Analytical models for fishery reference points

Transcription

1 515 Analytical models for fishery reference points Jon T. Schnute and Laura J. Richards Abstract: Fishery reference points are widely applied in formulating harvest management policies. We supply precise mathematical definitions for several reference points in common use. We then derive analytical expressions for these quantities from age-structured population models. In particular, we explain how the maximum sustainable harvest rate and catch (h*, C*), two quantities of management importance, can replace the classical recruitment parameters (α, β) in the Beverton Holt and Ricker recruitment curves. We also demonstrate dependencies of various reference points on subsets of model parameters. Although our analysis is restricted to special cases, our models still have general utility. For example, simple calculations from analytical formulas enable checks on the output from more complex models and guide the choice of reference points for fishery management. Résumé : Certains paramètres de référence sont couramment employés pour la formulation des politiques d exploitation des pêches. Nous donnons une définition mathématique précise de divers paramètres de référence d usage courant et nous développons les formules d analyse correspondantes à partir de modèles de populations à structure par âges. Plus particulièrement, nous expliquons comment deux paramètres importants en gestion, soit le taux d exploitation maximal et les captures (h*, C*), peuvent remplacer les paramètres de recrutement classiques (α, β) dans les fonctions de recrutement de Beverton Holt et de Ricker. Nous montrons aussi le lien de dépendance qui existe entre divers paramètres de référence et des sous-ensembles de paramètres de modélisation. Bien que notre analyse ne porte que sur des cas spéciaux, nos modèles sont quand même d une utilité générale. Par exemple, de simples calculs, effectués avec les formules d analyse, permettent de vérifier les résultats de modèles plus complexes et guident le choix des paramètres de référence à retenir pour la gestion d une pêche. [Traduit par la Rédaction] Introduction Mathematical models have been used extensively to estimate fish population parameters from fishery data. These parameters typically govern the dynamics of recruitment, mortality, and growth. From a management perspective, the underlying population dynamics must be translated into decision rules, such as the biomass of fish that can safely be removed by the fishery. Thus, management focus shifts from biological parameters to quantities that more directly guide the decision process. Such quantities are commonly termed reference points. International agreements (e.g., United Nations 1995) now include specific recommendations on the use of precautionary reference points for fish stock conservation. Caddy and Mahon (1995), Caddy and McGarvey (1996), and Garcia (1994, 1996) discuss practical applications of this approach. Any reference point ultimately depends on the parameters that govern population dynamics. For example, the catch that can safely be removed must somehow depend on the population s recruitment patterns. In this paper, we give precise mathematical definitions for several proposed reference points, based on concepts of extinction, maximum sustainable yield, yield per recruit (Deriso 1982, 1987), and spawners per recruit (Shepherd 1982; Clark 1991, 1993; Mace 1994). We then derive analytical expressions for these quantities from Received April 2, Accepted June 11, J13940 J.T. Schnute 1 and L.J. Richards. Fisheries and Oceans Canada, Science Branch, Pacific Biological Station, 3190 Hammond Bay Road, Nanaimo, BC V9R 5K6, Canada. 1 Author to whom all correspondence should be addressed. schnutej@dfo-mpo.gc.ca Can. J. Fish. Aquat. Sci. 55: (1998) age-structured population models. Thus, at least for certain special cases, we express modern reference points in terms of classical biological parameters. Our formulas provide an intuitive and quantitative link between biology and fishery management. This paper consolidates a number of significant developments in the history of fishery assessment models. It includes yield per recruit analyses comparable with those originally proposed by Beverton and Holt (1957), as well as the recruitment models described by Schaefer (1954), Beverton and Holt (1957), Ricker (1958), Deriso (1980), and Schnute (1985). The age-structured models here contain some of the features discussed by Gulland (1965), Pope (1972), Fournier and Archibald (1982), and Deriso et al. (1985). We also derive moment equations similar to those found by Deriso (1980), Schnute (1985, 1987), and Fournier and Doonan (1987), as reviewed by Hilborn and Walters (1992, chap. 9). Following Deriso (1982, 1987), we use these moment equations to obtain analytic expressions for modern reference points. In particular, our analysis extends recent work by Schnute and Kronlund (1996), who showed that classical recruitment curves could be written explicitly in terms of management parameters. They considered only the case motivated by salmonid species, in which adults die after spawning. Here, we model agestructured populations that spawn multiple times, and our result includes that of Schnute and Kronlund (1996) as a special case. Analytical formulas for reference points in age-structured models require somewhat restrictive assumptions. We present two underlying models. The first deals only with population numbers, while the second describes stock biomass and fish growth. Because the second model contains the first as a special case, we could theoretically have confined our discussion

