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DISCUSSION / DISCUSSION
Reply to the comment by Francis on ‘‘Differences in predicted catch composition between two widely used catch equation formulations’’1 Trevor A. Branch
Abstract: Francis (2010. Can. J. Fish. Aquat. Sci. 67: 763–765) writes a thoughtful response detailing concerns with my suggestion that the continuous (Baranov) catch formulation is preferable to the discrete catch formulation when fishing mortality is high (T.A. Branch. 2009. Can. J. Fish. Aquat. Sci. 66: 126–132). He suggests the discrete formulation allows for multiple gear encounters and that formulation choice should depend on which formulation better fits the data. Here I first distinguish between gear selectivity and availability and then show that our two views are complementary: the original assumes fish groups with differing gear selectivity but full availability, whereas Francis assumes fish groups fully selected by fishing gear but with differing availability. I maintain that the discrete formulation only models a single instantaneous interaction between fish and fishing gear and therefore only part of the population can be caught if fish groups have equal gear selectivity that is less than 100%, whereas under the same assumptions, the continuous formulation would allow the entire population to be caught. Finally, when the balance between gear selectivity and availability is unknown, I agree that formulation choice could be driven by model fits to the data, although formulation choice could also be based on how the fishery operates. Re´sume´ : Francis (2010. J. can. sci. halieut. aquat. 67 : 763–765) a re´pondu de fac¸on bien argumente´e en exprimant ses inquie´tudes concernant ma suggestion que la formulation continue (de Baranov) des captures est pre´fe´rable a` la formulation discontinue lorsque la mortalite´ due a` la peˆche est e´leve´e (T. A. Branch. 2009. J. can. sci. halieut. aquat. 66 : 126–132). Il indique que la formulation discontinue peut eˆtre utilise´e lorsque les poissons font face a` plusieurs engins de peˆche et que le choix de formulation devrait de´pendre de quelle formulation s’accorde mieux aux donne´es. Je distingue ici d’abord entre la se´lectivite´ et la disponibilite´ des engins et de´montre ensuite que nos vues sont comple´mentaires : l’original pre´suppose des groupes de poissons avec une se´lectivite´ diffe´rente des engins de peˆche, mais avec une pleine disponibilite´, alors que Francis pre´suppose des groupes de poissons avec une pleine se´lectivite´ des engins, mais avec diffe´rentes disponibilite´s. Je maintiens que la formulation discontinue ne mode´lise qu’une seule interaction instantane´e entre les poissons et les engins de peˆche et qu’en conse´quence seulement une partie de la population peut eˆtre capture´e si les se´lectivite´s des engins sont e´gales pour les groupes de poissons et infe´rieures a` 100 %; avec les meˆmes pre´suppositions, la formulation continue permettrait la capture de l’ensemble de la population. Enfin, lorsque l’e´quilibre entre la se´lectivite´ des engins et la disponibilite´ est inconnue, je suis d’accord que le choix de la formulation pourrait se baser sur l’ajustement du mode`le aux donne´es, bien que le choix de la formulation puisse aussi se baser sur le mode d’ope´ration de la peˆche. [Traduit par la Re´daction]
In my original paper (Branch 2009a, 2009b), I discussed how the choice of catch equation (continuous vs. discrete) has a marked impact on predicted catch composition when gear selectivity differs among groups of fish. These groups could be species, length, sex, or age groups, but in the general discussion that follows, I will refer to them as age groups for simplicity. Given differences between the two
catch formulations in both predicted catch composition and the proportion of the stock that can be caught at high fishing mortality, I suggested that the continuous formulation was preferable at high fishing mortality levels where multiple gear encounters could occur. Francis (2010) responds with two particular criticisms: first that the discrete formulation does not in fact preclude multiple gear encounters, and sec-
Received 2 January 2010. Accepted 8 March 2010. Published on the NRC Research Press Web site at cjfas.nrc.ca on 26 March 2010. J21596 Paper handled by Associate Editor Terrance Quinn II. T.A. Branch. School of Aquatic and Fishery Sciences, Box 355020, University of Washington, Seattle, WA 98195, USA (e-mail:
[email protected]). 1Appears
in Can. J. Fish. Aquat. Sci. 66: 126–132.
