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Recovering viable fisheries

Université Paris X-Nanterre

Maison Max Weber (bâtiments K et G)

200, Avenue de la République

92001 NANTERRE CEDEX

Tél et Fax : 33.(0)1.40.97.59.07

Email : secretariat-economix@u-paris10.fr

Document de Travail

Working Paper

2006-05

Vincent MARTINET

Olivier THEBAUD

Luc DOYEN

EconomiX

http://economix.u-paris10.fr/

Université Paris X Nanterre UMR 7166 CNRS

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Recovering viable fisheries

Vincent Martineta,b; Olivier Th´ ebaudb,c; Luc Doyend

aEconomiX-Universit´ e Paris X. E-mail: vincent.martinet@u-paris10.fr

bMarine Economics Department, IFREMER,

ZI Pointe du Diable BP 70, 29280 Plouzan´ e, France. Fax: +33 2 98 22 47 76

cUBO-CEDEM. E-mail: olivier.thebaud@ifremer.fr

dCERESP, Mus´ eum National d’Histoire Naturelle. E-mail: lucdoyen@mnhn.fr

Abstract

This paper develops a formal analysis of the recovery processes for a fishery, from

undesired to desired levels of sustainable exploitation, using the theoretical frame-

work of viability control. We define sustainability in terms of biological, economic

and social constraints which need to be met for a viable fishery to exist. Biological

constraints are based on the definition of a minimal resource stock to be preserved.

Economic constraints relate to the existence of a minimum profit per vessel. Social

constraints refer to the maintenance of a minimum size of the fleet, and to the max-

imum speed at which fleet adjustment can take place. Using fleet size and fishing

effort per vessel as control variables, we identify the states of this bioeconomic sys-

tem for which sustainable exploitation is possible, i.e. for which all constraints are

dynamically met. Such favorable states are called viable states. We then examine

possible transition phases, from non-viable to viable states. We characterize recov-

ery paths, with respect to the economic and social costs of limiting catches during

the recovery period, and to the duration of this transition period. Sensitivity of each

of the constraints to transition costs and time are analyzed. The analysis is applied

to a single stock fishery; preliminary results of an empirical application to the bay

of Biscay nephrops fishery are presented.

Key words: sustainable fishing, recovery, fishery policies, bio-economic modeling.

JEL Classifications: Q22, C61

Preprint submitted to EconomiX - working paper17 July 2006

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Contents

1 Introduction3

2 Defining a sustainable fishery5

2.1 A bio-economic model of the fishery5

2.2Sustainable exploitation patterns6

3 Dynamic analysis of the fishery9

3.1Open access versus optimal harvesting strategies9

3.2Viable harvesting strategies 12

3.3 Recovery paths15

4Recovering MSY production 18

4.1MSY and viability analysis18

4.2 Transition phases21

5 Conclusion 22

AAnnexe23

A.1 Parameters of the case study: the Bay of Biscay Nephrops fishery

(ICES area VIII)23

A.2 Individual economic behavior24

A.3 About the viability analysis25

Acknowledgments This paper was prepared as part of the CHALOUPE research

project, funded by the French National Research Agency under its Biodiversity

program. The authors would like to thank Olivier Guyader, Claire Macher, Michel

Bertignac and Fabienne Daurs for their assistance in the development of the simpli-

fied bioeconomic model of the bay of Biscay nephrops fishery used for the analysis,

and for the fruitful discussions regarding the application of viability analysis to the

problem of fisheries restoration.

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1Introduction

According to recent studies, the maximum production potential of marine

fisheries worldwide was reached at least two decades ago; since then, due

to the widespread development of excess harvesting capacity, there has been

an increase in the proportion of marine fish stocks which are exploited be-

yond levels at which they can produce their maximum (Garcia and Grainger,

2005; FAO, 2004). Hence, the problem of managing fisheries is increasingly

cast in terms of restoring them to both higher and sustainable levels of fish

stocks, catches, and revenues from fishing. Examples are the restoration plans

discussed and/or adopted by the European Commission in recent years for

several collapsed stocks in E.U. waters, or the international commitment to

return fisheries to levels allowing their maximum sustainable yield to be ex-

tracted by 2015, taken by countries present at the Johannesburg Summit on

Sustainable Development in 2002.

