MANAGEMENT PROCEDURE TRIALS AND STATISTICS. Geof H. Givens and David J. Thomas. 15 February Abstract

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1 BOWHEAD ABORIGINAL WHALING MANAGEMENT PROCEDURE TRIALS AND STATISTICS Geof H. Givens and David J. Thomas 15 February 1997 Abstract We address the terms of reference of the Aboriginal Whaling Management Procedure (AWMP) Steering Group (IWC, 1996): terminology, data availability, casespecicity, incorporation of need, performance evaluation statistics, testing framework, and simulation trials. Case-specic strike limit algorithms are most appropriate for aboriginal whaling regimes. Need and quotas should be expressed in terms of strikes. The goal of meeting aboriginal need has very dierent implications that the commercial goal of maximizing catch. A thorough testing framework is proposed, along with a diverse set of simulation evaluation trials, analogous to those used to evaluate the commercial Revised Management Procedure, that are appropriate and sucient for evaluation of an AWMP for the bowhead stock, and which are a specic example of a general framework. Eight performance statistics are also proposed to measure the extent to which the simulation performance of a candidate AWMP strike limit algorithm meets each of the aboriginal management objectives of the IWC. 1 INTRODUCTION The International Whaling Commission (IWC) has resolved that an Aboriginal Whaling Management Procedure (AWMP) be developed for whale stocks subject to aboriginal subsistence whaling. This decision was based on longstanding objectives for management of subsistence whaling that were adopted by the Commission at its 34th Annual Meeting (IWC, 1982) and most recently rearmed in aforementioned IWC Resolution (IWC, 1995a). The IWC currently has the framework of an AWMP, as described in Sub-paragraph 13(a) and footnote 1 of paragraph 13 of the Schedule (IWC, 1995b), adopted in 1982 (e.g. see review in Donovan, 1991). Givens et al. (1996) oer an interpretation and quantication of this Geof H. Givens is Assistant Professor of Statistics and David J. Thomas is Graduate Research Assistant, both at Colorado State University, Fort Collins, CO

2 AWMP CCP CLA PDM SLA RMP Table 1: Relevant acronyms. Aboriginal Whaling Management Procedure Common Control Program Catch Limit Algorithm Population Dynamics Model Strike Limit Algorithm Revised Management Procedure existing AWMP, and they show how it can be implemented for the Bering-Chukchi-Beaufort Seas stock of bowhead whales. Wade and Givens (1996) examine related methods. In the remainder of this paper, we refer to IWC Resolution as `the Resolution', to sub-paragraph 13(a) of the Schedule as `13(a)' and to its footnote as `the Footnote'; when we refer to `bowheads' we mean the Bering-Chukchi-Beaufort Seas stock of bowhead whales. Relevant acronyms are listed in Table 1. The principles of the existing AWMP, as given in 13(a) and the Footnote, are: 1. For stocks at or above MSYL (maximum sustainable yield level), aboriginal subsistence catches shall be permitted so long as total removals do not exceed 90% of MSY. 2. For stocks below MSYL but above a certain minimum level, aboriginal subsistence catches shall be permitted so long as they are set at levels which will allow whale stocks to move to the MSYL 1. The Footnote states: The Commission, upon advice of the Scientic Committee (SC), shall establish as far as possible (a) a minimum stock level below which whales shall not be taken, and (b) a rate of increase towards the MSYL for each stock. The SC shall advise on a minimum stock level and on a range of rates of increase toward the MSYL under dierent catch regimes. The objectives as given in the Resolution are: 1 In text. 1. Ensure that the risks of extinction to individual stocks are not seriously increased by subsistence whaling. 2. Enable aboriginal people to harvest whales in perpetuity at levels appropriate to their cultural and nutritional requirements, subject to the other objectives. 2

3 3. Maintain the status of stocks at or above the level giving the highest net recruitment and ensure that stocks below that level are moved towards it, so far as the environment permits. In response to the Resolution, the SC established an AWMP Steering Group, and recommended that it initially address seven issues (IWC, 1996): terminology, data availability, case-specicity, incorporation of need, performance evaluation criteria, testing framework, and simulation trials. In the remainder of this paper, we address all of these issues. Simulation evaluation of an AWMP strike limit algorithm is a daunting task; this paper proposes a concrete starting point. In the case of commercial management procedure development, substantial revisions to the simulation evaluation process were made throughout the duration of the trials. 1.1 Commercial and Aboriginal Management The IWC has recognized the dierence between aboriginal and commercial whaling since its inception. A major focus of the SC over the last decade has been the Revised Management Procedure (RMP) for commercial whaling (IWC, 1994a). The RMP relies on a general catch limit algorithm (the RMP CLA) that provides catch limits for a stock based on historical catch data and sighting survey data for that stock. The SC distinguishes between the RMP and the Revised Management Scheme (the former constitutes all scientic aspects of the latter), however this distinction is not necessary in our considerations. Thus, we consider the RMP to be a management strategy in the broadest scientic sense, and we consider the RMP CLA to be a key component of the RMP that provides quotas. Development of the general RMP CLA was based on both generic and stock-specic simulation evaluation. It is reasonable to expect that a candidate AWMP will consist of a broadly conceived management strategy, including a possibly stock-specic strike limit algorithm (AWMP SLA) that produces strike limits or quotas based on historical catch data, sighting survey data, and established aboriginal and subsistence need. At this point of AWMP development, an AWMP SLA has not been selected, and it is likely that several candidate AWMP SLAs may be evaluated before one is chosen. For simulation evaluation of the RMP CLA, a `true population' was simulated using a population dynamics model (PDM). Simulated sighting survey data were periodically generated from the `true population', and this information was passed to the RMP CLA. The RMP CLA returned a catch limit for each simulation year, regardless of whether simulated sighting survey data existed for the year. The sighting survey data were generated with random error, hence the RMP catch limits were also random. To assess the degree of variation in the catch limits, multiple simulations of each scenario were conducted. Many dierent scenarios, or `trials', were investigated to consider dierent assumptions about population, biological, environmental, and other variables. To ease the process of RMP CLA evaluation, a common control program (CCP) was written to carry out the simulation of the `true population', to simulate the sighting survey 3

