Recovering viable fisheries
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Recovering viable fisheries
Vincent Martineta,b; Olivier Th´ ebaudb,c; Luc Doyend
aEconomiX-Universit´ e Paris X. E-mail: firstname.lastname@example.org
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: email@example.com
dCERESP, Mus´ eum National d’Histoire Naturelle. E-mail: firstname.lastname@example.org
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
2 Defining a sustainable fishery5
2.1 A bio-economic model of the fishery5
2.2 Sustainable exploitation patterns6
3 Dynamic analysis of the fishery9
3.1 Open access versus optimal harvesting strategies9
3.2 Viable harvesting strategies12
3.3 Recovery paths15
4 Recovering MSY production18
4.1MSY and viability analysis18
4.2 Transition phases21
5 Conclusion 22
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.
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
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
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
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
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
2 Defining a sustainable fishery
2.1 A 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
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
The fleet is assumed homogenous. Each vessel has the same access to the
resource and the same harvesting characteristics. Global catches are defined
where q represents the catchability of the resource.
We thus get the dynamics of the resource combining eq. (1) and (2), following
St+1= St+ Rt− Ct= St+ rSt
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.
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
p(1 − τd)qStet
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
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.
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:
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.
2Taking λ = 1 means that the studied species is the only one exploited by the
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:
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
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
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
Proposition 1 The minimal resource stock for fishing activity to respect the
per vessel profit constraint (9) is
S =πmin+ (β1+ β2¯ e)
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.
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.
λ(1 − τd)qSt− β2
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
Minimal effort (day of sea per year) to satisfy profit constraint
05000 10000 15000 200002500030000
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).
4It tends towards infinity in the neighborhood of S?.
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
Parameters values and constraints levels are given in the following table.
Parameter valueConstraint level
Smin= 5,000 tons
K = 30800 tons
Xmin= 100 vessels
πmin= 130,000 euros
p =8,500 euros per tonsα1= 10
β1= 70,000 euros per year
¯ X =
377 euros per day of sea
¯ e = 220 days
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.
3Dynamic 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.
3.1 Open 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.
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
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).
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
1 + δ(t − t0)Xt
pqStet− (β1+ β2et)
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%
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
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.
Open_Access_Resource_StockOpen_Access_Resource_Stock Open_Access_Resource_StockOpen_Access_Resource_Stock Open_Access_Resource_StockOpen_Access_Resource_Stock 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_Stock Open_Access_Resource_StockOpen_Access_Resource_StockOpen_Access_Resource_Stock
(a) Resource stock StOpen Acces
05 1015 20253035404550
(b) Resource stock StCBA
Open_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_Size Open_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_Size Open_Access_Fleet_SizeOpen_Access_Fleet_SizeOpen_Access_Fleet_Size Open_Access_Fleet_Size
(c) Fleet size XtOpen Acces
(d) Fleet size XtCBA
0510 1520 25 3035404550
(e) Profit πtOpen Acces
051015 20 25303540 45 50
CBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_Profit CBA_Profit CBA_Profit CBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_ProfitCBA_Profit
051015 2025 303540 4550
(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)
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
We thus have ξt= 0 and
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.
maximal effort e
minimal effort e
Fig. 3. Viable stationary states (the upper limit corresponds to minimal effort e(S),
the lower limit to maximal effort ¯ e.
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.
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.
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-
sions are not necessarily viable and may lead the system outside the viability
The stationary states described in the previous section are particular cases of
viable trajectories (that are stationary trajectories, associated to ad hoc deci-
sions). If the initial state belongs to the left-bottom hand-side of the viability
kernel the resource stock will increase for any viable decision. On the contrary,
if the initial state is on the right-up hand-side, the resource stock will decrease
whatever viable decision applies.
From the very definition of this viability kernel, for any outside initial state,
there are no decisions that make it possible to satisfy the constraints in the
long run. For example, any trajectory strating from the upper area of the
state’s constraint set, i.e. for any (S0,X0) ∈ [Smin;K]×[Xmin;¯ X], at least one
of the constraint will be violated in a finite time, wathever decisions apply.
