© 2005 International Council for the Exploration of the Sea
Reference points and management strategies: lessons from quantum mechanics
Fisheries and Oceans Canada, Pacific Biological Station 3190 Hammond Bay Road, Nanaimo, B.C. V9T 6N7, Canada
*Correspondence to J. T. Schnute: tel: +1 250 756 7146; fax: +1 250 756 7053. e-mail: schnutej{at}pac.dfo-mpo.gc.ca.
Fisheries management often relies heavily on precautionary reference points estimated from complex statistical models. An alternative approach uses management strategies defined by mathematical algorithms that calculate controls, like catch quotas, directly from the observed data. We combine these two distinct paradigms into a common framework using arguments from the historical development of quantum mechanics. In fisheries, as in physics, the core of the argument lies in the technical details. We illustrate the process of designing a management algorithm similar to one actually used by the International Whaling Commission. Reference points and surplus production models play a conceptual role in defining management strategies, even if marine populations do not obey such simplistic rules. Physicists have encountered similar problems in formulating quantum theory, where mathematical objects with seemingly unrealistic properties generate results of great practical importance.
Keywords: fishery management, management model, policy, quantum mechanics, reference point, strategy, surplus production
Received 18 February 2005; accepted 27 July 2005.
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Introduction |
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 Top Introduction Example with catch and...Estimation modelDesign choicesLessons from quantum mechanicsDiscussionAppendix Dynamic model...References |
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Historically, fishery models started with simple descriptions of stocks that generate surplus production. During the past three decades, these key ideas have grown into complex statistical models that use all available data to estimate numerous quantities of biological interest, including biomass levels, selectivities, and mortalities. Policy often becomes linked to model constructs, such as the carrying capacity and maximum sustainable yield (MSY). For example, some implementations of the precautionary approach use these quantities as reference points in specifying targets and harvest levels.
During the 1980s, investigators began exploring a different paradigm that focuses directly on management procedures or strategies used to regulate fisheries (de la Mare, 1998; Sainsbury et al., 2000). For example, if the goal is to set a quota (i.e. total allowable catch) for removals from a fish stock, then a strategy boils down to an algorithm that computes the quota from the available data. Proposed strategies can be tested using simulation models that incorporate realistic biological complexity, such as age structure, spatial movement, climate change, or species interactions. The research focus shifts from elaborate estimation models to robust management algorithms.
From a management perspective, model properties and biological details often seem remote from the practical need to set policy. Butterworth (in press) argues that "fishery reference points miss the point" because they focus on hypothetical ideas like MSY, rather than quantities that can actually be observed. He cites a historical precedent in quantum mechanics, where Heisenberg (1925) advanced the field by linking theoretical physics with quantities that are directly measurable. In this paper, we use the mathematical details of a particular management strategy to extend Butterworth's analogy. Our example shows how the two distinct paradigms of reference points and management strategies combine to give a modern strategic theory. As in quantum mechanics, theoretical concepts gain relevance by their connection with actual observations.
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Example with catch and index data |
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TopIntroductionExample with catch and...Estimation modelDesign choicesLessons from quantum mechanicsDiscussionAppendix Dynamic model...References |
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Our example deals with a fishery where the available data include only two time-series: the annual catch Ct and a biomass index It for years t = 1, ..., n. What should be the total catch quota Qn+1 in year n + 1? Mathematically, a management strategy takes the form
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| (1) |
= (1, ...,k) of control parameters allows policy adjustments that reflect species biology and address various management objectives. Table 1 lists three empirical harvest policies (T1.1)–(T1.3) that are special cases of (1). The first maintains a constant catch, and the second adjusts the catch up or down according to the most recent observed change in the index. The third policy (T1.3) represents an average of the previous two, where a single control parameter (0
1) sets the relative weight assigned to a stable catch (T1.1) and a response to abundance changes (T1.2) . Such empirical policies have the practical advantage that stakeholders can understand them easily. Unfortunately, they capture nothing about biological limits. For example, assuming that the index tracks the population fairly well, (T1.2) essentially maintains constant fishing mortality, which might be too high (potentially destroying the stock) or too low (potentially forfeiting available production).