4 data, and to communicate with any candidate RMP CLA. We envision the same setup for evaluation of candidate AWMP SLAs. The specic simulation trials and statistics proposed in this paper to assess the performance of candidate AWMP SLAs are somewhat analogous to those used to evaluate the RMP CLA. As for commercial whaling, some AWMP SLAs may include the tting of a PDM to available data. The PDM used by an AWMP SLA may not be the same PDM as the one used by the CCP. 1.2 Generality and Stock-Specicity Unlike commercial whaling, where the number of stocks to be whaled (and whaling operations) can potentially be very large, the stocks to be hunted (and whaling operations) in the aboriginal regime are limited to those specically identied by the IWC. The potential for expansion is thus very limited. The small number of aboriginal whaling operations could allow for more stock-specic AWMP development than in the commercial case, if such speci- city was desired. Two aspects of AWMP development where questions of stock-specicity arise are (i) the AWMP SLA simulation evaluation framework, and (ii) the candidate SLAs considered for evaluation. Simulation evaluation framework Information and data availability dier greatly between stocks subject to aboriginal management (Breiwick and Smith, 1995; IWC, 1997a). Nevertheless, we propose that the simulation evaluation framework be generic at the fundamental, conceptual level. That is, we propose that a generic set of principles and reasoning processes be used to identify simulation trials for all stocks. However, the actual parameter values used in such trials should be stock-specic to most accurately evaluate performance for the stock in question. Although developers of candidate SLAs may nd generic trials useful in exploring and understanding SLA behavior, such trials are not relevant for SC evaluation of SLA performance. Aboriginal stocks are not hypothetical; nor is subsistence need. The use of only a generic, abstract simulated stock does not allow the SC to adequately consider risk avoidance and need satisfaction for the particular stocks the Commission must manage. There may be 22,000 eastern north Pacic grey whales, 8,000 bowheads, and 1,000 west Greenland n whales; moreover these species dier biologically. How can a single, hypothetical stock adequately and plausibly represent these diverse cases? The Resolution compels the SC to use stock-specic values for trial scenarios. Suppose that the same, generic parameter values were used in trials for all stocks, and that this development process yielded identication of an AWMP (with one or more SLAs) that had certain risk avoidance and need satisfaction characteristics. The relationship of these generic performance characteristics to particular stocks would be debatable. Moreover, suppose the SC was then also presented with the results of stock-specic trials that showed the AWMP to have dierent performance for dierent stocks. These stock-specic results would reasonably 4

5 override the generic results. The SC has previously recognized the critical importance of casespecic performance evaluation by its use of `implementation simulation trials' during RMP development (IWC, 1994b). In particular, the need objective must be interpreted in a stock-specic manner. It does not suce for total aboriginal need summed across all aboriginal stocks to be satised by some AWMP. Because need and harvest levels vary between aboriginal stocks, the average rate of need satisfaction can be very misleading. It must be determined whether need is met individually for each stock. Thus, we present in this paper a simulation evaluation framework that includes specic values for bowheads. We view our trial proposals as case-specic values chosen for trials formulated using reasoning that is generalizable to other stocks. For example, we propose bowhead trials that assume 7,200, 8,200, and 9,400 whales in the current stock. The general reasoning is to use low, central, and high values that receive at least a minimum level of support from the best available data. Candidate SLAs We do not propose any SLAs, but it is worthwhile considering whether SLAs might be generic or stock-specic. Extensive research and survey eort has been devoted to some stocks, but the status, dynamics, and even identity of other stocks are not well understood (Breiwick and Smith, 1995; International Whaling Commission, 1997b). For any stock, an AWMP SLA is very likely to be based on data relevant to the stock. If a single AWMP SLA is developed to use only those types of data available for all aboriginal stocks, it will ignore large amounts of data about some aboriginal stocks. All else being equal, when more information is known about a stock (particularly when sustainable yield and replacement yield can be estimated precisely), the reduced uncertainty should increase the ability of an AWMP SLA to satisfy need without increasing risk. In other words, when a stock is better understood, it can be managed more eciently. This is important because the Resolution clearly requires that an AWMP SLA must err conservatively by ensuring that any catch limit is suciently low to control risk. Therefore, an AWMP SLA that exploits all important data about a stock should provide a better opportunity for need to be satised while still controlling risk. Conversely, an AWMP SLA that is `dumbed down' in order to treat all stocks identically would be less able to satisfy the second Resolution objective for well understood stocks. Suppose the SC was presented with results for all aboriginal stocks from both the `best' generic SLA and the stock-specic SLAs that were `best' for each individual stock. Furthermore, suppose that for each stock risk was equal for the generic and specic alternatives, but that in at least one case need was more completely satised by a stock-specic SLA. In this case, the SC would be compelled to choose the stock-specic algorithm in order to satisfy the Resolution objectives. Since this situation could actually occur in practice, the SC is obligated not to prevent this possibility with an a priori declaration that all candidate SLAs must be generic. 5