The problems posed by fisheries restoration are dynamic in nature: beyond the

issue of selecting adequate objective levels for restored fisheries, a key ques-

tion is the identification and the selection of the possible paths towards these

objective levels. In practical situations, this question is crucial as it relates to

the feasibility (technical, economic, biological) and to the social and political

acceptability of the adjustments required for fisheries to be restored, hence to

the actual possibilities to drive fisheries back towards decided sustainability

objectives.

The definition of optimal strategies for the harvesting of marine fish stocks

has been widely studied in the literature on renewable resource management.

While most of the initial work focused on the comparative statics of the prob-

lem, analysis of the dynamics of bio-economic systems has developed as a

substantial body of literature. Different approaches have been proposed. In

the domain of fisheries, Clark (1985) described how to optimally drive a dy-

namic bioeconomic system towards a stationary state, based on a single com-

mand variable (fishing effort), and looking at a single optimization criteria (net

present value of the expected benefits derived from harvesting). Alternative

approaches have been based on simulations of specific adjustment trajectories

for given bioeconomic systems, according to predetermined scenarii, and on

their a posteriori evaluation with respect to various criteria (see e.g. Smith

(1969); Mardle and Pascoe (2002); Holland and Schnier (2006)).

In other areas of natural resource management, the maximin approach (Solow,

1974; Cairns and Long, 2006) has been used. Instead of maximizing the net

present value criteria, this approach considers the maximal level of profit (or

utility) that can be sustain forever, given the initial state of the economy. It

thus strongly includes an intergenerational equity concern. For example, Tian

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and Cairns (2006) examine the sustainable management of a renewable re-

source: the forest on Easter Island. They show that, depending on the initial

configuration of the bioeconomic system, a sustainable exploitation pattern

would have been possible on the island, with higher population levels and

welfare than actually observed (Brander and Taylor, 1998). But this approach

still considers only an economic criterion, when sustainable management of re-

newable resources may require to consider economic, social and environmental

criteria together.

In this paper, we develop a formal analysis of the recovery paths for a fishery,

based on viable control theory. This allows us to characterize of the dynamics

of a fishery in terms of its capacity to remain within pre-defined constraints,

beyond which its continued long-term existence would be jeopardized. The

constraints considered in the analysis relate to micro-economic, biological and

social factors. Following B´ en´ e et al. (2001), we use the mathematical concept of

viability kernel to identify the set of states of the fishery for which it is possible

to satisfy these constraints dynamically. This kernel represents the “target”

states for a perennial fishery. Our analysis focuses on the ways by which the

fishery can recover from states outside the kernel to viable states in general,

and to specific target states in particular. We use the concept of minimal time

of crisis (Doyen and Saint-Pierre, 1997) to consider the horizon at which such

targets can be reached, and examine transition paths considering transition

time and transition costs defined as the discounted sum of fleet profits during

the transition phase toward target states.

The analysis is applied to the case of the bay of Biscay (ICES area VIII)

nephrops fishery, and focuses on the implications of restoring this fishery to

levels allowing maximum sustainable yield to be extracted. We examine the

relationship between the viability of the fishery on the one hand, and the

possible transition phases towards this maximum yield objective on the other

hand.

The paper is organized as follows. The simplified model of the bay of Biscay

nephrops fishery used for the analysis is presented in section 2, as well as the

definition of the economic, biological and social constraints determining the

viability of the fishery. Section 3.2 then presents the analysis of the conditions

under which these constraints can be satisfied throughout time, and of the

possible transition paths towards these viable states from initially non-viable

situations 3.3. The specific issue of recovering MSY production levels without

jeopardizing the viability of the fishery is addressed in Section . Section 5

concludes.

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2Defining a sustainable fishery

2.1A bio-economic model of the fishery

In this paper, we consider a single stock fishery, characterized by the size of

the fleet Xt∈ [0,¯ X] ⊂ N+, which can evolve in time. The exploited resource

is represented by its stock St∈ [0,K], where K is the carrying capacity of the

ecosystem.