Can. J. Fish. Aquat. Sci. 67: 766–768 (2010)
doi:10.1139/F10-024
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Branch
ond, that the choice between the two formulations should be based on which one fits the data better. In this response, I first define gear selectivity and availability, show that the arguments advanced by Branch (2009a) and Francis (2010) are not in opposition but are complementary, and show how the two approaches can be combined by including parameters for both gear selectivity and availability. To understand the complementary nature of the two papers, some clarification is needed on the definition of ‘‘selectivity’’. Branch (2009a) defined ‘‘selectivity’’ to be synonymous with gear selectivity (the relative proportion of fish of different ages available to the fishing gear that are caught), whereas Francis (2010) defined ‘‘selectivity’’ as encompassing both gear selectivity and availability (where availability is the absolute proportion of fish of different ages that are in the same area as the fishing fleet and not separated spatiotemporally). Here I will specifically refer to these two components as ‘‘gear selectivity’’ and ‘‘availability’’, while calling their combined effects ‘‘catchability’’ (the relative proportion of fish of different ages in the entire population that are caught by the fishing gear). In doing so, I gloss over a long history of debate over these terms that is touched on by Marr (1951), Beverton and Holt (1957), MacCall (1986), and Millar and Fryer (1999). Having defined these terms, Branch (2009a) can be seen as focusing on the case in which gear selectivity (Sa) differs among ages, but all ages are fully available to the fishing fleet (Aa = 1), whereas Francis (2010) focuses on the case in which all fish are fully selected by the fishing gear (Sa = 1) but they differ in their availability to the fishing fleet (0 £ Aa £ 1). If gear selectivity differs among ages, then the continuous and discrete formulations give different predictions. However, if all ages have a gear selectivity of one and availability differs, then both formulations predict the same catch composition, as pointed out by Francis (2010), and he is correct in saying that the discrete formulation is a better choice than the continuous formulation. However, his first criticism, that the discrete formulation can model multiple interactions with the fishery within each time period, is incorrect. That the two formulations predict the same catch composition for full gear selectivity (Sa = 1 for all a) does not imply that the discrete formulation allows for multiple interactions with the fishery within each time period. On the contrary, the discrete formulation explicitly models the population first being affected by natural mortality, followed by an instantaneous single pulse of fishing at time t* (one interaction), and finally further natural mortality until the end of the time period. The effects of this assumption are most easily seen with a contrived example in which all ages are equally but not fully selected by the fishing gear (e.g., Sa = 0.5 for all a). In this case, the two formulations would predict identical catch compositions, but the discrete formulation would allow 50% of the available population to be harvested at the maximum exploitation rate (ut = 1), in contrast to the continuous formulation that would allow the entire available population to be harvested at maximum instantaneous fishing mortality (Ft ? ?). These predictions differ because the discrete formulation only allows for a single interaction between fish and fishery. The second criticism of Francis (2010) is that the choice of formulation should not be based on assumptions regard-
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ing the conduct of the fishery (i.e., whether multiple gear interactions occur), but on whichever formulation fits the data better. In light of the above discussion about gear selectivity and availability, our respective views are complementary. If gear selectivity is the most important facet of catchability, then the continuous catch formulation is preferable at high fishing mortalities (Branch 2009a), whereas if availability is the most important facet of catchability, then the discrete catch formulation should be used (Francis 2010). But what should we do when both gear selectivity and availability are important, or we do not know their relative importance? In Branch (2009a, 2009b), I proposed using a hybrid formulation from Megrey (1989) that combines the features of the continuous and discrete formulations, i.e., gear selectivity is exponentiated together with fishing mortality, whereas availability is not exponentiated, ð1Þ
Naþ1;tþ1 ¼ Na;t Aa eMSa Ft þ Na;t ð1 Aa ÞeM 0 Aa 1; 0 Sa
ð2Þ
Ca;t ¼
Sa Ft Aa Na;t ð1 eMSa Ft Þ M þ Sa Ft
which shows how numbers N at age a and time t change for given gear selectivity S, availability A, fishing mortality F, and natural mortality M, and how these quantities affect catches C over the time period from t to t + 1. Francis (2010) correctly points out that in practice, the gear selectivity and availability parameters in eqs. 1–2 are likely to be confounded and difficult to estimate separately and therefore recommends choosing whichever of the continuous or discrete formulations fits the data better. As an inveterate modeler, I generally support this view, which seems a reasonable way forward for most stock assessments given that some stock assessment programs such as CASAL (Bull et al. 2005) and Stock Synthesis II (Methot 2005) allow for easy switching between the discrete and continuous formulations. However, I have some sympathy for the alternative argument that formulation choice should arise from basic assumptions about how a particular fishery operates, not just model fits to the data. In a few instances, there may be data to model both gear selectivity and availability. Pribac et al. (2005) included a fixed (not estimated) gear selectivity curve based on previous gear experiments and then estimated availability from the data. The gillnets used in this fishery for gummy sharks (Mustelus antarcticus) are known to be highly size selective, and availability clearly also differs by age, given marked discrepancies between fishery length frequencies and survey length frequencies. Dorn and Methot (1990) and Methot and Dorn (1995) describe a complementary model for Pacific whiting (Merluccius productus) in which gear selectivity is estimated, whereas the relative availability of Pacific whiting to the Canadian and US fisheries in a given year is prespecified using data from acoustic surveys. Hybrid models of this kind may become more common in the future when, for example, gear selectivity is estimated externally to the model using gear experiments (Millar and Fryer 1999) and availability is estimated internally in the model, or alternatively availability is estimated externally and gear selectivity is estimated within the model. Published by NRC Research Press
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The logical end point of this progression is a model that internally estimates both gear selectivity and availability from available data. I have a paper in preparation that models both for run reconstructions of Bristol Bay sockeye salmon (Oncorhynchus nerka) in which gear selectivity can be estimated from differences between catch length frequencies and escapement length frequencies (Kendall and Quinn 2009; Kendall et al. 2009), and availability by stock (fish headed for different rivers) can be estimated from genetic data allocating catches to each stock (Dann et al. 2009).