6 Thus, the Resolution seems to require that stock-specic AWMP SLAs be evaluated. Nevertheless, an implicit assumption of the Resolution is that there be a single, cohesive AWMP for all stocks. Unication of stock-specic SLAs might be accomplished by relying on a common set of generic principles. Sub-paragraph 13(a) species one set of principles that eectively dene the current AWMP. These principles can be applied to all aboriginal stocks. In fact, the Resolution states that \the review [of potential AWMPs] should be based on the principles listed in sub-paragraph 13(a) of the Schedule, and shall also consider the footnote to that sub-paragraph: : :." Thus, the Resolution appears to mandate 13(a) as the set of general, unifying principles that can constitute an overarching general AWMP. Givens et al. (1996) demonstrate how these principles can be applied to the bowhead stock; Wade and Givens (1996) describe a family of SLAs that obey 13(a) and whose members include SLAs suitable for diverse aboriginal whaling situations, including both well- and poorly-understood stocks. A unied AWMP can be a quantitative, probabilistic interpretation of a set of common principles; the quantication would be the same for all stocks. For dierent stocks, the application of the common AWMP principles could be carried out using possibly dierent SLAs. Each SLA might be based on some sort of statistical estimation and/or stock assessment, the details of which depended on what data were available. 1.3 Need Aboriginal cultural and subsistence need is a xed exogenous quantity determined by the Commission, usually upon advice from its Technical Committee. The Commission is typically provided with information that includes needed number of landed whales, needed number of strikes to achieve these landings, and hunting eciency data. Although past practice has varied, we assume that Commission establishes for each year a number of strikes that it deems `appropriate' to the level of need, and a needed number of landings. The distinction between strikes and landings is very important. From 1973 through 1995, annual bowhead hunting eciency has averaged 65% (72% during ), with a maximum of 79% (Suydam et al., 1995a; Suydam et al., 1995b; Suydam et al., 1996). An AWMP might regulate either strikes, landings, or both. From the SC's perspective, it is wiser to develop a SLA that provides a strike limit, and to assume that all strikes are taken and result in the death of a whale for the following reasons: it obviates the need to model hunting eciency, survival rates of struck animals, and strike limit utilization; and it is a conservative approach with respect to the risk objective. The bowhead harvest is strongly aected by ice and weather conditions, and the whaling teams are spread over more than 2,000 kilometers of the bowhead's remote coastal migration path. Currently, the bowhead whaling villages distribute the quota set by the IWC, but sometimes some or many villages may not manage to fulll their portions of the quota 6

7 because of bad hunting conditions. Better conditions may provide other villages with an unexpected bounty. Similarly, communication diculties between whaling teams at distant villages may cause some variation in the total number of whales struck or landed. For these reasons, the overall quota may not always be met. In fact, total landed bowheads were only 71% of permitted landings from 1978{1995, total strikes were 97% of permitted, and no violations have occurred in 15 years. The IWC has in recent years allowed some year-to-year variation in catch by establishing multi-year quotas that limit the maximum numbers of bowheads landed and struck in each year, the maximum numbers landed and struck over the multi-year period, and, sometimes, the maximum amount of unused strikes that may be carried over from one year to the next. There are many ways for an AWMP SLA to permit and account for year-to-year variation in strikes and landings. We label the amount of variation permitted `carryover'. We believe the second objective of the Resolution requires that nonzero carryover be permitted, since this best enables the aboriginal villages to meet their cultural and subsistence need. In Section J below, we consider how AWMP SLA performance evaluation trials might be used to investigate dierent approaches for allowing carryover. 1.4 Available Evidence about Bowheads Substantial evidence exists about some bowhead biological and population parameters (see Givens et al. (1995), IWC (1995d), and IWC (1996) for recent references). This evidence is most concisely expressed in the form of posterior distributions obtained from the `projection' and `hitting' applications of the Bayesian synthesis methodology to relevant bowhead data (sometimes called `forwards' and `backwards', respectively). There has been substantial SC debate about the best analysis approach using these data. In this paper we introduce the concept of the mixed posterior, denoted IM(), which is the equally weighted mixture of the `projection' (Givens et al., 1995) and `hitting' (Butterworth and Punt, 1995; Givens and Thompson, 1996) posteriors obtained from the SC consensus pre-model distributions described by IWC (1995d), but replacing the distributions for current population size and rate of increase by those obtained by Zeh et al. (1995) and summarized by Givens and Bravington (1996). When, for any simulation, several quantities are to be drawn from IM, they should be sampled jointly, rather than marginally. A sample from IM can be obtained by accumulating and combining equally sized samples from the `projection' and `hitting' posteriors. We support the use of IM regardless of preferences for any particular assessment method because, for any parameter, IM should assign substantial mass to all of the diverse range of values favored by any particular assessment method. Punt and Butterworth (1996) have shown that in a general sense the `projection' and `hitting' Bayesian synthesis assessment methods bracket other assessments. Even when two dierent synthesis analyses favor dierent ranges for a parameter, IM assigns non-negligible probability to the most extreme values favored by either analysis. In other words, IM gives probability wherever any individual assessment does, and gives low/no probability only where all individual assessments do. If the 7

