Sustainability of fisheries through marine reserves: a robust modeling analysis

Journal of Environmental Management 69 (2003) 1–13 www.elsevier.com/locate/jenvman

Sustainability of fisheries through marine reserves: a robust modeling analysis L. Doyena, C. Be´ne´b,* a

CNRS, CIRED (Centre International de Recherche pour l’Environnement et le De´veloppement), Campus Jardin Tropical de Paris, 45bis Avenue de la Belle Gabrielle, 94736 Nogent sur Marne, France b Department of Economics, CEMARE (Center for the Economics and Management of Aquatic Resources), University of Portsmouth, Locksway Road, Portsmouth PO4 8JF, UK Received 8 February 2002; revised 20 November 2002; accepted 23 December 2002

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. q 2003 Elsevier Ltd. All rights reserved. Keywords: Renewable resource; Fisheries management; Marine protected area; Uncertainty; Invariance analysis

1. Introduction Over-exploitation of marine fisheries resources remains a serious problem worldwide. The Food and Agricultural Organization estimates for instance that at present 47– 50% of marine fish stocks are fully exploited, 15– 18% are overexploited and 9 – 10% have been depleted or are recovering from depletion (FAO, 2000). Among the many factors that contribute to fisheries management failures, the role played by uncertainty is important. Uncertainty in fisheries can be classified under three principal forms (Charles, 1998): random fluctuations such as those affecting fish survival rate in the ocean or fish price in the market; uncertainty in parameter estimates and states of nature such as uncertainty on the stock size or the fishing mortality; and structural uncertainty that reflects a basic lack of knowledge about the nature of the fishery system such as * Corresponding author. Fax: þ 44-23-9284-4037. E-mail addresses: [email protected] (C. Be´ne´), doyen@ centre-cired.fr (L. Doyen). 0301-4797/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0301-4797(03)00004-5

uncertainty on the multi-species interactions, for instance predator-prey effects. It is often suggested that uncertainty could be reduced if more research and data collection were to be undertaken. The principle is that more data will reduce some of the inherent uncertainty, particularly that characterizing parameters and states of the nature. Ulltang (1998) for instance argues that in general fish stock assessments do not fully utilize available oceanographic and biological knowledge and that we are far from any limits set by nature on predictability. While no one can dispute the fact that more information and sophisticated analyses can possibly reduce uncertainty, it must be recognized that such efforts carry a substantial price tag, and many feel that, relative to expense, the benefits could be marginal in many cases (Walters and Collie, 1988). Beyond this practical limitation, theoretical analysis from other disciplines (physics, mathematics, economics) have also demonstrated the unpredictable nature of certain complex systems, and therefore the difficulty to meet the need for prediction (May, 1972; Gleick, 1987). In fisheries, these conclusions question our ability to manage

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commercially valuable species in complex and stochastic marine environments and Lauck et al. (1998, p. 74) even suggest that ‘predictability of anything as complex (and, we should add, as unobservable) as marine ecosystem will forever remain a chimera’. An alternative approach may be, therefore, to recognize the irreducible and persistent nature of uncertainty (Ludwig et al., 1993). In this respect, there is a growing consensus among scientists from various disciplines that we must shift away part of our research effort from the attempts to reduce uncertainty to the search for the solutions to cope with it (Holling, 1978; Behnke and Sconnes, 1993; Harwood, 2000). In fisheries, it becomes now widely accepted that management must somehow allow for uncertainty and potential inaccuracy both in the design and implementation of management strategies (Gordon and Munro, 1996; Charles, 1998; Weeks and Berkeley, 2000). Since recently, a growing number of works have been suggesting that one possible way to integrate and to ‘manage’ this uncertainty may be through the implementation of marine reserves (Hall, 1998; Mangel, 2000). Indeed, an increasing number of—so far mainly theoretical—analyses suggest that, in the context of fisheries activities, marine reserves could prevent overfishing and depletion of the exploited stocks even in the presence of parameter uncertainty (Clark, 1996; Botsford et al., 1997).1 The principle is that the protection of a minimum spawning stock through the establishment of a reserve will act as a buffer against stock assessment failures and/or environmental uncertainty that may otherwise lead to the depletion of the stock and to the subsequent collapse of the fishery. In a recent article, Lauck et al. (1998) investigate more deeply this hypothesis. Recognizing that both fisheries management actions and data are variable and uncertain, they consider how the establishment of a Marine Fishery Reserve (MFR) might help to ensure the sustainability of the stock by maintaining the latter above a predefined Save Minimum Biomass Level (SMBL) over a given time horizon. For this, they model a single harvested stock and use a dynamic model embodied within a probabilistic approach. Their analysis shows that MFR can improve the chances of sustainability of the fish stock but does not 1 Marine reserves have been sometimes presented in the literature as a possible ‘global panacea’ for multi-nature degradation affecting the World’s oceans. However, as pointed out by Boersma and Parrish (1999), it seems that among the four main threats facing marine environment, namely commercial species over-exploitation, development of coastal areas, biological and chemical pollutions, solely conservation of exploited species may—under certain conditions—be partially ensured through marine reserves. In contrast, the use of marine reserve to prevent impacts of coastal development and/or pollution does not seem to be appropriate. In the present article, we will only consider and discuss this first aspect, i.e. the potential role of marine reserves to ensure the sustainability of exploited resources, and within this framework, we will concentrate on the question of marine reserve and uncertainty. No further comment will be made on the other potential role(s) of marine reserves.

