Recovering viable fisheries
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 framework 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 maximum 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 system 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 recovery paths, wih 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.
- SourceAvailable from: Christophe Béné[Show abstract] [Hide abstract]
ABSTRACT: Among the many factors that contribute to overexploitation of marine fisheries, the role played by uncertainty is important. This uncertainty includes both the scientific uncertainties related to the resource dynamics or assessments and the uncontrollability of catches. Some recent works advocate for the use of marine reserves as a central element of future stock management. In the present paper, we study the influence of protected areas upon fisheries sustainability through a simple dynamic model integrating non-stochastic harvesting uncertainty and a constraint of safe minimum biomass level. Using the mathematical concept of invariance kernel in a robust and worst-case context, we examine through a formal modeling analysis how marine reserves might guarantee viable fisheries. We also show how sustainability requirement is not necessarily conflicting with optimization of catches. Numerical simulations are provided to illustrate the main findings.Journal of Environmental Management 10/2003; 69(1):1-13. · 3.06 Impact Factor
- Bioscience. 01/1992; 43(1):570.
- [Show abstract] [Hide abstract]
ABSTRACT: Predicting global fisheries is a high-order challenge but predictions have been made and updates are needed. Past forecasts, present trends and perspectives of key parameters of the fisheries--including potential harvest, state of stocks, supply and demand, trade, fishing technology and governance--are reviewed in detail, as the basis for new forecasts and forecasting performance assessment. The future of marine capture fisheries will be conditioned by the political, social and economic evolution of the world within which they operate. Consequently, recent global scenarios for the future world are reviewed, with the emphasis on fisheries. The main driving forces (e.g. global economic development, demography, environment, public awareness, information technology, energy, ethics) including aquaculture are described. Outlooks are provided for each aspect of the fishery sector. The conclusion puts these elements in perspective and offers the authors' personal interpretation of the possible future pathway of fisheries, the uncertainty about it and the still unanswered questions of direct relevance in shaping that future.Philosophical Transactions of The Royal Society B Biological Sciences 02/2005; 360(1453):21-46. · 6.23 Impact Factor
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)188.8.131.52.07
Email : firstname.lastname@example.org
Document de Travail
Université Paris X Nanterre UMR 7166 CNRS
Recovering viable fisheries
Vincent Martineta,b; Olivier Th´ ebaudb,c; Luc Doyend
aEconomiX-Universit´ e Paris X. E-mail: email@example.com
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: firstname.lastname@example.org
dCERESP, Mus´ eum National d’Histoire Naturelle. E-mail: email@example.com
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
2.2Sustainable 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
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
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
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
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.
K =30800 tons
p =8,500 euros per tonsα1=10
β1= 70,000 euros per year
¯ X =
377 euros per day of sea
¯ e = 220 days
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.
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_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
(b) Resource stock StCBA
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
(d) Fleet size XtCBA
(e) Profit πtOpen Acces
(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-