8 Table 2: Rough deconstruction of Resolution objectives into variables and conditions. Variable Performance Criterion risk of extinction is not seriously increased quota satises need stock size moves towards MSYL stock size is eventually maintained above MSYL SC obtains other, substantially dierent bowhead assessment results from using methodologies, and if it believes that these results have important implications for the values of parameters that IM should support, then these assessment results can easily be incorporated as additional components of the mixture IM. Trials of a bowhead AWMP SLA should be diverse and challenging, and should investigate both likely and unlikely scenarios. However, it does not make sense to consider trials that are outside the range of plausibility for the bowhead stock. Punt and Polacheck (1995) also tried to limit their focus to bowhead-plausible SLA trials. In fact, implausible trials contravene the Resolution objectives because need and risk may be improperly balanced for the bowhead stock if the AWMP SLA is chosen on the basis of implausible trials. Plausibility should be considered a scale, not a yes/no issue. Statistical probabilities are the units of this scale, and thus statistical evaluation of the available bowhead evidence can measure bowhead-plausibility. 2 PERFORMANCE EVALUATION STATISTICS To evaluate the performance of any AWMP SLA, statistics must be used to measure the extent to which the SLA satises the management objectives of the Resolution. The objectives specify that certain variables satisfy certain performance criteria. In a very rough sense, these variables and criteria can be summarized as shown in Table 2. Simulation cannot conclusively determine whether a criterion will be met; it can only estimate the probability that a criterion will be met. Thus, performance evaluation must be probabilistic. Further, evaluation must be somewhat subjective because words like `seriously' and `moves towards' are not absolute. How much increased risk is `serious'? Can a temporary need shortfall be tolerated if harvest exceeds need some other times? How slowly can a population move towards MSYL without being intolerably slow? Even if answers to questions like these can be provided, one must ask: what is the greatest tolerable estimated probability that a criterion is not met? The political, decision-theoretic, and epistemological issues raised by questions like these are beyond the scope of this paper. We take the approach of developing statistics that are sensitive to variations in the extent to which objectives may be satised. Our proposed 8

9 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 Table 3: Summary of performance evaluation statistics. nal depletion of stock most severe depletion of stock total strike need satised longest time period with annual 5% strike need shortfall % of simulated stocks that reach MSYL during simulated management proportion of time stock stays above MSYL after rst attaining MSYL time until stock reaches MSYL % of simulated stocks that reach MSYL or grow 50% of the way towards MSYL statistics are `refutative' in the sense that they can be used to identify a poor SLA, but they cannot conclusively determine that a SLA is wholly acceptable. Scientists and policymakers with diverse priorities and viewpoints should all be able to assess the performance of a SLA from these statistics. In other words, we intend that individuals may disagree about the conclusions formed from these statistics while agreeing that these statistics contain the necessary and sucient information to form conclusions. 2.1 Notation We propose in Section 3 that the simulation performance evaluation of an AWMP SLA be conducted in a manner similar to the evaluation of the commercial RMP CLA. Let N random replicate simulations be performed for any scenario. Let these simulations be T years in length. Let t = 0 at the beginning of the rst year of simulated management. Let the strike need in year t as established by the Commission be denoted N t. Let the AWMP SLA strike limit quota in year t be denoted Q t, and the true 1 + stock size at the start of year t be denoted Pt 1+. The statistics we propose are summarized in Table Extinction Risk It is important to know the status of the stock at the end of the simulated management period in order to determine whether the management procedure led to increased depletion or extinction of the stock. Our rst proposed statistic measures the depletion of the stock at the end of simulated management: T 1 = P 1+ final K 1+ (1) where P 1+ final is the size of the 1+ component of the stock in the nal year of simulated management, and K 1+ is the pre-exploitation size of the 1+ component. 9

10 It is also important to know whether the stock ever reached a dangerously low level during the simulated management period. We propose the statistic: T 2 = min t P 1+ t (2) K 1+ where the minimum is taken over all years subject to simulated management. T 1 and T 2 were previously used by the SC (IWC, 1991) to assess a virtually identical objective for commercial whaling (IWC, 1988) during the development of the RMP. In Section P we propose that N = 100 random replicates of each management period be simulated. We propose that the 1st, 5th, 25th, 50th, 76th, 96th, and 100th smallest values of T 1 and of T 2 be reported to summarize these 100 replicates. 2.3 Need As noted in Section 1.3, we believe that the SC should evaluate whether candidate AWMP SLAs satisfy strike need. We consider the level of need met and the sustainability of the harvest separately. Our rst proposed statistic measures the total proportion of strike need met over the simulated management period: T 3 = P t Q t t N t (3) where Q t is the quota given by the SLA in year t of the simulated management period, and N t is the corresponding strike need for that year. Since SLAs may cope with carryover in a variety of ways, it is possible that Q t > N t, or even T 3 > 1, though this is very unlikely since the hunters would seem to have no reason to kill more than N t whales and SLAs are likely to bound Q t above by N t, except possibly for carryover. T 3 measures only the long-run proportion of need satised; it is possible that T 3 might be high although N t might exceed Q t in a large majority of cases. For this reason, T 3 1 does not imply that need was met. T 3 compares the cumulative quota during the simulated management period to the the cumulative need. To measure the short-term level of need met, we propose the statistic: T 4 = longest run of Q t < 0:95N t : (4) A run of Q t < 0:95N t is a consecutive sequence of years for which Q t < 0:95N t, and the length of a run is the number of consecutive years in the run. We scale N t by 0.95 to account for hunting variability and carryover. This reects the idea that a modest need shortfall in a few years may be tolerable (particularly when carryover is permitted) but prolonged severe shortage is unacceptable. T 3 and T 4 do not measure whether the satisfaction of need is sustainable in perpetuity. We propose that sustainability be measured with the statistics proposed for the other two management objectives. By using the growth and recovery statistics, we are subjecting the satisfaction of need to the other Resolution objectives, as is required by the Resolution. 10

