Modification of the biological intercept model to account for ontogenetic effects in laboratory-reared delta smelt

Fishery Bulletin, Jan, 2007 by James A. Hobbs, William A. Bennett, Jessica E. Burton, Bradd Baskerville-Bridges

Statistical analyses, similar to those of Hare and Cowen (1995), were used to evaluate age-independent variability in the OS-FS relationship, as well as growth rate and ontogenetic effects. First, regression models (linear and polynomial) of standard length and otolith radius were regressed on age and otolith radius and age on standard length. Second, Pearson correlation coefficients of the otolith radius and standard length-on-age residuals were quantified to estimate the amount of age-independent variability. If no age-independent variability existed in the OS-FS relationship, the residuals of otolith radius-on-age and standard lengthon-age should be perfectly correlated. The unexplained variability in the correlation between the residuals of the two models can be considered the degree to which age-independent variability can influence the OS-FS relationship. To account for growth rate effects, Pearson correlation coefficients were quantified for the residuals of age-on-length and otolith radius-on-length. Moreover, significant growth effects were detected with a positive correlation. Lastly, the slopes of the OS-FS relationship for each life stage were compared to account for ontogenetic effects. If otolith growth and fish somatic growth are in constant proportion throughout the life stages in question, the slopes between otolith-fish size should not be significantly different (Cock, 1966). The slope of otolith size to fish size was calculated with an allometric model of the form y = a[x.sup.b], where log transformation results in the formula log(y) = log(a) bxlog(x) and the parameter b is equal to the slope (Gould, 1966).

Back-calculations

Size-at-age was back-calculated by using three models (Table 2). First, we examined the TVG model. This model accounts for variability in the underlying assumption of constant proportionality of otolith size to fish size by adjusting the contribution of increment widths by a growth factor. Second, we applied the MF model, which accounts for a nonlinear relationship between otolith size and fish size by directly estimating the shape parameters with a simple allometric model. Finally, a modified stage-specific form of the BI model was applied. Although this model depends on a constant proportion between otolith and fish size, we mediated this effect by applying the model to each specific life stage. To account for our stage-specific differences in the OS-FS relationship, we back-calculated size-at-age for the larval stage (5.4 mm SL to 12 mm SL) and juvenile stage (>12 mm SL) with 12 mm SL as the biological intercept. The mean back-calculated size-at-age trajectory was compared to measured standard lengths at time of sampling. We evaluated the fit of each model to the standard length-at-sampling data by comparing the [r.sup.2] values, as well as the minimum and maximum percent deviation of the mean back-calculated size-at-age value from the mean length-at-sampling.

Results

Validation of daily otolith increment formation

The relationship between the number of increments and days after hatching of delta smelt larvae are shown in (Table 1; Fig. 1). The slope of the regression of increment count on known-age was not significantly different from one and thus indicated that increment formation occurred daily. However, the intercept was significantly different from zero (P<0.001), indicating that the first increment was not laid at hatching, rather that ring formation began 6 dah. This observation was confirmed by examination of larvae sampled at one and five dah (Table 1).

[FIGURE 1 OMITTED]

Mean somatic and otolith growth

All somatic otolith-growth relationships were best described by life-stage-specific linear regression models, where larval (0-20 dah, 5-12 mm SL) and juvenile (>20 dah, >12 mm SL) life stages were considered separately. Calculated Akaike information criterion (Sokal and Rohlf, 1973) for the linear models were lower than polynomial models ranging from the 2nd to 9th orders. Somatic growth showed variations in growth over time: fast growth occurred from hatching to 40 dah, followed by a period of slowed growth from 40 dah up to 80 dah. After 80 dah, fish experienced a period of rapid somatic growth associated with the juvenile stage (Fig. 2, A and C). Otolith growth showed a different trend. Otolith growth was slow from hatching to 40 dah, which then increased exponentially from 40 dah to 100 dah, indicating that the relationship between otolith growth and fish growth changes abruptly around 40 dah with the completion of caudal flexing (Fig. 2B). Finally, the relationship between otolith size and fish size was best described by a stage-specific linear regression (Fig. 2D), which accounted for the lack of constant linear proportionality of otolith growth to fish growth. It is important to note that some patterns in the residuals were apparent in the early larval stages. However, we do not consider these slight deviations to have a significant effect on further residual analyses.