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Reference point advocates might suggest very different strategies based on theoretical concepts like the carrying capacity K, maximum sustainable catch C*, or the associated MSY biomass B* and harvest rate h* = C*/B*. For example, the precautionary harvest policy (T1.4) depends on two reference points (K, h*), the current biomass B, and two control parameters (
, ß) with 0 1 and 0 < ß 1. In this scenario, the fishery is closed when the current biomass drops below a specified fraction of the carrying capacity (B K). If the biomass reaches or exceeds the carrying capacity (B 
K), the quota Q = ßh*B represents harvesting at a specified fraction ß of the MSY harvest rate h*. When the biomass lies between the protected level and the carrying capacity (K < B < K), the harvest rate increases linearly from 0 to ßh*. The controls (, ß) play a precautionary role in setting a minimum stock size K and maximum harvest rate ßh*.
Obviously, (T1.4) does not define a management procedure of the form (1). It involves two theoretical quantities (K, h*) and an unmeasured current biomass B, but not the available data. However, it can readily be converted to a data-driven procedure by using a model that generates estimates 
from the available data. Then the calculation
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| (2) |
= (, ß). Any model that generates the required estimates will suffice for this definition, but a simple model would certainly make the algorithm more tractable and possibly more robust. Regardless of the model chosen for parameter estimation, this argument demonstrates that reference points can be used to design a management strategy. |
Estimation model |
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TopIntroductionExample with catch and...Estimation modelDesign choicesLessons from quantum mechanicsDiscussionAppendix Dynamic model...References |
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We illustrate one possible approach with the estimation model (T1.5)–(T1.6). The dynamic equation (T1.5) describes a removal process (Bt+1 = Bt – Ct) for the annual biomass Bt, with an additional production term that is positive when Bt < K and negative when Bt > K. The model has biological meaning for any exponent p > –1. Although (T1.5) is not defined when p = 0, due to the presence of a factor p in the denominator, it takes the limiting form (T1.9) as p
0. In the Appendix, we compare our formulation with the historical model proposed by Pella and Tomlinson (1969), and we give mathematical proofs that (T1.5) has the properties described here. Our version emphasizes the direct role in the system dynamics of the reference parameters K and h* used by the policy (T1.4) , as well as continuity when p = 0. If the population is not harvested (Ct = 0), then (T1.5) expresses Bt+1 as a concave function of Bt with a fixed point at K (Figure 1a). Thus, if Bt = K, then Bt+1 = K when Ct = 0. However, sequential processes associated with dome-shaped curves like those in Figure 1a sometimes exhibit bifurcation and chaos (Kuznetsov, 2004). In (T1.5) with Ct = 0, the fixed point K is stable only if h* < 2/(1 + p). The upper limit (T1.8) for h* expresses this constraint, as well as the biological fact that the harvest rate h* must take values between 0 and 1. For example, if p = 5, then stability occurs when h* < 1/3. The case p = 5 in Figure 1a violates this constraint because h* = 0.5 > 1/3. The corresponding dome-shaped curve has a very steep descending limb, and this geometric feature leads to unstable behaviour (Appendix).
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A population maintained at biomass B generates the annual catch
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| (3) |
For p > 0, the curves in Figure 1a are dome-shaped. Each curve intersects the Bt-axis at two points: the origin and a high value
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| (4) |
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| (5) |
, and finally only the left limb of the dome remains as an unbounded curve.
The model defined by (T1.5)–(T1.6) assumes that the biomass Bt follows a deterministic trajectory. Stochastic properties enter the index It, which is proportional to Bt with an unknown coefficient q and is influenced by lognormal measurement error with unknown standard deviation 
. The resulting posterior distribution (T1.11) depends on K, h*, q, , and an unknown biomass B1 needed to initialize the dynamic process (T1.5) . In the formulation here, we do not include p in the list of parameters to be estimated from the data. Instead we treat p as a fixed control parameter, chosen for biological or other reasons, where p determines the ratio B*/K from (T1.7) . Any particular set of parameters (B1, K, h*, q, p) automatically determines Bn+1 after n iterations through the deterministic process (T1.5) . Consequently, a posterior sample drawn from (T1.11) gives a distribution of vectors (Bn+1, K, h*) and a corresponding distribution of quota values Qn+1 (Bn+1, K, h*; ,ß) calculated from (T1.4) .