11 Finally, to summarize N = 100 random replicate simulations, the 1st, 5th, 25th, 50th, 76th, 96th, and 100th smallest values of T 3 and of T 4 should be reported. 2.4 Recovery The last Resolution objective is to `maintain the status of stocks at or above the level giving the highest net recruitment and ensure that stocks below that level are moved towards it, so far as the environment permits.' This objective consists of several dierent goals conditional on stock status. We rst consider the maintenance of stocks at or above the level giving the highest net recruitment. This level is the maximum sustainable yield level, MSYL. We propose T 5 NX = 1 (5) N i=1 1 P 1+ final K 1+ MSY L 1+ where 1 (ab) is 1 if a b and 0 otherwise, N is the total number of replicate simulations, and MSY L 1+ is MSYL expressed in terms of the 1+ population component. Thus T 5 measures the proportion of replicate simulated management periods that ended with the stock at or above MSYL. It is also important to consider maintenance above MSYL individually for any simulation. A straightforward measure would be the proportion of time the simulated stock was at least MSYL. However, it is possible that this proportion might be large, but the stock was declining so that the stock was rarely above MSYL during the later years of the simulation management period. Further, this proportion would not distinguish between two dierent cases: a stock that was below MSYL for the rst half of the simulated management period and above MSYL for the second half; and a stock that alternated between above and below MSYL on a yearly basis. To account for these concerns, we instead propose 1 T 6 TX = (6) where ( t MSY L = mint ft : T + 1 P 1+ t T? t MSY L 1 P t 1+ t=t MSY L K 1+ MSY L 1+ MSY L K g if stock ever reached MSY L 1+ by time T otherwise and T is the nal year of the simulated management period. Thus T 6 measures the time that the stock was at least MSYL, as a proportion of the time since it rst achieved MSYL. When t MSY L exceeds T, T 6 = 0. A severely depleted stock that grew steadily towards MSYL without reaching MSYL by the end of the simulated management period would have T 5 = 0 and T 6 =0 yet may nevertheless satisfy the Resolution objectives because of its steady growth. Therefore, We next focus on measuring the extent to which depleted stocks grow towards MSYL. 11 ) (7)

12 We propose T 7 = t MSY L (8) to measure the speed with which a stock grows to rst achieve MSYL. By reporting quantiles of T 7, one can set t MSY L = T + 1 for stocks which never reach MSYL without aecting the reported quantiles (except for quantiles that are greater than T, which should be interpreted as `never reaching MSYL during the simulation period', regardless of value). Since t MSY L depends on biological parameters, its value should be compared across candidate SLAs within a single trial, but not across trials. It would be very appealing to nd a statistic that simultaneously measured all aspects of the recovery objective. After long weeks of consideration and vivid imagination of stock trajectories with wild oscillations and other bizarre pathologies, we have concluded that any such statistic would be exceedingly complex. Instead, we have chosen a simpler statistic that is sensitive to the combined goals of the recovery objective for most straightforward stock trajectories. Let T 8 = 1 N NX i=1 1 (P 1+ final MSY L 1+ K 1+ or P 1+ final (MSY L 1+K 1+ +P 1+ 0 )=2) : (9) T 8 therefore measures the proportion of simulations for which the stock either (i) ends above MSYL, or (ii) grows by at least 50% of the amount needed to achieve MSYL from P If either criterion is met, the stock is considered successfully managed. Note that as premanagement depletion, P 1+ 0 =K 1+ becomes more severe, (ii) requires more rapid population growth over the 100 year simulated management period. The two conditions in (9) are not completely redundant because their eects vary as P 1+ 0 =K 1+ and P final =K are above or below MSY L 1+. Finally, to summarize 100 random replicate simulations, the 1st, 5th, 25th, 50th, 76th, 96th, and 100th smallest values of T 6 and of T 7 should be reported. Note that T 5 and T 8 are proportions achieved across simulations, so we propose reporting their single values obtained from the 100 replicate simulations. It is possible that the recovery statistics described in this section would provide very little information when a stock would fail to achieve MSYL before the end of simulation even when no catch is taken (t MSY L;no catch > T ). This raises the question of whether it would be worthwhile to set T on the basis of t MSY L;no catch or whether it would be benecial to scale our statistics by their values when no catch is taken. For simplicity, we ignore these concerns, noting that in practice t MSY L;no catch > 100 is very unlikely for the bowhead stock. 3 SIMULATION TRIALS We propose to use a modied version of the single stock common control program (version 8) that has been used for simulation testing of the commercial RMP CLA, along with a set of trial specications that are modeled after the RMP trial specications. The justications for these specications may be applied to other aboriginal stocks, but we present particular 12

