Yield–mortality models: a precautionary bioeconomic approach

Fisheries Research 40 (1999) 7±16 Yield±mortality models: a precautionary bioeconomic approach Omar Defeoa,b,1,*, Juan Carlos Seijoa,c,2 a Cinvestav-IPN Unidad MeÂrida, AP 73 Cordemex, 97310 MeÂrida, YucataÂn, Mexico b Instituto Nacional de Pesca, Constituyente 1497, 11200 Montevideo, Uruguay c Centro Marista de Estudios Superiories, Km 7 Carretera a Progreso por Avda. M. Champagnat, 97110 MeÂrida, YucataÂn, Mexico Received 18 November 1997; accepted 4 October 1998 Abstract This paper develops a bioeconomic approach to yield±mortality models. The usefulness of the conceptual background of these models, and the application of the resulting reference points (RPs), are analysed in a precautionary management framework. Quanti®cation of uncertainty by bootstrapping provided a more realistic comparison of the relative performance of RPs for management advice. Sensitivity analysis, performed under wide initial variations in the natural mortality coef®cient M, displayed two important features of this bioeconomic model: (1) the yield at maximum economic yield (YMEY) and the associated mortality levels ZMEY and FMEY were always the most conservative RPs. This takes utmost importance at low M values, where maximum sustainable yield (MSY) and the yield at maximum biological production (YMBP) tended to coincide, whereas MEY and mortality-related RPs were substantially lower, thus constituting useful target RPs for precautionary management, (2) The RPs derived from the biological production curve were always lower than those derived from the sustainable yield curve. The former could be used as target RPs, whereas both MSY and FMSY should be considered as upper exploitation limits. Sensitivity analysis conducted with the economic input of the system, the unit cost of effort (c), revealed consistency of the bioeconomic performance of the model with accepted theory, i.e., YMEY approaches MSY with decreasing c values. A simple approach to the formulation of risk-averse management strategies was explored using decision theory. For this purpose, the maximax, maximin and minimax regret criterion were used to evaluate alternative management decisions under uncertainty, without the need for explicit statements of probabilities on alternative hypotheses. Results con®rmed the need to consider economic and MBP related RPs for precautionary management. Guidelines for future work are suggested. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Yield±mortality model; Bioeconomic model; Uncertainty; Risk; Precautionary ®shery management; Reference points 1. Introduction The recent collapse of major exploited ®sh stocks emphasises the urgent need to develop management *Corresponding author. Tel.: +52-99-812903; fax: +5299812917; e-mail: [email protected] 1 E-mail: [email protected] 2 E-mail: [email protected] strategies that minimise the risk of falling below an undesirable threshold (FAO, 1993; Mace, 1994). For this purpose, reference points (RPs) are employed under the precautionary approach concept to set limits for protecting stocks against overexploitation and collapse (Caddy and Mahon, 1995). As the precautionary approach is composed not only of RPs but also of the risk of exceeding these, it is essential to provide estimates of risk and uncertainty, something that has 0165-7836/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S0165-7836(98)00220-3 8 O. Defeo, J.C. Seijo / Fisheries Research 40 (1999) 7±16 not been routinely done in many management systems (Seijo et al., 1997). RPs such as maximum sustainable yield (MSY) and ®shing mortality at MSY (FMSY), mainly derived from conventional catch±effort surplus production models, have been intensively studied and applied for management advice since the classic paper of Schaefer (1954). However, the use of catch±effort surplus production models, either dynamic or static (see Punt and Hilborn, 1996), is problematic (Caddy, 1996; Caddy and Defeo, 1996): the input variable, ®shing effort, is strongly dependent on the catchability coef®cient q, which in turn is extremely sensitive to environmental and technological variables (Caddy, 1979). Indeed, the impressive increase in ®shing power of industrial vessels that occurred over the last two decades determined progressive and yet unmeasured changes in q, which is also subjected to variations in ®shing intensity and stock biomass. These interactions have resulted in a poor ability to calibrate ®shing effort, and hence, in an important