Baba et al.: An environmentally based growth model for juvenile Corbicula /aponica 



17 



r,=(P, 



Pi-i)IPi=v 



(4) 



where r t = relative increase rate of a parameter in the /th 

 time step; and 

 P l = parameter value after the /th time step. 



Relationships between the parameter value and the rela- 

 tive increase rate of the parameter can be expressed by 



P, = P ( ,( !+/-,) 



P 2 =P i a+r 2 ) = P (l+r 1 Kl + r 2 ) 

 P 3 =P 2 (l+r 3 ) = P (l + r 1 )(l + 7- 2 )(l + r ! ) 



(5) 



P.=P fl (1+r ' ) " 



where P (l = parameter value at the first sampling; 



P, = parameter value after the /'th time step; and 

 r i = relative increase rate of the parameter in the 

 /th time step. 



We used one day as the time step in this study In our envi- 

 ronmentally based growth model, we assumed that the 

 daily relative increase rate of location parameter (dRIRL) 

 depends on the age of the animal and on environmental 

 factors for each day. Sigmoid functions that take values 

 between and a certain maximum are empirically appro- 

 priate for expressing the relationships between the dRIRL 

 and independent variables, especially for measures such as 

 shell length that do not show negative growth. Therefore, 

 using categorical variables indicating animal ages and envi- 

 ronmental factors for each day as independent variables, we 

 express the dRIRL by the multivariate logistic function 



related because the dRIRS is larger when the dRIRL is 

 larger. Therefore, we estimated the dRIRS from an equa- 

 tion expressing the relationship to the dRIRL. We tested 

 two functions, 



Yl + Y 2 S i (7l+/2 S , >0) 



(yi + 7 2 s, <0) 



and 



t, = 







s, - 7i > 0) 

 (s, -/, <0) 



(7) 



(8) 



where t i = dRIRS on the /th day from the first sampling; 

 Yv Yz = coefficients of the equations; and 



Sj = dRIRL on the /th day from the first sampling. 



Model estimation 



Likelihood function The location and scale parameters 

 at the first sampling (o and fe ), the coefficients of Equa- 

 tion 6 (s max , a, and p k ), and the coefficients of Equations 7 

 and 8 (y-j and y 2 ) are estimated as values that maximize 

 total log-likelihood. The total log-likelihood is evaluated 

 by the adequate probability density function selected in 

 the first step. The log-likelihood functions take the follow- 

 ing forms: 



Normal distribution 



log, L„ ormal (a Q ,b„, s max , a j , p k , y v y 2 ) 

 = X2>g* -A r exp[-(Z <7i -a,)/24 2 ] 



2nb 



(9) 



Largest extreme value distribution 



s, =s max / 1 + exp 



2>a+£a b * 



(61 



where s i = dRIRL on the /th day from the first sampling; 

 s max = potential maximum dRIRL of the animal; 

 a., P k = coefficients of each independent variable; 

 A = categorical variable ( a dummy variable indi- 

 cating animal ages ) that takes the value 1 

 orO; 

 E kl = the kt\\ environmental factor on the /th day 



from the first sampling; 

 n A = number of age categories; and 

 n E = number of environmental factors. 



The categorical variable takes the value of 1 when the 

 animal is the category, otherwise it takes 0. The multivari- 

 ate logistic function with s max = 1 is used for logistic regres- 

 sions (Sokal and Rohlf, 1995). A method of giving a value to 

 the categorical variable is described by Zar ( 1999). 



Modeling the change in scale The daily relative increase 

 rate of scale parameter (dRIRS) and dRIRL must be cor- 



log e -L,ar gcs /o ,fe ,s max ,a,,^„7 1 ,)' 2 ) 



N n q 



=XZ 1 °g«{ (1/ V ex p[-^-«,> / 4] 



xexp{-exp[-(Z 9i -<5 9 )/feJU, 



(10) 



where a , 6 = values of the location and scale param- 

 eters, respectively, at the first sampling; 

 s max> a j> Pk = coefficients of Equation 6; 



Y v y 2 - coefficients of Equations 7 and 8; 

 N = number of samplings; 

 n q = number of data at the qth sampling; 

 a q = location parameter at the qth sampling 



estimated by Equation 5 (r,=s, ); 

 b q = scale parameter at the qth sampling esti- 

 mated by Equation 5 (r~^); and 

 / = length of the /th individual at the <?th 

 sampling. 



AIC is used to select significant environmental factors, 

 the age categorization, and the equation to express the 