References Beverton, R.J.H., and Holt, S.J. 1957. On the dynamics of exploited fish populations. Fisheries Investigations, Ministry of Agriculture, Fisheries and Food (G.B.), Series II, Vol. 19. Branch, T.A. 2009a. Differences in predicted catch composition between two widely used catch equation formulations. Can. J. Fish. Aquat. Sci. 66: 126–132. doi:10.1139/F08-196. Branch, T.A. 2009b. Corrigendum: Differences in predicted catch composition between two widely used catch equation formulations. Can. J. Fish. Aquat. Sci. 66: 1631. doi:10.1139/F09-107. Bull, B., Francis, R.I.C.C., Dunn, A., McKenzie, A., Gilbert, D.J., and Smith, M.H. 2005. CASAL (C++ algorithmic stock assessment laboratory): CASAL user manual v2.07–2005/06/23. NIWA Technical Report No. 126. Dann, T.H., Habicht, C., Jasper, J.R., Hoyt, H.A., Barclay, A.W., Templin, W.D., Baker, T.T., West, F.W., and Fair, L.F. 2009. Genetic stock composition of the commercial harvest of sockeye salmon in Bristol Bay, Alaska, 2006–2008. Alaska Department of Fish and Game Fishery Manuscript Series No. 09-06. Dorn, M.W., and Methot, R.D. 1990. Status of the Pacific whiting resource in 1989 and recommendations to management in 1990. Alaska Fisheries Science Center, National Marine Fisheries Service, National Oceanic and Atmospheric Administration, NOAA Tech. Memo. NMFS F/NWC-182. Francis, R.I.C.C. 2010. Comment on ‘‘Differences in predicted
Can. J. Fish. Aquat. Sci. Vol. 66, 2010 catch composition between two widely used catch equation formulations’’. Can. J. Fish. Aquat. Sci. 67: 763–765. Kendall, N.W., and Quinn, T.P. 2009. Effects of populationspecific variation in age and length on fishery selection and exploitation rates of sockeye salmon (Oncorhynchus nerka). Can. J. Fish. Aquat. Sci. 66: 896–908. doi:10.1139/F09-047. Kendall, N.W., Hard, J.J., and Quinn, T.P. 2009. Quantifying six decades of fishery selection for size and age at maturity in sockeye salmon. Evol. Appl. 2: 523–536. doi:10.1111/j.1752-4571. 2009.00086.x. MacCall, A.D. 1986. Virtual population analysis (VPA) equations for nonhomogeneous populations, and a family of approximations including improvements on Pope’s cohort analysis. Can. J. Fish. Aquat. Sci. 43: 2406–2409. Marr, J.C. 1951. On the use of the terms abundance, availability and apparent abundance in fishery biology. Copeia, 1951: 163– 169. doi:10.2307/1437549. Megrey, B.A. 1989. Review and comparison of age-structured stock assessment models from theoretical and applied points of view. In Mathematical analysis of fish stock dynamics. Edited by E.F. Edwards and B.A. Megrey. American Fisheries Society, Bethesda, Maryland. pp. 8–48. Methot, R. 2005. Technical description of the Stock Synthesis II assessment program. NOAA Fisheries, Seattle, Washington, SEDAR 16-AW-04. Methot, R.D., and Dorn, M.W. 1995. Biology and fisheries of North Pacific hake (M. productus). In Hake: biology, fisheries and markets. Edited by J. Alheit and T.J. Pitcher. Chapman & Hall, London, UK. pp. 389–414. Millar, R.B., and Fryer, R.J. 1999. Estimating the size-selection curves of towed gears, traps, nets and hooks. Rev. Fish Biol. Fish. 9: 89–116. doi:10.1023/A:1008838220001. Pribac, F., Punt, A.E., Taylor, B.L., and Walker, T.I. 2005. Using length, age and tagging data in a stock assessment of a length selective fishery for gummy shark (Mustelus antarcticus). J. Northwest Atl. Fish. Sci. 35: 267–290. doi:10.2960/J.v35.m521.
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