Smith and Kostlan: Growth of Etelis carbunculus from sagittal otolith radius 



463 



Estimates of otolith growth rate as a function of 

 radius were fit by SAS (1985) nonlinear regression to 

 a modified Gompertz rate curve (Gompertz 1825, Win- 

 sor 1932) of the form 



axe 



-bx 



+ C 



(1) 



where 



y = 



X = 



a and b 

 c 



estimated density of daily increments or 



otolith growth rate (^m/day), 



radial distance (fim) at which each density 



was recorded, 

 = shape parameters, 

 = a constant representing the asymptotic 



otolith growth rate. 



The Gompertz rate curve was chosen for its general 

 shape and was then modified in form to provide a better 

 fit to observed data. The constant was added because 

 recorded microincrement densities did not subside to 

 zero, instead reaching a low positive value. 



Substituting dx/dt for y in equation (1) gives a clear 

 picture of the otolith growth-rate function. Since the 

 growth rate was estimated in days, the integral of the 

 reciprocal of the righthand side of this function pro- 

 vides an estimate of the fish age in days as a function 

 of otolith radius in microns, as follows: 



n - m 



kok 



k=l 



(-l) k a' 

 b k c k k k '' 



f(k + 1, kmb) 



kpk 



k=l 



(-l) k a' 



b k c k k kH 



r(k + l, knb) 



(4) 



where r is the incomplete gamma function and k is a 

 dummy variable. The first term, representing the 

 asymptotic growth rate, is solved separately. Using the 

 ratio test for convergence and an estimate based on 

 the Gaussian continued fraction 



V k e- V < T(k + 1, V)< 



(V 



k+l. 



V-k 



(5) 



(in Wall 1948), the first series converges for mb>M and 

 the second for nb>M, where M is a solution of 



2 -M 



M 2 e 



M-l 



bc/va and v is any number >1. (6) 



Since 



then 



dx/dt = axe _bx + c 



t x 



;" 



dx 



(2) 



where m and n are any two distances along the otolith 

 radius. Substituting u = bx and du = bdx gives 



tx 



mb 



du/b 



(au/b)e 



(3) 



- 1 r 



cb J 



nb 



+ C 



du 



cb J (a/bc)ue~ u + 1 



mb 



Thus, for each set of values of a, b, and c, a distinct 

 point can be identified at which the expanded series 

 converges quickly. For speed of computation, we chose 

 v = 2, to ensure that the ratio of consecutive integrals 

 would be less than 1/2. Age in days was estimated for 

 each otolith radius by a Fortran program, adapted from 

 Davis and Rabinowitz's (1984) routine for Romberg in- 

 tegration. Given values of x (otolith radius in microns), 

 a, b, and c, the program first solves equation (6) by 

 Newton's method (in Atkinson 1989) and then either 

 uses the expanded series from equation (4), if it con- 

 verges quickly, or evaluates the integral given in equa- 

 tion (3) by Romberg integration. Both estimates were 

 made with a tolerance of 10" 15 . 



Fork lengths and estimated ages of the fish sampled 

 were fit to the von Bertalanffy (1957) growth curve 



Lt = L^ (1 



-K(t- 



to)), 



1 r nh °° 



{£ ((a/bc)ue- u ) k (-l) k } du 



cb 



mb 



k = 



1 °° r nh 



X (-l) k I [(a k /b k c k )u k e~ ku ] du 



cb k=o J 



mb 



where L^ and t represent theoretical values of 

 asymptotic length at infinite age and age at zero length, 

 respectively; K is a growth constant; and t is fish age 

 in years. The Marquardt method of iterative least- 

 squares fitting was used, under SAS (1985) nonlinear 

 fitting procedure. For comparison, the data were also 

 fit by von Bertalanffy 's method of log-linear regres- 

 sion, using a BASIC program by Gaschiitz et al. (1988). 



