Natanson et a\ Age and growth of Lamnus nmtJi in the western North Atlantic 



269 



where o 



tn = ^ 



= the age estimated by MULTIFAN for the 

 youngest age class at the time it first appeared 

 in the length-frequency samples; and 



= the time elapsed in years between the theoret- 

 ical birthday and the first appearance of the 

 youngest year class in the samples. 



The theoretical birthday was defined as 1 April based on 

 the gestation period and the time of mating (Jensen et 

 al.^). The first appearance of the youngest age class in the 

 samples was 1 July. 



The model with seasonal growth components required 

 the use of a modified von Bertalanffy equation to incorpo- 

 rate the amplitude and phase of the seasonal growth: 



L. = LJ 1 



khere 



-K'((-(„+(»/2;rlsin(2;r((12/+l>/12)-02) 



/)j = amplitude; and 



&2 = (MULTIFAN phase) -^ t^ (from Eq. 1) 



(1) The Gulland and Holt ( 1959) and Francis (1988a) mod- 

 els were used to generate VBGFs from the tag-recapture 

 data. The Gulland and Holt ( 1959) method uses graphical 

 interpretation of the recapture data to produce estimates 

 of L_ and K. Specifically, annual growth rate (cm/jT) was 

 plotted against average FL (cm) between tagging and 

 recapture to calculate linear regression coefficients. The 

 slope of the line is equal to -K and the x-axis intercept is 

 equal to L^. 



The Francis ( 1988a) method (GROTAG) uses maximum 

 likelihood techniques to estimate growth parameters and 

 variability from tagging data. With this method, a coeffi- 

 cient of variation of growth variability iv), the mean and 

 standard deviation of measurement errors (m and s), and 

 outlier contamination (p), are estimated as well as growth 

 rates at two user-selected lengths (a and /3). The reference 

 lengths, a and /3, were chosen to lie within the range of 

 tagged individuals. The form of the von Bertalanffy equa- 



(2) tion becomes 



M 



Pea - ag,i 



ga-Sp 



-L, 



1 + 



g„ - gp 

 a-p 



(3) 



Tag-Recapture analysis 



Data from three independent tagging studies from the 

 western North Atlantic Ocean were combined for tag and 

 recapture analysis. In the 1960s, 542 porbeagles were 

 tagged and 53 recaptured as part of a Noi-wegian study 

 of the unfished population. In 1994 through 1996, the 

 Canadian Department of Fisheries and Oceans (DFO) con- 

 ducted a tagging program in which 256 porbeagles were 

 tagged and 25 recaptured. Between 1979 and 1999, mem- 

 bers of the National Marine Fisheries Services (NMFS) 

 Cooperative Shark Tagging Program tagged 1034 and 

 recaptured 119 porbeagles. Sharks were tagged and recap- 

 tured by biologists and commercial and recreational fish- 

 ermen in the United States and Canada and by biologists 

 in the Norwegian study. All measurements were con- 

 verted to FL by using the morphometric conversions 

 reported in Campana.' Where Norwegian measurements 

 were reported as Aasen's (1963) total length, they were 

 converted to FL with the equation: 



FL = Q.93TL. 



Only those sharks reliably measured at the time of tagging 

 and recapture were used in the analyses. Reliability was 

 based on prior knowledge of the individual's expertise in 

 measuring the shark or on detailed questioning of those indi- 

 viduals as to the method used. The majority of sharks were 

 measured by NMFS biologists or their representatives. 



^ Jensen, C, L. J. Natanson, H. L. Pratt Jr., N. E. Kohler, 

 and S. Campana. 2001. The reproductive biology of the por- 

 beagle shark, Lamna nasus, in the western North Atlantic 

 Ocean. Unpubl. manuscript. Apex Predators Program, NMFS, 

 28 Tarzwell Dr, Narragansett, RI 02882. 



where Lj = length at tagging; 



AL and AT = increments in length and time, respecitively; 



and 

 g^, and gp = mean annual growth rates at the arbitrary 



lengths a and /3. 



The simplest model, with minimal parameters (a and /3), 

 was used initially with additional parameters added to suc- 

 cessively increase model complexity. Significant improve- 

 ment in the model results were determined by using log 

 likelihood ratio tests (Francis, 1988a). The modeling was 

 carried out by using a Solver-based spreadsheet in MS 

 Excel (Simpfendorfer''). 



The value of t^ cannot be estimated from tagging data 

 alone; rather it requires an estimate of absolute size at 

 age, such as size at birth, and was calculated with the VB- 

 GF by solving for fg such that 



:f-Kl/A:)[ln{(L. -L,)/L,]], 



(4) 



where L, = known length at age (size at birth); 



The tfy values were calculated from an average size at birth 

 of 67 FL cm (Aasen, 1963) with t = 0. 



Longevity 



Several methods were used to estimate longevity. The 

 oldest fish aged from the vertebral method provides an 

 initial value, but is likely to be underestimated in a 

 fished population. Taylor (1958) defined the life span of 

 a teleost species as the time required to attain 95'7( of 



^ Simpfendorfer. C. 2000. Unpubl. data. Mote Marine Labo- 

 ratory, 1600 City Island Park, Sarasota, FL 33577. 