2 516 Table 1. Deterministic age-structured model for fish numbers, assuming knife-edged recruitment and maturity, with age at recruitment equal to age at maturity. Parameters (T1.1) F = (α, β, γ, δ) (T1.2) β >0;0<δ 1 (T1.3) α > δ Model 1: Age-structured population (T1.4) N 1t =αs t r (1 βγs t r ) 1/γ (T1.5) N at = (1 δ)(n a 1, t 1 C a 1, t 1 ); a 2 (T1.6) C at = h t N at (T1.7) (T1.8) (T1.9) (T1.10) S t = (N at C at ) Moments R t = N 1t P t = Nat C t = Cat Model 1A: Moment equations (T1.11) R t =αs t r (1 βγs t r ) 1/γ (T1.12) P t = R t + (1 δ)s t 1 (T1.13) (T1.14) C t = h t P t S t = P t C t entirely to the second. Unfortunately, the algebra and resulting formulas are much more complex in the general case, and we use the first model to develop intuition about the problem. Furthermore, by considering two models, we can investigate the role of key assumptions in the analytical results. To keep the paper compact and give the reader perspective on our analyses, we present most of our assumptions and results in tables. We distinguish equations in tables by using the prefix T. Thus, the symbols (2.3) and (T2.3) denote the third equation in section 2 and Table 2, respectively. We confine all mathematical proofs to Appendices A E. Section 1 defines our first model and presents corresponding moment equations and equilibrium conditions. Section 2 provides mathematical definitions for various reference points and expresses them analytically in terms of parameters for model 1. Section 3 extends these results to model 2. We discuss further implications of our analysis in section Age-structured model Our first model (model 1 in Table 1) consists of the four equations (T1.4) (T1.7). These represent an age-structured population that experiences recruitment, harvest, and survival. The matrix N at denotes the number of fish in age-class a at the start of year t. Similarly, C at indicates the number caught from ageclass a during year t. Fish are fully recruited into age-class 1, which corresponds to the actual age r. Thus, the spawning Can. J. Fish. Aquat. Sci. Vol. 55, 1998 stock S t r from year t rgives birth to recruits N 1t in year t. The first equation (T1.4) expresses this relationship with the recruitment function (1.1) f (S; α, β, γ) =αs(1 βγs) 1/γ. As discussed by Schnute (1985), (1.1) reduces to functions proposed by Beverton and Holt (1957), Ricker (1958), or Schaefer (1954) when γ is fixed at 1, 0, or 1, respectively. The parameters (α, β, γ) in (T1.4) have simple biological interpretations: α measures productivity at low stock sizes, β is a population scale parameter, and γ determines the overall shape of the recruitment curve. In particular, α and γ are dimensionless. The units of β relate to population measurements. For example, if S is measured in thousands of fish, then β has units 10 3 fish 1. For age-classes a 2, the second equation (T1.5) expresses the population N at as survivors from the population N a 1, t 1 in the preceding year. First, the catch C a 1, t 1 is removed; then the remaining population experiences a death rate δ, so that the fraction 1 δ survives. Fishery literature often uses a parameter M to quantify natural mortality. In this context, δ can be interpreted as (1.2) δ=1 e M. For small values of M, (1.2) implies the approximation δ M. The third equation (T1.6) relates the catch C at to the harvest rate h t in year t and the population N at. Thus, the fraction h t acts as a control variable in this model, related to the classical fishing mortality F t by (1.3) h t = 1 e F t. Again, h t F t for small F t. The final equation (T1.7) assumes that the stock S t available for future recruitment consists of all survivors from the fishery. Model 1 contains several simplifying assumptions. It represents a population with knife-edged selectivity by the fishery. New recruits at age r (age-class 1) enter the population at the start of each year t. Along with survivors from the previous year, these are equally vulnerable to the fishery, where the catch at age a is proportional to abundance at age a. After the fishery, the recruited population also comprises the spawning stock, and spawning occurs prior to natural mortality. Thus, the model assumes a regular annual cycle of recruitment, fishing, spawning, and natural mortality. In particular, fishing and natural mortality do not occur simultaneously. We make these assumptions to keep the model as simple as possible, although we could develop similar analyses starting from different assumptions about the population life cycle. Conceptually, model 1 specifies the process component of a state space model (Schnute 1994). The four equations recursively define the state variables N at, C at, and S t in terms of parameters (α, β, γ, δ) and controls h t. The controls must be fractions (1.4) 0 h t 1. Similarly, the parameters (T1.1) are subject to the constraints (T1.2). Also, the constraint (T1.3) is required to guarantee that the population can survive (Appendix D). For example, in the salmonid case contemplated by Ricker (1958), adults die when they spawn (δ =1). The condition α > 1 ensures that each adult produces at least one juvenile for the next generation. When

3 Schnute and Richards 517 some adults survive spawning (δ < 1), α can be less than 1, as long as the production rate dominates the death rate (α > δ). As shown in Appendix A, model 1 implies a corresponding model 1A (Table 1) for the four annual quantities S t, R t, P t, and C t, defined by (T1.7) (T1.10), respectively. The four equations (T1.11) (T1.14) give a sequential algorithm for computing the recruitment R t, population P t before fishing, catch C t, and spawning stock S t after the fishery. Both models 1 and 1A involve the four parameters (T1.1) and the harvest rate controls h t. Results that can be proved for model 1A automatically apply to model 1. Furthermore, model 1A has an obvious biological interpretation independent of its derivation from model 1. Thus, the simpler moment model 1A could also be considered a starting point for our analysis. If the harvest rate sequence h t is fixed at a constant value h, then both models 1 and 1A have equilibrium solutions that can be obtained by dropping the index t from the model equations. For example, model 1A then reduces to four simultaneous equations in the four constant states R, P, C, and S. Solving these equations (Appendix B) gives (T2.3) (T2.6) in Table 2. We simplify these results by introducing two new variables, the annual survival rate (1.5) σ t = (1 δ)(1 h t ), which is the product of survivals from natural and fishing mortality, and the recruit per spawner ratio (1.6) ρ t = R t /S t. For a given harvest rate h, equations in Table 2 can be applied sequentially to obtain analytical expressions for equilibrium values of all quantities in models 1 and 1A. These include the equilibrium rates (σ, ρ) obtained from (1.5) (1.6), the four states (R, P, C, S) in model 1A, and the age-structured population (N a ) in model Reference points The equilibrium results in Table 2 can be used to investigate how a population governed by model 1 or 1A responds to a long-term harvest rate h. For example, if (2.1) (α, β, γ, δ) = (1.42, 