Natural mortality and body size in fish populations

Highlights

• Within- and among- fish population scaling of natural mortality with size should be clearly distinguished.

• Natural mortality rates in fish populations consistently scale with body length to the power of approximately − 1.

• We provide empirical models to predict the intercept of the mortality-length relationship from growth parameters.

Abstract

Fisheries stock assessments increasingly account for size-dependence in natural mortality rates, usually by modeling mortality as a power function of body length. Various empirical studies have indicated a scaling of mortality with length in the range of − 0.84 to − 1.11, but substantially different scaling exponents ranging from − 0.75 to − 1.5 have been proposed on theoretical grounds or derived from some empirical models. To resolve these controversies and provide a well-supported default estimate of scaling for stock assessments, we re-analyzed two major data sets used in previous studies that supported different scaling exponents, and a combined data set. Both original data sets and the combined data yielded within-population exponents close to − 1 when analyzed using joint-slope mixed-effects models with population as a random effect. When population effects were disregarded, regression models yielded exponents that did not correctly reflect within-population scaling. The greatest deviations from the correct within-population scaling of approximately − 1 occurred in multiple regression models of mortality, size, and growth parameters. We conclude that within- and among-population scaling of natural mortality should be clearly distinguished, and that within-population scaling of natural mortality with length in fish populations is highly consistent at approximately − 1. We also explored empirical models for predicting the intercept of the mortality-length relationship for a given population from growth parameters.

Introduction

Modeling of natural mortality M forms part of all age and size-based fisheries assessment methods, from Beverton and Holt’s (1957) yield-per-recruit model to today’s integrated assessment models (Methot and Wetzel, 2013, Brodziak et al., 2011). Traditionally, natural mortality has been assumed to be constant (independent of size and age, and time invariant) within the recruited stock. Natural mortality is regarded as difficult to estimate within stock assessment models and it is common practice to fix M or estimate a prior for M from empirical models relating M in the recruited stock to growth parameters, environmental temperature, or longevity (Pauly, 1980, Then et al., 2015, Beverton and Holt, 1959, Hoenig, 1983).

More complex and realistic mortality models are increasingly being used in fisheries models and stock assessments. In particular, size-dependent and equivalent age-dependent patterns are often incorporated into assessments. Accounting for such patterns is particularly important for example when juvenile fish are harvested or stocked into the population (Lorenzen, 2005) and has become increasingly common practice in assessments. Lorenzen, 1996, Lorenzen, 2000, Lorenzen, 2005 conducted an extensive meta-analysis of mortality-size relationships in juvenile and adult fishes and pioneered the use of the resulting relationships with a mortality-length scaling of approximately − 1 in fish population modeling and assessment. Such ‘Lorenzen M′ natural mortality models, often converted to age-based mortality relationships using a stock-specific growth function and scaled to constant M estimates for the recruited stock, have found wide application in fisheries stock assessments (e.g. McKechnie et al. (2017); ICCAT International Commission for the Conservation of Atlantic Tunas (2018), SEDAR (Southeast Data, Assessment and Reviews (2018)) and other applications such as mark-recapture studies (Coggins et al., 2006, Lorenzen, 2006).

The scaling of M with body size in fishes has been investigated at the population and community level (Peterson and Wroblewski, 1984, McGurk, 1986, Lorenzen, 1996). From a theoretical perspective, multiple authors have suggested a ‘metabolic’ scaling exponent for mortality with length of − 0.75 (Peterson and Wroblewski, 1984, Andersen, 2019). Major empirical studies have demonstrated a broadly consistent allometric scaling with weight exponents between − 0.28 and − 0.37 and/or corresponding to length exponents between − 0.84 and − 1.11 assuming isometric growth (Table 1; except for two studies based on the same data set, Gislason et al., 2010 and Charnov et al., 2013).

Gislason et al. (2010) and Charnov et al. (2013) aimed to place the size-dependence of M within the wider context of fish life histories, in effect unifying concepts of life history correlates commonly applied to constant M values for mature fish with size-dependent mortality patterns. For this purpose, Gislason et al. (2010) assembled a data set comprising M-at-size estimates together with growth parameters for the respective populations. Some populations are represented by multiple M estimates and some by only one. Using multiple regression of M against growth parameters, Gislason et al. (2010) estimated a steep scaling of natural mortality with length of − 1.61 and Charnov et al. (2013) proposed a simplified general model for size-dependent mortality of M(L)= (L/L_∞)^−1.5 K. This model and the implied -1.5 scaling of M with length, colloquially known as ‘Charnov M′ has since been used occasionally in stock assessments (e.g. SEDAR (Southeast Data, Assessment and Reviews (2020)) as an alternative to the “Lorenzen M” models with a more moderate length scaling of mortality of around − 1.

Assessment results are often sensitive to the scaling and intercept parameters of the mortality-length relationship, which are fixed a priori in most assessments. Considerable discussion has therefore ensued in stock assessment panels about the reasons for and implications of the substantially different scaling relationships implied by the Lorenzen and Charnov M models. Here we revisit evidence for population-level scaling in both data sets from which the different models were derived and a combined data set. We also re-evaluate patterns of population-level and ensemble-level scaling and their relationships with life history traits. We conclude by providing robust estimates for within-population scaling of natural mortality with body length for use in stock assessments and explore empirical models for predicting the intercept of mortality-length relationships from growth parameters.