The system thus faces a crisis situation if the bioeconoic state is outside the
kernel or if the intertemporal path leaves it.
In this section, we use the above framework of analysis to characterize recov-
ery processes, from situations outside the viability kernel which we call crisis
situations, to viable situations. A crisis situation corresponds to configurations
that do not make it possible to respect the viability constraints, namely to sat-
isfy (St,Xt,ξt,πt) / ∈ K, in the long run. We define the characteristic function
0 If (St,Xt,ξt,πt) ∈ K
1 If (St,Xt,ξt,πt) / ∈ K
3.3.1Achieving viability in the future: the concept of minimal time of crisis
In the context of this analysis, the issue is to reach the viability kernel in
the future. From a theoretical point of view, the characteristic function (14)
counts the number of period when viability constraints do not hold true. It
can be interpreted as the time spent outside the kernel. A transition phase is
then characterised by a time of transition, corresponding to this time. Starting
from a given bioeconomic state, various transition phases exist, that reach the
kernel more or less quickly.
We define the minimal time of crisis as the time spent outside the kernel by the
fastest transition phase starting from a given bioeconomic state (the minimal
time to reach the target).
Based on this notion of minimal time of crisis we are able to define the notion
of viability at time T, which is the set of states that make it possible to
belong to the viability kernel after T. For example, the set of states that are
viable at time 2 is composed of all states for which the minimal time of crisis
is lower than or equal to 2. In particular, the viability kernel defined in the
previous section corresponds to viability at time t0. The formal link between
the viability at scale T and the minimal time of crisis is developed in Doyen
and Saint-Pierre (1997).
More formally, the minimal time of crisis, i.e. the minimal time spent outside
of K by trajectories starting at (S,X), is defined by the map
This map is represented for the nephrops fishery by figure 5.
0 50 100150200 250 300 350
Fig. 5. Scale of viability and minimal time of crisis.
By the very definition of the viability kernel, any state outside the kernel (crisis
situation) does not make it possible to respect the constraints. All viability
constraints thus cannot be respected during the transition phase.
In particular, the recovery strategy associated with the minimal time of crisis
may require to close the fishery for a while (there is no effort, i.e. no fishing
activity: the capital is not used), along with reducing the fleet size as quickly
as possible (given the inertia constraint 7). This entails a strong violation of
the minimum profit constraint. As noted before, due to economic and social
requirements, transition phases may need to ensure a minimum level of revenue
to vessels, even if it is lower than the minimum viable profit.
3.3.2Recovery paths under constraint
Even if the optimal recovery strategy requires closing the fishery for a while
(Clark, 1985), this is not always possible because it neglects fisher’s needs
to cover some fixed costs or to ensure a minimal activity and revenue. One
may thus require a minimum activity during the transition phase, or more
specifically, a minimum remuneration of labour and capital.
To take such requirements into account leads to “softening” one or several
viability constraints during the transition phase. In particular, it is possible
to accept that the fishery can face periods where profits from the activity in
excess of the opportunity costs of capital and labour are negative, without
inducing the definitive shutdown of the activity.
In our model, this possibility is defined by introducing constraints on transi-
tion decisions, i.e. by restricting the set of admissible choices such that e(t)
ensures a minimal profit constraint during the transition phase. We define this
constraint ˜ π.
The map representing the transition phases under constraint for the nephrops
fishery is represented in figure 6.
050 100150200250 300 350
Fig. 6. Transition phases under constraint.
We can compare the various areas with respect to the minimal time of crisis
without constraint defined in the previous section. As the admissible deci-
sion set is restricted during the transition phase under constraint, it is longer
to reach the target (the viability kernel) from any given crisis situation. This
means that a same initial state will stand in a farther area of the map (charac-
terized by a greater minimal time of crisis) with the ˜ π constraint on transition
Moreover, with this constraint, an area appears on the map, from which it is
not possible to achieve recovery (the white area on the left hand side on Figure
6). Thus, for any given initial state, there is a maximum profit constraint on
the transition phase for which it is possible to reach the viability kernel in a
4Recovering MSY production
In this section, we apply the general framework previously developed to ad-
dress a particular issue. We examine how to reach a sustainable goal defined
as a production in the neighborhood of the MSY, starting from the situation
of the Nephrops fishery in 2003.