0.56, 1.0, 0.2), then Figs. 1A 1D represent P, C, C/R, and S/R, respectively, as functions of h. We choose δ=0.2 in (2.1) as typical for a marine species that is available to the fishery for years after recruitment. The parameter γ= 1 selects a Beverton Holt recruitment curve, and we explain below the significance of our choice for (α, β). Case 1 in Table 3 lists all parameters and other quantities associated with Fig. 1. From Fig. 1A, the population P declines with increasing harvest rate h, and extinction occurs when h = The total catch C in Fig. 1B increases to a maximum value 1 when h = 0.25 and then declines to zero at the extinction rate h = The parameters (α, β) in (2.1) have been chosen so that (h, C) = (0.25, 1.0) is the maximum point in Fig. 1B. The maximum C = 1 can be interpreted as one unit of population, which might be measured in thousands or millions of fish. Figure 1C portrays the yield per recruit (YPR) C/R as an increasing function of h, although the curve applies only to the range 0 h < 0.55 in which h is kept below the extinction level. Similarly, Fig. 1D shows that the number of spawners per Table 2. Equilibrium population states expressed as a function of constant harvest rate h, based on the model in Table 1 with ρ=r/s. (T2.1) σ(h) = (1 δ)(1 h) (T2.2) ρ(h)= 1 σ 1 h (T2.3) R(h) = ρ 1 ρ γ βγ α (T3.4) P(h) = R 1 σ (T2.5) (T2.6) (T2.7) C(h) = hp S(h) = (1 h)p N a (h) =σ a 1 R recruit (SPR) S/R declines steadily across the valid range of h. Because S/R is the reciprocal of the equilibrium value ρ from (1.6), it follows that ρ is an increasing function of h. The population trends illustrated in Fig. 1 apply generally to models 1 and 1A, given parameters (α, β, γ, δ) that meet the constraints (T1.2) (T1.3). Intuitively, the total population declines to an extinction level with increasing harvest rate (Fig. 1A). The catch, necessarily zero when h is zero or the extinction rate, follows a dome-shaped curve between these two points (Fig. 1B). The number caught per available recruit steadily increases as h increases (Fig. 1C). Similarly, a higher catch per recruit leaves fewer spawners per recruit (Fig. 1D). Based on these simple ideas, a variety of reference points have been proposed as measures for comparison with possible management actions. Table 4 presents mathematical definitions for common reference values of harvest rate h and catch C. For consistency with the framework being developed here, we use somewhat different notation from that in other literature. Equation (T4.1) defines h as the harvest rate that drives the population to extinction. For example, h = 0.55 in Fig. 1A (Table 3, case 1). The dagger symbol in this notation serves as a reminder of mortality and danger. We use an asterisk in (T4.2) to denote the harvest rate h* that gives the maximum catch C*, which is also termed the maximum sustainable yield (MSY). For example, (h*, C*) = (0.25,1.0) in Fig. 1B (Table 3, case 1). Similarly, the prime symbol in (T4.3) indicates the harvest rate h that maximizes the yield per recruit C/R. The corresponding catch C is defined by (T4.4). In both (T4.2) and (T4.3), the maximum must be taken across the range 0 h h because the equilibrium population P(h) = 0 for h h. Although h* <h (Fig. 1B), models 1 and 1A imply that h occurs at the upper limit h (Fig. 1C). Thus, for these models, h =h and C =0, and the maximum YPR has no interesting biological significance. Later in this paper, we show that the concepts in (T4.3) (T4.4) have greater relevance for biomass models that include fish growth. The remaining reference values (T4.5) (T4.8) in Table 4 are defined relative to an equilibrium population that experiences no harvest. The YPR curve in Fig. 1C has a slope that declines steadily as h increases from zero. Given a fraction x, the harvest rate h x in (T4.5) corresponds to the value h where the slope of this curve equals x times the slope at h = 0. Line segments in Fig. 1C illustrate this concept when x = 0.2. The reference point (h 0.2, C 0.2 ) = (0.31, 0.96) from (T4.5) (T4.6)

4 518 Can. J. Fish. Aquat. Sci. Vol. 55, 1998 Fig. 1. Equilibrium quantities P, C, C/R, and S/R from Table 2 expressed as functions of h with fixed parameters (C*, h*, δ, γ) = (1, 0.25, 0.2, 1), as in Table 3, case 1. Dotted lines correspond to (A) P(h*) and h*, (B) C* and h*, (C) h 0.2, and (D) (S/R) h and h Line segments in 0.35 Fig. 1C indicate slopes of the YPR relationship at h = 0 and h = h 0.2. Broken curves in Figs. 1C and 1D indicate values of h > h. corresponds to a harvest level somewhat higher than the MSY rate h* (Table 3, case 1). Similarly, the SPR curve in Fig. 1D declines steadily as h increases from zero. Given a fraction y, the harvest rate h y in (T4.7) corresponds to the value h at which the ratio S/R equals y times its value at h = 0. Figure 1D illustrates the choice y = 0.35, where the corresponding reference point (h 0.35, C 0.35 ) = (0.27, 1.00) from (T4.7) (T4.8) approximates the MSY point (h*, C*) (Table 3, case 1). The relationship h = 1 e F from (1.3) associates each harvest rate defined in Table 4 with a fishing mortality rate F. The corresponding F-value typically has a precedent in existing fishery literature. For example, the extinction rate h corresponds to F crash (ICES 1996; Cook et al. 1997). Similarly, h*, h, and h y are equivalent to F msy, F max, and F 100y% (Caddy and Mahon 1995). Our reference value h x, like F x in Deriso (1987), is computed from a ratio of slopes on the YPR curve. However, due to a technical difference (described below) in defining the slopes, h x and F x are not related by (1.3). For this reason, we use the reference value h 0.2 in Fig. 1C, rather than the common choice F 0.1 elsewhere. A close examination of definitions clarifies the relationship between h x and F z for different fractions x and z. We obtain h x by plotting YPR C/R in relation to h whereas F z comes from a plot of C/R versus F. Because h is a nonlinear function of F, these two graphs do not have proportional slopes. Define (2.2) h z = 1 e F z to be the harvest rate corresponding to the conventional reference value F z. Then (Appendix E) h x = h z if and only if either of the following two equivalent conditions hold: z (2.3) x =, z = x(1 h 1 h x ). z We show below that our reference value h x leads to simpler analytical results than h z, although (2.3) can be used to convert z to x and vice versa. The definitions in Table 4 apply to any fishery model for which the equilibrium catch C(h), recruitment R(h), and spawning stock S(h) can be expressed as functions of a constant harvest rate h. In general, these can be obtained numerically

