447 



Abstract — We propose a new equa- 

 tion to describe the relation between 

 otolith length (OL) and somatic length 

 (fork length [FL]) offish for the entire 

 lifespan of the fish. The equation was 

 developed by applying a mathematical 

 smoothing method based on an allo- 

 metric equation with a constant term 

 for walleye pollock (Theragra chalco- 

 gramma) — a species that shows an 

 extended longevity (>20 years). The 

 most appropriate equation for defining 

 the relation between OL and FL was a 

 four-phase allometric smoothing func- 

 tion with three inflection points. The 

 inflection points correspond to the 

 timing of settlement of walleye pol- 

 lock, changes in sexual maturity, and 

 direction of otolith growth. Allometric 

 smoothing functions describing the 

 relation between short otolith radius 

 and FL, long otolith radius and FL, 

 and FL and body weight were also 

 developed. The proposed allometric 

 smoothing functions cover the entire 

 lifespan of walleye pollock. We term 

 these equations "allometric smooth- 

 ing functions for otolith and somatic 

 growth over the lifespan of walleye 

 pollock." 



An allometric smoothing function 



to describe the relation between otolith 



and somatic growth over the lifespan 



of walleye pollock (Theragra chalcogramma) 



Seiji Katakura (contact author)^ 



Hisatoshi Ikeda^ 



Akira Nishimura^ 



Tsuneo Nishiyama^ 



Yasunori Sakurai^ 



Email address for S, Katakura: sei|ika(Sifra.affrc.go.ip 



' Hokkaido National Fisheries Research Institute 

 116, Katsurakoi, Kushiro 

 Hokkaido 085-0802, Japan 



^ Graduate School ol Fisheries Sciences 

 Hokkaido University, 3-1-1, MInato-cho, Hakodate 

 Hokkaido 041-8611, Japan 



^ Department of Marine Sciences and Technology 

 Hokkaido Tokai University, 5-1-1-1, Minamisawa, Minami-ku 

 Sapporo, Hokkaido 005-8601, Japan 



Manuscript submitted 21 August 2006 

 to the Scientific Editor's Office. 



Manuscript approved for publication 

 30 March 2007 by the Scientific Editor. 



Fish. Bull. 105:447-456 (2007). 



The power function v = ax^, used as 

 an allometric equation (Huxley, 1924), 

 is a useful tool for growth analysis of 

 organisms. Equations that describe 

 the relation between fish otolith 

 length (OL: distance between the tip 

 of rostrum and tip of postrostrum) 

 and somatic length (e.g., fork length; 

 FL: distance between the tip of head 

 and fork of tail fin) have been widely 

 used in fishery biology and ecological 

 studies to estimate somatic length at 

 younger ages with back-calculation 

 methods. These methods are based 

 on linear equations, log-transformed 

 allometric equations, and quadratic 

 equations (reviewed by Francis, 

 1990). However, these previous equa- 

 tions do not adequately reflect the 

 complex changes in growth over the 

 lifetime of a fish, especially for long- 

 lived species. 



Walleye pollock {Theragra chalco- 

 gramma (Pallas)) is the most abun- 

 dant fish in the Bering Sea, consti- 

 tutes the majority of the commercial 

 catches from this area (Wespestad, 

 1993), and is a long-lived species. The 

 oldest recorded age for this species is 

 28 years (McFarlane and Beamish, 

 1990). Juvenile walleye pollock serve 



as a substantial prey source for older 

 walleye pollock, other fish species, 

 marine mammals, and sea birds. 

 Thus, the year-class strength and 

 population dynamics of walleye pol- 

 lock have a significant influence on 

 the entire ecosystem (Springer, 1992; 

 Hunt et al,, 2002). Estimations of so- 

 matic length and growth analyses at 

 particular ages or life stages are im- 

 perative for fishery biology and eco- 

 logical studies of walleye pollock. 



In studies of the growth of walleye 

 pollock, the equation that describes 

 the relation between OL and somatic 

 length (i,e,, fork length) (referred to 

 as the "OL-FL equation" in this ar- 

 ticle) is required in order to recon- 

 struct the size of walleye pollock from 

 otoliths collected from the stomachs 

 of predators. Frost and Lowry (1981) 

 applied two linear equations, with 

 an inflection point at 10 mm OL, 

 corresponding to 220 mm FL, in a 

 total size range of 60-570 mm FL. 

 Nishimura and Yamada (1988) ap- 

 plied the log-transformed allometric 

 equation of the three linear equa- 

 tions with log-transformed OL and 

 total length (TL: distance between 

 the tip of head and tip of tail fin; 4.6- 