uncertainty component when estimating this variable as the input of catch± effort models. Another problem with these models is their inability to capture the dynamics of the resource, due to the lack of age structure. Moreover, there is a time lag between the ®shing effort exerted and the corresponding biomass ¯uctuations resulting from variations in the input variable. Yield±mortality (Y±Z) models (Caddy and Csirke, 1983; Csirke and Caddy, 1983) constitute a valid and perhaps better alternative to the above input±output models. They link two main outputs of the ®shery system: yield Y (dependent variable) and the instantaneous total mortality coef®cient Z, and as such have been referred to as ``output±output models'' or ``control curves'' (sensu Caddy, 1996). Fitting Y against Z generates a biological production curve, which includes natural deaths plus harvested yield for the population as a whole. Caddy and Defeo (1996) (see also Caddy and Mahon, 1995; Caddy, 1996; Die and Caddy, 1997) suggested that none of the above criticisms to catch±effort models applied to the same extent to Y±Z models for several reasons: 1. Y can be measured with a relatively narrow margin of error and Z can be easily estimated by different approaches, including the classical age (Sparre and Venema, 1992) and length-converted (Pauly et al., 1995) catch curves. The possibility of easily acquiring unbiased estimation of mortality rates from length-converted catch curves makes the present type of model highly relevant. Z values represent the impact of ®shing on all harvested year classes and thus implicitly contain information about the age structure of the population. 2. As both Y and Z constitute outputs of the biological and economic fishery subsystems (Caddy, 1996), sources of uncertainty are likely to be relatively well known and quantified, or at least easier to estimate than in catch±effort methods. 3. Y±Z models provide alternative benchmarks to MSY, based on the maximum biological production (MBP) concept (Caddy and Csirke, 1983), such as the yield at maximum biological production (YMBP) and the corresponding mortality rates at which the total biological production of the system is maximised (ZMBP and FMBP). These RPs have been suggested as more conservative than their ``maximum sustainable'' counterparts, thus constituting precautionary RPs for management advice (Die and Caddy, 1997). Concerning this, Caddy and Defeo (1996) demonstrated that ZMBP, composed by yield and predation, tends to fall in the low percentiles of the ZMSY cumulative distribution. 4. Unless considerable annual changes in fishing effort occur, the successive annual values in a yield±mortality plot tend to show a degree of serial autocorrelation with trend. Moreover, they do not show sharp jumps from left to right-hand sides of the yield curve, characteristic of many catch±effort production models with wide departures from equilibrium. The economic literature which refers to surplus production models is based on the classic Gordon model (Gordon, 1954), which has been derived from the catch±effort Schaefer model, and thus the criticisms mentioned above could be equally applied to the economic model. However, it is recognised that the bioeconomic RPs, MEY and FMEY, occur at lower levels than MSY and FMSY, which enable their use as precautionary RPs (see Caddy and Mahon, 1995). Thus, it seems justi®ed to develop bioeconomic Y±Z models in order to evaluate the feasibility of using the resulting RPs in a precautionary management context. 9 O. Defeo, J.C. Seijo / Fisheries Research 40 (1999) 7±16 In this paper we develop a bioeconomic model to extend the theory of production modelling with mortality rates, based on the exponential Y±Z model described by Caddy and Defeo (1996). Uncertainty in RPs, estimated by ``bootstrapping'', and risk analysis, using decision theory without mathematical probabilities, are also included for choosing between alternative management decisions. A comparison of model performance under different values of input parameters is also performed. The model is offered as a precautionary, bioeconomic approach for ®sheries management. 