5 Schnute and Richards 519 Table 3. Parameters and reference values for the Beverton Holt (γ= 1) and Ricker (γ=0) stock recruitment functions with C* = 1, two choices for (h*, δ), and additional parameters (, w, κ) = (1, 5, 0) for model 2. δ=0.2 δ=0.4 Model 1 Model 2 Model 1 Model 2 γ= 1 γ=0 γ= 1 γ=0 γ= 1 γ=0 γ= 1 γ=0 Case 1 a 2 3 b 4 c α β h* h h h h C* C C C a Case associated with Fig. 1. b Case associated with Fig. 2. c Case associated with Fig. 3. Table 4. Definitions of fishery reference values. Extinction (T4.1) P(h ) = 0 Maximum sustainable yield (MSY) (T4.2) (T4.3) (T4.4) C* = C(h*) = max C(h) 0 h h Yield per recruit (YPR) C(h ) R(h ) = max C(h) 0 h h R(h) C =C(h ) (T4.5) d(c/r) dh = x d(c/r) h=h x dh ; 0 x 1 h=0 (T4.6) C x = C(h x ) Spawners per recruit (SPR) (T4.7) (T4.8) (S/R) h=h y = y(s/r) h=0; 0 y 1 C y = C(h y ) from simulations. For example, with given parameters (T1.1) and constant harvest rate h, the dynamic equations (T1.4) (T1.7) of model 1 can be run sequentially until the moments (T1.7) (T1.10) reach equilibrium. Repeating this process for multiple values of h gives equilibrium functions that can be graphed to obtain the reference points illustrated in Fig. 1. In this paper, we avoid simulation by deriving analytical formulas for the reference points in Table 4. Table 5 relates h*, C*, h, h x, and h y to the parameters (α, β, γ, δ) in model 1. Although the MSY point (h*, C*) cannot be expressed analytically in terms of (α, β), equations (T5.2) (T5.3) give the opposite relationship: (2.4) (h*, C*) (α, β). For simplicity, these equations include the survival rate σ* in Table 5. Key equations for biological reference values derived from the equilibrium model in Table 2. (T5.1) (T5.2) (T5.3) β= σ* = (1 δ)(1 h*) α= 1 σ 1 + γh 1 h 1 σ (T5.1) associated with the MSY harvest h*. The example in Fig. 1 illustrates their application, where parameters (α, β) in (2.1) are computed by (T5.2) (T5.3) from the MSY point (h*, C*) = (0.25,1.0). More generally, this approach makes it possible to express the recruitment function (1.1) for model 1 directly in terms of (h*, C*). Schnute and Kronlund (1996) discuss the management implications of this function in the context of salmon populations, where adults die after spawning. When δ=1, our results (T5.2) (T5.3) reduce to theirs (Schnute and Kronlund 1996, equations (T3.3) (T3.4), p. 1283). Conversely, our analysis extends theirs to marine populations in which adults may survive to spawn multiple times. Reference values h, h, h x, and h y for model 1 can be computed from the simple analytic expressions (T5.4) (T5.6). The extinction rate h depends only on the productivity parameter α and the death rate δ. In particular, the condition (T1.3) guarantees that h > 0. Because h x and h y are derived from yield and spawners per recruit, these values depend only on δ and 1/γ h 2 (1 h )(1 σ +γh )C (T5.4) h = h = α δ 1 +α δ (T5.5) (T5.6) h x = δ x 1/2 1 x 1/2 1 δ ; δ α (1 +α δ)<x1/2 < 1 δ(1 y) h y = δ+(1 δ)y ; δ α < y < 1

6 520 Table 6. Explicit formulas for quantities associated with MSY, based on the equilibrium model in Table 2 when γ= 1 (Beverton Holt recruitment curve). (T6.1) ρ =(αδ) 1/2 (T6.2) h = (αδ)1/2 δ 1 +(αδ) 1/2 δ (T6.3) C = 1 β (α1/2 δ 1/2 ) 2 (T6.4) R = α1/2 β (α1/2 δ 1/2 ) (T6.5) P = 1 βδ 1/2 (α1/2 δ 1/2 )(1 +(αδ) 1/2 δ) (T6.6) S = 1 βδ 1/2 (α1/2 δ 1/2 ) the choice of fraction x or y, independent of the recruitment function (1.1). However, because the extinction rate depends on α, lower bounds for x and y also depend on α. The fractions x and y can always be adjusted to give equivalent reference points. For example, given x, h y = h x if (2.5) y = δ(1 h x). δ+(1 δ)h x The classical reference value h z in (2.2) can be expressed analytically for model 1 as (2.6) h z = δ 1 δ [δ 2 + 4z(1 δ)] 1/2 δ 2z(1 δ). 2z(1 δ) This result comes from a quadratic equation in h z (Appendix E). By contrast, a linear equation in h x gives the simpler expression (T5.5), which we prefer to (2.6). As stated earlier, the transformations (2.3) can be used to convert between x and z corresponding to equivalent reference values h x = h z. For the Beverton Holt recruitment curve (γ = 1) in model 1, simple analytic formulas can be obtained for all population variables associated with MSY (Table 6). Consistent with our earlier notation (h*, C*) for the MSY point, we denote other MSY state variables with an asterisk. In particular, the recruit to spawner ratio ρ* = R*/S* in this case is simply the geometric mean (T6.1) of the production rate α and death rate δ. Equations (T6.2) (T6.3) provide an inverse of the transformation (2.4) in (T5.2) (T5.3) when γ= 1. Thus, in the Beverton Holt context, h* and C* can be written explicitly in terms of (α, β, δ). The role of the dimensioned parameter β becomes clear in Table 6, where all population variables (C*, R*, P*, S*) are scaled to 1/β. 3. Weight-structured model Our second model (model 2 in Table 7) extends the first by including population weight structure. We assume that the weight w a of fish at age a conforms to the von Bertalanffy growth function (Ricker 1975, p. 221) (3.1) w a = w[1 e K(a a 0 ) ] with three parameters (w, K, a 0 ). This function can also be Table 7. Deterministic age-structured model that incorporates fish growth, where all population variables (3.3) have units of biomass. Parameters (T7.1) F = (α, β, γ, δ,, w, κ) (T7.2) β >0;0<δ 1; w ;0 κ<1 (T7.3) α (1 κ)(1 δ) δ+(1 κ)(1 δ) w >δ 1 Weights (T7.4) w a = w (w ) κ a 1 (T7.5) w a = (1 κ)w +κw a 1 written in the equivalent form (T7.4) with parameters (, w, κ), where is the weight of the first age-class from (3.1) and (3.2) κ=e K. Thus, the parameters (T7.1) for model 2 include the four parameters (T1.1) in model 1, plus the three growth parameters in (T7.4). Furthermore, (T7.4) implies the recursive relationship (T7.5), in which w a is an average of w and w a 1, weighted by 1 κ and κ, respectively. The four equations (T7.6) (T7.9) that define model 2 are identical to (T1.4) (T1.7) for model 1, except that the survival equation (T7.7) contains the weight adjustment factor w a /w a 1. w Model 2: Age-structured biomass (T7.6) N 1t =αs t r (1 βγs t r ) 1/γ (T7.7) (T7.8) (T7.9) (T7.10) (T7.11) (T7.12) N at = w a w a 1 (1 δ)(n a 1,t 1 C a 1,t 1 ) ; a 2 C at = h t N at S t = (N at C at ) Moments R t = N 1t (T7.13) W t = P t = Nat C t = Cat P t (N at /w a ) Model 2A: Moment equations (T7.14) R t =αs t r (1 βγs t r ) 1/γ (T7.15) (T7.16) (T7.17) (T7.18) P t = R t + (1 κ)w +κw t 1 W t 1 R t W t = P t +(1 δ) S 1 t 1 W t 1 C t = h t P t S t = P t C t Can. J. Fish. Aquat. Sci. Vol. 55, 1998 (1 δ)s t 1