The dynamics of the bio-economic system is controlled by the effort et ∈

[0, ¯ e] ⊂ N+(day of sea per period and per vessel1) and the change in the

fleet size ξt ∈ [−α1,α2] ⊂ N+, the number of boats entering or exiting the

fleet. We assume that the admissible controls belong to a set (e,ξ) ∈ U, where

U = [0, ¯ e] × [−α1,α2].

We use a discreet time version of the “logistic model” of Schaefer (1954)

to represent the fish stocks renewal function. Hence, the regeneration of the

resource stock is given by

Rt(St) = rSt

?

1 −St

K

?

.(1)

The fleet is assumed homogenous. Each vessel has the same access to the

resource and the same harvesting characteristics. Global catches are defined

by

Ct= qStetXt

(2)

where q represents the catchability of the resource.

We thus get the dynamics of the resource combining eq. (1) and (2), following

Gordon (1954)

St+1= St+ Rt− Ct= St+ rSt

?

1 −St

K

?

− qStetXt

(3)

The economic dynamics are characterized by the per vessel profit. This profit

depends on the landings Lt of the resource defined with respect to the per

vessel catches ct= Ct/Xt= qStetand a discard rate τd

Lt= (1 − τd)qStet.(4)

1The maximal day of sea per period ¯ e represents a technical constraints. In any

case, it cannot be more than 365 days per year.

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These landings give the gross return for the targeted species which is a part

λ of the vessel’s total gross return.2Vessel profit thus reads

πt=

?

p(1 − τd)qStet

?1

λ− (β1+ β2et)(5)

where p is an exogenous resource price that is considered constant. β1repre-

sents fixed costs and β2a per effort unit cost.

We consider that the production structure is (slowly) flexible, in terms of both

capital and labour. The size of the fleet evolves according to a decision control

ξt,

Xt+1= Xt+ ξt. (6)

To take into account the inertia of capital, the change of the fleet size is

limited. A maximum number α2of vessels can enter the fishery in any time

period, due to technical constraints. The number of vessels exiting the fleet in

any time period can not exceed α1, due to social and political constraints (see

below). Its reads

−α1≤ ξt≤ α2.

This means that levels of capital in the fishery (number of vessels) cannot

change quickly. On the other hand, fleet activity (effort per period et) can

change, and even be set to nil.

(7)

2.2Sustainable exploitation patterns

We define sustainability of the exploitation with respect to a set of biological,

economic and social constraints that have to be respected throughout time for

a viable fishery to exist.

Biological constraints: In order to preserve the natural renewable resource,

we consider a minimal resource stock Sminwhich is the minimal biomass en-

suring the regeneration of the stock:

St≥ Smin

(8)

Economic constraints: We consider an individual economic constraint on

the vessel performance: profit per vessel is required to be greater than a thresh-

old πminfor economic units to be viable.

πt≥ πmin

(9)

2Taking λ = 1 means that the studied species is the only one exploited by the

fleet.

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This minimal profit is defined such as to ensure remuneration of both cap-

ital and labour, at least at their opportunity costs. It can also be set as a

sustainability goal ensuring level of economic performance greater than those

ensuring strict economic viability.

Social constraints: To take into account social concerns, the viability of the

fishery is described by a constraint on the fleet size. We require the number

of vessels to be greater than a threshold Xmin:

Xt≥ Xmin

(10)

ensuring a minimal 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. The constraint on the adjustment

possibilities regarding the fleet size (eq. 7) can be interpreted as a social and

political constraint limiting the number of vessels (and employment) leaving

the fleet.3

We call this set of constraint K, in the sense that states and controls respecting

the entire set of constraints belong to K, which reads

Viability constraint (8), (9) and (10) are respected ⇔ (St,Xt,ξt,πt) ∈ K

Considering the state constraints (8) and (10), viability requires as a necessary

condition (St,Kt) ∈ [Smin;K] × [Xmin;¯ X]. Nevertheless, it does not mean

that this whole state constraint set make it possible to satisfy the profitability

constraint.

In particular, the biological configuration for which it is possible to have a prof-

itable fishing activity can be determined. It appears that the profit constraint

(9) induces stronger limitations on stock size than the biological constraint (8).

