16 



Fishery Bulletin 102(1) 



E14CT E142 



N44 



N42° 



Lake Abashiri 



Figure 1 



Location of sampling site for Corbicula japonica juveniles in Lake Abashiri. Japan (#i. Envi- 

 ronmental factors — water temperature, water fluorescence, salinity, and turbidity — were 

 measured at four sites, designated by ©■ 



for subsequent analysis with the environmentally based 

 growth model. The water fluorescence reflects the density 

 of phytoplankton. 



Model structure 



Modeling the distribution of a single sample Normal 

 distribution is usually used to describe a single cohort in 

 fishes and aquatic invertebrates (e.g. Pauly, 1987; Founier 

 and Sibert, 1990; Yamakawa and Matsumiya, 1997). How- 

 ever, an adequate function to describe a single cohort of 

 each animal should be selected to avoid biases caused 

 by any inadequacies of the function. Probability density 

 functions of many distributions are applicable for that 

 purpose, and the appropriate can be selected among easily 

 calculable functions to ensure convergence of the model. 

 Characteristics of many distributions are well described 

 by Evans et al. (1993). We used two distributions: normal 

 distribution and largest extreme value distribution. The 

 normal distribution is symmetric. The largest extreme 

 value distribution is asymmetric with a longer tail toward 

 the larger side. These are expressed by a location param- 

 eter and a scale parameter. 



To use all the information inherent in data, parameters 

 of the distribution functions are estimated from raw data 

 (e.g. lengths I, not from summarized data such as length fre- 

 quency. This estimation method is described by Sakamoto 

 et al. ( 1983 1. The most adequate distribution is selected by 

 AIC. Log-likelihood functions of the distributions take the 

 following forms: 



Normal distribution 



1 



log e L m)rmal (a,b) = £log ( . T =!=exp[-(/,-a) 2 /2& 2 ] , (1) 



(2) 



3 Abashiri Local Office of the Hokkaido Development Bureau. 

 2-6-1 Shinmachi, Abashiri, Hokkaido 093-0046, Japan. 



Largest extreme value distribution 



log,. L,ar gt -Ja,b) = £log,.{< 1/ 6)exp[-(/, -a) lb] 



x expj- exp[-( /, - a ) / b]\\ , 



where n = number of data; 



/, = length of (th individual; 



a = location parameter; and 



b = scale parameter. 



The location parameter is a mean in the normal distri- 

 bution. The location parameter is a mode in the largest 

 extreme distributions. The scale parameter is a standard 

 deviation in the normal distribution. 

 The AIC is calculated by 



AIC = -2 log tma.ximum likelihood) + 2m. (3) 



where m = number of parameters to be estimated. 



The model with the minimum AIC is the best model. A 

 difference of more than 1 or 2 is regarded as significant in 

 terms of AIC (Sakamoto et al., 1983). 



Modeling the change in the location Values of the location 

 and scale parameters usually increase with the growth of 

 an animal. The relative increase rate in a certain time step 

 is defined as 