necessary disadvantage the fishery in the sense that the catches might be optimized through reserve size. In the present paper, we attempt to deal with similar concerns by studying the sustainability of a protected area under uncertainty. More specifically, we address the following three questions, using a theoretical approach. 1. How do marine reserves promote resource protection? 2. How does uncertainty level modify the reserves protection capacity? 3. What is the nature of the trade-off between the stock protection and the level of catch? However, instead of using a probabilistic framework such as that used by Lauck et al. (1998), we adopt another alternative to treat uncertainty, that is a robust, or worstcase, approach. There are several ways to tackle uncertainty. The first one is through the concept of ‘risk’, where the probability of occurrence of the event under consideration is assumed to be known from experience and quantifiable through statistics and probability. Within the framework of dynamic decision under uncertainty, the usual way to deal with risk is through stochastic control approaches. However, there are also situations of ‘ambiguity’, or ‘radical uncertainty’, where the probability of occurrence of the event is not quantifiable or remains unknown. Many environmental or precaution problems involve ambiguity in the sense of controversies, beliefs, or irreducible scientific uncertainties. In this case, one solution to cope with such uncertainty is through multiprior, or Choquet capacity models (Cohen and Tallon, 2000). Alternatively, pessimistic, worst-case, minimax, total risk-averse, guaranteed or robust control frameworks may also provide useful analytical frameworks. To quote a few, we refer to Barron and Jensen (1989), Basar and Bernhard (1995) and Freeman and Kokotowicz (1996) or for an economic perspective Hansen and Sargent (2000). In the present case, we adopt a robust approach and we assume that the uncertainty affecting the fishery relates to the harvesting rate rather than to the dynamics of the exploited species. Furthermore we consider that the uncertainty level, which remains unknown over time, evolves within a given bounded set. This set provides the possible scenarios of catches. We then use the mathematical framework of viability analysis (Aubin, 1991) to investigate how marine reserves can guarantee or increase the sustainability of the fishery. In brief, viability approach is a mathematical framework which aims at analyzing the compatibility between the dynamics (eventually uncertain) of a system and a set of constraints representing safe, tolerable, effective, feasible or admissible situations affecting the system. The viability approach, therefore, allows coping with multi-criteria constraints and may offer the opportunity to reconcile economical and environmental issues within a sustainability perspective. This viability approach has already been applied to fishery issues in Be´ne´ and Doyen (2000) and Be´ne´ et al. (2001). In

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the present case, since we address conservation issues, the constraint set only refers to a safe minimum biomass threshold. From a mathematical point of view, the viability approach focuses on the set of conditions (states or decisions) that allow dynamical systems to remain viable over time, i.e. to stay within their viability domains defined by the sets of constraints. Within this framework, we specifically use the concept of invariance kernel to address fishery sustainability from a robust, or worst-case, perspective. The invariance kernel represents the set of initial conditions inside the viability domain such that, for any perturbation or uncertainty imposed on the system, the associated trajectories remain viable for every time. In this sense, the invariance kernel plays the role of indicator of robust sustainability. For instance, in the present case, if the invariance kernel appears to be empty, this indicates the existence of a possible catastrophic harvesting scenario leading to the collapse of the fishery due to the violation of the viability minimum biomass threshold. In the rest of the paper, the different elements of the article are organized as follows. In Section 2, we present the main hypotheses and mathematical model developed for the analysis. In Section 3, we use the concept of invariance kernel combined with guaranteed catch to answer the three questions related to marine reserves, conservation and catch trade-off. Then Section 4 provides numerical illustrations, and Section 5 discusses and summarizes the major findings. To restrict the mathematical content to a minimum in the core of the text, formal propositions, lemmas, and proofs of the different results are presented in Appendix A at the end of the article.

equal to u
, namely luðtÞ 2 u
l # s;

t ¼ 0; 1; …; n; …

ð1Þ

where the value s represents the uncertainty degree within the present frame.2 The case of certainty corresponds to the particular value s ¼ 0 where target u
and observed harvesting levels uðtÞ coincide. Taking into account this uncertainty and assuming that harvesting takes place before reproduction,3 we obtain the following uncertain dynamics for the renewable resource: Nðt þ 1Þ ¼ f ðNðtÞð1 2 AuðtÞÞÞ; uðtÞ [ ½
u 2 s; u
þ s;

ð2Þ

t ¼ 0; 1; …; n; …

At this stage, it is important to emphasize that the structure of the model does not allow us to account for two crucial aspects of marine population dynamics in relation to marine reserves: (1) the spatial dynamics of the species targeted by the fishery: the movements of the individuals between the reserve and the fished area are assumed to be instantaneous; and (2) the existence of different life stages for the targeted species.4 2.2. The viability and conservation constraint A second step in this analysis is to introduce a state variable constraint considered from a regulating (say a government) agency viewpoint. In particular we assume that the agency requires the adoption of a Safe Minimum Biomass Level (SMBL) Nmin for the resource, also possibly termed viability threshold,5 such that, at any time, the stock biomass NðtÞ should not be below that threshold Nmin : 0 , Nmin # NðtÞ:

ð3Þ

Hereafter we shall say that the evolution Nð·Þ ¼ ðNð0Þ; Nð1Þ; …; NðtÞ; …Þ of the stock is viable if the conservation condition (3) holds true for every period t:

2. The model 2.1. The uncertain dynamics Following Lauck et al. (1998) the model represents the dynamics of a single stock, using a discrete time approach. In absence of harvest, the natural growth rate of the stock N is represented by a law f through the relation Nðt þ 1Þ ¼ f ðNðtÞÞ;

3

t ¼ 0; 1; …; n; …

To model the effect of a MFR, it is assumed that only a fraction A [ ½0; 1 of the area in which the stock is located is available for harvesting. In other words, the MFR is ð1 2 AÞ: Under this assumption, if NðtÞ is the overall stock level in the entire area, the fraction of the stock protected within the MFR is ð1 2 AÞNðtÞ and the fraction available for harvest is ANðtÞ: We further assume that the harvest rate in the overall area is targeted at some level u
. However, the effective harvesting level turns out to be uncontrollable in the sense that the actual harvest rate uðtÞ [ ½0; 1 is close but not

2 Observe that the condition uðtÞ [ ½0; 1 induces a bound on s; namely s # minð
u; 1 2 u
Þ # 1: 3 The total stock remaining after fishing is the sum of the survivals in the fishing area and the protected individuals in the reserve: ð1 2 uðtÞÞANðtÞ þ ð1 2 AÞNðtÞ ¼ ð1 2 AuðtÞÞNðtÞ: 4 These two aspects play a central role in the question of the potential success of MFR with respect to fishery exploitation, since they intervene in the two major mechanisms ensuring the movement of individuals between inside and outside the reserve: the active diffusion of adult (spillover) or the passive diffusion of larvae (larval dispersal). The extension of the present paper to include these dimensions is the main objective of our future research. 5 Such minimum level might be identified by biologists. This SMBL constraint is usually introduced in relation to Population Viability Analysis (PVA). PVA involves predicting the probability of extinction (or quasiextinction-when numbers fall below some pre-specified threshold) within a particular time interval for endangered species or populations, using either analytical or more often computer simulation techniques (Harwood, 2000). Uncertainty is central to this process because extinction is an uncertain event and because extinction probabilities depend on the magnitude and nature of the variability which is incorporated into the survival and reproduction rates, as well as the rate of increase or decrease in the population.