2. Material and methods 2.1. Theory The relative performance of the logistic (Schaefer, 1954) and exponential (Fox, 1970) surplus production models shows that they can provide substantially different predictions while using the same data. It has been recognised that, after some equilibrium approximation, the plot of catch rate against effort tends to fall off at a progressively diminishing rate, thus making application of exponential yield models more attractive than the logistic ones. Concerning Y±Z models, Caddy and Defeo (1996) concluded that the use of the quadratic form of the logistic should be avoided because of theoretical and statistical considerations. Instead, they suggested using the exponential version, once an estimate of M has been obtained, either independently, or from the logistic ®t. The exponential type of model for catch±effort data as described by Fox (1970) may be summarised mathematically by3: model, by transforming effort into analytical mortality rates as follows: Yi ˆ Zi ÿ M†B1 exp ÿb0 Zi ÿ M††; where M is the instantaneous natural mortality coef®cient, B1 is the virgin population or carrying capacity of the system, b0 ˆb/q, q being the catchability coef®cient, and the remaining parameters as de®ned above. Then Yi ˆ B1 exp ÿb0 Zi ÿ M††; Zi ÿ M 3 Theory and approaches to fitting nonlinear and linearised Y±Z models have been fully described elsewhere (Caddy and Csirke, 1983; Csirke and Caddy, 1983; Caddy and Defeo, 1996) and thus will not be described in detail here. (2) where B1 and b0 can be estimated by non-linear regression techniques. The model is ®tted for different trial values of M, selecting those that maximise a goodness of ®t criterion (Caddy, 1986; see below). Alternatively, an initial M estimate given by ®tting the logistic Y±Z model could be used. Once a ``best'' M value has been found, the remaining management parameters can be computed by differentiating Eq. (1) with respect to the instantaneous rate of ®shing mortality (F) and thus setting it equal to zero at MSY (Caddy and Defeo, 1996): dYi ˆ ÿb0 Fi B1 exp ÿb0 FMSY † ‡ B1 exp ÿb0 FMSY † dF ˆ 0: (3) So FMSY ˆ ÿ 1 : b0 (4) The above authors demonstrated that MSY is estimated as follows: MSY ˆ FMSY B1 exp b0 FMSY †: (5) Substituting Eq. (4) in Eq. (5) it is noticeable that Yi ˆ fi U1 exp ÿbfi †; where yield (Y) and ®shing effort (f) in year i are exponentially related, b a parameter and U1 is the catch rate corresponding to the virgin stock. Caddy and Defeo (1996) extended the theory of production modelling with mortality rates to include the Fox (1) MSY ˆ FMSY B1 eÿ1 (6) MSY ˆ 0:37FMSY B1 : (7) and thus, The linearised version of Eq. (2) is given by   Yi ˆ ‰ln B1 ‡ b0 MŠ ÿ b0 Zi ; ln Zi ÿ M (8) which can be ®tted by linear regression, with ln(Yi/ ZiÿM) and Zi as the dependent and independent variables, respectively. This model is also ®tted for 10 O. Defeo, J.C. Seijo / Fisheries Research 40 (1999) 7±16 different trial values of M, selecting those that maximise a goodness of ®t criterion. The yield at MBP (YMBP) and the corresponding mortality rates FMBP and ZMBP are estimated as described in Caddy and Csirke (1983) and PeÂrez and Defeo (1996). To obtain bioeconomic RPs, we developed an equation for estimating the economic rent () of a stock from the exponential version of the Y±Z model in its linearised form:  ˆ pFB1 exp ÿb0 F† ÿ cF ; q (9) where p is the average price per unit yield of the target species, c the unit cost of ®shing effort and the remaining parameters as de®ned above. Differentiating Eq. (9), an expression that yields the marginal rent (m) with changes in the instantaneous rate of ®shing mortality (F) is obtained:   d F 0 pFB1 exp b F† ÿ c m F† ˆ dF q and pB1 exp ÿb0 F† ÿ pFB1 b0 exp ÿb0 F† ÿ c=q ˆ 0; FMEY ˆ ‰ÿW e=pqB1 c† ‡ 1Š ; b0 (10) where W is a function W[a] with the property aˆW exp(ÿW), de®ning a as e c: aˆ pB1 q† Many statistical packages have the capability to assist in solution for this equation. In this case, we used MathCad 54. Once the parameters a and b0 of Eq. (8) are obtained, and given a known M value, B1 is estimated as (Caddy and Defeo, 1996): B1 ˆ exp a ÿ b0 M†: Given p, q and c constants, and knowing B1, FMEY is calculated as follows: FMEY ˆ ÿW ‡ 1 : b0 (11) Yield at MEY (YMEY) was obtained by YMEY ˆ FMEY B1 exp b0 FMEY †: 4 Mathcad 5.0 for Windows. 1994. Mathsoft, Inc. (12) The bioeconomic model developed here assumes ``pseudo-equilibrium conditions'' (sensu Caddy, 1996, p. 219). However, Z values derived from a multi-age group represent more closely both past and present impacts of ®shing on all harvested year classes than do annual values of ®shing effort, thus providing robustness with respect to departures from equilibrium. For this reason, at least as a ®rst approach, it was considered reasonable to ®t the model without further equilibrium adjustment (see Caddy (1986) and Caddy and Defeo (1996) for details on the subject). 