This result is stated in proposition 1 below, which is proven in the appendix

A.2.

Proposition 1 The minimal resource stock for fishing activity to respect the

per vessel profit constraint (9) is

S =πmin+ (β1+ β2¯ e)

pq¯ e

. (11)

3This interpretation is somewhat different from that encountered in the literature

regarding capital inertia, which is assumed to result mainly from the lack of pos-

sibilities to quickly reallocate specific fishing assets to alternative uses, a technical,

rather than social constraint.

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This induced constraint comes from the fact that a minimal effort is required

to satisfy the profitability constraint (9), as proved in the appendix A.2 by

the lemma 2. This minimal effort depends on the resource stock as the catches

increase with the resource stock.

e(St) =

πmin+ β1

λ(1 − τd)qSt− β2

p

(12)

This minimum effort is represented in Figure 1 with respect to the resource

stock. The vertical asymptote represents the limit S?from which fishing effort

has a positive marginal value (Appendix A.2, Lemma 1). The fishing effort

required to satisfy the profit constraint is a decreasing function of the stock

size.4The more fishes there are, the less fishing effort to catch them is needed.

The horizontal line represents the maximal days of sea per period. One can

see that there is a stock threshold S for which the number of days required

to satisfy the profit constraint is greater than the possible number of days per

period.

Minimal effort (day of sea per year) to satisfy profit constraint

0500010000 15000200002500030000

0

50

100

150

200

250

300

350

Fig. 1. Minimal effort e(St) required to satisfy profit constraint (9), with respect to

the stock size.

We thus have an induced constraint for the fishing activity to generate suffi-

cient profits. This constraint level is greater than the initial resource constraint

Smin, which means that ensuring the economic profitability of the fishery in

the long run implies that the resource constraint is also satisfied. This result

was also derived by B´ en´ e et al. (2001).

Model parameters

4It tends towards infinity in the neighborhood of S?.

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The analysis is applied to a case study: the Bay of Biscay Nephrops fish-

ery (ICES area VIII). Parameters estimation procedures are detailed in the

appendix A.1.

Parameters values and constraints levels are given in the following table.

ParametervalueConstraintlevel

r =0.78

Smin=5,000 tons

K =30800 tons

Xmin=100 vessels

q =72.10−7j−1

πmin=130,000 euros

p =8,500 euros per tonsα1=10

β1= 70,000 euros per year

α2= 10

β2=

¯ X =

377 euros per day of sea

500 boats

¯ e = 220 days

τd= 33%

λ =43%

In 2003, the fleet was composed by 235 vessels, of an average profit of 165 000

euros. The resource stock was about 18 600 tons. The average number of days

of sea was 203. The catches were about 5769 tons.

3 Dynamic analysis of the fishery

In this section, we examine the consistency between the dynamics of the ex-

ploited system and the viability constraints, focusing on conditions for fishing

activity to be viable in the long run.

3.1Open access versus optimal harvesting strategies

Following Clark (1990), the dynamics of the system can be compared un-

der two exploitation patterns: open access, and a policy guided exploitation

maximizing the intertemporal profit derived from the fishery, which we call

Cost-Benefit Analysis (CBA). We analyze the possible dynamics of the fishery

under these two scenarios. The dynamics are illustrated in Figure 2.

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Figure 2 represents the intertemporal dynamics of resource stock, fleet size

and per vessel profit for the two exploitation patterns on a 50 years period.

We compute 16 intretemporal paths representing various representative initial

states.

Open Access

sels can freely enter and exit the fishery, subject to the inertia constraints de-

scribed above, and choose their individual effort level. In that case, as claimed

in lemma 1, the individual effort will be maximum.

The open access case corresponds to situations in which ves-

We consider that, if individual profit is greater than the minimal profit πmin,

the fleet size increases as new vessels enter the fishery. On the contrary, if profit

are less than 90% of the πminlevel, vessels leave the fleet. This represents the

fact that negative profits often occur transitionally in fisheries: some negative

profits may be supported for short periods.