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3. Invariance analysis 3.1. The invariance kernel as an indicator of robust sustainability A question that arises now is whether the uncertain dynamics of the renewable resource (2) is compatible with the constraint (3) imposed by the existence of the SMBL. In other words, an objective here is to identify the levels of resource that are associated with viable trajectories whatever the uncertain sequential harvests uðtÞ: In a formal way, this requires to compute the set of initial levels of stock N0 such that whatever the future exploitation rate scenario uð·Þ ¼ ðuð0Þ; uð1Þ; …; uðtÞ; …Þ; the fishery remains viable, i.e. the stock NðtÞ stays above the SMBL Nmin for an infinite period of time. This set of initial levels of stock N0 is called the invariance kernel associated with the dynamics (2) and constraint (3). Since it depends on both the harvesting fraction area A and the uncertainty degree s; we denote this invariance kernel by InvðA; sÞ: In our context, considering conservation issue within an uncertain evolution, this invariance kernel can be considered as an indicator of robust sustainability and viability of the fishery.6 3.2. Re-formulation of the questions on reserve and uncertainty This concept of invariance kernel is useful because it allow us to reformulate the three questions identified in the introduction under a more formal—and therefore testable— form. The questions now become 1. Does a relationship exist between the invariance kernel Inv(A,s) and the area (1 2 A) allocated to the MFR? Identifying this relationship is equivalent to determining formally whether or not a MFR can ensure the conservation of the stock. In particular, a ‘positive’ relation between Inv(A,s) and (1 2 A) would induce that the presence of the MFR does promote the sustainability of the fishery. 2. What is the degree of sensibility of the invariance kernel Inv(A,s) to the level of uncertainty s affecting the harvesting rate? This second question is equivalent of 6

In a more formal way, in our case, the invariance kernel InvðA; sÞ reads as follows: InvðA; sÞ ¼ {N0 $ Nmin lfor every uð·Þ satisfying Eq: ð1Þ; the solution Nð·Þ of system ð2Þ starting from N0 satisfies NðtÞ $ Nmin }: In the general mathematical framework, this set can alternatively be empty, or be the whole constraint set or even a strict part of the initial constraint domain. In the present case, the analysis will reveal that the kernel reduces to the two extreme cases, i.e. the empty set and the whole domain depending on conditions on the parameters (Section 3.4).

determining whether uncertainty affects the ability of the MFR to guarantee the sustainability of the stock. For instance it could be observed that a high level of uncertainty reduces the capacity of a MFR to ensure the viability. 3. How does the presence of a MFR affect the catch levels? In other words the question here is whether the implementation of a MFR of area (1 2 A) can have a beneficial effect on the catch CðtÞ ¼ AuðtÞNðtÞ: Given the uncertainty on the effective harvesting rate s, a robust approach leads us to study the ‘guaranteed catches’ C(A) defined as the minimum catch volumes obtained over time for any harvest scenario, i.e. CðAÞ ¼ min{CðtÞlt; uðtÞ}: The search for these guaranteed catches makes sense, however, only if the sustainability of the stock is ensured or equivalently if the invariance kernel is not empty. 3.3. Assumptions and notations We assume that the endogenous dynamics of the resource follows the familiar Beverton –Holt stock –recruitment relation, namely f ðNÞ ¼

cN ; 1 þ dN

with parameters7 c . 1 and d . 0: To ensure the viability of the stock in absence of harvest we also need to assume that the viability threshold Nmin is smaller than the carrying capacity defined in the case of Beverton – Holt dynamics by ðc 2 1Þ=d: We have, therefore, the following conditions: 0 , Nmin ,

c21 ; d

1 , c:

ð4Þ

We also introduce † The stock’s biomass function FðA; uÞ formalized within the Beverton – Holt framework by8 FðA; uÞ ¼

c 1 2 : d dð1 2 AuÞ

† The harvesting fraction threshold Asus defined by9
 1 1 Asus ðsÞ ¼ 1þ : u
þ s dNmin 2 c

7

ð5Þ

ð6Þ

In the literature, c and d are used to parameterize the degree of density dependence characterizing the stock recruitment relationship of the species. 8 The biomass FðA; uÞ represents the stock levels at equilibrium, i.e. for which Nðt þ 1Þ ¼ NðtÞ with constant harvesting uðtÞ ¼ u and A: 9 This threshold Asus ; originally defined by the equation FðAsus ; u
þ sÞ ¼ Nmin ; is the area which ensures, for a given exploitation level ð
u þ sÞ; the equilibrium of the biomass at a level equal to the minimum threshold Nmin : This threshold Asus exists by assumption (4). The subscript notation ‘sus’ in Asus stands for ‘sustainability’.

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3.4. Do marine reserves promote sustainability? The first fundamental question about marine reserve in the context of extractive activities by a fishery is whether the creation of a no-take area can help promoting the conservation or preservation of the species targeted by the fishery. The invariance analysis confirms that the answer is Yes: marine reserves can ensure fisheries sustainability, even in the context of uncertainty. The basic idea is that a low harvesting area A; by reducing fishing pressure, promotes the resource NðtÞ and thus sustainability concerns. This result is formally enounced through Proposition A.1 presented and proved in Appendix A. This proposition claims that, given harvesting parameters u and s; the invariance kernel is empty if the harvesting fraction threshold A is greater than the threshold Asus ðsÞ defined above in Eq. (6). Conversely, the invariance kernel is the whole interval of possible biomass levels ½Nmin ; þ1½ whenever the harvesting fraction threshold is smaller than Asus : We have thus ( InvðA; sÞ ¼

B;

if Asus ðsÞ , A;

½Nmin ; þ1½; if Asus ðsÞ $ A:

