Stephenson and Hall; Timing of otolith ring formation in marine teleosts from northwestern Australia 



901 



pooled age classes (Barger, 1985; Manickchand-Heileman 

 and Kenny, 1990; Murphy and Taylor, 1990; Ross et al., 

 1995; Pearson, 1996; Morales-Nin and Moranta, 1997; Van 

 der Walt and Beekley, 1997) or on a restricted number of 

 age classes (Sainsbury and Whitelaw, 1984; McPherson and 

 Squire, 1992). 



Analysis with pooled data has limited value because 

 there may be different patterns of gi'owth-ring formation 

 at different life stages (e.g. at sexual maturity) and pooled 

 data may have interage differences masked by dominant 

 age groups (Beamish and McFarlane, 1983; Hyndes et al. , 

 1992). Studies in which data were pooled only for young 

 and old fish, due to low fish numbers, reduced these prob- 

 lems and improved the credibility of the results (Hyndes et 

 al., 1992; Fletcher and Blight, 1996; Hasp et al., 2002). 



Accounts of statistical analysis of the marginal incre- 

 ment data are rare. Davis and West ( 1992) used AN OVA to 

 show that there were differences in the marginal increment 

 of urohyal bones of L. vitta with time of year. As this sea- 

 sonal pattern was the same for age classes 1 to 6, Davis and 

 West ( 1992) pooled the data and used a graphical represen- 

 tation to show the time of formation of the annual rings. 



This article describes a method for modeling changes 

 in the index of completion of an otolith growth increment 

 over time. This method enables quantitative determination 

 of the most probable time of growth-ring formation (with 

 confidence intervals) and is illustrated for the species L. 

 vitta, L. sp. 3, N. furcosus, and L. sebae, from the Pilbara 

 fish trawl fishery. 



Materials and methods 



Between October and November 1993 and between October 

 and November 1994, samples of 30 fish of each species were 

 randomly selected each month from fishery-independent 

 trawl surveys. For the other months between January 1994 

 and March 1995, samples of 30 fish of each species were 

 randomly selected each month from commercial catches. 

 The samples came from an area between 115°30'E longi- 

 tude and 120°E longitude; between the 50 meter and 100 

 meter depth isobaths. 



The sagittal otoliths were extracted from each sampled 

 fish and the right otolith was embedded in epoxy resin and 

 then sectioned transversely through the otolith core to a 

 thickness of 0.4 mm. A Gemmaster high speed saw with a 

 100 mm by 0.1 mm diamond tipped saw blade was used for 

 sectioning. The otolith sections were set on 76 mm by 50 

 mm glass slides with casting resin and covered with cover 

 slips. The sections were viewed with a dissecting microscope 

 with an attached color video camera connected to a per- 

 sonal computer and a color monitor. Transmitted light 

 revealed alternating wide opaque and narrow translucent 

 zones. The translucent zones, referred to in the present 

 study as growth rings, were counted to determine fish ages. 



The distance from the outer extremity of the last wide, 

 dark band to the otolith edge, u',, is referred to as the mar- 

 ginal increment and the distance between the outer edges 

 of the second to last and the last dark band is denoted by 

 «',_[. The distances w^ and w^^ on the portion of the otolith 



ventral to the sulcus towards the proximal margin were 

 measured on the computer screen. The index of comple- 

 tion, c,, was determined by using the formula of Tanaka 

 etal. (1981) 



(1) 



and written to a file by using a computer program written 

 in the programming language "HiSoft Basic" (version 2.0. 

 MichTron, Auburn Hills, MI). 



The index of completion, c,, we expect to increase over 

 time, and then decrease abruptly when a new growth ring 

 is formed. The timing of formation of a new ring would occur 

 at the same time for a fish species with the same number of 

 rings, but there would be considerable variability in timing 

 and detection between species and individuals (Fig. 1). 



The increase in the index of completion over time, t, is 

 modeled as a strictly increasing function /■(<, a, b, d) with the 

 following parameters: maximum value, a, rate of increase, 

 6, and horizontal translation, d. 



For our study, data were collected over a period of 18 

 months (October 1993 to March 1995) and the relation 

 between the index of completion and time was expressed 

 as two functions, denoted Fj and Fg 



Fj : f-j = fit, a. b, c/,) and F., : i .-. = f\t, a, b, d.J, 



(2) 



where c ^ and t ,, = the estimates of the index of comple- 

 tion; 

 t = the time in months from t = il Octo- 

 ber 1993) to < = 18 (31 March 1995); 

 and 

 d^, d.2 = the translation parameters for func- 

 tions Fj and F.2 respectively. 



Ifthe point (c,, t) is associated with function Fj, the value 

 of the normal probability density function of the observed 

 deviation from Fj, evaluated at observation, i, is given by 



A,. = — p=exp 



and similarly the value of the normal probability density 

 function of the observed deviation from F.^, evaluated at 

 observation, i, is given by 



'^., =• 



a\!lK 



^exp 



(c-t\)- 



where a- is the variance of the residuals when Fj is fitted 

 to the data and where it is assumed to be equal to the vari- 

 ance of the residuals when the function F^ is fitted. 



To ensure the tractability of the subsequent analysis, 

 we assume that the probability, P,, of a point with index of 

 completion c,, at time t, being represented by Fj, is given 

 by the logistic function 



P.= 



(3) 