It appears that in this open access case, whatever the initial stock configu-

ration, the system reaches a limit cycle in both the resource stock and the

fleet size.5In the case considered, the stationary state is characterized by

263 vessels and a resource stock of 14 400 tons. The individual profit at this

stationary state is minimum (πmin).

Cost-Benefit Analysis

order to maximize profit derived from the fishery, it is possible to establish

the sets of optimal decisions concerning effort levels and changes in fleet size,

for different initial conditions in the fishery.

Considering that the fishing fleet is regulated in

At fleet level, the optimal behaviour is determined by maximizing the in-

tertemporal sum of discounted fleet profits, with respect to the allocation of

the fishing effort through time, which reads

max

e(.)

∞

?

t=t0

1

1 + δ(t − t0)Xt

?

pqStet− (β1+ β2et)

?

(13)

where δ represents the social discount rate or, from a microeconomic perspec-

tive, the opportunity cost of capital.6In the general framework, the optimal

5The periodic behavior observed here is linked to the choice of the adjustment

possibilities in vessel numbers. We consider that, in a free access configuration, a

maximal number of vessels will enter the fishery if there are some positive profits. A

finer adjustment parameter (a slower entry) would lead to smaller variations. The

extreme case of a continuous capital stock adjustment would lead to a steady state

point. Note that the resource can not been exhausted as the catches per unit of

effort become very small when the stock is reduced to low levels.

6For the numerical application, we set an interest rate equals to 5%

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solution of such a problem (Clark, 1990) is to reach an optimal steady state

following a “bang-bang” strategy (or most rapid approach). In our model, we

can see that, whatever the initial state, the system reaches such a stationary

state, with a high stock level and a lower fleet size, allowing to maximize the

catches while minimizing harvesting costs. The stationary state is reached as

quickly as possible. In the case considered, this stationary state is character-

ized by 190 vessels and a resource stock of 19 000 tons. The individual profit

is greater than the minimum viability profit πmin. It is of 222 000 euros per

vessel.

Nevertheless, there is no “bang-bang” strategy as there is an inertia in the

capital (fleet size) adjustement. When the fleet size is smaller than the targeted

size, the stock size and the profit level evolve smoothly, with increasing or

decreasing profit, depending on the resource stock (the larger the stock, the

more harvesting and profit). On the contrary, when the fleet size is large and

require a long time period for adjustement, we observe an alternance of nil

harvesting and maximal one. The larger the fleet, the larger the variations of

profit and stock size during the transition phase.7

In both exploitation scenarii, a minimum profit is guaranteed after a transition

phase. When a steady state is achieved, both the minimum profit per vessel

and the resource stock are lower in the open access case then in the regulated

case, while the fleet in the open access regime is larger. Open access profit

is characterized by periodical oscillations around the profitability constraint,

ensuring a minimum profit lower than the economic viability constraint.

The time of transition can be long : it is between 10 to 25 years in the open

access configuration, and between 3 to 25 years in the regulated case. During

this transition phase there is no guaranteed profit for the fleet.

The two harvesting scenarios considered here thus lead to paths that do not

respect the viability constraints, as defined in the previous section, at least

until the transition phase is achieved (and periodically for the open access

strategy from then on). If these constraints apply, it is possible that some of

the trajectories represented above may actually lead to situations of crisis due

to a collapse of the stock, the economic extinction of the fishery, or to social

unrest associated with the adjustment paths considered. We propose to analyse

the viability of the fishery by defining intertemporal paths of harvesting that

satisfy all the constraints defined in the previous section simultaneously.

7The observed amplitude of variations does not depend on the size of the resource

stock, just on that of the fleet.