2.2. Data application and main inputs The example to be given below is based on a hypothetical data set used by Caddy (1986), p. 387, which seems adapted to the methodology proposed (Table 1). We emphasise that a real data set of bioeconomic information which would allow the calculations reported here is not available to us, so the following results (and the estimates of mortality used) are only intended to illustrate the bioeconomic model developed and the ®tting procedure. Eq. (8) was used to ®t the Y±Z model. FMEY was obtained after numerically estimating a value for W from Eq. (11). Eq. (12) was used to estimate YMEY. It should be emphasised that a non-linear ®tting of the exponential model could alternatively be used for this purpose (Eqs. (1) and (2): see Caddy and Defeo, 1996). Table 1 Hypothetical data used in the present paper to fit the bioeconomic Y±Z model (adapted from Caddy, 1986) Year Yield (t) Z (1/yr) 1 2 3 4 5 6 7 8 9 10 11 12 7.5 12.5 19.0 35.0 40.5 39.5 30.5 20.0 26.0 29.5 27.5 29.0 0.175 0.170 0.250 0.440 0.610 0.795 1.080 1.170 0.900 0.790 0.710 0.470 11 O. Defeo, J.C. Seijo / Fisheries Research 40 (1999) 7±16 Input data to run the model were pˆ$3000; qˆ0.0001 and cˆ$25. The M value used as input for the model was found by iterating Eq. (8) and maximising the goodness-of-®t-criteria; the highest R2 corresponded to Mˆ0.13/yr. Table 2 Schematic representation of a decision table without mathematical probabilities 2.3. Uncertainty estimates and risk analysis D1 D2 D3 The bootstrap method (Efron, 1982; Manly, 1991) was employed to estimate con®dence limits for the model parameters by randomly resampling with replacement the original set of data pairs. A total of 300 bootstrap simulations were performed to obtain 300 estimates of regression parameters, which in turn allowed an estimate of the mean, coef®cient of variation (CV) and 95% con®dence intervals for the RPs associated with sustainable yield and biological production estimators. CV values were used to provide standardised means of comparing uncertainty in RPs. Sensitivity analysis was performed by introducing uncertainty in the unit cost of effort (c) and natural mortality (M). Two scenarios of c ($25 and $15 per unit of effort) and four of M (0.05, 0.09, 0.13 and 0.15/ yr) were used to quantify the resulting rates of change in the RPs estimated by bootstrapping. Risk was estimated with the aid of decision analysis without mathematical probabilities, in order to represent different degrees of management caution (Francis, 1992; Cordue and Francis, 1994). The maximin, minimax and maximax criteria (Schmid, 1989; FAO, 1995) were used for this purpose. Maximin is a risk-averse approach that consists in selecting the management decision that involves the maximum value of the observed minimum outcome. The minimax regret criterion is a less cautious approach that selects the management action that minimises the maximum regret, de®ned as the difference between the real bene®t and the one that could have been obtained if the correct decision had been taken. Finally, an optimist and risk prone policy maker could use the maximax approach, by selecting the management option with the higher value of a given RP resulting from the comparison of alternative management schemes (PeÂrez and Defeo, 1996; Seijo et al., 1997). The key elements of the decision table are schematically shown in Table 2, and explained below: Decision Alternative states of nature S1 S2 Y11 Y12 Y13 Y21 Y22 Y13 Criterion value C1 C2 C3 D1±D3 represent alternative management decisions; S1 and S2 represent alternative hypotheses about a parameter of the stock or other state of nature; Yij represents the value of the outcome of a fishery performance variable resulting from a decision Dj as applied to a given state of nature Si; and Cj is the value of each action across all alternative hypotheses estimated by the maximin, minimax and maximax criteria. 1. Columns represent alternative hypotheses about a parameter of the stock or other state of nature. In our example we used three scenarios of M (0.09, 0.13 and 0.15/yr) that indirectly re¯ect alternative hypotheses about resource productivity. 2. Rows represent three alternative management decisions (D1±D3). Here we used the total mortality rate Z expected for a given management action as the control variable providing feedback on the impact of fishing. Each management decision was selected to reflect different degrees of risk aversion. Thus, we used ZMEY, ZMBP and ZMSY to represent, respectively, risk averse, risk neutral and risk prone attitudes of the policy-maker. 