7 Schnute and Richards 521 This factor accounts for biomass growth from age a 1 to age a. We redefine all population variables (3.3) N at, P t, R t, S t, C at, C t in model 2 to be measures of biomass, rather than numbers as in model 1. In particular, we use N at to denote biomass at age. Similarly, P t denotes total population biomass in place of the common notation B t (Hilborn and Walters 1992). We make this choice to avoid prolific notation, with replacement letters for each variable in (3.3). Three of the seven model parameters (T7.1) are dimensioned: β has units of inverse biomass, and and w have units of fish weight. Biomass and weight typically have different units. For example, individual fish and population biomass might be measured in kilograms and tonnes, respectively. Model parameters must conform to the obvious constraints (T7.2). In particular, because K > 0 in (3.1), κ in (3.2) must be a fraction. We allow the special case κ=0 (3.4) w a =, a = 1 w, a 2 which corresponds to the limit in (T7.4) as κ 0. In effect, fish reach the asymptotic weight w at age 2, so that only two weight-classes are present in the population. Incidentally, the case κ=1 is degenerate in (T7.4) because the free parameter w then cancels from the equation. Model 2 reduces to model 1 when (3.5) = w. In this case, (T7.4) implies constant weight w a = w ; consequently, w a /w a 1 = 1, and equations (T1.5) and (T7.7) become identical. The constraint (T7.3) for model 2 also reduces to the simpler constraint (T1.3) when = w. Intuitively, (T1.3) guarantees that a model 1 population can survive because recruitment exceeds losses from natural mortality. More generally (Appendix D), the constraint (T7.3) ensures that a model 2 population can survive because the combined effects of recruitment and growth exceed the loss of biomass through natural mortality. To construct moment equations for model 2, we define moments (T7.10) (T7.12) analogous to those for model 1. We also define the mean weight W t in (T7.13) as the ratio of total biomass P t to total population numbers. In particular, because N at denotes biomass, the total population at the start of year t is Σ a (N at /w a ). The assumptions and definitions (T7.6) (T7.13) of model 2 imply model 2A: five moment equations (T7.14) (T7.18) that recursively update the five population variables R t, P t, W t, C t, and S t (Appendix A). Intuitively, the survival equation (T7.15) comes partly from the growth relationship (T7.5). After 1 year, the mean size W t 1 grows to the new size (1 κ)w +κw t 1. The weight ratio in (T7.15) accounts for the growth of survivors (1 δ)s t 1 from year t 1. In the compact form listed in Table 7, model 2A is new to fisheries literature. It has the desirable property that the population mean weight W t responds naturally to the effects of recruitment R t and harvest rate h t. Logical connections between models 2 and 2A mirror those between models 1 and 1A. Results from model 2A automatically apply to model 2, and the simpler model 2A could be taken as a starting point for our analysis. Given a constant harvest rate h, equilibrium versions of the Table 8. Equilibrium population states expressed as a function of constant harvest rate h, based on the model in Table 7. (T8.1) σ(h) = (1 δ)(1 h) (T8.2) W(h)= (1 σ) +(1 κ)σw 1 κσ (T8.3) ρ(h) = 1 σ W 1 h (T8.4) R(h)= ρ 1 ρ γ βγ α (T8.5) P(h)= W R 1 σ (T8.6) (T8.7) (T8.8) C(h) = hp S(h) = (1 h)p N a (h)= w a σ a 1 R five equations in model 2A can be solved to obtain equilibrium functions W(h), R(h), P(h), C(h), and S(h) (Appendix B). Results of this analysis are summarized in Table 8, where we also include equilibrium versions of σ t and ρ t defined in (1.5) (1.6) and the age-structured biomass N at from model 2. In particular, the mean weight W(h) can be expressed as an average of and w, weighted in proportion to 1 σ and (1 κ)σ, respectively. Table 8 makes it easy to produce graphs for model 2 similar to those devised for model 1. For example, Fig. 2 corresponds to case 3 in Table 3. The parameters (h*, C*, γ, δ) = (0.25, 1.0, 1, 0.2) are identical to those used in Fig. 1 (Table 3, case 1); however, the choice (, w,, κ) = (1, 5, 0) corresponds to a growth relationship (3.4) in which fish older than age-class 1 have five times the weight of new recruits. At least two features distinguish Fig. 2 from Fig. 1. First, the extinction rate h = 0.61 is higher (Figs. 2A and 2B), due to the benefits of biomass growth. Second, the YPR curve (Fig. 2C) is dome shaped, with a maximum YPR achieved when h =0.38. A harvest rate higher than h reduces the proportion of large fish and, consequently, reduces the yield available from fish growth. Model 1, which does not consider growth, inherently lacks this feature (e.g., Fig. 1C). Equilibrium equations for model 2 can be used to derive analytic expressions for reference points, similar to those obtained for model 1 (Appendices C E). Table 9 illustrates some of the possibilities. Equations (T9.1) (T9.5) extend (T5.1) (T5.3) from model 1 to model 2, based on the MSY mean weight W* and another quantity Q* (discussed below). Thus, as in (2.4), α and β can be computed explicitly from the parameter vector (3.6) F* = (h*, C*, γ, δ,, w, κ) by the sequential algorithm (T9.1) (T9.5). In effect, (h*, C*) can replace (α, β) in the parameter vector (T7.1) for model 2. For example, these equations have been used to compute (α, β) for each case in Table 3 from specified MSY parameters (C* = 1; h* = 0.25 or 0.35). To avoid excessive complexity, the remaining reference points in Table 9 are obtained analytically only for the special