To finalize a policy of the form (1), set the quota at a specified quantile level 
(0 < < 1) in the distribution of quota values Qn+1. For example, the choice = 0.4 defines a more conservative policy than the median quota with = 0.5. Because the posterior sample depends directly on the input data, this gives a data-driven policy that varies with the choice of control vector
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| (6) |
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Design choices |
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TopIntroductionExample with catch and...Estimation modelDesign choicesLessons from quantum mechanicsDiscussionAppendix Dynamic model...References |
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The technical details summarized in Table 1 highlight four interrelated choices in designing a management strategy. First, data sources must be identified, like the sequence (Ct, It) here. This list might be altered to include age or size distributions, climate indicators, or indices for other species. Second, a harvest policy must be represented as an explicit formula that depends directly on the data or on hypothetical reference points, illustrated here by (T1.1)–(T1.4). Strategies can be mixed, like the compromise (T1.3) between (T1.1) and (T1.2) . Similarly, a policy might blend theoretical and empirical approaches by assigning relative weights to (T1.1) , (T1.2) , and (T1.4) in a single formula. The calculation could also involve other reference points thought useful for guiding policy, such as a hypothetical natural mortality M or fishing mortality F associated with some feature of the population dynamics (not necessarily MSY, as in the example here). Analysts can take a creative approach, allowing free speculation about hypothetical properties of the population dynamics.
The third step links the data to the harvest policy. It requires a model to generate estimates of unknown parameters in step 2 from the data identified in step 1. Our model (T1.5) emphasizes this linkage with explicit reference to the parameters (Bn+1, K, h*) used in the policy rule (T1.4) . The example in Table 1 corresponds roughly to a policy adopted by the International Whaling Commission (IWCSC, 1994a, b). We include a general exponent p, where they implicitly choose p = 2, and our harvest policy (T1.4) captures the spirit of their control law (IWCSC, 1994a [sections 4.2 and 4.4], 1994b [p. 154 – equation [4] and Figure 3]). Other surplus production models might also be used, such as those developed by Schnute and Richards (1998, 2002) to give analytical formulae for fishery reference points. These would adapt well to situations with additional data on mean age and/or weight. They also include asymptotic production curves potentially more realistic at large stock sizes than dome-shaped versions of model (T1.5) , which imply that large stock sizes can produce extinction (Figure 1a).
As we discussed earlier in regard to the calculation (2), any model qualifies as a candidate for the third step, as long as it produces an algorithm to estimate all parameters in the control rule. But exactly how do the estimates follow from the model? This question leads to the fourth step of defining an associated likelihood function or Bayes posterior, like (T1.11) here. Our algorithm incorporates an unknown initial biomass B1 to begin a recursion through the deterministic dynamics (T1.5) . A variant of the model, such as one that includes process error in (T1.5) , would lead to a different posterior and consequently a different estimation algorithm. In these circumstances, a Kalman filter (Schnute, 1994; Walters, 2004) or a similar iterative procedure might lead to a better policy.
Our example illustrates how each step in this process potentially introduces parameters to the control vector . We have not used special parameters at the data compilation phase, but they might enter the analysis through pre-processing algorithms that select or alter the input data. For example, a smoothing algorithm might require weight factors that influence the degree of smoothing. In the second step, our formulae (T1.3) and (T1.4) show how control parameters like or (, ß) come into the control rule itself. Our model (T1.5) introduces another control parameter p at the third step, and our estimation algorithm calls for still another parameter to select a quantile in the posterior distribution for Qn+1.
After all these choices have been made, the final algorithm boils down to a function (1) that determines the quota from available data. This has to be tested with simulations to see how well it would work under various realistic scenarios. Which choices in the four-step design process really matter? Can particular choices be matched with scenarios where they work well or poorly? We explore these questions more fully in the Discussion.
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Lessons from quantum mechanics |
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TopIntroductionExample with catch and...Estimation modelDesign choicesLessons from quantum mechanicsDiscussionAppendix Dynamic model...References |
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Problems reconciling reference points and management strategies bear a striking resemblance to issues that accompanied the development of quantum mechanics. Herbert (1987, pp. 41–53) describes four historical versions of quantum theory known as matrix mechanics (by Heisenberg), wave mechanics (by SchrÖdinger), transformation theory (by Dirac), and the sum over histories approach (by Feynman). All have a strong focus on actual observations, stemming from the original work by Heisenberg. The tricky part deals with the mathematical details of unobservable processes that create these observations. In Feynman's theory, events that actually happen are influenced by all possible events that might have happened. Herbert (1987, p. 53) quotes the well-known physicist Freeman Dysan (a colleague of Feynman): "Thirty-one years ago, Dick Feynman told me about his ‘sum over histories’ version of quantum mechanics. ‘The electron does anything it likes,’ he said. ‘It just goes in any direction, at any speed, forward or backward in time, however it likes, and then you add up the amplitudes, and it gives you the wave function.’ I said to him ‘You're crazy.’ But he isn't."