3. Each cell Yij within the table represents the value of the outcome of a management decision Dj as applied to a given value of M. In our example, the performance fishery variable is the expected biomass at the three Z levels mentioned in (b). Simple biomass estimates were obtained by dividing the estimated yield by the corresponding F level expected at each management action. 4. Finally, Cj is the criterion value of each action across all alternative hypotheses estimated by maximin, minimax and maximax criteria. 3. Results Fig. 1(a) shows the relationship between Y and Z for the hypothetical data set ®tted by the linearised 12 O. Defeo, J.C. Seijo / Fisheries Research 40 (1999) 7±16 Fig. 1. Bioeconomic Y±Z model: yield and biological production curves fitted to a hypothetical yield±mortality data set under two different natural mortality scenarios: (a) Mˆ0.13/yr, which maximised the goodness-of-fit criterion in Eq. (8); and (b) Mˆ0.05/yr. The positions of MSY, YMEY, MBP and YMBP and the corresponding mortality rates are shown. exponential model (Mˆ0.13/yr), and Table 3 shows the mean estimates of the RPs and the respective 95% con®dence interval obtained by bootstrap simulations. Bioeconomic RPs fell below the other two, in the following order: YMEY<YMBP<MSY, and the same remained true for mortality-based RPs. The coef®cient of variation for bioeconomic RPs based on mortality indicators was also considerably smaller than in the other RPs (Table 3). It is noteworthy that YMPB, as well as the corresponding mortality rates at MBP (FMBP and ZMBP), were consistently lower than those corresponding to MSY (Fig. 1(a)). Sensitivity analysis involving percent changes of ÿ62%, ÿ31% and ‡15% in the natural mortality coef®cient (Mˆ0.05, 0.09, and 0.15/yr, respectively), also determined drastic rates of change in the RPs estimated by bootstrapping (Table 4). However, in all cases, changes in M did not affect the consistent decreasing trend from MSY to YMEY mentioned above, i.e., for a given M value, bioeconomic RPs were always lower than MSY-based ones, with those derived from MBP at intermediate levels (Fig. 2). The same happened for mortality-based RPs. This takes utmost importance at low M values, where MSY and YMBP were very similar (Fig. 1(b)), whereas MEY and mortality-related RPs were substantially lower (Fig. 2). Rates of change were, however, highest for YMEY and FMBP, suggesting higher sensibility to marginal changes in natural mortality (Table 4). Simulations involving changes in the unit cost of ®shing effort (c) resulted, as expected, in variations in the bioeconomic RPs derived from the Y±Z model (Table 3). A 40% reduction in c (from $25 to $15 per unit of effort) resulted in a concomitant increase in the mean bootstrap estimates of bioeconomic RPs in the order of 14% for YMEY, 38% for FMEY, and 22% for ZMEY. Minimum differences in MSY and mortality-related RPs obeyed to the random nature of the resampling technique and could not be caused by a potential effect of variability in c. Fig. 3 shows the empirical distributions of YMEY and MSY obtained by bootstrapping under the two selected input values of c. In both cases, YMEY fell below MSY, but got closer to each other under a lower cost scenario; the same trend was true for the remaining bioeconomic RPs when compared with the biological ones (Table 3). Coef®cients of variation and con®dence intervals of RPs tended to be lower with the higher c scenario. Decision analysis without mathematical probabilities, systematically identi®ed ZMEY as the most cautious approach to management. This occurred with all the three criteria used for guiding management decisions (minimax, maximin and maximax: Table 5). This means that the outcomes given by biomass estimates were best when the management decision was based on ZMEY as the control ®shery variable. The second best alternative was de®ned by ZMBP and thus constitutes a safer management target when compared with ZMSY. This is especially important when economic information is lacking or highly variable. 