Viability theory, introduced byAubin(1990), aims at identifying decision rules or controls for dynamical systems (in particular non-linear control problems), such that the systems are main- tained at each instant inside a given set of admissible states of diverse nature, called the viability constraints set. Although dynamic optimisation problems are usually formulated under constraints, the role played by the constraints poses difficult technical problems and is generally not tackled as a specific issue (De Lara and Doyen,2008). Furthermore, the optimization procedure reduces the diversity of feasible forms of evolution by, in general, selecting a single trajectory (De Lara and Doyen,2008). In contrast, viability analysis, instead of maximizing an objective function, focuses on the role of constraints and on characterizing the safe paths and decisions.
1.3. Viability models S. Gourguet
1.3.1
Non-linear dynamical system and viability constraints
Solving the viability problem relies on the consistency between a controlled dynamic and ac- ceptability constraints applying both to states and decisions of the system. Viability constraints for a non linear dynamic system x are represented by:
x(t+1)= ft,x(t),u(t), t= t0, . . . ,T, u(t)∈Bt,x(t), ∀t=t0, . . . ,T, x(t)∈A(t), ∀t=t0, . . . ,T. (1.8)
where u(t) is the control or decision,Bt,x(t)is the set of admissible and ‘a priori’ feasible deci- sions, andA(t) corresponds to a non empty state or target (according the objective of the analysis) constraint domain and represents the safety, the admissibility or the effectiveness of the state for the system x at timet.
This approach has been applied to a number of environmental management and sustainability issues (Bene et al.,2001;Bonneuil,2003;Eisenack et al.,2006;Rapaport et al., 2006;Aubin and Saint-Pierre, 2007;Martinet and Doyen, 2007;Tichit et al.,2007;Baumgärtner and Quaas, 2009; Bene and Doyen,2008;De Lara and Doyen,2008;De Lara et al.,2011;Doyen and Martinet,2012; Doyen and Péreau, 2012) and to fisheries management problems in particular (Béné and Doyen, 2000;Mullon et al.,2004;Cury et al.,2005;Doyen et al.,2007;Martinet et al.,2007;De Lara et al., 2007;Chapel et al.,2008;Bendor et al.,2009;Martinet et al.,2010;Doyen and Péreau,2012).
1.3.2
Co-viability analysis
As fisheries management is characterized by multiple management objectives, the co-viability approach combines multiples constraints, like biological and economic viability constraints, as defined by the set of equations (1.9) inspired byBene et al.(2001):
x(t)≥ xmin≥ 0 ∀t =t0, . . . ,T, π x(t),E(t)≥ πmin ≥0 ∀t =t0, . . . ,T. (1.9)
where xmin is the resource minimum level to maintain, π
x(t),E(t) represents the net benefit (or profit) from the harvesting of the resource x andπminis the minimum profit to guarantee at all time
(typically, the lower viable bound on the profit can be zero).
Co-viability analysis shows the ability of management actions to maintain natural and economic capital stocks above some minimum levels. This approach (Le Fur et al., 1999;Bene et al., 2001; Eisenack et al.,2006;Martinet et al.,2007) can be particularly useful in multi-criteria management problems, as it can highlight the domain of possibilities, feasibility and trade-offs between poten- tially conflicting objectives or constraints that are required to be fulfilled both in present and future time periods.
1.3.3
Stochastic co-viability
Risk and uncertainty of fishery systems constitute major issues in fisheries management, and acceptability constraints in a co-viability context have to be articulated with uncertainty in a prob- abilistic or stochastic sense. The probability of co-viability CVA of a fishery system, regarding control or decision u(t) and considering the multiple constraints defined in section 1.3.2, is ex- pressed by:
CVAu(t0), . . . ,u(T)
=P
constraints (1.3.2) are satisfied fort= t0, . . . ,T
. (1.10)
The idea underlying stochastic viability is to require the respect of the constraints at a given confidence level. Therefore, of particular interest are the set of control or decision u(t) such that the probability of co-viability CVA is above a certain level as:
CVAu(t0), . . . ,u(T)
≥β. (1.11)
Withβsome confidence level (typically 90%, 95% or 100%).
By compiling ecological and economic goals from stochastic simulation models, stochastic co- viability analysis (De Lara and Doyen,2008;Baumgärtner and Quaas,2009;Doyen and De Lara, 2010) can be used to address important issues of vulnerability, risk, safety and precaution, and to determine the ability of a particular resource system to achieve specified multiple sustainability ob- jectives with sufficiently high probability. In contrast to the MSE approach, stochastic co-viability analysis proposes a ‘sustainability metric’ to rank alternative management strategies through the co-viability probability. Indeed, as viable management requires all constraints (and hence objec- tives) to be satisfied, the approach is not based on arbitrary weights that may reflect priorities in
1.3. Viability models S. Gourguet
the objectives. This approach therefore provides a useful tool to inform policy makers about the trade-offs involved in managing fisheries under multiple constraints in a stochastic environment. As depicted in box 3, the viability kernel plays a major mathematical role in the viability analysis.
Box 3: Viability kernel.
A major mathematical tool to study the whole viability of the system is provided by the so-called viability kernel, denoted by Viab. It corresponds to the set of all initial conditions such that there exists at least one trajectory starting from the initial conditions that stays in the set of constraintsA. Figure 1.6 gives a graphical representation of the viability kernel.
Figure 1.6: The state constraint setAdefined by a set of ecological and economic viability constraints
corresponds to the large blue set. It includes the smaller viability kernel Viab (in dark blue).
For decision makers, knowing the viability kernel has practical interest since it describes the states from which controls can be found that maintain the system in a desirable configuration until the horizon timeT.
Only a few applied studies (Béné and Doyen, 2000;De Lara et al.,2007; Doyen et al., 2007; Martinet et al.,2007;De Lara and Martinet,2009;Martinet et al.,2010;De Lara et al.,2011;Péreau et al., 2012) have made use of the co-viability approach to integrate economic and ecological ob- jectives for fisheries management. Among these studies, onlyDoyen et al.(2007) andDe Lara and Martinet(2009) integrate uncertainty affecting biological dynamics. While these studies integrate
several species in their analysis, the diversity among the fishery industry is not taken into account.