13 values that are reasonable for bowheads. The modied CCP permits many sorts of possible trials. For each factor that may vary in the CCP, we list a baseline option and a set of additional trials that we believe are appropriate and sucient for the evaluation of an AWMP SLA for bowheads. The baseline trial is considered to be the most interpretable starting point for comparison with alternatives. Table 4 summarizes the simulation trials we propose. A: Years of Pre-management Catch and Protection Baseline: Bowhead catch history, no protection We propose that the entire bowhead catch history, from 1848 to the start of AWMP management be used, and that there be no initial period of protection. This is the only reasonable choice for the bowheads, as it fully exploits the available data, and it does not interrupt the aboriginal subsistence hunt by requiring a temporary halt before AWMP management begins. Generic application of the CCP has used the historic catch to determine the pre-management stock trajectory, including initial carrying capacity and stock size at the beginning of the rst year of simulated management. Our proposal is more similar to case-specic RMP `implementation simulation trials', and represents a critical departure from the generic RMP trials where `true stock abundance' was not stored in absolute terms during simulation. Early generic trials set initial carrying capacity, K, at an arbitrary 10,000 whales, and all removals were considered relative to this value. Later trials found a relative value for K by assuming a series of pre-management catches of one whale per year, and assuming a pre-management depletion level. In neither case could K be interpreted as an actual number of whales. Neither case works for an aboriginal management situation because aboriginal cultural and nutritional requirements are expressed in an absolute number of whales. Suppose an early RMP trial (where K = 10; 000) gave a quota of 100 whales in a certain year. This was interpreted as a quota of 1% of K, or 1 D t % of current abundance, where D t is current depletion. The actual number of whales to be removed depended on stock size. In the aboriginal case, if need is 100 whales and if the quota depends on need in some way, then the quota cannot be calculated without an absolute population abundance. In addition, SLA performance cannot be reliably evaluated without an absolute population number because the need objective cannot be directly assessed. The results obtained for any indirect SLA evaluation using a relative index may depend on the relative index so the candidate AWMP SLA with the best simulation performance may not perform best on a stock with the true absolute abundance. One reason that generic RMP trials avoided reliance on the full catch history and on absolute abundances was concern that CLA developers would reverse-engineer important information about the `true population' and CCP PDM from the data passed to the candidate CLA routine. This concern is valid, but the bowhead situation appears to require a dierent solution to this problem. Other trials: No other trials 13

14 Table 4: Summary of proposed simulation trials. OPTIONS RMP TRIALS AWMP TRIALS SEE Biological Parameters and Models MSYR 1%, 2.5%, 4%, 7% 1%, 2.5%, 4% D MSYR y Unif(0.1%, 5%) IM E age of maturity various always random G MSYL 0.4, 0.6, 0.8 same H mortality rates various always random F age of recruitment various 1 year N PDM various age struc., SC choices O Stock Condition and Historic Catches Pre-management depletion 0.05, 0.2, 0.3, 0.4, 0.6, 0.99 not directly specied B, C Pre-management depletion Unif(0.01,0.99) not directly specied B; C Current stock size not directly specied 7,200, 8,200, 9,400 C Current stock size y not directly specied IM E Historic catch bias factor none or 0.5 None, 0.5, 1.5, random M Historic catch period various 1848 to present A Protection period various none A Surveys Survey frequency, years 2, 5, 7, 10, 20, Unif(1,9) 5, 10, 20 Q End of surveys 0, 20, 50 years, or never never Q Abundance estimate bias 1, 0.5, 1.5, 0:5! 1, 1:5! 1 same L Process error factor, z 0, 0.5, 1, 3.65 Table 5 K Abundance estimate CV z est 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2 Table 5 K Strategic survey timing none, or strategic same if needed T Environment K changes various similar R MSYR changes various similar R K and MSYR change jointly 50% linear reduction similar R Episodic events 2% prob. of 50% reduction similar S Additional AWMP Issues Need not applicable constant at 68 or 150, or I 68 doubling at year 50 Carryover not considered none, greedy, strategic J y Random current stock size and MSYR should be sampled jointly with each other and with mortality and maturity parameters for the random parameters trial; see E and the Appendix. z CV est and should be set jointly according to Table 5. 14

