose theirindividual effort level.7 In that case, as claimed in Lemma 3(Appendix A.2), the individual effort will be maximum asthe individual's optimal behavior is to have the maximumadmissible effort. This reads

eOAt ¼ esup: ð15Þ

Following standard microeconomic theory, we considerthat if individual profit is greater than the minimum profitπmin, the fleet size increases as new vessels enter the fishery,leading to rent dissipation. On the contrary, if the individualprofit is lower than the πmin level, vessels leave the fleet asfishing activity does not cover opportunity costs of productionfactors. In our representation of the Open Access regime, thedynamics of capital, i.e. that of the fleet size, can be describedas follows

nOAt ¼ nsup if ptzpmin;−ninf if ptbpmin:

�ð16Þ

4.1.2. Discounted dynamic maximum economic yieldAs a second scenario, we consider a regulated fishery wherethe decision maker optimizes the discounted intertemporalprofit of the fleet.

At fleet level, the optimal behavior is determined bymaximizing the intertemporal sum of discounted fleet profits,with respect to the allocation of the fishing effort through timeand the management of the fleet size, which reads

maxeð:Þ;nð:Þ

Xlt¼0

1ð1þ dÞt Kt pqBtet−ðxf þ xvetÞð Þ; ð17Þ

where δ represents the social discount rate or, from a micro-economic perspective, the opportunity cost of capital.8 In thegeneral framework, the optimal solution of such a problem(Clark, 1990) is to reach an optimal steady state following a“bang–bang” strategy (or most rapid approach). In the casepresented here, there is no “bang–bang” strategy, given theinertia in capital (fleet size) adjustment.

4.1.3. Minimum time of crisisThe two harvesting scenarios considered above lead to pathsthat do not take the viability constraints into account. Ifthese constraints apply, it is possible that some of thesetrajectories may actually lead to situations of crisis due to the

7 Given that some constraints exist on the adjustment of fleet size,resulting in particular from regulations and social norms, this pathcouldmoreprecisely bedescribed asa caseof regulatedopenaccess.8 For the numerical application, the interest rate is set to 5%.

Fig. 3 –Recovery trajectories and the historical path from the estimated 1994 crisis situation. Historical path (—), Open Accessregime (– - – -), Optimal Economic Intertemporal path (- - - -), and Minimal Time of Crisis (– – –).


collapse of the stock, the economic extinction of the fishery,or to social unrest triggered by too ‘painful’ adjustments.Each time a viability constraint is not respected, the fishery isabout to collapse, unless some external help occurs (financialsupport when the economic constraint is not achieved, orsocial compromise when the fleet size adjustment is toostrong).9

Adjustment towards viable states of the fishery could alsobe characterized in terms of the capacity of intertemporalrecovery paths to respect the constraints defined in Section2.2. An adjustment path of particular interest is thus the one

9 Vessels are often retired from the fishery with financialcompensation.

that ensures a minimum time of crisis as defined in theprevious section. We thus compute the minimum time ofcrisis associated with the 1994 bioeconomic state, and therecovery path that minimizes the time of crisis.10

4.2. Compared trajectories

The computed trajectories are presented in Figs. 3 and4. Fig. 3(a)shows the evolutions of the resource stockBtwhile the fleet size

10 In this simulation, we strengthen constraint (7) and limit thespeed of the fleet size adjustment to 5 boats per year, whichappears to be a “softer” adjustment speed than that observed forthe historical path.

Fig. 4 –Recovery trajectories and thehistorical path from the 1994 crisis situation.Historical path (—), OpenAccess regime (– - – -),Optimal Economic Intertemporal path (- - - -), and Minimal Time of Crisis (– – –).


Kt is depicted in Fig. 3(b). Thepaths of fishing effort et is shown inFig. 4(a) and the per vessel profit πt in Fig. 4(b).