Like Feynman contemplating the paths an electron might take, fishery managers need to consider the possible trajectories of marine stocks. The exact behaviour of an electron between two observed events remains unknowable in quantum mechanics, just as the exact dynamics of a population cannot be known in the past or predicted in the future. Feynman invented a mathematical technique called the path integral to compute a sum over histories (Feynman and Hibbs, 1965). In fisheries, Bayesian posterior samples have become a standard method of representing the range of histories and predictions. As we have shown, this leads to a distribution of possible quotas corresponding to a stated policy. Physicists continue to debate the reality of ideas embedded in their mathematical models, which can seem crazy and illogical. As Feynman (1985, p. 10) says: "They've learned to realize that whether they like a theory or they don't like a theory is not the essential question. Rather, it is whether or not the theory gives predictions that agree with experiment." Similarly, the issue in fisheries is not whether reference points such as those in (T1.4) exist or particular models such as (T1.5) accurately represent stock biomass, but whether or not policies developed from these ideas define good management strategies.
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Discussion |
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TopIntroductionExample with catch and...Estimation modelDesign choicesLessons from quantum mechanicsDiscussionAppendix Dynamic model...References |
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Our example illustrates a framework for designing management strategies based conceptually on reference points. We focus primarily on the algorithm (1) used to calculate a fishery quota from available data. The model in Table 1 gives technical details for a potential strategy similar to a method used by the International Whaling Commission (IWCSC, 1994a, b). We make no claims about its effectiveness in practice. Also, we have not included details about two other key components in a complete management strategy evaluation (MSE): (i) simulation models with realistic biological complexity to test proposed strategies, and (ii) objective functions used to evaluate the results. In practice, MSE usually focuses on pragmatic (rather than mathematically optimal) policies, robust to a spectrum of biological scenarios and acceptable to a community of stakeholders.
Readers of the literature will find a diverse vocabulary associated with MSE. For example, some authors refer to a management procedure, rather than to a management strategy as we have termed the concept represented by (1). Complex simulation models to test management strategies are sometimes called operating models. On a more general level, our quota for an entire fishery might instead be called a total allowable catch (TAC). We have tried to define precisely all terms used in this paper, but they may not agree with terminology used elsewhere for similar ideas.
At the risk of proliferating acronyms, we want to distinguish MSE from classical stock assessment (CSA), in which a complex biological model is used to estimate reference points that guide policy. Strict advocates of CSA take these reference points as legitimate aspects of reality and alter management plans accordingly. By contrast, strict advocates of MSE use reference points estimated from simple models as mathematical artefacts in the definition of an algorithm. These quantities may or may not have meaning in reality, and they almost certainly differ from seemingly comparable reference points estimated from a CSA model. However, that does not matter because the goal is a robust management algorithm. Here lies the link with quantum mechanics, where intermediate calculations might seem meaningless, but still produce useful outcomes. From the perspective of MSE, the complex CSA model might be just one of many operating models used to test a proposed management strategy. Alternatively, the CSA model itself might be formalized as a strategy that can be tested against other operating models. In this way, MSE can become a testing framework for CSA.
We recognize that fishery scientists may not be strict advocates of MSE, CSA, or any other approach. Given the uncertainties inherent in most fishery data, analysts usually like to compare results from several different viewpoints. Although MSE sounds attractive, its success depends on the discovery of a good strategy, a goal that might prove elusive.
