9711

• UCREFRP Catalog Number: 9711
• Author: Coggins, L.G.
• Title: Abundance trends and status of the Little Colorado River population of humpback chub
• USFW Year: 2008.
• USFW - Doc Type: an update considering 1989-2006 data.
• Copyright Material: NO

8 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989-2006

equation (7). This is accomplished by defining temperature-
dependent multipliers of (X, and K as:

(19)

1; (7') = Q, wand

(20)

1m (r)= Qm\\)'

where Ie tr) is the temperature-dependent multi plier of a,
and 1m (T) is the temperature-dependent multiplier of K. The
consumption and metabolism coefficients (Qe and Qm) of a
Q I 0 relationship allow these multipliers (fc tr) and fm tr) )
to increase or decrease with temperature (T). One can think
of these constants as the amount that the anabolic or catabolic
processes will ch.mge with an increase in temperature from
I DoC to 20ne. Inclusion of these temperature-dependent multi-
pliers into equation (7) yields:

(21)

dL "" aLof;(r)- Kfl j~,,(r).
dt

Equation (21) accounts for growth rate differences as a
function of temperature but does not account for movement
between the two thermal habitats. Thus, a logistic function was
used to model the probability of occupancy in either the LCR
or the mainstem Colorado River as a function of fish length.
The probability ofLCR occupancy is assumed to be:

PLCR=l-

0.8
(LL,)

(22)

l+e

20

where L is fish total length and L, is the fish total length where
the probability of year-round residence in the LCR is 0.6. The
behavior of this model is such that the probability of year-
round LCR residency approaches unity at lengths much less
than LJ and decreases to 0.2 at lengths much larger than LJ" The
number 20 in the denominator of the exponent governs the rate
at which the probability changes from near unity to near 0.2.
The asymptote at 0.2 requires at least some LCR residency
for even the largest fish and is consistent with the observa-
tion that adult HBC use the LCR for spawning (Gorman
and Stone, 1999).
A weighted temperanlre function experienced by fish of a
particular length can then be defined as:

(23)

1'(')= (PLCR)T;cu(1)+(1- PLCR)7~fS(1),

where TLCR ~) is the time-dependent water temperature in the
LCR and T\)s (I) is the time-dependent water temperature in the
mainstem Colorado River. This overall temperature experi-

enced by a fish of a given length is then used in equation (21)
to predict growth rate considering time-dependent changes in
water temperature and size-dependent changes in LCR versus
mainstem Colorado River occupancy.
To model the time-dependent water temperature in the
LCR, data reported by Voichick and Wright (2007) were used
to predict average monthly water temperature. A sine curve
was tit to these data as:

(24)

TWH (t)::: 1:",< + (T,'\O\ - 1;,;,( )sin (2ff(t + t 1"<>" )),

where t is time in fraction of a year starting April 1 , t peak is
a phase shift allowing predicted peak temperature to align
temporally with the observed peak temperature, Tm'e is the
VHlmplitude temperature .md roughly con'esponds to the
average annual temperature, and T.uax is the maximum annual
temperature. Values for t peak' Taw, and Till" were estimated
by minimizing the squared difference between observed and
predicted average monthly temperature.
Water temperature variation in the mainstem Colorado
River near the confluence of the LCR is much less variable
than in the LCR (Voichick and Wright, 2007). Thus, a constant
water temperature in the mainstem Colorado River of 100C
was assumed. This value couesponds roughly to the average
water temperature in the LCR inflow reach of the Colorado
River during much of the time the when growth increments
were observed (1989-2006),
The parameter vector fj ::: {H, d, In, n, Qc, L, } was esti-
mated by maximizing the log-likelihood equation (18). With
this more complex model. predicted recapture lengths were
found by integrating the temperature-dependent growth model
(equation 21) with respect to time. These predictions were
then used in the second term of equation (J 7) to compute the
deviations between observed and predicted growth. Walters
and Essington (written COITlll1un., 2007) recommend constrain-
ing 1.8 < Qm < 2.4 following guidance from a meta-analysis
by Clark and Johnson (1999). TIlerefore. Qm was specified as
2. The weighting term for the log-likelihood penalties were
specified equal to those found to be optimum for the previous
analysis. To further reduce the parameter set, cr2 L was specitied
as 2,000 to correspond with a coefficient of variation of about
10%, as is typically observed in fish populations (S. Martell
and e. Walters, University of British Columbia, oral commun.,
2006). The relative fit to the data for the temperature-indepen-
dent growth model (TIGM) and the temperature-dependent
growth lTIode (TDGM) were evaluated using AIC techniques
(Burnham and Andersen, 2002).

Section3-lncorporation of Ageing Error in
ASMR Assessments

As mentioned above, Coggins and others (2006a)
assigned age to individual fish su"ictly as an inverse von