This result means that the threshold Asus represents the maximum fraction of the area which can be opened to exploitation without possibly jeopardizing the stock viability. The reserve size 1 2 Asus is thus a critical value of sustainability for the fishery. Furthermore, Proposition A.1 implicitly enounces that the invariance kernel increases10 when the harvested area fraction A decreases, or symmetrically when the protected fraction increases. Since the kernel stands for an indicator of sustainability, this demonstrates again how marine reserves contribute to fisheries sustainability. 3.5. Does uncertainty affect marine reserve sustainability? The above analysis shows that marine reserve is a way to cope with uncertainty. The analysis also shows that the sustainability condition depends critically upon the size of the reserve ð1 2 AÞ compared to the threshold area 1 2 Asus : From definition (6) of Asus ; it turns out that this harvesting area threshold is inversely related to s. This implies that the higher the uncertainty, the lower the capacity of the MFR to ensure the sustainability of the stock. Thus the capacity of marine reserves to promote fisheries sustainability is negatively related to the level of uncertainty affecting the fishery. This result is formally enounced in Proposition A.2 in Appendix A. Within the favorable case of sustainability, we can focus furthermore on the extreme theoretical situation where Asus $ 1: In that case, a robust viability occurs whatever 10 In the sense of inclusion of sets, namely, A1 # A2 ) InvðA2 ; sÞ , InvðA1 ; sÞ:

5

the MFR fraction is. This situation describes the case where the sustainability of the fishery is ensured even without marine reserve. The detailed analysis provided by Corollary A.3 shows that this specific situation holds when the safe biomass minimum level is sufficiently small, i.e. Nmin # Fð1; u
þ sÞ: This last requirement is associated to the condition cð1 2 u
2 sÞ . 1 which is obtained for a high endogenous parameter c or symmetrically for a low uncertainty degree s. This last result also points out a critical degree of uncertainty sp ¼ 1 2 u
2 ð1=cÞ above which the implementation of a reserve becomes necessary to maintain the viability of the fishery for any conservation threshold Nmin : Note that this critical level s p is negatively related to the exploitation rate u but positively related to the resource parameter c. A more restrictive uncertainty threshold induced by Corollary A.3 is defined by

s^ ¼ 1 2 u
2

1 : c 2 dNmin

ð7Þ

Given the conservation stock value Nmin ; the effort variability s^ is the maximal uncertainty degree below which the reserve would not be necessary from a sustainability point of view. 3.6. Do marine reserves promote catches ? Another important question that needs attention concerns the influence of MFR on catches. Does the creation of a marine reserve impact negatively the catches of the fishery or can it help to increase these catches? The answer is not straightforward because two opposite effects occur. On one hand, a low harvesting area A promotes the stock N(t) and thus the catches CðtÞ ¼ AuðtÞNðtÞ as they depend positively on resource level NðtÞ: On the other hand, a too weak harvesting area A directly reduces the captures CðtÞ or even stops them in the particular case A ¼ 0 where the fishery is closed. This means that, from a catch point of view, a fine tuning for A is required to reconcile the two previous opposite effects. This is revealed by the optimal reserve size 1 2 Ap for guaranteed catches that we define and compute herafter. The interesting case is where sustainability exists in the sense that the invariance kernel InvðA; sÞ is not empty and 0 , A # Asus ðsÞ: 3.6.1. Guaranteed catches In this context, using a robust (i.e. worst-case) approach, we focus on the case of lowest catch over time, termed ‘guaranteed’ catch. For this we consider the function CðA; N0 Þ defined for any N0 $ Nmin by CðA; N0 Þ ¼

inf

t;NðtÞ;uðtÞ

NðtÞAuðtÞ

where NðtÞ; uðtÞ satisfy dynamics (2) and initial condition Nð0Þ ¼ N0 : It can be proved (Proposition A.4) that the shape of the function CðA; N0 Þ depends on the initial level N0 of the stock with respect to the equilibrium biomass function

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FðA; u
þ sÞ: In effect, the function CðA; N0 Þ has the following behavior: ( if N0 # FðA; u
þ sÞ; N0 ð
u 2 sÞA; CðA; N0 Þ ¼ FðA; u
þ sÞð
u 2 sÞA; if N0 $ FðA; u
þ sÞ:

3.6.2. Maximal guaranteed catches Let us now compute the maximal (local) value Ap ðsÞ of the worst catch function C(A,N0) with respect to A, defined by CðAp ðsÞ; N0 Þ ¼ max CðA; N0 Þ: A[½0;1

We consider the non-trivial case11 where N0 $ FðA; u
þ sÞ: It turns out that the maximum value of the function—in other terms the maximum guaranteed catches—is obtained for the following area threshold: pffiffi c21 p pffiffi : A ð sÞ ¼ ð8Þ ð
u þ sÞ c This result is formally proved in Proposition A.5. 3.6.3. Consistency between sustainability and optimality Now let us study the compatibility between sustainability and optimality requirements. We need to compare the optimal and sustainable reserve size rate. The analysis shows that when the safe minimum biomass level Nmin is set up such that pffiffi c2 c Nmin # ; ð9Þ d we have Asus ðsÞ $ Ap ðsÞ; which means that the maximal guaranteed catches associated to Ap ðsÞ is sustainable (Proposition A.6). In that situation, a very interesting outcome is that the function CðAÞ increases when A decreases from Asus ðsÞ to Ap ðsÞ: In other words, an increase in the share of the total area included in the marine reserve ensures a larger guaranteed catch volume, while it still maintains the sustainability of the stock. This finding indicates that the presence of larger reserve may have positive impacts on the catch while achieving conservation and viability goal. Illustration of this interesting result is provided in Section 4 and discussed in detail in Section 4. We also observe that the requirement (9) on Nmin to achieve sustainable optimal guaranteed catches is more restrictive than the initial one (Eq. (4)) imposed by the endogenous dynamics of the stock. In other words, the quantification of an appropriate SBML Nmin might be more stringent than expected. 11 In the case where N0 # FðA; u
þ sÞ; the guaranteed catch function CðA; N0 Þ is simply linear with respect to A and consequently the maximal value is Ap ðsÞ ¼ 1: Corollary A.3 provides conditions for A ¼ 1 to be sustainable.

