451 



Dynamic age-length keys 



Are Salthaug 



Institute of Marine Research 



Nordnesgaten 50 



PO Box 1870 



N-5817 Bergen, Norway 



E-mail address arese'imrno 



Information about age composition is 

 important when analyzing fish popu- 

 lation dynamics. Age determination 

 of individual fish is more difficult and 

 time consuming than the recording 

 of length measurements but by using 

 age-length keys, age distributions can 

 be estimated without much difficulty 

 from length distributions (Fridrikson, 

 1934). Knowledge of the age-length 

 composition in the population or in 

 a given subgroup of the population 

 is required for constructing adequate 

 age-length keys. Various methods 

 for construction and evaluation of 

 age-length keys are described in the 

 literature (see e.g. Fridrikson, 1934; 

 Macdonald and Pitcher, 1979; Schnute 

 and Fournier, 1980; Kimura and Chi- 

 kuni, 1987; Hayes, 1993; Terceiro and 

 Ross, 1993; Goodyear, 1997). Because 

 of individual variation in growth rates 

 and the variation in mortality rates at 

 different ages and sizes, the age and 

 length composition of a fish stock are 

 constantly changing. With sufficient 

 information about a fish stock, the 

 change in the age-length composition 

 can be modeled and theoretical age- 

 length keys can be constructed for 

 specific time periods. Age distributions 

 can then be estimated from length 

 distributions taken at different times 

 of the season. In this work, a simple 

 but useful modeling approach for con- 

 structing dynamic age-length keys is 

 described and applied to data from the 

 Atlantic cod [Gadus morhua) stock in 

 the Barents Sea. 



probability of an individual being a 

 certain length (/) within an age group 

 (a ) at a given time is assumed to follow 

 a normal probability density function 

 (Fig. lA), Niji^, fj^), with expectation 

 s^ and standard deviation o^. When 

 lengths of individual fish are recorded, 

 they are normally classified as discrete 

 length groups (e.g. 1-cm or 5-cm length 

 intervals). The probability (P) for an 

 individual in age group a to belong in a 

 discrete length group, s, at a given time 

 is then given by 



P.S ] N{^i,,<Jjdl, (1) 



where l^^^ ^ and /„,„ ^ are the upper 

 and lower length limits of length group 

 s, respectively. 



Because the normal distribution is 

 defined on the interval (-o<5,~), it has 

 mass below zero which may not be 

 negligible for distributions centered 

 near zero or with large variance. Thus, 

 the Pjj/s should be normalized across 

 length groups for each age, i.e. 



PJln., 



so that 



X'^^^ 



The theoretical number of individu- 

 als in age group a and length group s, 

 N^^^, can then be found (Fig. IB): 



of individuals in age group a and length 

 group s by the total number of indi- 

 viduals in the length group (Fig. IC): 



a, 



Jn. 



(3) 



The total number of individuals in 

 length group s (denominator) is found 

 by summarizing the individuals from 

 all age groups (a) in the length group. 

 Note that an index of abundance (i.e. 

 a relative measure) can be used as the 

 estimated number of individuals in an 

 age group (A^„). The expectation (|i^) 

 and standard deviation (cr^) increase 

 with time as the individuals grow 

 larger at different growth rates. By 

 analyzing age and length data from a 

 fish stock, n^ and o^ can be estimated 

 from observed data or models. 



The method was applied to data on 

 the Atlantic cod (Gadus morhua ) stock 

 in the Barents Sea from the period 1981 

 to 2000. Data from the annual bottom 

 trawl surveys in the Barents Sea, which 

 is conducted by the Institute of Marine 

 Research in Bergen (Norway) around 

 February (see e.g. Jakobsen et al.'), was 

 used to estimate the parameters in the 

 model (A''^, n^, and a J for each month 

 by interpolating between annual esti- 

 mates (described later). Monthly age- 

 length keys (Q„,) from the model were 

 then tested by comparing predicted and 

 observed age distributions in samples 

 from commercial catches where the 

 individual fish were both age and size 

 measured. Note that the data used to 

 estimate parameters and the data used 

 to test the model were from different 

 sources. 



Equation 1 and 4 in Pennington et 

 al. (2002) were used to estimate the 

 average length (jj-J in February and 



N =P N 



(2) 



Jakobsen, T., K. Korsbrekke, S. Mehl. and 

 O. Nakken. 1997. Norwegian combined 

 acoustic and bottom trawl surveys for 

 demersal fish in the Barents Sea during 

 winter. ICES CM 1997A':17, 26 p. 



Material and methods 



The model is based on principles 

 described by Schnute and Fournier 

 (1980) and Fournier et al. (1990). In 

 an age-structured fish population, the 



where A^^ = the number of individuals 

 in age group a. 



The proportion of individuals from age 

 group a in length group ,s, Q^^, is conse- 

 quently found by dividing the number 



Manuscript accepted 22 October 2002. 



Manuscript received 9 January 2003 at 

 NMFS Scientific Publications Office. 



Fish. Bull. 101:451-456 (2003). 