Based on the model developed, the historical dynamics ofthe Nephrops fishery from the 1994 situation (lowest estimat-ed biomass) are characterized by a strong reduction of the fleetsize, along with a recovery of the resource stock. From 1994 to2001, profit was lower than the viability thresholdπmin=130,000 euros. During these years, even if the bioeco-nomic state reached the viability kernel quickly (and thuswould have made it possible to satisfy the profit constraint),profit remained low. Starting from 1994, the time of crisisassociated to the historical path appears to be 7 years.

The simulations show that an open access exploitationfrom the 1994 situationwould have led to an initial decrease ofthe resource stock, the fleet size and the per vessel profit, witha recovery at the end of the simulation period. The associatedtime of crisis is 7 years, which is the same as the historicaltrajectory.

Maximizing the intertemporal economic profit wouldhave led to an alternance of oscillating exploitation levels,tending towards the stationary state characterizing theMaximum Economic Yield. Such a scenario leads to 4 crisisperiods (1994, 1996, 1998 and 2001), each of them one yearlong, and there is a transitionary period of 7 years, during


which the system oscillates between viable and unviablesituations.

Viable recovery path defined using the minimal time ofcrisis framework developed in this article leads to somewhatsofter recovery approaches. The reduction of the fleet size isless stringent and the recovery time is 1 year, which isshorter than in the other scenarios. The recovery strategyassociated with the minimum time of crisis requires a shutdown of the fishery during one time period (with a negativeprofit) in order to restore the stock, and then an exploitationpattern making it possible to provide the minimum profit tothe whole fleet.

5. Conclusion

In this paper, we examine the viability of a fishery with respectto economic, social and biological constraints. The main con-straint is a minimum profit per vessel that must be guaranteedat each time period.

We use the viability approach to determine the set ofbioeconomic states and decisions that make it possible tosatisfy the constraints dynamically. This set is called theviability kernel. Any trajectory leaving this set will violate theconstraints in a finite time, whatever decisions apply. Thesystem then faces a crisis situation.

We then study transition phases from crisis situations,i.e. states outside the viability kernel, to viable exploitationconfigurations. These transition phases are characterized bythe time of crisis which is the number of period duringwhich the viability constraints are not met. Of particularinterest is the minimal time of crisis with respect toadmissible control paths. Such an optimal control problemis original and informative as it combines viability andtarget issues and consequently highlights the sustainabilityissues. Nevertheless, this optimal control problem is diffi-cult to address mathematically imposing the use ofnumerical methods.

To illustrate the approach, we compute recovery pathsfrom the estimated historical crisis of 1994 in the Bay ofBiscay Nephrops fishery, based on a simplified model of thefishery. We compare the recovery path given by our approach

of minimum time of crisis with the estimated historical path,a simulated open access exploitation regime and an eco-nomically optimal intertemporal harvesting strategy. Thesimulations show that the minimum time of crisis approach,defined as the most rapid recovery path toward a viablefishery, entails a less stringent reduction in fleet size but thestrongest short-term cost to the remaining fishing vessels.Final levels of economic performance are however belowthose observed with the other transition paths, whichindicates that adaptation of harvesting strategies accordingto the status of the fishery may be preferable to singleapproaches to management in practice. Interestingly, theestimated historical path shows a quite radical reduction infleet size, with levels of effort per vessel remaining relativelystable in the initial years of the transition phase. Thisprobably illustrates the fact that social constraints as definedin this analysis (in terms of the acceptable speed of fleetreduction) were in fact less stringent than economic viabilityconstraints, with respect to the possible adjustment pathsfor this modeled fishery.

While these results serve to illustrate the purpose andusefulness of the viability framework for the analysis offisheries restoration processes, the model does not claim torepresent fully the management issues involved in the Bay ofBiscay Nephrops fishery. A study of the viability of this fisherythrough a more realistic and detailed model taking intoaccount the age structure of the resource, environmentaluncertainty on recruitment and ecological interactions, shouldbe considered in future research.