4. Numerical illustrations In this section we provide numerical examples for the major results presented in Sections 3.5 and 3.6. In particular we focus on the question of consistency between sustainability and optimality objectives. 4.1. Consistency between sustainability and optimality: the consequence of uncertainty on the harvesting level To focus on uncertainty and the role played by s, we fix the other parameters: Nmin ¼ 4;

c ¼ 2;

d ¼ 0:1:

4.1.1. Without uncertainty We assume first a situation of certainty, i.e. s ¼ 0; and we consider two targeted harvesting levels: † First, u
¼ 0:33: The numerical application yields Asus ð0Þ ¼ 9=8;

Ap ð0Þ < 0:87:

Since Asus ð0Þ $ 1; the implementation of a MFR is unjustified from a conservation point of view for this first targeted harvesting level. † Second, u
¼ 0:50: In that case, the numerical results are Asus ð0Þ ¼ 0:75;

Ap ð0Þ < 0:57;

and we recover conditions and results of Proposition A.6. In that case the implementation of a MFR is necessarily to guarantee the sustainability of the stock. It is also possible to identify an ‘optimal’ reserve size ð1 2 Ap ð0ÞÞ which yields maximal guaranteed catches. In the present numerical example, this optimal reserve size is about 43% of the total area. Thus, even in the case of no uncertainty, MFR may turn out to be an effective management tool from both the points of view of sustainability and optimization of catches. 4.1.2. With uncertainty We now allow for uncertainty, i.e. s . 0: We still consider the targeted harvesting level u
¼ 0:33: Fig. 1(a) illustrates the situation for two specific levels of uncertainty s ¼ 1=6 and 1/4. For comparison purpose, the case with no uncertainty s ¼ 0 is also represented. The figure shows the evolution of the guaranteed catches CðAÞ; along with the associated sustainable thresholds Asus ð·Þ: Note first that, in contrast with the situation with no uncertainty, the creation of the MFR is now necessary to ensure the sustainability of the stock. This is emphasized by the critical uncertainty value s^ introduced in Eq. (7) which is here equal to s^ ¼ 1=24 < 4%: This means that uncertain degree s higher than 4% requires the implementation of a reserve for conservation issue. For instance for s ¼ 1=6; we obtain Asus ðsÞ ¼ 0:75: This last result means that the minimal fraction to be

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protected to guarantee the sustainability of the stock in that case must be 25%. We also observe that for values of A diminishing from Asus to Ap ; or symmetrically for a protected area increasing from ð1 2 Asus Þ to ð1 2 Ap Þ; the associated guaranteed catch volumes increase. This illustrates the fact that in situation of uncertainty the creation of MFR may also be an effective tool to optimize catches. Fig. 1(b) is a generalization of Fig. 1(a). Still for a given harvest level u
¼ 0:33; it represents the values of Asus ð·Þ associated to different levels of uncertainty s. Also represented are the corresponding Ap ð·Þ values. One simple

7

way to interpret Fig. 1(b) is to say that the vertical distance between the curve labeled Asus and the straight line A ¼ 1 represents the fraction of the fishing ground that should be protected to ensure the sustainability of the stock. Similarly the distance between the curve Ap and the line A ¼ 1 represents the fraction of the fishing ground that should be protected to ensure maximal guaranteed catches. In agreement with results enounced pffiffi in Proposition A.6, since the condition Nmin # ðc 2 cÞ=d holds true, note that for any level s; the curve Ap is below Asus : This indicates that a protected fraction ð1 2 Ap Þ also ensures the sustainability of the stock.

Fig. 1. Numerical illustrations for the consistency between sustainability and optimality issues. We compare sustainable Asus and optimal Ap area rate thresholds for several uncertainty degrees s. Fixed parameter values are Nmin ¼ 4; c ¼ 2; d ¼ 0:1; u
¼ 33%:

8

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4.2. Consistency between sustainability and optimality: the impact of the resource dynamics In Fig. 2(a)– (f), we compare the evolutions over the time horizon t ¼ 1; …; 100 of the biomass NðtÞ (dashed lines) and captures CðtÞ (solid lines) obtained for a fishing area set at the optimal level Ap or sustainable level Asus under 3 different values of the parameter c. Several combinations of admissible harvesting scenarios u(t) and initial biomass N(0) are used to illustrate the general character of the results. The others parameter values are set constant: Nmin ¼ 1; d ¼ 0:1; u ¼ 33% and s ¼ 1=6: The straight dotted line in each figure represents the Safe Minimum Biomass Level Nmin set here equal to 1. The three

sets of figures illustrate distinct situations with respect to this SMBL. In Fig. 2(a) and (b) obtained for c ¼ 1:25; both sustainable Asus and optimal Ap areas are viable: N(t) remains well above the SMBL in both cases. We have therefore here compatibility between sustainability and optimality. This compatibility remains as long as the pffiffi condition Nmin # ðc 2 cÞ=d holds true as claimed by Proposition A.6. The critical value of c for such a configuration is the solution of the equation Ap ¼ Asus : In the present case, this critical value is c^ < 1:19: Fig. 2(c) and (d) represents situations where c ¼ 1:14 # c^ : In accordance with Proposition A.6, we observe that the optimal area Ap is not sustainable any longer (Fig. 2(c)). The stock passes below the safe threshold Nmin ¼ 1 at many occasions,

Fig. 2. Comparison over time [0,100] between biomass N(t) (dashed lines) and captures C(t) (solid lines) for the sustainable area Asus (right-hand side) and the optimal area Ap (left-hand side) under 3 values of resource parameter c. The straight line represents the safe minimum biomass level Nmin ¼ 1: Others parameters are d ¼ 0:1; u
¼ 33% and s ¼ 1=6: Different harvesting scenarios u(t) and initial stock levels N(0) are used. Cases (a) and (b) illustrates a situation of compatibility between sustainability and optimality. Cases (d) and (f) are still sustainable while cases (c) and (e) illustrate situations of crisis where the stock passes below the SMBL.

