Some explorations of the life history ratios to describe length composition, spawning-per-recruit, and the spawning potential ratio

ICES Journal of Marine Science ICES Journal of Marine Science (2015), 72(1), 204– 216. doi:10.1093/icesjms/fst235 Original Article Some explorations of the life history ratios to describe length composition, spawning-per-recruit, and the spawning potential ratio Adrian Hordyk 1*, Kotaro Ono 2, Keith Sainsbury 3, Neil Loneragan1, and Jeremy Prince 1,4 1 Centre for Fish, Fisheries and Aquatic Ecosystems Research, Murdoch University, 90 South Street, Murdoch, WA 6150, Australia School of Aquatic and Fisheries Science, University of Washington, Seattle, Washington, USA 3 University of Tasmania, Tasmania, Australia 4 Biospherics, P/L, South Fremantle, Western Australia, Australia 2 *Corresponding author: tel: +61 893606685; e-mail: . Hordyk, A., Ono, K., Sainsbury, K., Loneragan, N., and Prince, J. Some explorations of the life history ratios to describe length composition, spawning-per-recruit, and the spawning potential ratio. – ICES Journal of Marine Science, 72: 204 –216. Received 29 August 2013; accepted 10 December 2013; advance access publication 15 January 2014. Evaluating the status of data-poor fish stocks is often limited by incomplete knowledge of the basic life history parameters: the natural mortality rate (M), the von Bertalanffy growth parameters (L1 and k), and the length at maturity (Lm). A common approach to estimate these individual parameters has been to use the Beverton– Holt life history invariants, the ratios M/k and Lm/L1, especially for estimating M. In this study, we assumed no knowledge of the individual parameters, and explored how the information on life history strategy contained in these ratios can be applied to assessing data-poor stocks. We developed analytical models to develop a relationship between M/k and the von Bertalanffy growth curve, and demonstrate the link between the life history ratios and yield- and spawning-per-recruit. We further developed the previously recognized relationship between M/k and yield- and spawning-per-recruit by using information on Lm/L1, knife-edge selectivity (Lc/L1), and the ratio of fishing to natural mortality (F/M), to demonstrate the link between an exploited stock’s expected length composition, and its spawning potential ratio (SPR), an internationally recognized measurement of stock status. Variation in length-at-age and logistic selectivity patterns were incorporated in the model to demonstrate how SPR can be calculated from the observed size composition of the catch; an advance which has potential as a cost-effective method for assessing data-poor stocks. A companion paper investigates the effects of deviations in the main assumptions of the model on the application of the analytical models developed in this study as a cost-effective method for stock assessment [Hordyk, A. R., Ono, K., Valencia, S., Loneragan, N. R., and Prince, J. D. 2015. A novel length based empirical estimation method of spawning potential ratio (SPR), and tests of its performance, for small-scale, data-poor fisheries. ICES Journal of Marine Science, 72: 217 – 231]. Keywords: Beverton–Holt invariants, dimensionless numbers, fish growth, life history, mortality. Introduction Studies of the life history information on the growth, mortality, and maturity schedules of fish stocks form the basis of fisheries science, and such information is essential for the effective management of exploited populations (Beverton and Holt, 1957; Hilborn and Walters, 1992). However, this information is only available for about 1200 of the 7000 or so exploited fish species, and it is not always available for all regions where the species are fished (Froese and Binohlan, 2000). Patterns of variation between life history parameters, particularly between natural mortality (M), and the von Bertalanffy growth rate and asymptotic size (k and L1 ), have long been observed for many fish populations, and a large amount of work has been done to document and understand these relationships, often with the aim of providing cost-effective “short-cuts” to estimate the parameters, such as natural mortality, that are difficult or expensive to obtain (Beverton and Holt, 1959; Pauly, 1980; Roff, 1984; Beverton, 1992; Charnov, 1993). Typically, these studies have explored the relationships between single # International Council for the Exploration of the Sea 2014. All rights reserved. For Permissions, please email: 205 Some explorations of the life history ratios variables, such as M or k, that are commonly used in stock assessments. The M/k ratio is one of the life history ratios that is known to be relatively consistent between closely related stocks (Beverton, 1992). The results of bio-energetic modelling by Jensen (1996) suggest that an optimal value for M/k is 1.5. Assuming von Bertalanffy growth, the relationship between the biomass and mean length of a cohort can be derived from the M/k ratio (Beverton, 1992; Jensen, 1996). Using this relationship, life history theory predicts that length at maturity (Lm) will be found at the length of maximum biomass in the population. This theoretical relationship provides the basis for the second life history ratio Lm /L1 , with an optimal value of 0.66 (Jensen, 1996). Supported by the other empirical work (Pauly, 1980; Charnov, 1993), these values for the ratios of M/k and Lm /L1 have become known as the Beverton–Holt life history invariants (BH-LHI), and are often used as proxies for estimates of M/k (1.5) and Lm /L1 (0.66) in data-poor stocks. For example, it is not uncommon for studies to estimate the notoriously difficult parameter M by simply using the Beverton–Holt invariant M ¼ 1.5k (e.g. see many papers citing Jensen, 1996), or by estimating the fishing mortality rate F when the total mortality (Z ¼ F + M) and k are known, by assuming that M/k ¼ 1.5 (Beddington and Kirkwood, 2005). Extending their earlier seminal work (Beverton and Holt, 1957), Beverton and Holt (1964) were the first to show that yield and biomass equations could be restated using the dimensionless ratios M/k and Lm /L1 . Since then, others have built on their work to incorporate the dimensionless numbers into fisheries science (Jensen, 1996; Mangel, 1996; Williams and Shertzer, 2003; Beddington and Kirkwood, 2005). However, it has often been assumed that these ratios are invariant, which is evidently not the case. For example, a recent study of the Kyphosid Scorpis aequipinnis, in southwest Australia, has found that the M/k for the species is 0.33 (Coulson et al., 2012) much lower than the BH-LHI of 1.5, and reminiscent of a life history more commonly associated with marine mammals (Prince et al., 2015). Furthermore, a meta-analysis of 123 marine species of teleosts, chondrichthyes, invertebrates, and marine mammals found that the ratio of M/k ranged from 0.12 to 3.52, indicating that the life history ratios often vary considerably from the BH-LHI (Prince et al., 2015). Others have also acknowledged this variability in M/k. For example, t (...truncated)