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<br />Deviations between observed and predicted growth increment
<br />for individual fish (i) are given as:
<br />
<br />(15)
<br />
<br />D, == L{t + /),t )i - L{t)j - (L:c - L{t),Xt - ).
<br />
<br />Walters <md Essington (written commun., 2007) note that
<br />it ~sth,en possible to estimate the parameter vector 1;1 = {L"" k,
<br />(7;, (7,;'} by maximizing the log-likelihood function:
<br />(16)
<br />
<br />In L(BIL(t ),L(t +!J.t ))= -%:t tD;
<br />- i Jql+MP
<br />
<br />1 s ,
<br />-2' ~ J[(I+/Vl' I
<br />
<br />where s is the number of growth increments. This is essentially
<br />.ID inverse-variance weighting strategy. However, they further
<br />note that the variance components of the measurement error
<br />and L~ are typically not separately estimable, such that, in
<br />practice, it is usually necessary to specify one of them a priori.
<br />Though this procedure is applicable assuming fish growth
<br />is described by equation (12), if fish growth is desClibed by
<br />equation (7), there is no analytical solution of (7;(I+j,'/, as in
<br />equation (14). However, Walters and Essington (wlitten com-
<br />mun., 2007) note that if one first estimates {I. imd in particular
<br />k, using equations (14-16), and assuming that the individual
<br />variances computed using equation (14) are adequate, then
<br />deviations from the general model (equation 7) can be used in
<br />the log-likelihood. These deviations are computed as;
<br />
<br />(7)
<br />
<br />"'}"'At
<br />
<br />D, == L(t + !J.t), -J(~j;~ - ,d} )dt .
<br />
<br />hi,
<br />
<br />After specifying the parameters a and b from equation (6),
<br />estimation proceeds as above with the parameter vector r,J =
<br />{H, d, /11, 11, a2L' a2m}.
<br />The above procedure was implemented using both Excel
<br />Solver tool (Lasdon and Allan, 2002) and AD Model Builder
<br />(ADMB: Fournier, 2000) to obtain estimates of t). The param-
<br />eter set was reduced by specifying d", = 31.8 mm2, based on
<br />an analysis of the observed error between consecutive measure-
<br />ments of identical fish within 10 days. The a and b parameters
<br />for equation (6) were specified as 0.01 and 3, respectively.
<br />To calculate the conditional variance of eachL(t + M), k was
<br />specified as 0.145, based on previous analyses. Additionally,
<br />penalty terms were included in the log-likelihood equation (16)
<br />to constrain d and n so that they did not deviate too far from the
<br />theoretical values. assuming standard von Bertalanffy growth
<br />of 2/3 and 1, respectively. Alternative weight values on these
<br />penalty ternlS were evaluated to find .ID appropriate u"Udeoff
<br />between minimum weights and decreased log-likelihood.
<br />
<br />7
<br />
<br />Because all the information contained in the mark-
<br />recapture data are for fish larger than 150 mm TL, extrapo-
<br />lating results to the growth rate of smaller fish could be
<br />problematic. Though this is not necessarily a concern related
<br />to assignment of age to fish greater than or equal to 150
<br />mm TL as required by ASMR, an accurate growth curve
<br />across all sizes would be desirable. Fortunately, Robinson
<br />and Childs (2001) conducted rigorous monthly sampling of
<br />juvenile HBC in the LCR during 1991-94. They used these
<br />data to estimate average monthly length from age-O months
<br />to age-32 months. These estimates were used in an additional
<br />log-likelihood term to constrain the predicted lengths from
<br />the geneml model to be similar to those reported by Robin-
<br />son and Childs (2001). Using these auxiliary data and assum-
<br />ing normal deviations allowed the incorporation of informa-
<br />tion on the growth rate of fish before they are large enough to
<br />be implanted with PIT tags. With these constraints in place,
<br />the full log-likelihood was:
<br />
<br />(18)
<br />
<br />. . ~'
<br />In L(611.(/),.I.(1 + ;\/},)= -2 L
<br />
<br />I
<br />
<br />1 ~ '
<br />2 ~ tYt".r+ v}
<br />
<br />vt
<br />
<br />_+:(d_~)l -f(I1-I)7- m~,Yrln(L(i)-/(i)f,
<br />_It ,) _A - "1
<br />
<br />where A is the weighting value for the penalty terms, LV) is
<br />the predicted length in month i from the general model, and
<br />tV) is the prydicted length over /110S = 32 months, as reported
<br />by Robinson and Childs (2001). The weighting ternl can be
<br />interpreted as the prior vari.illce on the standard von Berta-
<br />lanffy parameters (d = 2/3 and II = 1).
<br />A logical extension of the general model is to assume
<br />temperature dependence in growth rate. Accounting for
<br />changes in growth rate as a function of temperature is likely
<br />to be very impOltant for the analysis of this dataset for two
<br />reasons. The first is to account for the growth rate differences
<br />associated with occupancy in either the LCR or the main stem
<br />Colorado River. The second is to account for seasonal changes
<br />in water temperature within the LCR. The importance of this
<br />second consideration is further magnified by the temporal
<br />distlibution of sampling within the LCR. Sampling in the LCR
<br />typically occurs in the spring and fall. Therefore, much of the
<br />observed growth increment data corresponds to either summer
<br />growth (i.e., observations of fish captured in spring and again
<br />in fall) or winter growth (i.e., observations of fish captured
<br />in fall and the following spring). Because growth varies with
<br />temperature (Paloheimo and Dickie, 19(6), one would expect
<br />slower growth rates dming winter than during summer. This
<br />geneml prediction is also consistent with both field (Robinson
<br />and Childs, 2(01) and laboratory (Clarkson and Childs, 20(0)
<br />observations of HBC.
<br />Walters and Essington (written commun., 2007) pres-
<br />ent a method to allow temperature dependence in a and K in
<br />
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