L. Doyen, C. Be´ne´ / Journal of Environmental Management 69 (2003) 1–13

indicating the occurrence of catastrophic scenarios and a danger of collapse. In contrast, fishing activity restricted to an area Asus ¼ 8% is still viable in a robust way as the stock remains above the SMBL in every scenario (Fig. 2(d)). Finally, Fig. 2(e) and (f) represent the limit situation from which the fishery has to be totally closed to ensure the sustainability of the stock, i.e. Asus ¼ 0%: This extreme situation, obtained for c ¼ 1:11; corresponds to the limit case where the stock is just able to sustain itself at the SMBL (Fig. 2(f)). In that case, any fishing pressure induced by the opening of a fishing zone drives the stock below the SMBL, as illustrated for instance in Fig. 2(e) in the specific case of the optimal area Ap < 9%:

5. Discussion and concluding remarks As recalled during the Technical Consultation on the Precautionary Approach to Capture Fisheries, it must be recognized that ‘changes in fisheries systems are slowly reversible, difficult to control, not well understood, and subject to changing environment and human values’ (FAO 1995, p. 36). Underlying this assertion is the recognition that fisheries are characterized by an extremely high degree of uncertainty. This uncertainty can take different forms and derives from different mechanisms of various natures. There is now a wide consensus among scholars but also—to a less extent perhaps—policy-makers to recognize that failures to integrate this uncertainty have caused in the past many management malfunctions or even collapses of fisheries. Recently the concept of MFR has been proposed as a solution to tackle this uncertainty. Historically the first theoretical works on MFRs (see Table 1 in Mangel (2000), p. 549 for a review) were not specifically addressing this uncertainty issue. They were instead mainly deterministic models aimed at demonstrating the ability of marine reserves to satisfy two objectives which are usually considered antagonistic, namely (a) the reduction of overexploitation, i.e. a conservation objective and (b) the enhancement of fishery yields, i.e. a production/economic objective. It is only recently that one started to realize that marine reserve may also be potentially a powerful tool to implement a precautionary management and to address the problem of uncertainty in fisheries. One pioneer paper on this aspect is Lauck et al. (1998). With respect to this relationship between uncertainty and MFRs, it is interesting to note the recent evolution in the major questions that emerged in the literature. These can be summarized into three main steps: (1) As mentioned just above, MFRs were initially assumed to increase the ‘sustainability’ of the stock. This was historically the first argument used to justified reserve implementation. For instance Polunin et al. (1983, p. 188) claim: ‘it has long been supposed that marine protected areas have beneficial consequences for the sustainable

9

exploitation of populations. It is increasingly clear that they could play a substantial role in the resuscitation of overexploited stocks, and they could also assist the maintenance of depletion-prone species’. However, as was also pointed out above, the models used to investigate this assumption were generally deterministic and did not consider the uncertainty dimension attached to fisheries. (2) With respect to catch, it was assumed that MFRs could possibly (a) compensate for the reduction of the fishing grounds (following the implementation of the reserve) and (b) reduce the variability of the catch. The mechanisms responsible for these effects are not straightforward and appear to be, in fact, partly artifacts of the types of model used to test these effects. For instance, while Hasting and Bostford’s analysis (1999) suggests that management of fisheries through reserves and through effort control can produce identical yields, in contrast Mangel (2000) concludes that catch is a declining function of the reserve size. In fact no clear-cut conclusion has been achieved so far on this question. As far as the potential effect of MFR on catch variability is concerned, the basic interrogation is whether the presence of protected areas can lead to a certain decrease in the variability of the catch. Very few studies have been conducted on this aspect. Mangel (2000), using a model of open-population under environmental uncertainty, investigated the mean-variance trade-off of the catch level as a function of the fraction of habitat protected (see his Fig. 5, p. 554). He concluded in page 555 that “The predictions are that implementing a reserve in this case will not increase catch, but will decrease variability in catch and consequently will be advantageous in avoiding boom-and-bust cycles”. Given the high dependence of the results on the type/nature of the models used to investigate this question, further research is still required to determine whether Mangel’s conclusion can be generalized. (3) Recently the list of potential advantages of MFRs has been extended under the increasing awareness of uncertainty in fisheries. In particular, the capacity of MFRs to play the role of buffer in this context of uncertainty and variability is now seriously envisaged. This third issue is in fact the extended version of step 1 above in the specific context of uncertainty. However, as Lauck et al. (1998) notice, this question is a very new topic that has been hardly tackled so far. In this general context, our main objective in this paper was to address, from a theoretical perspective, the third question above, i.e. the relationship between uncertainty and MFRs. The major findings of our study can be presented through a list of key-points. First, through Proposition A.6 applied in the specific case of certainty, it was demonstrated formally that the creation of MFRs promotes the sustainability of exploited stocks. The second numerical example in Section 4.1.1 with u
¼ 50% is one illustration of this result. With no MFR the viability of the stock is not ensured, while the implementation of a reserve covering at least 25% of the fishery size

10

L. Doyen, C. Be´ne´ / Journal of Environmental Management 69 (2003) 1–13

ensures that the stock remains always above the safe minimum biomass level. This assertion represents therefore an answer to the questions underlying step 1 above. Second, through Proposition A.1, we proved that the creation of MFR can still promote and ensure the sustainability of the stock exploited, even under uncertainty. In particular we showed that, whatever the level of uncertainty, the fishery is sustainable if the fraction of the fishing grounds protected is greater than the threshold 1 2 Asus ðsÞ: However, we also proved through Proposition A.2 that this sustainable reserve threshold depends on the level of uncertainty and in particular that the higher the uncertainty, the larger the protected area must be. This result, which seems to be intuitively straightforward, represents a major finding for the questions underpinning the step 3 identified above. Finally the last worthwhile result related to this question of uncertainty and marine reserve is the existence of ‘minimal’ uncertainty levels s p or s^ under which the implementation of a MFR is not justified from a ‘pure’ conservation point of view. In other words, it seems that it is possible to quantify the degree of uncertainty for which the creation of a reserve may become necessary. Interestingly we observe that this critical uncertainty value is function of the level of exploitation u
and the parameter c redlecting the endogenous dynamics of the resource. Third, with reference to the catch issue, we showed formally using a robust approach that, under both certainty and uncertainty, it is possible to identify an ‘optimal’ reserve size which would ensure a maximum guaranteed catch level. In particular, for the no-uncertainty situation presented in Fig. 1(a), although the implementation of a MFR revealed unjustified from a sustainability/conservation perspective, the analysis indicated that the creation of a MFR covering approximately 13% of the original fishing grounds would ensure a higher guaranteed catch level in comparison with a situation without MFR. Equally interestingly, in the case of uncertainty, we were able to demonstrate that under the conditions defined in Proposition A.6 the creation of a MFR can simultaneously ensure the sustainability of the exploited stock and increase the guaranteed catches of the fishing industry. This last result reconciles economic and environmental considerations within the specific context of radical uncertainty. To conclude, it is important to keep in mind that the validity of our results is related to the initial assumptions adopted in the model. In particular the choice of a Beverton– Holt type of recruitment strongly conditions the conclusions of this study. It would be relevant to pursue this work using other population dynamics models. In this respect, we may wonder whether the Ricker stockrecruitment model, for instance, would complexify the analysis or even alter some of the conclusions achieved here. Another interesting direction of research would be to combine into the model both an uncertainty affecting the catch with an uncertainty affecting the resource’s biological parameters.