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Open_Access_Resource_StockOpen_Access_Resource_Stock Open_Access_Resource_Stock Open_Access_Resource_StockOpen_Access_Resource_Stock Open_Access_Resource_Stock Open_Access_Resource_StockOpen_Access_Resource_StockOpen_Access_Resource_Stock Open_Access_Resource_Stock Open_Access_Resource_StockOpen_Access_Resource_Stock Open_Access_Resource_StockOpen_Access_Resource_StockOpen_Access_Resource_Stock Open_Access_Resource_Stock

(a) Resource stock StOpen Acces

05 101520 2530 35404550

0

5000

10000

15000

20000

25000

30000

CBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_StockCBA_Resource_Stock

(b) Resource stock StCBA

05 101520253035404550

0

5000

10000

15000

20000

25000

30000

Open_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_Size Open_Access_Fleet_SizeOpen_Access_Fleet_Size Open_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_Size Open_Access_Fleet_Size Open_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_Size

(c) Fleet size XtOpen Acces

05101520253035404550

0

50

100

150

200

250

300

350

400

450

500

CBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_SizeCBA_Fleet_Size

(d) Fleet size XtCBA

0510 1520253035404550

0

50

100

150

200

250

300

350

400

450

500

Open_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_ProfitOpen_Access_Profit

(e) Profit πtOpen Acces

05101520253035404550

-100

-50

0

50

100

150

200

250

300

350

400

CBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_Profit

05101520253035404550

-100

-50

0

50

100

150

200

250

300

350

400

(f) Profit πtCBA

Fig. 2. Initial conditions are combining four stock levels (5 000, 12 500, 20 000 and

27 500 tons) and four fleet size (100, 200, 350 and 450 vessels).

3.2Viable harvesting strategies

The aim of this section is to define state configurations (resource stock and

fleet size) which are compatible with our viability constraints. The question is

to determine whether the dynamics (Eq. 3 and 6) is compatible with the set

of constraints. For this purpose, we use the viable control approach and study

the consistence between dynamics (3) and (6) and the constraints (7), (8), (9)

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and (10).

The set of bioeconomic states from which there exist intertemporal paths

respecting the whole constraint is called the viability kernel of the problem. A

description of the viability theory and viable control approach is provided in

the appendix A.3, along with a formal definition of this viability kernel.

Viable stationary states

stationary states. These states are characterized by

A first analysis relies on the definition of viable

St+1=St

Xt+1=Xt

We thus have ξt= 0 and

Rt= Ct

⇔

etXt=r

q

?

1 −St

K

?

We can determine admissible pairs (Xss,ess) with respect to resource stock S.

Extreme cases correspond to maximun effort ¯ e on the one hand (which leads

to a linear relationship between the fleet size and the resource stock), and

minimum effort e(St) on the other hand. These two frontiers are represented

on Figure 3. The inner area corresponds to possible stationary states that

satisfy all the constraints, including the profitability constraint.

Profit_Max

Profit_Min

050100150

S

200250300350

0

50

100

150

200

250

300

K

X

Fleet size

Resource Stock

00

000000 0000

maximal effort e

minimal effort e

min

Fig. 3. Viable stationary states (the upper limit corresponds to minimal effort e(S),

the lower limit to maximal effort ¯ e.

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We can see on Figure 3 that there is a maximal sustainable size for the fleet,

when the effort is minimum. It is interesting to note that the stock level

associated with the maximal sustainable size of the fleet is not equal to the

stock producing the MSY. This is due to the fact that a greater stock produces

less yield but ensures less costly catches.

In our illustrative case, the maximum sustained fleet size is 279 vessels. It is

associated with a resource stock of 17 600 tons and a per vessel effort of 166

days of sea.

Viable states

the stationary set described above, but that however make it possible to satisfy

the constraints. We now describe the set of all states that satisfy all of the

constraints in a dynamic perspective, including non stationary trajectories.

General statements on this viability kernel and the way it is computed are

presented in appendix A.3.

In our problem, there are some states that do not belong to

The viability kernel for the nephrops fishery, using parameter values presented

in section 4 is represented in Figure 4.

050001000015000200002500030000 350

stock

0

50

100

100

200

250

300

350

400

450

500

K

Fleet size

Resource

Fig. 4. Viability kernel.

This viability kernel represents the “goal” of the recovery paths, i.e. the set

of states the system must reach to ensure a viable exploitation. For any given

initial state (S0,X0) in the viability kernel, there exists at least one intertem-

poral decision series (e(.),ξ(.)) for which the associated trajectory starting

from (S0,X0) respects all of the constraints forever. Note that there may exist

several viable decisions. Another important point is that all admissible deci-

14