Carlson and Baremore: Growth dynamics of Carcharhinus brevipmna 



289 



2001). Other methods have been used recently to report 

 yearly band formation in sharks, including oxytetra- 

 cycline marking (Simpfendorfer et al., 2002; Skomal 

 and Natanson, 2003; Driggers et al., 2004) and bomb 

 radiocarbon methods (Campana et al., 2002). However, 

 validation exists for relatively few elasmobranch species 

 (Cortes, 2000). 



Two-phase growth models may be more appropriate 

 for describing the growth of sharks, especially those 

 that are longer lived. Soriano et al. (1992) developed a 

 biphasic growth model which they applied to the long- 

 lived Nile perch (Lates niloticus) to better describe their 

 change in growth from zooplanktivores as juveniles to 

 piscivores as adults. Growth by sharks could be regard- 

 ed as being found in two phases: a rapid juvenile growth 

 followed by a slower adult growth. From a bioenergetic 

 perspective, this would follow a change from energy 

 devoted to growth to energy devoted to reproduction. 

 The logistic model could be regarded as a two-phase 

 model and may help to describe this change. The shift 

 from juvenile to adult would correspond to the inflection 

 point (f u ) of the curve, which approximates biological 

 age-at-maturity. In spinner sharks, age at maturity 

 was reported to be about 6-7 years for males and 7-8 

 years for females (Branstetter, 1987). This estimate of 

 age-at-maturity is similar to the inflection points from 

 our logistic model of 6.75 and 7.62 years for males and 

 females, respectively. Although each species should be 

 evaluated separately, future studies should investigate 

 the use of two-phase models to provide a more accurate 

 description of the growth of elasmobranchs. 



There have been few other examples of fitting alter- 

 native growth models to size-at-age data when results 

 from the von Bertalanffy model were biologically in- 

 correct or when models did not fit the data well. The 

 present study represents the first attempt to dp so for a 

 species of shark. Comparison of age and growth models 

 by Mollet et al. (2002) and Neer and Cailliet (2001) for 

 two species of rays revealed that the Gompertz model 

 best described their respective data although all models 

 they tested fitted the data fairly well. For pelagic sting- 

 ray (Dasyatis violacea) the Gompertz model predicted a 

 more reasonable size-at-birth and growth rate than the 

 von Bertalanffy growth model (Mollet et al., 2002). Neer 

 and Cailliet (2001) reported a slightly better statistical 

 fit for the Pacific electric ray {Torpedo californica) when 

 using the Gompertz model. However, because the differ- 

 ence in model parameters was negligible, results were 

 reported only for the von Bertalanffy model. 



The von Bertalanffy growth model is still the most 

 common model used to describe growth in fisheries 

 literature, despite criticism by Roff (1980) who recom- 

 mended its retirement. As pointed out by Roff (1980), 

 the choice of using another equation should be deter- 

 mined by the variables that are being investigated and 

 the results that are produced by the equation; for exam- 

 ple, if the results appear to be biologically unrealistic. 

 Our analysis of the growth of the spinner shark clearly 

 demonstrates the value of this approach. Use of the von 

 Bertalanffy growth model should continue because it 



permits comparison of growth curves to information al- 

 ready published and in some cases adequately describes 

 the growth of a given organism. However, the variety of 

 statistical techniques and quality of each study make 

 comparisons of von Bertalanffy growth curves between 

 different populations difficult and results should be in- 

 terpreted with caution regardless of what growth model 

 is used (Roff, 1980). 



Acknowledgments 



We thank Enric Cortes, Pete Sheridan (NOAA Fisheries, 

 Panama City Laboratory), and Miguel Arraya (Universi- 

 dad Arturo Prat, Chile) for providing comments on ear- 

 lier versions of this manuscript. Ken Goldman (Jackson 

 State University) was especially helpful in discussion 

 on precision and bias in age estimation, Miguel Arraya 

 on the validity of the comparison of growth models, and 

 Henry Mollet (Monterey Bay Aquarium) with the Gomp- 

 ertz model. Many different laboratories and institutions 

 aided with the collection of vertebrae. George Burgess 

 and Matt Callahan (University of Florida) provided 

 samples from the directed shark longline fishery. Lisa 

 Natanson (NOAA Fisheries, Narragansett Laboratory) 

 obtained samples during their longline surveys from the 

 U.S. south Atlantic Ocean. Observers Armando de ron 

 Santiago, Carl Greene, Matt Rayl, Bill Habich, Mike 

 Farni, Jacques Hill, and Jeff Pulver collected samples 

 from the directed shark gillnet fishery. Mark Grace and 

 Lisa Jones (NOAA Fisheries, Pascagoula Laboratory) 

 provided samples from fishery-independent longline 

 surveys. We also thank Linda Lombardi, Lori Hale, and 

 numerous interns who assisted with the cleaning and 

 processing of vertebrae samples. 



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