Finally, it must be recalled that this research was based on a mono-species model with no spatial dimension intrinsically attached to it. As such it was not intended to address questions such as the role of marine protected area in the conservation of multi-species communities, or to tackle the crucial question of the relation between the capacity of protection of marine reserve and the mobility of the species to be protected. As clearly pointed out by Boersma and Parrish (1999), the degree of protection of an MPA is closely dependent on the species’ mobility through, e.g., dispersal pattern and site fidelity. However, although these biological and behavioral characteristics were not included in the present model, it is worth noting that the analysis gives some interesting insight into this spatial issue. The results show basically that, for one given species, holding the other parameters such as its dispersal pattern constant, it is possible to infer some conclusion regarding the relation between the level of uncertainty affecting the fishery, the size of the protected area and the potential effect of the protected area on the sustainability of the stock and associated catch volume. From a broader point of view, we believe that the mathematical framework of viability provides interesting analytical tools to cope with sustainability or conservation issues. In particular, the notion of invariance kernel may bring important insight whenever ‘persistent, irreducible’ or no probabilistic uncertainty appears to play a crucial role in the evolution of environmental systems. Appendix A We first enumerate the main results in the forms of proposition and corollary. In a second part, the associated proofs are detailed. A.1. The main results A.1.1. The invariance kernel We compute the invariance kernel as follows: Proposition A.1. Assume condition (4). Consider Asus defined by Eq. (6). The invariance kernel satisfies ( B; if Asus ðsÞ , A; InvðA; sÞ ¼ ½Nmin ; þ1½; if Asus ðsÞ $ A: We derive directly from the very definition of Asus in Eq. (6) the monotonicity result: Proposition A.2. Assume Eq. (4). The invariance kernel InvðA; sÞ decreases with respect to the level of uncertainty s:

s1 # s2 ) InvðA; s2 Þ , InvðA; s1 Þ: Writing the condition Asus $ 1; we deduce the following corollary indicating how the fishery is sustainable without reserve despite the uncertainty.

L. Doyen, C. Be´ne´ / Journal of Environmental Management 69 (2003) 1–13

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Corollary A.3. Assume that cð1 2 u
2 sÞ . 1: If Nmin # ðc=dÞ 2 1=ðdð1 2 u
2 sÞÞ; then sustainability always applies, namely for any A [ ½0; 1; any s $ 0; we have

Lemma A.7. Consider N . 0: The following assertions are equivalent:

InvðA; sÞ ¼ ½Nmin ; þ1½:

(a) minu[U f ðNð1 2 AuÞÞ $ N; (b) g2 ðNÞ $ 0; (c) N # Fðuþ ; AÞ

A.1.2. Guaranteed catches. We compute CðA; N0 Þ the lowest possible catch over time termed guaranteed catch. Proposition A.4. Assume Eq. (4) and consider 0 , A # 1. Consider any initial condition N0 [ S: Then the guaranteed catch function satisfies ( if N0 # FðA; u
þ sÞ; N0 ð
u 2 sÞA; CðA; N0 Þ ¼ FðA; u
þ sÞð
u 2 sÞA; if N0 $ FðA; u
þ sÞ: We then compute the maximal value Ap ðsÞ of the worst catch function CðA; N0 Þ: A first order condition provides the optimal reserve size as follows: pffiffi Proposition A.5. Assume Eq. (4) and cð1 2 u
2 sÞ # 1: Consider N0 $ FðA; u
þ sÞ: Then the maximal value of the guaranteed catches is obtained for the harvesting area: pffiffi c21 pffiffi : Ap ðsÞ ¼ ðA:1Þ ð
u þ sÞ c Consistency between conservation and optimality issues occurs under the following conditions on parameters: pffiffi Proposition A.6. Assume Eq. (4) and cð1 2 u
2 sÞ # 1: Consider N0 $ FðA; u
þ sÞ: If the safe biomass level satisfies pffiffi c2 c Nmin # ; d the maximal guaranteed harvesting catches associated to Ap ðsÞ is sustainable, namely Ap ðsÞ # Asus ðsÞ:

Lemma A.8. The set ½FðA; uþ Þ; þ1½ is invariant or equivalently, for any N $ FðA; uþ Þ and any u [ U; we have f ðNð1 2 AuÞÞ $ FðA; uþ Þ: Proof of Lemma A.7. By the very definition of the function g2 ; assertions (a) and (b) are equivalent. It is straightforward to prove that (a) , (c) by replacing f by its value. Indeed f ðNð1 2 Auþ ÞÞ $ N ,

, cð1 2 Auþ Þ 2 1 $ Ndð1 2 Auþ Þ , Fðuþ ; AÞ $ N: A Proof of Lemma A.8. Assume that N $ Fðuþ ; AÞ: Since f is increasing, we claim that, for any u [ U; f ðNð1 2 AuÞÞ $ f ðFðuþ ; AÞð1 2 AuÞÞ: By virtue of Lemma A.7, we have also f ðFðuþ ; AÞð1 2 AuÞÞ $ Fðuþ ; AÞ: Therefore, for any admissible harvesting u [ U; we can write f ðNð1 2 AuÞÞ $ Fðuþ ; AÞ and we conclude. A

A.2.3. Proof of Proposition A.1 Assume that A , Asus ðsÞ: The function Fð·; ·Þ defined by Eq. (5) is clearly decreasing with respect to A: Consequently FðA; uþ Þ , FðAsus ðsÞ; uþ Þ ¼ Nmin : By Lemma A.7, we deduce that g2 ðNÞ , 0

A.2. The proofs A.2.1. Notations. For sake of clarity, we introduce the notations † The set of admissible or viable states S ¼ ½Nmin ; þ1½: † The set of harvesting rates U ¼ ½u2 ; uþ  with uþ ¼ u
þ s and u2 ¼ u
2 s: † The resource growth function g2 ðNÞ ¼ min ðf ðNð1 2 AuÞÞ 2 NÞ ¼ f ðNð1 2 Auþ ÞÞ 2 N: u[U