13 O. Defeo, J.C. Seijo / Fisheries Research 40 (1999) 7±16 Table 3 Mean, coefficient of variation (CV) and 95% confidence intervals (CI: percentile approach) of the RPs derived from the bioeconomic Y±Z model, estimated by the bootstrap technique for two values of the unit cost of fishing effort (c) Parameter B1 MBP YMEY YMBP MSY FMEY FMBP FMSY ZMEY ZMBP ZMSY cˆ$15 cˆ$25 Mean CV (%) 2.5 CI 97.5 CI Mean CV (%) 2.5 CI 97.5 CI 225 48 32 35 36 0.258 0.375 0.440 0.388 0.505 0.570 14.01 7.93 8.61 6.13 6.25 4.55 11.80 10.05 3.03 8.76 7.76 160 40 26 31 31 0.234 0.283 0.348 0.364 0.413 0.478 291 56 38 40 41 0.283 0.466 0.531 0.413 0.596 0.661 228 49 28 35 36 0.187 0.352 0.435 0.317 0.482 0.565 13.18 7.67 11.50 5.83 6.02 4.57 11.20 9.08 2.69 8.18 6.99 185 42 22 32 32 0.168 0.281 0.363 0.298 0.493 0.493 296 57 34 40 41 0.203 0.429 0.511 0.333 0.559 0.641 B1, MBP, MSY, YMBP and YMEY are given in tonnes, while mortality parameters are given on an annual basis. 4. Discussion This paper shows, for the ®rst time, a bioeconomic approach for yield±mortality models. Means and con®dence intervals of bioeconomic RPs fell in the lower bound of those corresponding to the biological model, suggesting that they constitute cautious RPs to management. Moreover, the RPs derived from the biological production curve, YMBP, ZMBP and FMBP, resulted safer targets than the corresponding mortality rates at Table 4 Mean estimates and relative changes (between parentheses) in RPs for three M values (0.05, 0.09 and 0.15), representing, respectively, changes in ÿ62%, ÿ31% and ‡15% with respect to Mˆ0.13/yr, considered as the most likely value Parameter B1 MBP YMEY YMBP MSY FMEY FMBP FMSY ZMEY ZMBP ZMSY Natural mortality (M) 0.05 0.09 0.15 121 (ÿ47) 32 (ÿ34) 12 (ÿ55) 30 (ÿ16) 31 (ÿ13) 0.123 (ÿ34) 0.661 (‡88) 0.692 (‡59) 0.173 (ÿ45) 0.711 (‡47) 0.742 (‡32) 156 (ÿ32) 38 (ÿ22) 19 (ÿ32) 32 (ÿ9) 32 (ÿ11) 0.161 (ÿ14) 0.502 (‡43) 0.559 (‡29) 0.263 (ÿ17) 0.592 (‡23) 0.649 (‡15) 339 (‡49) 67 (‡38) 39 (‡41) 43 (‡20) 45 (‡23) 0.201 (‡7) 0.265 (ÿ25) 0. 360 (ÿ17) 0.351 (‡11) 0.415 (ÿ14) 0.510 (ÿ10) B1, MBP, MSY, YMBP and YMEY are given in tonnes, while mortality parameters are given on an annual basis. MSY. These results are consistent with previous work of Caddy and Csirke (1983), Caddy and Mahon (1995), Caddy and Defeo (1996) and Die and Caddy (1997), who suggested MSY and mortality-related parameters as limit RPs, and those related to MEY and MBP as safer RPs than MSY. Table 5 Results of the decision table without mathematical probabilities Management decision Alternative states of nature Mˆ0.09 Mˆ0.13 Criterion value Mˆ0.15 Minimum value 146 60 57 Maximin ZMEY ZMBP ZMSY 146 60 57 148 102 84 178 169 124 Minimax ZMEY ZMBP ZMSY 0 86 89 0 46 64 0 9 54 Maximum regret 0 86 89 Maximax ZMEY ZMBP ZMSY 146 60 57 148 102 84 178 169 124 Maximum value 178 169 184 Biomass estimates (fishery performance variable) under three management decisions given by annual mortality levels (control variable) defined by ZMEY, ZMBP and ZMSY. Three states of nature were defined by natural mortality (M) values. In each case, the selected management decisions are highlighted. 14 O. Defeo, J.C. Seijo / Fisheries Research 40 (1999) 7±16 Fig. 3. Effect of two different input values of c in 300 bootstrap estimates of YMEY and MSY derived from the bioeconomic Y±Z model: both RPs are presented against B1 (see also Table 3). Fig. 2. Effect of different assumed values of natural mortality M, on the estimated distribution functions for YMEY, YMBP and MSY obtained by bootstrapping. Mˆ0.13/yr is considered the most likely value. Sensitivity analysis performed under wide initial variations in the natural mortality coef®cient M, lead to two main conclusions. First, YMEY and the associated mortality levels ZMEY and FMEY were always the most conservative RPs, and thus constitute useful targets for precautionary management. This takes utmost importance at low M values, where MSY and YMBP tended to coincide. Second, the RPs derived from the biological production curve were always lower than ``sustainable yield'' estimators. The former could be used as target RPs, whereas both MSY and FMSY should be considered as upper exploitation limits. This reassures the notion that RPs based on the concept of maximum biological production have useful properties for developing ®sheries and also for managing already developed ®sheries (Die and Caddy, 1997). We showed in this paper that the above statement also applies to bioeconomic RPs estimated from Y±Z models. O. Defeo, J.C. Seijo / Fisheries Research 40 (1999) 7±16 The inclusion of economic data (cost of ®shing) allowed us to express the uncertainty and to select management strategies on both biological and economic grounds. Sensitivity analysis of the model to a single variation in unit costs showed that bioeconomic RPs increased with decreasing costs and got closer to the ``maximum sustainable'' RPs. This