Acknowledgments

This paper was prepared as part of the CHALOUPE researchproject, funded by the French National Research Agencyunder its Biodiversity program. The authors would like tothank Olivier Guyader, Claire Macher, Michel Bertignac andFabienne Daurès for their assistance in the development ofthe simplified bioeconomic model of the bay of Biscaynephrops fishery used for the analysis, and for the fruitfuldiscussions regarding the application of viability analysis tothe problem of fisheries restoration.

Appendix A

A.1. Parameters of the case study: the Bay of Biscay Nephrops fishery (ICES area VIII)

The analysis is applied to a case study: the Bay of Biscay Nephrops fishery (ICES area VIII). The numerical model has beencalibrated with time-series data available for the fishery.

Biological parameters are estimated using LPUE series (landings per unit of effort) as an index of abundance. We usednonlinear parameter estimation techniques to find the best fit of the predicted LPUE, given the observed LPUE. The fitting criterionis theminimization of the squared deviation between observed and predicted LPUE (Hilborn andWalters, 1992). Fig. A.1 representsobserved and predicted LPUE. As already stressed, the deterministic surplus production model estimated here does not capturefluctuations in recruitment, and their consequences in terms of the age structure of the stock. In particular, years of highrecruitment observed in the late 1980s and early 2000s have entailed somewhat more variable LPUEs throughout the period, andled to delays in the adjustments observed in the fishery, as compared to those estimated via the model. The estimated dynamicsof the bioeconomic system are however qualitatively close to those observed over the period.

Economic parameters are estimated using costs and earnings data collected by the Fisheries Information System of Ifremer viasurveys of individual vessel owners.

Parameters values are as follows.


Fig. A.1 –Fitting of observed and predicted LPUE i

Parameter
Value
n the biologica

Constraint

l parameters estimation model.

Level
r 0.78 Bmin 5000 tons Bsup 30,800 tons Kmin 100 vessels q 72 ·10−7 j−1 πmin 130,000 euros p 8500 euros per ton ξinf 10 vessels ωf 70,000 euros per year ξsup 10 vessels ωv 377 euros per day of sea esup 220 days τd 33% λ 43%
The historical path (with the estimated biomass) is summarized in the following table.

1994
1995 1996 1997 1998 1999 2000 2001 2002 2003 Estimated resource stock (tons) 14,281 15,054 15,482 16,328 16,871 18,082 19,471 20,721 20,728 18,600 Observed fleet size (vessels) 309 303 291 287 282 270 252 259 245 235 Observed fishing effort (days at sea per vessel — mean) 164 170 159 161 139 126 123 137 147 163 Profit (k-euros per vessel — mean) 78 96 91 105 88 87 98 133 148 165
A.2. Proofs

An induced effort constraint: It turns out that a minimum level of effort is required to satisfy the profitability constraint (10). Sincecatches per unit of effort are proportional to the resource stock under the Schaefermodel considered here, thisminimumeffort leveldepends on the size of the resource stock.

Lemma 1. The minimum effort et insuring profit πmin at a given level of stock Bt is given by

PeðBtÞ ¼ pmin þ xfpk ð1−sdÞqBt−xv

; ðA:1Þ

which leads to the induced constraint

etzPeðBtÞ: ðA:2Þ

Proof of Lemma 1. At a given level of stock biomass Bt at time t, for constraint (10) to be satisfied, we must have

ðpð1−sdÞqBtetÞ1k −ðxf þ xvetÞzpmin

which leads to

etzpmin þ xf

pk ð1−sdÞqBt−xv

ðA:3Þ

Hence the minimum effort e(Bt). □


Note that it is not admissible for thisminimumeffort to be greater that themaximal effort esup. As theminimal effort increasesas the resource stock decreases, it induces a minimal resource stock.

An induced stock constraint: It turns out that the profit constraint (10) and the maximal effort condition (Eq. (8)) also generatestronger limitations on stock size than the biological constraint (9).

