A technical note on seasonal growth models
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- García-Berthou, E., Carmona-Catot, G., Merciai, R. et al. Rev Fish Biol Fisheries (2012) 22: 635. doi:10.1007/s11160-012-9262-x
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The growth of many organisms is seasonal, with a dependence on variation in temperature, light, and food availability. A growth model proposed by Somers (Fishbyte 6:8–11, 1988) is one of the most widely used models to describe seasonal growth. We point out that three different formulae (beyond numerous typographical errors) have been used in the literature referring to Somers (Fishbyte 6:8–11, 1988). These formulae correspond to different curves and yield different parameter estimates with different biological interpretations. These inconsistencies have led to the wrong identification of the period of lowest growth rate (winter point) in some papers of the literature. We urge authors to carefully edit their formulae to assure use of the original definition in Somers (Fishbyte 6:8–11, 1988).
Keywordsvon Bertalanffy growth functionSeasonalityTemperatureFishery modelsSomers’ (1988) growth model
The growth of most plant species and ectothermic animals, such as fish, reptiles, crustaceans, and many invertebrate taxa, is strongly seasonal, with a dependence on temperature, light, and food supply (e.g., Pauly 1990; Alcoverro et al. 1995; Adolph and Porter 1996; Coma et al. 2000; Böhlenius et al. 2006). Even at tropical latitudes, the growth of fish and other organisms is often seasonal, depending on minor variations in temperature (Pauly 1990; Pauly et al. 1992) or increased food availability during the rainy season (Bayley 1988). Therefore, understanding and accounting for seasonality in growth is essential for understanding the ecology, evolution, and management of fish and many organisms.
Number of citations, by February 2012, for the most widely used seasonal growth models, according to Google Scholar and ISI Web of Knowledge (ISI WOK)
Somers’ (1988) growth model
Different formulae incorrectly referred to as Somers’ model
The formulae referred to as Somers’ (1988) model in the literature, plus a fourth possible one for comparison. See the text for interpretation and further details on the parameters
References using the formula
L(t) = L∞(1 − exp(−K(t − t0) − S(t) + S(t0)))
L(t) = L∞(1 − exp(−K(t − t0) + S(t) − S(t0)))
L(t) = L∞(1 − exp(−K(t − t0) − S(t) − S(t0)))
L(t) = L∞(1 − exp(−K(t − t0) + S(t) + S(t0)))
Formulae 1 and 2 share the positive attributes of having L(t0) = 0 and mean lengths at age N + t0 that are equivalent to the traditional von Bertalanffy growth model (Fig. 1). However, inspection of Fig. 1 shows that the growth trajectories between N + t0 and N + 1 + t0 are quite different for the two formulae. The WP, where seasonal growth is at a minimum (or growth is 0 if C = 1 as in Fig. 1), should be found half-way between the start of consecutive oscillation periods. The value of WP only equals ts + 0.5 for formula 1; in fact, the WP for formula 2 is at ts (Fig. 1).
To further illustrate the differences between these four formulae, we used the non-linear least squares fitting function “nls” in the R environment (R Development Core Team 2011) to fit each formulae to length and age data for anchoveta (Engraulis ringens), from Fig. 9 in Cubillos et al. (2001) and available in the FSAdata package (Ogle 2011a). The data and script for fitting these models and reproducing the illustrations of this paper are available in the supplementary information to this article. Simpler instructions on how to fit Somers’ growth model with R are available elsewhere (Ogle 2011b).
Parameter estimates for the anchoveta data using the four published formula for Somers’ model
Gayanilo et al. (2005) and other authors state that WP = ts + 0.5 while reporting formula 4, which are not consistent. Because the FiSAT software seems to provide correct results and Gayanilo and Pauly (1997) use formulae 1, we suspect that this is a typographical error. However, the literature contains some examples of wrong interpretations of the parameters derived from these inconsistencies. For instance, the top graph of Fig. 4 of Chatzinikolaou and Richardson (2008) reports ts = 0.08 and WP = 0.58, which would correspond to a WP in early July, whereas the fitted curve clearly shows that WP is around early January (0.08); the same inconsistency can be observed in their bottom graph of Fig. 4. Similarly, Bilgin et al. (2009b) state “for males, however, the slow growth period started in May (WP = 0.407)” whereas their Fig. 5a clearly shows that May is the time with the highest growth rate and the minimum is rather in November (WP = 0.407 + 0.5). Our results for the anchoveta data also suggest a similar inconsistency in Cubillos et al. (2001).
We have shown important errors in the formulae reported by many papers that use Somers’ (1988) seasonal growth model. While all formulae provide equivalent fits to the length-at-age data, the differences in the formula have implications in the estimates obtained for the t0 and ts parameters and, more importantly, their biological interpretation. These inconsistencies have led to the wrong identification of the period of lowest growth rate (winter point) in some papers in the literature. The original formula proposed by Somers (1988) should be used in all cases for the sake of priority and correct interpretation of parameter estimates. The differences between the formulae are subtle (i.e., differences in signs); thus, we urge authors to carefully edit their formulae to assure that they use and report Somers’ (1988) original formulation.
We thank Luis A. Cubillos for sharing the anchoveta data and Professor Daniel Pauly and an anonymous reviewer for helpful comments. Financial support for this research was provided by the Spanish Ministry of Science (projects CGL2009-12877-C02-01 and Consolider-Ingenio 2010 CSD2009-00065). GCC and RM held doctoral fellowships, from the University of Girona (BR2010/10) and the Spanish Ministry of Education (AP2010-4025) respectively, during the preparation of the manuscript.