In particular, we examine the consequences of this production objective on
the fleet configuration (number of vessels and profit), and the time needed to
reach it. First, we define the viability kernel associated to the MSY production
objective, and the associated time of crisis. We then examine how to minimize
the transition cost towards this objective state of the fishery.
4.1MSY and viability analysis
We first determine the bioeconomic states that are compatible with MSY.
We then examine if these configurations are viable, i.e. respect the viability
constraints defined in previous sections.
The maximum sustainable Yield is a particular stationary state where the
resource regeneration is maximum, such that the sustainable harvesting is
maximal too. We consider a production constraint defined as
C(t) ≥ CMSY
The production of the stock at MSY level is
For the application case, we set CMSY = 6000 tons.
Assuming such a minimal production constraint induces a posteriori con-
straints on the resource stock, and on the minimal fleet size.8The induced
constraint on the resource stock is
S(t) ≥ SMSY =K
In a static perspective, the minimal size of the fleet to ensure MSY production
is derived from the defintion of catches (eq. 2). We get
⇒ XMSY(S) =rK
Thus, at SMSY we have XMSY=
number of vessels equal to 247 boats at the MSY stationary state. With such
a fleet size, the per vessel profit is limited to 150 000 euros. It is lower than
the observed average profit in 2003 (about 165 000 euros). Having the MSY
production as an objective requires to increase the fleet size and to reduce the
per vessel profit with respect to the present situation.
2q¯ e. In our application, we get a minimum
Viability kernel of the MSY production
The first step of the analysis is to define bio-economic states that make it
possible to respect the production constraint (16). For this purpose, we apply
the viability approach to the system described by dynamics (3) and (6) and
constraints (7) and we use the production constraint (16) instead of the eco-
nomic and social constraints (10) and (9) on the minimal profit and minimal
fleet size. We apply the framework proposed in section 3.3.
The viability kernel associated with this viability problem is given in Figure
7. It represents the combinations of stock and fleet size for which there exist
viable decisions compatible with the minimum production level.
We see in figure 7 that a minimum size of the fleet is required to produce at
this level. As an increasing stock ensures increasing catches for a given effort
level, the bigger the stock is, the less boats are needed.
We now turn to analyse the economic viability of such a production objective.
We consider once again the profit constraint πmindefined by equation (9). We
consider the viability kernel associated with all of the constraints, including
the minimal profit per vessel and the minimal production of the ecosystem.
This viability kernel and the associated times of crisis are presented in Figure
8A higher production level requires higher capital use.
050 100 150 200 250 300 350
Fig. 7. Biological and economic states allowing to sustain the MSY production.
0 50 100150 200 250300350
Fig. 8. Viability kernel and minimal time of crisis for viability constraints and MSY
Comparing Figure 7 and Figure 8, we can see that the profit constraint reduces
the set of viable states that make it possible to produce MSY. Stationary states
correpond to the resource stock producing MSY. The minimal effort satisfying
profit constraint (eq. 12) at the MSY stock is
λ(1 − τd)qK
Using relationship (19), we get the maximal number of vessels sustainable at
¯ X(SMSY) =
λ(1 − τd)qK
In our case study, we get¯ X = 373 vessels, having an individual effort of 145
days of sea. Combining this result with the minimal fleet size to ensure MSY,
we have several possible stationary states, with a number of vessels between
247 and 373 vessels.
4.2 Transition phases
As an application of the results of section 3.3, we consider in this section the
following sustainability goals :
Thus, viability constraints are associated with the MSY production (eq. 16)
along with a constraint on the individual profit, in order to get it as close as
possible to the observed profit.
The viability kernel and minimal time of crisis associated with this viability
problem is represented in Figure 9.