The problem posed here represents perhaps the simplest challenge for MSE, one that goes back to early literature on catch and cpue data (e.g. Schaefer, 1954, 1957; Pella and Tomlinson, 1969). How should a history of catches and indices be used to determine the next fishery quota? So far, fishery science has not agreed on a sensible default answer (or spectrum of answers) to this question. Other fields, such as economics, have arrived at decision rules that routinely appear in textbooks. For example, a formula devised by Black and Scholes (1973) gives a rule for determining a fair call price C for an option contract that gives its owner the right, without obligation, to buy a stock at a future time T for a specified strike price K. If the current stock price is S, the prevailing interest rate is r, and the stock price moves lognormally in small time-steps dt with variance 2 dt, then a continuous-time stochastic model gives a formula for the call price (Luenberger, 1998, pp. 355–356)
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| (7) |
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denotes the standard cumulative normal distribution function with (–) = 0 and (+) = 1. Schnute (2005) discusses this result in the context of fishery research and cites an example where the formula apparently failed in practice (Lowenstein, 2001). In 1997, after Black died, Scholes and another collaborator Merton won the Nobel prize in economics "for a new method to determine the value of derivatives", including (7). This mathematical formula illustrates the goal of MSE for fishery management: an explicit algorithm to make a decision based on available data. It might not always be a good decision, but it would provide a standard against which to evaluate other possibilities. In the current fishery literature, examples of MSE often carry the burden of details associated with a particular context. Consequently, they lack the generality needed to address questions such as that raised in the quota calculation (1).
What will students learn from fishery management textbooks in 10, 20, or even 50 years? If MSE progresses successfully, they may find a specific formula for calculating a quota from catch and index data, extensively tested by many independent investigators using a large number of simulation models. Or perhaps they will learn several formulae appropriate for different circumstances. The textbooks might also prove that under some conditions no algorithm can be trusted. In the meantime, investigators can continue the search for quota algorithms that work well in specified contexts and gain a broad level of acceptance in practice.
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Appendix Dynamic model properties |
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TopIntroductionExample with catch and...Estimation modelDesign choicesLessons from quantum mechanicsDiscussionAppendix Dynamic model...References |
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Model (T1.5) has historic roots in the differential equation proposed by Pella and Tomlinson (1969, p. 423)
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| (A.1) |
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| (A.2) |
Following Schnute and Richards (2002), we use a superscript asterisk to denote quantities associated with MSY conditions. To show that h* in (T1.5) corresponds to the MSY harvest rate h* = C*/B*, consider the equilibrium catch equation (3) that expresses catch C in terms of biomass B. The derivative condition dC/dB = 0 gives the formula (T1.7) for the biomass B* that produces maximum catch. Substituting the value B = B* into (3) gives the maximum catch C* = h*B*. This proves that h* has its intended biological meaning.
The special case p = 0 in (T1.9) follows from (T1.5) and the limit formula limp 0(1 – xp/p = – log x, which can be verified using standard arguments from calculus. Similarly, (T1.10) follows from (T1.7) and the formula limp0(1 + p)1/p = e.
To find conditions under which the value B = K is a stable fixed point for the dynamic equation (T1.5) , define
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| (A.3) |
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| (A.4) |
. This condition reduces to h* < 2/(1 + p) when p > –1 and 0 < h* < 1. Thus, the upper bound in (T1.8) guarantees a stable carrying capacity K. One case in Figure 1a (p = 5, h* = 0.5) has 
, and this steep negative slope would produce unstable dynamic behaviour around B = K.
For p > –1, the function f (B) is concave, as demonstrated by the derivative condition
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0, Bmax e1/h* K, consistent with (T1.9) .
For –1 < p < 0, the shape of the curve depends on the value of r in (A.2), which is negative in this case. If r < –1, Bmax is properly defined by (4), and the curve is dome-shaped. If r –1, a condition equivalent to (5), the curve has no maximum, and f (B) as B . Also, the slope (A.4) satisfies the condition 
, so that B = K is a stable equilibrium point.
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Acknowledgements |
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We thank Doug Butterworth for his interesting talk and subsequent discussions at the Fourth World Fisheries Congress in May 2004. He also gave us a preprint of his related paper (Butterworth, in press) and, acting as a reviewer of this article, made excellent suggestions for improving the final draft. Sean Cox, Bill de la Mare, Eva Plagányi, André Punt, Laura Richards, and Keith Sainsbury have also assisted and encouraged our work on management strategies. In acknowledging valuable contributions from these colleagues, we recognize that they may not agree fully with the views expressed here.
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References |
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TopIntroductionExample with catch and...Estimation modelDesign choicesLessons from quantum mechanicsDiscussionAppendix Dynamic model...References |
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