8 522 Can. J. Fish. Aquat. Sci. Vol. 55, 1998 Fig. 2. Equilibrium quantities P, C, C/R, and S/R from Table 8 expressed as functions of h with fixed parameters (C*, h*, δ, γ,, w, κ) = (1, 0.25, 0.2, 1, 1, 5, 0), as in Table 3, case 3. Dotted lines correspond to (A) P(h*) and h*, (B) C* and h*, (C) h 0.2, and (D) (S/R) h 0.35 Line segments in Fig. 2C indicate slopes of the YPR relationship at h = 0 and h = h 0.2. Broken curves in Figs. 2C and 2D indicate values of h > h. and h case κ=0 in (3.4). Furthermore, under the constant-weight condition (3.5), model 2 reduces to model 1, and the results in Table 9 simplify to those in Table 5. When (3.5) holds, W* = in (T9.2) and any quantity denoted Q in Table 9 becomes zero. From these facts, it is fairly easy to see by inspection that, given (3.5), expressions for α, β, h, h x, and h y in Table 9 reduce to their counterparts in Table 5. When κ=0, equations (T9.6) (T9.12) in Table 9 show the dependencies of various reference points on the remaining six parameters (α, β, γ, δ,, w ) from (T7.1). In particular, h depends only on (α, δ,, w ). Furthermore, as in Table 5, h, h x, and h y are independent of the recruitment parameters (α, β, γ). A modal YPR harvest rate h exists only if the condition w > /(1 δ) in (T9.8) holds. This condition can be given biological meaning in light of the growth function (3.4). The population consists of recruits with weight and survivors with weight w. The survivor weight must be large enough to compensate for biomass lost by the death of recruits. So far, our discussion of Table 3 has been confined to cases 1 and 3, used to construct Figs. 1 and 2. More generally, we examine two values δ, corresponding to relatively longlived (δ =0.2) and short-lived (δ =0.4) populations. We use values of h* close to δ, based on the concept that populations can sustain harvest rates near the natural mortality rate. Thus, (h*, δ) = (0.25, 0.2) in cases 1 4 and (h*, δ) = (0.35, 0.4) in cases 5 8. We also investigate Beverton Holt and Ricker recruitment functions, where the former has been used in both Figs. 1 and 2. For comparison, case 4 corresponds to a Ricker function with other parameters (3.6) the same as in case 3. Figure 3 illustrates equilibrium plots for case 4, which are similar to those in Fig. 2, except that the convexity of P(h) has changed. Reference values h 0.2 and h 0.35 for the examples in Table 3 are consistently more precautionary than h* for model 2 and less precautionary than h* for model 1. In particular, when δ=0.4, the value x = 0.2 in h x for model 1 (cases 5 and 6) violates the constraint in (T5.5). As discussed above, reference values h x and h y are independent of the recruitment parameters

9 Schnute and Richards 523 (α, β, γ). However, corresponding catch values C x and C y vary slightly with the choice of Beverton Holt or Ricker function. Furthermore, the extinction rate h is consistently lower for the Ricker function. 4. Discussion Models 1 and 2 provide a conceptual framework for modern stock assessment, including the analysis of fishery reference points. These models represent the basic processes of recruitment, fishing, natural mortality, and growth from which more realistic models can be developed. Although models 1 and 2 contain simplifying assumptions, we believe that they still have general utility. In particular, simple models serve to highlight important model features, such as dependencies of reference points on subsets of model parameters. For example, we have illustrated how reference values h x and h y in Tables 5 and 9 do not depend on the recruitment function, although admissible values x and y depend (through h ) on the productivity parameter α. Our appendices illustrate the derivation of analytical reference points from basic theoretical concepts. Thus, simple model assumptions lead to many interesting analytical results, such as the relationship (2.5) between h x and h y for a given x. Our proofs can be used to guide the exploration of other reference point definitions in the context of models similar to those discussed here. In addition, the logical framework here ties together a disparate literature on fisheries stock assessment. For simplicity, our models assume that natural mortality occurs after the fishery, in contrast with the typical Baranov assumption of concurrent fishing and natural mortality. As in (1.3), we express results in terms of harvest rates h, rather than fishing mortalities F. We distinguish reference values with mathematical accents to achieve compactness. Other less cryptic notation could be devised, such as h MSY = h*, h YPR, x = h x, and h SPR, y = h y. With appropriate choices of model parameters, our analytical results can be used to check the output from more realistic models. For example, although we have assumed knife-edged selectivity, age a = 1 could approximate the average age at recruitment for a population in which the fishery selectivity varies with age. Similarly, given the simplifying assumption κ=0 in (3.4) and Table 9, the weight parameter w could be selected to approximate the average population weight, exclusive of new recruits. The same approximation can be used for growth laws other than the von Bertalanffy relationship (T7.4). The classical YPR analysis of Beverton and Holt (1957) involves two control parameters: fishing mortality rate F and the first recruitment age r. We have considered models with only one control parameter, the harvest rate h = 1 e F. However, our models could easily be extended to include age r as a control in the classical sense. This could be achieved by including a death rate parameter in the recruitment equations (T1.4) and (T7.6) and by redefining as a function of r. Schnute and Kronlund (1996) argued that the transition (2.4) resulted in parameters (h*, C*) that had greater relevance to management policy and that possessed improved statistical properties relative to (α, β). However, they considered a simplified version of model 1 in which fish die after spawning. We have extended their analysis to multiple spawning Table 9. Key equations for biological reference points derived from the equilibrium model in Table 8. (T9.1) (T9.2) σ* = (1 δ)(1 h*) W = (1 σ ) +(1 κ)σ w 1 κσ (T9.3) Q = 1 1 σ W 1 κσ (T9.4) α= 1 σ 1 + γ(1 + Q )h W 1 h 1 σ (T9.5) β= (T9.6) Q = populations that experience von Bertalanffy growth. For these more general models, we have shown that the transition (2.4) can still be expressed analytically, so that (h*, C*, γ) canbe used as the set of recruitment parameters. One benefit of (2.4) is obvious from Table 3. With fixed values of (h*, C*, γ), the resulting values (α, β) are highly variable and difficult to interpret. In contrast, (h*, C*) represent reference points with clear biological meaning and widespread interest in fishery management. In the existing literature, some results have