Lemma 2. The minimum resource stock for fishing activity to respect the per vessel profit constraint (10) is

PB ¼ pmin þ ðxf þ xvesupÞpk ð1−sdÞqesup

ðA:4Þ

which leads to the induced constraint

BtzPB ðA:5Þ

Proof of Lemma 2. Given the profit equation

pt ¼ pqBtet−ðxf þ xvetÞzpmin

and combining the minimal effort from Lemma 1 along with the maximum effort bound esup (Eq. (8)), which reads e (Bt)≤esup, we get

Btzpmin þ ðxf þ xvesupÞ

pk ð1−sdÞqesup

: ðA:6Þ

Hence B. □Note that at this stock level B, we have e(B)=esup, which means that the minimum effort to satisfy the constraint is the

maximum effort.Optimal fishing effort at vessel's level: We determine here the effort level that maximizes the profit of vessels.

Lemma 3. If the resource stock is greater than a level BV ¼xv

pk ð1−sdÞq

, the optimal fishing effort of a vessel is its maximum possible effortet= esup. Else, the optimal effort is 0.

Proof of Lemma 3. The profit, defined by Eq. (5) is

pt ¼ ðpð1−sdÞqBtetÞ1k −ðxf þ xvetÞ:

At a given time t, and for the resource stock Bt, taking the profit derivative with respect to the effort level et leads to

ApAe

¼ pkð1−sdÞqBt−xv

which is positive if the resource stock Bt is greater than a threshold B♭ such that

BV ¼xv

pk ð1−sdÞq

:

□

The maximization of the individual instantaneous profit thus would lead to an effort following a “bang–bang” strategy: nofishing if BtbB♭ and a maximum activity esup if BtNB♭. In our illustrative case, this value is B♭=4075 tons, which is lower than theresource constraint Bmin.Wewill thus consider that it would always be optimal to fish asmuch as possible if there is no regulationof the fishery.

A.3. Stationary states

Stationary states: Among all viable states, particular states allow the dynamics to follow stationary trajectories for associated adhoc exploitation decisions. These stationary states are characterized by Bt+1=Bt andKt+1=Ktwhich leads to ξt=0 and Rt=Ct. This laststatement is equivalent to etKt ¼ r

q 1− BtBsup

� �. Such a relation induces admissible stationary stateswhenever all the constraints hold

true. Extreme cases correspond tomaximumeffort esup on the one hand (which leads to a linear relationship between the fleet sizeand the resource stock), and minimum effort e(Bt) on the other hand. Hence we obtain the conditions on fleet size and resourcestock

rqPeðBÞ

1−B

Bsup

� �V K V

rqesup

1−B

Bsup

� �ðA:7Þ

which can occur if stock B is larger than B. These two frontiers are represented on Fig. 1. The inner area corresponds to possiblestationary states that satisfy all the constraints, including the profitability constraint.


Among the set of stationary states, some are of particular interest as usual reference points in the literature:

• The Open Access Equilibrium (OAE) corresponds to the equilibrium stationary state reached by a fishery under open access, withvessels able to freely enter or leave the fishery, and choose their effort level, in response to changes in profit levels in comparisonto constraint (10).

• The Maximum Economic Yield (MEY) corresponds to the stationary state in which the total profit of the fleet is maximum. Thisimplies a resource stock forwhich themarginal productivity in valueof the resource stock equals themarginal costs of catch, andto a production structure that minimizes costs, i.e. a minimum fleet size given that there are fixed costs and a positiveproductivity of effort.

• The Maximum Sustainable Yield (MSY) corresponds to the resource stock associated with the largest stock regeneration R(BMSY).Various stationary states are possible here, depending on fleet size and effort, and the associated per vessel catches (totalcatches being constant).

Note that there is a maximum sustainable size for the fleet. This corresponds to the same stock size as in the MEY state, butwith a maximum number of vessels sharing the global effort, the minimum profit constraint (10) being respected.

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