15 B: Pre-management Depletion Baseline: Eliminated In Section A we discuss why reliance on pre-management depletion is problematic. It also turns out that some very realistic depletion values cannot be achieved using any carrying capacity, for some values of biological and production parameters that the SC will want to investigate. Furthermore, it does not make sense to establish K 1+ and P 1+ 0 on the basis of depletion, a statistic that is not well known for bowheads. Knowledge of depletion suggests knowledge of carrying capacity, but specication of bowhead carrying capacity has been a major point of debate in the SC recently (Butterworth and Punt, 1995; IWC, 1996). In contrast, evidence about current stock size is some of the best, most precise, and most widely accepted evidence about the stock. Therefore, we propose that K 1+ and pre-management depletion be determined from P 1+ 0, catch history and biological/production parameters. For RMP CLA trials, the CCP had reversed the roles of depletion and P 1+ 0, determining a relative K 1+ and a relative P 1+ 0 using input values for biological/production parameters, an articial catch history, and depletion at the start of simulated management. Our proposal has three strengths: it eliminates potential inconsistencies between trial specications, it allows the best bowhead evidence to drive the simulation, and it continues our focus on running trials with actual stock sizes rather than relative indices, so that need issues can be addressed appropriately. This proposal eliminates specication of pre-management depletion. C: Pre-management Stock Size Baseline: 8,200 As mentioned in Section B, we propose that the CCP establish carrying capacity, K 1+, and depletion at the start of simulated management using input values for biological/production parameters, the historic bowhead catch history, and the stock size at the start of simulated management, P As a baseline value for the stock size at the start of simulated management, we propose 8,200, the mode of the 1993 Bayes empirical Bayes posterior (Zeh et al., 1995). Other trials: 7,200, 9,400 As alternatives we propose the 2.5% and 97.5% points of the 1993 Bayes empirical Bayes posterior (Zeh et al., 1995). D: MSYR Value and Population Component Baseline: 2.5%, 1+ population 15

16 The 1+ component is chosen to reect the current CCP (version 8) implementation and because it seems simpler and more interpretable than the mature component. Butterworth and Punt (1992) argue that MSYR 1+ should be preferred over MSYR (mat) because it is \most closely related to quantities which are directly measurable" (p. 583). They also note that MSYR values of 1%, 2.5%, 4%, and 7% relative to the mature population component played a prominent role in evaluation of the commercial RMP CLA. We choose MSYR 1+ = 2:5% as the baseline scenario, because it is a central value supported by the reasoning below. Other trials: 1%, 4% The choice of values should be driven by the available bowhead data. Several recent assessments (Butterworth and Punt, 1995; Givens et al., 1995; Punt and Butterworth, 1995) have yielded interval estimates for MSYR (mat) from 1.0% to 6.9%, with a region of common overlap from 3.1% to 5.4%. For a variety of assumptions about life history parameters and population models, it is widely known that MSYR (mat) is usually larger than MSYR 1+, often very roughly by a factor of 1.5{2. For example, Butterworth and Punt (1992) show results in their Table 4 where MSYR (mat) is 1.2 to 2.5 times as large as MSYR 1+, under assumptions for a stock that may be more productive than bowheads. In a study of various models for bowheads, Punt (1996) found MSYR (mat) to be 1.5{2.2 times as large. Thus, we view the SC's use of 1%{7% for MSYR (mat) in RMP CLA trials as a reason to consider limiting the range of MSYR 1+ values to some range roughly like 0.5% to 3.5%. Similarly, interval estimates of MSYR 1+ (Raftery et al., 1995; Punt and Butterworth, 1995) range from 0.4% to 3.4% with a region of common overlap from 1.7% to 2.5%, though both the 0.4% and 2.5% values come from an assessment with priors more conservative with respect to MSYR 1+ than have been recently supported by the SC. There is a common overlap between SC conventional wisdom and the assessment results, supporting a range of MSYR 1+ from 1% to 4%. A MSYR 1+ value of 2.5% is given high probability by all the above assessments and is in the middle of the plausible range, so we take it as baseline. These assessments estimate that there is very little probability that MSYR 1+ is less than 1% or greater than 4%, therefore we limit our trials to this plausible range. More extreme, implausible trials would be inappropriate because they could bias the AWMP SLA performance evaluation and interfere with the selection of an AWMP that satises the Resolution objectives. See E for discussion of a trial with random values of MSYR drawn from IM, which would be the most data-based, plausible approach for setting MSYR. A SC decision to eliminate the xed MSYR trials and focus only on the IM trial would be scientically and statistically supportable. E : Random Parameters Option Baseline: O This option allowed for MSYR, the pre-management catch history, and the depletion level immediately prior to management to vary randomly between simulation trials. We 16

17 propose that these parameters be set in dierent manners, as described in D, A, and B for the baseline case. Other trials: Modied Random Parameters We propose changing the nature of the random trials controlled by this option. Evaluation of the RMP CLA included a trial for which MSYR had a Unif(0.1%, 5%) distribution. Although these boundaries may be appropriate for a generic commercial stock, they are not relevant for bowheads. In particular, the RMP lower limit of 0.1% is well below the range of plausibility for bowheads. We propose instead that MSYR 1+ be drawn from IM(MSYR 1+ ). One RMP CLA trial assumed that pre-management 1 + depletion had a Unif(0.01,0.99) distribution. This range completely contradicts all known bowhead data. To the best of our knowledge, no bowhead assessment has produced a 1 + depletion estimate outside the range of (0.07,0.55) 2 (Tillman, 1980; Givens et al (1995)), and the estimate of 0.07 is probably biased severely downwards since it was based on estimates that are now eectively discredited. Furthermore, bowhead depletion is much less well known than is current stock size. Since in Section C we propose modifying the CCP so that population trajectories are established on the basis of current stock size rather than depletion, we propose replacing the random parameters depletion option with a corresponding random parameters option for pre-management stock size. We propose that the stock size at the start of simulated management be drawn from IM. The original random parameters option also randomly set the number of years of an articial catch of one whale per year prior to management. Our proposals above obviate the need for any random parameter option for the number of years of articial pre-management catch. To summarize, we propose that the modied random parameters trial sample MSYR and pre-management stock size jointly from IM. See the Appendix for technical details about trials that use random sampling strategies. F : Mortality Parameters Baseline: Modied random parameters We propose that baseline trials treat annual juvenile and adult mortality rates as random parameters, drawn from IM, and that the transition between rates be knife-edged at a random age, also drawn from IM. The median and 95% intervals for these parameters are: juvenile rate (0.13, 0.016), adult rate (0.031, ), nal age at juvenile rate 5 (1, 9). We propose that the annual survival rate (one minus the mortality rate) for newborn calves be the product of the juvenile and adult survival rates. These mortality parameters are uncertain, unknown quantities, but not parameters that determine fundamentally dierent management scenarios in the same way as parameters such as survey frequency, epidemic rates, or MSYR. Past analyses and assessments have 2 Mature depletion estimates have been considerably higher (e.g. Butterworth and Punt, 1995). 17