been obtained from stochastic differential equations for continuous processes (Beddington and May 1977; Thompson 1994). To achieve the analytical results here, we have confined our analysis to discrete deterministic process equations. These can be expanded into full state space models, which include stochastic error and observation equations (Schnute 1994; Schnute and Richards 1995; Richards et al. 1997). Thus, our analytical results might provide initial approximations to reference points and harvest strategies obtained from elaborate stochastic simulation models. One unique feature of model 2A is the mean weight moment equation (T7.16). This equation appears attractive for actual analyses because mean weights are commonly 1/γ (1 + Q )h 2 (1 h )[1 σ +γ(1 + Q )h ]C Special case k50 4α(1 δ) (1 +α δ) 2 w 1 (T9.7) h 1 = 1 +α δ (T9.8) h = δ 1 δ (T9.9) h x = δ 1 δ (T9.10) (T9.11) (T9.12) α δ+ w δ(w ) Q [1 +(1 + Q ) 1/2 ] 2 1/2 1 ; w > 1 1 δ 1/2 w δ(1 x)(w )+xw 1 ω= δ +(1 δ)w Q y = 4δ(w / 1)(1 δ)ωy [δ+(1 δ)ωy] 2 h y = 1 δ+(1 δ)ωy δ(1 ωy) + ωyq y [1 +(1 + Q y ) 1/2 ] 2

10 524 Can. J. Fish. Aquat. Sci. Vol. 55, 1998 Fig. 3. Equilibrium quantities P, C, C/R, and S/R from Table 8 expressed as functions of h with fixed parameters (C*, h*, δ, γ,, w, κ) = (1, 0.25, 0.2, 0, 1, 5, 0), as in Table 3, case 4. Dotted lines correspond to (A) P(h*) and h*, (B) C* and h*, (C) h 0.2, and (D) (S/R) h 0.35 Line segments in Fig. 3C indicate slopes of the YPR relationship at h = 0 and h =h 0.2. Broken curves in Figs. 3C and 3D indicate values of h > h. and h measured in fisheries data. Previous analyses (Deriso 1980; Schnute 1985) used second difference equations in P t rather than adding a weight moment (Hilborn and Walters 1992, p. 335). Although Schnute (1987) and Schnute et al. (1989) examined various weight-based moment measures, they did not include an explicit update equation analogous to (T7.16) for the mean weight W t. Here, we treat W t simply as another state variable in the model. Model 1A can similarly be extended to include an update equation analogous to (T7.16) for the mean age A t in year t, where (4.1) A t = anat /P t. It can be proved (Appendix A) from model 1 that (4.2) A t = 1 +(1 δ) S t 1 A P t 1. t To extend model 1A, insert (4.2) between equations (T1.12) and (T1.13). The resulting system sequentially updates the five variables R t, P t, A t, C t, and S t. This simple model could be used as a starting point for analysis of a fish stock with available age data. Our reference point definitions in Table 4 include only a few of the common approaches applied in fishery management. We have attempted to generalize our results with generic fractions x, y, and z, rather than specific choices, as in F 0.1. Furthermore, the fractions x, y, and z can be adjusted to match almost any reference point based on a constant harvest rate. Thus, our relationships provide a method for evaluating sensible choices. Other common reference points could also be defined and analyzed by our methods. However, some reference points such as F low, F med or F rep, and F high (Sissenwine and Shepherd 1987; Jakobsen 1992) are obtained from the observed ratio of recruits to spawners. These depend on estimates from observed data and so are not amenable to analytic expressions. The definitions in Table 4 are derived from equilibrium models. Much current fisheries research is directed to how fish

11 Schnute and Richards 525 populations respond to varying climatic conditions, including regime shifts (Beamish 1995). Clearly, future work should consider stochastic uncertainty associated with reference point definitions. For example, climatic-induced changes in weight at age could alter the growth parameters (, w, κ). In this case, reference points from Table 9 could take very different values under different climatic regimes. More generally, our analytical expressions provide a framework for exploring how stochasticity might affect reference point calculations. Thus, our methods can help guide the choice of reference points and link biology to fishery management. Acknowledgements We thank Grant Thompson and Carl Walters for helpful reviews. We dedicate this paper to David Cook on the occasion of his retirement as Editor of this journal. For more than a decade, our papers have benefited from his insights and guidance. References Beamish, R.J. (Editor) Climate change and northern fish populations. Can. Spec. Publ. Fish. Aquat. Sci. No Beddington, J.R., and May, R.M Harvesting natural populations in a randomly fluctuating environment. Science (Washington, D.C.), 197: Beverton, R.J.H., and Holt, S.J On the dynamics of exploited fish populations. Fish. Invest. Ser. 2 No. 19. Caddy, J.F., and Mahon, R Reference points for fisheries management. FAO Fish. Tech. Pap. No Caddy, J.F., and McGarvey, R Targets or limits for management of fisheries. N. Am. J. Fish. Manage. 16: Clark, W.G Groundfish exploitation rates based on life history parameters. Can. J. Fish. Aquat. Sci. 48: Clark, W.G The effect of recruitment variability on the choice of a target level of spawning biomass per recruit. Proceedings of the International Symposium on Management of Exploited Fish Populations. Alaska Sea Grant Rep pp Cook, R.M., Sinclair, A., and Stefánsson, G Potential collapse of North Sea cod stocks. Nature (Lond.), 385: Deriso, R.B Harvesting strategies and parameter estimation for an age-structured model. Can. J. Fish. Aquat. Sci. 37: Deriso, R.B Relationship of fishing mortality to natural mortality and growth at the level of maximum sustainable yield. Can. J. Fish. Aquat. Sci. 39: Deriso, R.B Optimal F 0.1 criteria and their relationship to maximum sustainable yield. Can. J. Fish. Aquat. Sci. 44(Suppl. 2): Deriso, R.B., Quinn, T.J., II, and Neal, P.R Catch age analysis with auxiliary information. Can. J. Fish. Aquat. Sci. 42: Fournier, D., and Archibald, C.P A general theory for analyzing catch at age data. Can. J. Fish. Aquat. Sci. 39: Fournier, D.A., and Doonan, I.J A length-based stock assessment method utilizing a generalized delay-difference model. Can. J. Fish. Aquat. Sci. 44: Garcia, S.M The precautionary principle: its implications in capture fisheries management. Ocean Coastal Manage. 