could be important for managing artisanal ®sheries with relatively low total costs and high unit value of harvested stocks, such as shell®shes. In these ®sheries, the bioeconomic equilibrium is reached at high levels of ®shing effort (Seijo and Defeo, 1994) and the corresponding FMEY approaches FMSY. Therefore, at very low levels of unit cost per effort, FMBP could be a more conservative RP than FMEY. The biomass-rent trade-off can be estimated to re¯ect the societal cost of adopting a highly risk-averse management option that departs from the rent-maximising paradigm. The simple approach to the formulation of riskaverse management strategies, explored using decision theory (Schmid, 1989) in conjunction with the bioeconomic Y±Z model, con®rmed the results reported above. The application of the concepts maximax, maximin and maximum regret criterion indicated that ZMEY should be the preferred option for management advice. In consistency with production modelling theory, mortality rates at MBP should also be selected as precautionary RPs. This methodology might be considered as a powerful tool for choice under uncertainty in a precautionary ®sheries management framework (FAO, 1995; PeÂrez and Defeo, 1996). The selection of the ®shery performance variable to conduct decision analysis is not trivial in ®sheries, because of the existence of common mutually-exclusive, multiple and often con¯icting objective criteria (Seijo et al., 1994; Francis and Shotton, 1997). For example, catch was also used as performance variable in decision analysis. As expected, it proved ineffective in providing a cautious approach to management, i.e., minimax, maximin and maximax unambiguously selected MSY and ZMSY as the appropriate management decisions. Thus, catch maximisation should not be used as a performance variable in decision analysis, under a precautionary management framework. The above trends, when compared with our results shown in Table 5, constitute a clear example of two clearly con¯icting management objectives: to maximise catch and to minimise the probability of low biomass. As 15 mentioned by Francis and Shotton (1997), the task is to decide the appropriate trade-off between objectives. To this end, a multiple criterion optimisation approach could be developed for one or more sets of policy goals and management targets, in order to re¯ect the willingness of the decision maker to allow for tradeoffs among performance variables (DõÂaz de LeoÂn and Seijo, 1992; Seijo et al., 1994). An alternative risk analysis could be performed by using the probability density functions of YMEY and MSY generated by bootstrapping against the corresponding mortality rates used as control variables (Caddy and Defeo, 1996). The reader is referred to Francis (1992), Cordue and Francis (1994) and Francis and Shotton (1997) for detailed methodological accounts of risk as applied to ®sheries management. We are aware that the bioeconomic model developed here assumes ``pseudo-equilibrium'' conditions, and that a dynamic approach should be preferred over the static one. However, as Die and Caddy (1997) noted, steady static yield±mortality models, and surplus production models in general, should be judged by their capacity for providing useful advice and not by their ability to fully replace dynamic stock assessments (see also Caddy, 1996). Laloe (1995) reached a similar conclusion in his review of the usefulness of the production models that form the basis for the method presented here. A stochastic dynamic model, following the systems science approach (Seijo, 1986; Seijo and Defeo, 1994), could be alternatively formulated to compare the performance of both dynamic and static approaches and evaluate, in the light of model assumptions, which of them will provide the most effective and useful management advice. Acknowledgements We thank Jack Frazier, Anita de Alava and two anonymous reviewers for critical reading of and valuable suggestions on the ®nal manuscript. References Caddy, J.F., 1979. Some considerations underlying definitions of catchability and fishing effort in shellfish fisheries, and their relevance for stock assessment purposes. Fish. Mar. Serv., Manuscript Report 1489, 1±18. 16 O. Defeo, J.C. Seijo / Fisheries Research 40 (1999) 7±16 Caddy, J.F., 1986. Stock assessment in data-limited situations ± the experience in tropical fisheries and its possible relevance to evaluation of invertebrate resources. In: Jamieson, G.S., Bourne, N. 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