0 50100 150 200 250300350
Fig. 9. Minimal time of crisis associated with MSY production level and maximal
per vessel profit.
We can see on Figure 9 that the viability kernel is smaller than the one ensuring
Here again, the minimal time of crisis is associated with a shutdown of fishing
activity. To take into account social and economic constraints, we consider a
minimal profit constraint during the transition phase. We set it at the level
of opportunity costs (which is the level we used in the previous section as a
minimal profit for economic viability): ˜ π = 130000 euros.
The problem is thus to reach viable states, where sustainability goals are to
produce at MSY level, with a maximum individual profits, and limiting the
loss during the transition phase (minimal profit covering opportunity costs of
the producing factors).
The map on Figure 10 represents the time of crisis under transition constraints.
Fig. 10. Transition phases under economic viability constraint to reach MSY pro-
Such a study show that the set of viable states is reduces and that transitions
phases would be longer if a viability constraint on the profit is required during
the transition phase.
In this paper, we examine the viability of a fishery with respect to economic,
social and biological constraints. The main constraint is a minimal profit per
vessel that must be guaranteed at each time period. We show that requiring
such a minimal profit induces a minimal threshold for the natural resource,
and thus a stronger constraint on the resource stock than the initial biological
constraint. As has already been demonstrated, it is thus possible to reconcile
economic and ecological objectives.
We use the viability approach to determine the set of bioeconomic states that
make it possible to satisfy the constraints dynamically. This set is called the
viability kernel of the problem. Any trajectory leaving this set will violate the
constraints in a finite time, whatever decisions apply. The system then faces
a crisis situation.
We then study transition phases from crisis situation, i.e. states outside the
viability kernel, to viable exploitation configurations. These transitions phases
are characterized by the time of transition on the one hand, and the cost of
the transition on the other hand. This cost is defined as the difference between
a minimum profit ensuring economic viability and the observed profit during
the transition phase. We show that the shorter the transition phase is, the
higher the transition costs are.
Using this general framework of analysis, we focus on a particular issue, exam-
ining how to ensure MSY production for the bay of Biscay nephrops fishery.
We show that such a production requirement implies increasing the fleet size
while reducing the per vessel profit. We then characterize transition phases
toward desired exploitation patterns. We define how to reach viable states
without jeopardizing the economic viability of the fleet.
A.1Parameters of the case study: the Bay of Biscay Nephrops fishery (ICES
All along the paper, numerical illustrations are provided, based on an empirical
application to the bay of Biscay nephrops fishery. The numerical model has
been calibrated with commercial time-series.
Biological parameters are estimated using CPUE series (catches per unit of
effort) as an index of abundance. We used nonlinear parameter estimation
techniques to find the best fit of the predicted biomass, given the observed
catches. The fitting criterion is the minimization of the squared deviation
between observed and predicted CPUE (Hilborn and Walters, 1992). Figure
A.1 represents observed and predicted CPUE.
Fig. A.1. Fitting of observed and predicted CPUE in the biological parameters
Economic parameters are estimated using costs and earnings data collected
by the Fisheries Information System of Ifremer via surveys of individual vessel
A.2Individual economic behavior
In this appendix, we detail individual optimal behavior. We first determine
the effort level that maximizes the profit of vessels.
Lemma 1 If the resource stock is greater than a level S?=
mal fishing effort of a vessel is its maximum possible effort e(t) = ¯ e. Else, the
optimal effort is 0.
λ(1−τd)q, the opti-
Proof of Lemma 1 The profit, defined by eq. (5) is
p(1 − τd)qStet
λ− (β1+ β2et).
At a given time t, and for the resource stock St, taking the profit derivative
with respect to the effort level etleads to
λ(1 − τd)qSt− β2
which is positive if the resource stock St is greater than a threshold S?such
λ(1 − τd)q.
The optimal individual effort thus follows a “bang-bang” strategy : no fishing
if St< S?and a maximum activity ¯ e if St> S?. In our illustrative case, this
value is S?= 4,075 tonnes, which is lower than the resource constraint Smin.
We will thus consider that it is always optimal to fish as much as possible.