† The maximum resource growth value 2

M ¼ max g ðNÞ: N$Nmin

A.2.2. Two useful lemmas. For sake of clarity, we also introduce the two following lemmas:

cNð1 2 Auþ Þ $N 1 þ dNð1 2 Auþ Þ

;N [ S:

This implies that M the maximal value of g2 on S is negative, i.e. M , 0: Now assume that the invariance kernel is non-empty and consider N0 ¼ Nð0Þ [ InvðA; sÞ: Consider the harvesting uðtÞ equal to uþ at all time and NðtÞ the associated solution. Since N0 belongs to the invariance kernel InvðA; sÞ; it means that NðtÞ remains in the set S. Consequently, g2 ðNðtÞÞ # M , 0 and we obtain Nðt þ 1Þ ¼ f ðNðtÞð1 2 AuðtÞÞ # NðtÞ þ g2 ðNðtÞÞ # NðtÞ þ M: Therefore, we deduce recursively that NðtÞ # N0 þ Mt: Since M is negative, there exists a time T [ N such that 0 , NðTÞ , Nmin : Therefore N0  InvðA; sÞ and we derive a contradiction. We conclude that when A , Asus ðsÞ; the invariant kernel InvðA; sÞ is empty.

L. Doyen, C. Be´ne´ / Journal of Environmental Management 69 (2003) 1–13

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Now let us assume that A $ Asus ðsÞ: Since the function Fð·; ·Þ is decreasing with respect to A, we deduce that FðA; uþ Þ $ FðAsus ðsÞ; uþ Þ ¼ Nmin :

ðA:2Þ

We need to prove that the set S is invariant or equivalently that, for every N [ S and every u [ U; we have f ðNð1 2 AuÞÞ [ S: To achieve this, we distinguish two cases:

But Nn ðnÞ ¼ N þ ðnÞ where N þ ð·Þ is the sequential solution associated with fixed uþ : Since, by standard arguments, this sequence N þ ð·Þ converges to the fixed point N ¼ FðA; uþ Þ; we use the limit in Eq. (A.3) to deduce CðA; N0 Þ # FðA; uþ ÞAu2 ; and to conclude.

A.2.5. Proof of Proposition A.5 † First consider the case, consistent with condition (A.2), where Nmin # N # Fðuþ ; AÞ: Then by Lemma A.7, we have inf u[U f ðNð1 2 AuÞÞ $ N $ Nmin : † Now assume that N $ Fðuþ ; AÞ: By virtue of Lemma A.8, we have also inf u[U f ðNð1 2 AuÞÞ $ Fðuþ ; AÞ $ Nmin : Consequently the set S is invariant. A recursive argument below provides the desired result NðtÞ $ Nmin for any solution Nð·Þ of Eq. (2) starting from Nð0Þ $ Nmin : Indeed, † The desired condition clearly holds for t ¼ 0: † Now assume NðtÞ $ Nmin holds true for some t. The invariance of S implies that, for any uðtÞ [ U; f ðNðtÞð1 2 AuðtÞÞÞ $ Nmin : This yields Nðt þ 1Þ $ Nmin as expected. Therefore, the invariance kernel InvðA; sÞ is the whole interval S ¼ ½Nmin ; 1½:

We study the catch function CðAÞ ¼ FðA; uþ ÞAu2 : We compute the derivative C 0 ðAÞ ¼

›C u2 cð1 2 Auþ Þ2 2 1 ðAÞ ¼ : ›A d ð1 2 Auþ Þ2

The solutions of the equation associated with pffiffi pffiffi first order 0 p optimalitypcondition C ðAÞ ¼ 0 are A ð s Þ ¼ ð c 2 1Þ=uþ c ffi ffi p ffi ffi and A~ ¼ ð c þ 1Þ=uþ c: We clearly have A~ $ Ap ðpsffiffiÞ and A~ $ 1: Since c . 1; we claim Ap ðsÞ . 0: Since cð1 2 u
2 sÞ # 1; we then have ~ 0 # Ap ðsÞ # 1 # A: Therefore, we obtain the behaviour of the function C with respect to rate A A

···

Ap ð sÞ

C 0 ðAÞ CðAÞ

þ b

0

2 d

A~

···

0

þ b

A.2.4. Proof of Proposition A.4

We deduce that Ap ðsÞ is the maximal argument of the catch function CðAÞ within the interval [0,1] namely CðAp ðsÞÞ ¼ maxA[½0;1 CðAÞ:

First assume N0 # FðA; uþ Þ: A recursive argument derived from Lemma A.7 similar to the one above ensures that NðtÞ $ N0 : Thus for any admissible uðtÞ and any time t; we can write

A.2.6. Proof of Proposition A.6

NðtÞAuðtÞ $ NðtÞAu2 $ N0 Au2 : Thus, taking the infimum, we obtain N0 Au2 # minðNðtÞAuðtÞÞ ¼ CðA; N0 Þ: Since N0 Au2 corresponds to some Nð0ÞAuð0Þ; we have CðA; N0 Þ # N0 Au2 and we conclude.We proceed similarly in the case N0 $ FðA; uþ Þ: Using again Lemma A.8, we prove recursively that NðtÞ $ FðA; uþ Þ: Consequently, NðtÞAuðtÞ $ NðtÞAu2 $ FðA; uþ ÞAu2 : Therefore, FðA; uþ ÞAu2 # CðA; N0 Þ: On the other hand, for any n, consider the solution Nn ðtÞ for ( 2 u ; if t ¼ n un ðtÞ ¼ uþ ; if t – n: We have CðA; N0 Þ # Nn ðnÞAun ðnÞ ¼ Nn ðnÞAu2 :

ðA:3Þ

We use the definitions of Ap and Asus to write pffiffi
 c21 1 1 p A ðsÞ 2 Asus ðsÞ ¼ þ pffiffi 2 þ 1 þ u c u dNmin 2 c
pffiffi
 c21 1 1 pffiffi 2 1 þ ¼ þ : c u dNmin 2 c pffiffi Since, by assumption, Nmin # ðc 2 cÞ=d; we deduce that
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