18 shown that key management quantities are less inuenced by choices for these parameters than choices for yield-related parameters (de la Mare, 1986; de la Mare, 1990; Lankester and Cooke, 1987; Butterworth and Punt, 1995; Givens et al., 1995; Givens and Bravington, 1996). For the sake of parsimony, we propose treating these mortality parameters as random so that all performance evaluation trials are eectively averaged across a range of values. See the Appendix for technical details about trials that use random sampling strategies. Other trials: No other trials The baseline trial suces, as it averages over plausible mortality schedules. G: Maturity Parameters Baseline: Modied random parameters, knife-edged We propose that baseline trials treat the age of sexual maturity as a random parameter, drawn from IM. Further, we propose that transition to maturity be treated as knife-edged. The median and 95% interval for age at rst parturition are 19 (15,24) in IM. The rationale for averaging across a range of maturity ages in all trials is the same as in Section F. See the Appendix for technical details about trials that use random sampling strategies. Other trials: No other trials We think the baseline trial suces, as it averages over plausible maturity ages. H: MSYL Value and Population Component Baseline: 0.60, 1+ population We propose a baseline value of 0.60 for MSYL, relative to the 1+ population component. The 1+ component is chosen because the CCP (version 8) implements the recruited component, which is 1+ for bowheads, and because it seems simpler and more interpretable than the mature component. Other trials: 0.4, 0.8 Recent bowhead assessments have not been able to determine the range of plausible values for MSYL. Therefore, in the absence of evidence, we propose retaining the possibilities used in RMP trials. I : Need Baseline: 68 strikes per year Strike need is aected by many factors including: changes in needed landings due to Eskimo population sizes, modernization, cultural priorities, and nutrition; changes in hunting eciency; and changes in how the Commission assesses the `appropriate' relationship between needed strikes and landings. Nearly all these factors are outside the expertise of the SC, which clearly would benet from some guidance from the Commission on this matter. We propose 68 strikes per year as the baseline because, except for carryover issues, it is the current strike need (IWC, 1995b). 18

19 Other trials: 68 strikes, doubling to 136 in year T=2, or 150 strikes every year Other trials can investigate the change from baseline performance if need were to increase. Rather than consider a large number of options for this dicult issue, we propose the approach of considering two extreme cases. The rst extreme proposal is that the strike need would eventually double to 136, and that this change would occur with the most extreme abruptness, occurring in a single year T =2 years from the start of simulated management. In Section P we propose that T = 100 so T=2 = 50. The second extreme proposal is that need is xed at 150 strikes. This level is intended to bound all plausible need scenarios. J : Carryover Baseline: none We propose that the baseline scenario consider zero carryover. We have argued in Section 1.3 that nonzero carryover provides the best opportunity for satisfaction of the Resolution objectives, however this baseline proposal is the simplest and provides the clearest basis for comparison. Bowhead hunting eciency is substantially less than 1.0, therefore needed strikes is greater than needed landings and a SLA quota is often likely to be greater than needed landings. We have proposed in Section 1.3 that that the entire SLA quota should be removed from the true population. Any carryover removals that exceed the needed landings but do not exceed the strike quota are therefore subsumed by our conservative removal assumption. Thus, our baseline proposal is suciently conservative to bound the likely SLA performance under modest carryover scenarios. Other trials: greedy and strategic One must distinguish between the carryover `rule' and the actual removals resulting from hunting that is consistent with the rule. The carryover rule is the specication of how much carryover is allowed in any year. This amount is calculated by a SLA, and dierent SLAs may have very dierent carryover rules. Since the rule is a component of the SLA, it is a target for simulation evaluation, not a variable for which dierent simulation scenarios need be developed. In contrast, the actual number of `excess' whales removed under a carryover rule is a variable that could dier between simulation scenarios. The specication of scenarios must be very general, since carryover rules may be quite diverse and complex. Until specic SLAs and their corresponding carryover rules have been identied, it is dicult to precisely specify appropriate trials. However, we believe that carryover can probably be suciently investigated with two trials: a greedy trial and a strategic trial. The greedy trial would specify that in any year, removals are as large as permitted by the SLA, including any carryover excess, regardless of the consequences of such behavior on future quotas. Suppose that two options were presented to hunters: (i) remove 50 whales this year and 50 next year, or (ii) remove up to 10% above 50 this year but next year remove 50 less twice the excess taken this year. Of these carryover options, (i) yields the greater 19