We then define the minimum effort level ensuring the minimum profit πmin.
For this purpose, we examine instantaneous condition on the effort et for
constraint (9) to be satified at time t, given stock St.
Lemma 2 The minimum effort etinsuring profit πminat a given level of stock
Stis given by
λ(1 − τd)qSt− β2
Proof of Lemma 2 At a given level of stock Stat time t, for constraint (9)
to be satisfied, we must have
p(1 − τd)qStet
λ− (β1+ β2et) ≥ πmin
which leads to
λ(1 − τd)qSt− β2
Hence the minimum effort e(St).
We are now able to prove proposition 1
Proof of Proposition 1 Given the profit equation
πt= pqStet− (β1+ β2et) ≥ πmin
and combinig the optimal effort from Lemma 1 along with the maximum effort
bound ¯ e, we get
St≥πmin+ (β1+ β2¯ e)
Note that at this stock level S, we have e(S) = ¯ e, which means that the
minimum effort to satisfy the constraint is the maximum effort.
A.3About the viability analysis
The viability approach and viable control framework (Aubin, 1991) focuses on
intertemporal feasible paths. It consists in the definition of a set constraints
that represents the “good health” or by extension the effectiveness of the
system at any moment, and in the study of conditions which allow these
constraints to be satisfied along time including both present and future. So,
it leaves the optimality framework and focus on the respect of constraints of
controlled dynamic systems; in this way, various criteria are considered instead
of an unique optimization criterion.
More formally, the viability approach deals with dynamic systems under state
and control constraints. The aim of the viability method is to analyse com-
patibility between the possibly uncertain dynamics of a system and state or
control constraints, and then to determine the set of controls or decisions
that would prevent this system from going into crises i.e. from violating these
constraints. We refer for instance to Martinet and Doyen (2006); B´ en´ e et al.
(2001); Doyen and B´ en´ e (2003) for some stylised models in other contexts.
More specifically, in the environmental context, viability may imply the sat-
isfaction of both economic and environmental constraints. In this sense, it is
a multi-criteria approach sometimes known as “co-viability”. Moreover, since
the viability constraints are the same at any moment and the term horizon is
infinite, an intergenerational equity feature is naturally integrated within this
The viability kernel: a definition
economic states that make it possible to satifsy the constraints throughout
time, given the dynamics. It is defined as the set of all states (St,Xt) from
which there is at least one feasable path (S(.),X(.)), associated with admissi-
ble decisions (e(.),ξ(.)), that satisfies all the constraints along time. It means
that, starting from a state outside the viability kernel, there are no decisions
that make it possible to satisfy the constraints forever (at least one of the
constraints won’t be respected after some finite time T, whatever decisions
The viability kernel is the set of bio-
Formally, for our problem, the viability kernel is defined by
V iab =
∃(e(.),ξ(.)) and (S(.),X(.)), starting from (S0,X0)
satisfying dynamics (3) and (6)
and constraints (7),(8),(9) and (10) for any t ∈ N+
In the general mathematical framework, this set can alternatively be empty,
the whole state constraint set K, or even a strict part of the initial state
constraint domain. The viability kernel captures an irreversibility mechanism.
Indeed, from the very definition of this kernel, every state lying outside the
viability kernel violates the constraints in finite time, no matter what the
decisions applied. This situation means that crisis is unavoidable. For instance,
the extreme case where the viability kernel is empty corresponds to a hopeless
configuration. An empty kernel means that there are no sustainable paths
for the described system, i.e. no paths that respect the set of constraints
representing sustainability. It is then necessary to “modify” the problem (to
consider a different set of sustainability constraints) to be able to define some
sustainable decisions or states.
This kernel is the set of initial fleet configurations and resource stocks from
which it is possible to define acceptable regimes of exploitation satisfying all of
the constraints throughout time. Therefore, the viability kernel provides the
ex post viability constraints that are the “true” constraints to be satisfied by
the bioeconomic system to be sustainable, in the sense that if a state is out this
kernel, there are no decisions that make it possible to satisfy the constraints
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