22: Garcia, S.M Stock recruitment relationships and the precautionary approach to management of tropical shrimp fisheries. Mar. Freshwater Res. 47: Gulland, J.A Estimation of mortality rates. Annex to Arctic Fisheries Working Group Report. ICES C.M Doc. No. 3. Hilborn, R., and Walters, C.J Quantitative fisheries stock assessment: choice, dynamics, and uncertainty. Chapman and Hall, New York. ICES Report of the comprehensive fishery evaluation working group. ICES CM 1996/Asses:20. Jakobsen, T Biological reference points for North-East Arctic cod and haddock. ICES J. Mar. Sci. 49: Mace, P.M Relationships between common biological reference points used as thresholds and targets of fisheries management strategies. Can. J. Fish. Aquat. Sci. 51: Pope, J.G An investigation of the accuracy of virtual population analysis using cohort analysis. ICNAF Res. Bull. 9: Richards, L.J., Schnute, J.T., and Olsen, N Visualizing catch age analysis: a case study. Can. J. Fish. Aquat. Sci. 54: Ricker, W.E Handbook of computations for biological statistics of fish populations. Bull. Fish. Res. Board Can. No Ricker, W.E Computation and interpretation of biological statistics of fish populations. Bull. Fish. Res. Board Can. No Schaefer, M.B Some aspects of the dynamics of populations important to the management of the commercial marine fisheries. Inter-Am. Trop. Tuna Comm. Bull. 1: Schnute, J A general theory for the analysis of catch and effort data. Can. J. Fish. Aquat. Sci. 42: Schnute, J A general fishery model for a size-structured population. Can. J. Fish. Aquat. Sci. 44: Schnute, J.T A general framework for developing sequential fisheries models. Can. J. Fish. Aquat. Sci. 51: Schnute, J.T., and Kronlund, A.R A management oriented approach to stock recruitment analysis. Can. J. Fish. Aquat. Sci. 53: Schnute, J.T., and Richards, L.J The influence of error on population estimates from catch age analysis. Can. J. Fish. Aquat. Sci. 52: Schnute, J.T., Richards, L.J., and Cass, A.J Fish growth: investigations based on a size-structured model. Can. J. Fish. Aquat. Sci. 46: Shepherd, J.G A versatile new stock recruitment relationship for fisheries, and the construction of sustainable yield curves. J. Cons. Int. Explor. Mer, 40: Sissenwine, M.P., and Shepherd, J.G An alternative perspective on recruitment overfishing and biological reference points. Can. J. Fish. Aquat. Sci. 44: Thompson, G.G A general diffusion model of stock recruitment systems with stochastic mortality. ICES C.M. 1994/T:18. United Nations Agreement for the implementation of the provisions of the United Nations Convention on the Law of the Sea of 10 December 1982 relating to the conservation and management of straddling fish stocks and highly migratory fish stocks. U.N. Gen. Assembly Doc. A/CONF.164/37. Appendix A. Moment equations The paper s assumptions are contained in the age-structured models 1 and 2. From these, we derive moment equations (Appendix A), equilibrium conditions (Appendix B), and reference points for MSY (Appendix C), extinction (Appendix D), and YPR and SPR (Appendix E). We focus primarily on model 2 because results for model 1 usually follow as special cases from the additional assumption (3.5). We begin by proving the equations (T7.14) (T7.18) of model 2A from the assumptions (T7.6) (T7.9) and moment definitions (T7.10) (T7.13) of model 2. Equations (T7.14), (T7.17), and (T7.18) are straightforward consequences of (T7.6), (T7.8), and (T7.9), respectively, given the definitions (T7.10) (T7.12). The calculation

12 526 Can. J. Fish. Aquat. Sci. Vol. 55, 1998 (A.1) (A.2) (A.3) P t = N 1t + Na+1,t P t = R t + w a+1 (1 δ)(n w a,t 1 C a,t 1 ) a (1 κ)w +κw a P t = R t +(1 δ)(1 h t 1 ) N w a,t 1 a P t = R t +(1 δ)(1 h t 1 ) (1 κ)w P t 1 W t 1 +κp t 1 P t = R t + (1 κ)w +κw t 1 W t 1 (1 δ)s t 1 proves (T7.15). In this calculation, (A.1) uses (T7.4) and (T7.8), (A.2) incorporates the result Σ a (N at /w a ) = P t /W t from (T7.13), and (A.3) uses the equality (1 h t )P t = S t from (T7.8) (T7.9). Similarly, the calculation (A.4) (A.5) W t = P t N at w a W t = P t R t 1 = P R t t + +(1 δ)(1 h t 1 ) N a+1,t w a+1 N a,t 1 w a W t = P t R t +(1 δ)(1 h t 1 ) P t 1 W t 1 R t W t = P t +(1 δ) S 1 t 1 W t 1 proves (T7.16), where (A.4) and (A.5) depend on the definition (T7.13) of mean weight. This completes the proof of model 2A. Given the equality = w in (3.5), assumptions (T7.6) (T7.9) of model 2 become identical to (T1.4) (T1.7) in model 1. Furthermore, (T7.4) then implies that w a = w for every age a. Thus, from (T7.13), (A.6) W t = P t w 1 N at a = w. Substituting (A.6) into (T7.15) gives (T1.12). Because (3.5) and model 2 are equivalent to model 1, this completes the proof that model 1 implies model 1A. The additional moment equation (4.2) for model 1A follows from the calculation A t = 1 P t anat = 1 P R t + t (a + 1)Na+1,t A t = 1 P R t +(1 δ)(1 h t 1 ) t (a + 1)Na,t 1 W t = 1 [R P t +(1 δ)(1 h t 1 )(P t 1 A t 1 + P t 1 )] t W t = 1 P t [R t +(1 δ)s t 1 (1 + A t 1 )] W t = 1 +(1 δ) S t 1 A P t 1, t based on equations (4.1) and (T1.5) (T1.8). Appendix B. Equilibrium With a constant harvest rate h, equilibrium versions of (T7.14) (T7.18) give five equations in R, P, W, C, and S. We use these to prove the results in Table 8. Equations (T8.6) (T8.7) are immediate consequences of (T7.17) (T7.18). Define ρ=r/s, so that (T7.16) can be written as (B.1) 1 W = 1 P = ρs + w 1 ρ(1 h) + (1 δ)s W (1 δ)(1 h) W where S/P = 1 h from (T8.7). Solve (B.1) for ρ and apply the definition (T8.1) of σ to prove (T8.3). Next, divide the equilibrium version of (T7.15) by S to obtain (B.2) 1 1 h =ρ+(1 κ)w +κw (1 δ) W = W 1 σ 1 h + (1 κ)w +κw W, σ 1 h, based on the expressions (T8.1) and (T8.3) for σ and ρ, respectively. Solving (B.2) for W gives (T8.2). Similarly, divide the equilibrium version of (T7.14) by S to obtain (B.3) ρ=α(1 βγr/ρ) 1/γ, and solve (B.3) for R to prove (T8.4). Starting from (T8.7), the calculation (B.4) P = S 1 h = R ρ(1 h) = W 1 h R 1 σ1 h = W R 1 σ proves (T8.5), where (B.4) uses the result (T8.3). Finally, the equilibrium equation (T7.7), that is, (B.5) N a = w a (1 δ)(1 h)n w a 1 = w a σn a 1 w a 1, a 1 inductively implies (T8.8), starting from the condition N 1 = R. This completes the proof of all equations in Table 8. If = w, as in (3.5), then W = in (T8.2); consequently, equations (T2.2) (T2.6) in Table 2 follow from their counterparts (T8.3) (T8.7) in Table 8. Also, (3.5) implies that w a = in (T7.4); thus, (T8.8) proves (T2.7) in this case and completes the proof of Table 2. Appendix C. MSY We begin by proving (T9.4) (T9.5) in Table 9 from definitions (T9.1) (T9.3) and equilibrium results in Table 8. Functions of