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

Table 1 Summary of sample size, number of increment at designated days
after hatching, and mean size-at-age.

                                   Mean
           No. of      Days       no. of
           Larvae     after     increments
Date       sampled   batching    counted

March 24      10          0           0
March 25      10          1           0
March 29      10          5         0.4
March 31      10          7         1.4
April 3       10         10         4.2
April 7        5         14         7.8
April 11       5         18        12.4
April 19       5         26          20
April 25       5         32        25.2
May 1          5         38        31.8
May 11         5         48        42.8
May 21         5         58        52.4
May 28         5         65        59.8
May 31         5         68          61
June 4         5         72        65.8
June 8         5         76        69.8
June 12        5         80        73.8
June 15        5         84        78.2
June 24        5         92        84.4
July 2         5        100          94

           Difference between
           days after hatching   Mean standard
           and mean no. of        length (mm)
Date       increments counted

March 24             0                 5.2
March 25             1                 5.2
March 29           4.6                 5.8
March 31           5.6                 6.1
April 3            5.8                 6.6
April 7            6.2                 7.3
April 11           5.6                 7.4
April 19             6                 8.7
April 25           6.8                 9.6
May 1              6.2                 9.9
May 11             5.2                12.2
May 21             5.6                14.6
May 28             5.2                16.5
May 31               7                16.2
June 4             6.2                17.1
June 8             6.2                17.2
June 12            6.2                19.4
June 15            5.8                19.6
June 24            7.6                20.9
July 2               6                23.5

Table 2
Summary of the three back-calculation models examined in this study:
the time-varying growth (TVG) model (Sirois et al. 1998), modified Fry
(MF) model (Vigliola et al. 2000), and the biological intercept (BI)
model (Campana 1990). L=standard length; R=otolith radius;
[L.sub.Op]=standard length-at-biological intercept; [L.sub.i]=standard
length-at-age i; [L.sub.cpt]=standard length-at-capture;
[R.sub.Op]=otolith radius-at-biological intercept; [R.sub.i]=otolith
radius-at-age i; [R.sub.cpt]=otolith radius-at-capture; W=mean otolith
increment width during each life stage; [W.sub.i]=otolith increment
width at i; [G.sub.e]=growth effect; and a=allometric shape parameter.

Back-calculation models     Equation

Time-varying growth (TVG)   [[L.sub.i] = [L.sub.Op] j ([W.sub.i]  
                            [G.sub.e] ([W.sub.i] - W))([L.sub.cpt] -
                            [L.sub.Op])([R.sub.cpt] -
                            [R.sub.Op]).sup.-1]

Modified Fry (MF)           [[L.sub.i] = a   exp (ln ([L.sub.0p] - a)  
                            ln([L.sub.cpt] - a) - ln ([L.sub.0p] - a))
                            (ln([R.sub.i]) - ln([R.sub.Op]))
                            (ln([R.sub.cpt]) - ln([R.sub.Op])).sup.-1)]

Biological intercept (BI)   [[L.sub.i] = [L.sub.cpt]   ([R.sub.i] -
                            [R.sub.cpt) ([L.sub.cpt] - [L.sub.Op])
                            ([R.sub.cpt] - [R.sub.Op]).sup.-1]

Back-calculation models     Reference

Time-varying growth (TVG)   Sirois et al. (1998)
Modified Fry (MF)
                            Vigliola et al. (2000)
Biological intercept (BI)   Campana (1990)

Table 3
Comparison of the allometric slope (b) values (log (otolith
radius) = bxlog (standard length). P values are from Student's
t-statistic.

Life   stage

         I       II      III       IV      V    VI
I        --
II     <0.001    --
III    <0.001    ns       --
IV       ns     <0.01   <0.001     --
V      <0.001    ns       ns     <0.001   --
VI     <0.001    ns       ns     <0.001   ns    --

Table 4
Comparison of growth curves generated from otolith size-at-age
data. [r.sup.2]= coefficient of determination for estimates
of size-at-age and known size-at-age for each age group.
"% range of mean size-at-capture" represents the variability
between estimated size and known mean size at time of sampling.

Model                  [r.sup.2]       % range of
                                   mean size at capture

Biological intercept     0.994         -3.5 to  6.4
Modified Fry             0.935          2.4 to  13.1
Time-varying growth      0.038        -10.3 to  0.90