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

6 Abundance Trends and the Status of the little Colorado River Population of Humpback Chub 1989-2006

Sainsbury, 1980; Kirkwood and Somers, 1984; Francis, 1988).
Typically, k will be negatively biased and L~ will be posi-
tively biased using this technique. This, in turn, has prompted
others to develop models to account and adjust for these biases
(e.g., James, 1991; Wang and others, 1995; Laslett and others,
2002). Van Bertalanffy growth functions were estimated with
these most recent methods. In general, however, poor results
were obtained primarily because of the inability of the models
to predict growth increments exhibited by small fish and large
fish simultaneously (U.S. Geological Survey, unpub. data,
2007). These results suggest a "kink" in the growth curve, as
would be found if fish grew along one curve when small and
then switched to another when larger.
Because water temperature is a dominant driver of
metabolic rate, and hence the von Bertalanffy k parameter
(Paloheimo and Dickie, 1966; Essington and others, 2001), the
"kink" hypothesis is consistent with a fish that is demonstrat-
ing an ontogenetic shift among habitats that have different
water temperatures, as is the case with the LCR and mainstem
Colorado River. The current reproductive ecology paradigm
for the LCR population of HBC is that most successful recruit-
ment occurs when fish are spu'N"Iled and reared within the LCR
(Valdez and Ryel, 1995; Gorman and Stone, 1999). However,
as fish approach some critical size, it is thought that they begin
to engage in a potaclromous migration between the Colorado
River and the LCR (Gorman and Stone, 1999). This ontoge-
netic shift in primary occupancy between two thermal habitats,
and therefore two basal metabolic rates, could induce a pattern
of shifting growth rate.
To account for this apparent pattern of shifting growth
rate, methods proposed by Walters and Essington (University
of British Columbia, University of Washington, respectively;
written commun.; 20(7) (hereafter Walters and Essington,
written commun., 20(7) to tit growth increment data to a gen-
eral growth model (Paloheimo and Dickie, 1965) were used to
describe the rate of change in weight as:

d1V m. va .1.T..TYI
"- - IlfY -111 'f'
dt -. ,

where the first term describes anabolic (i.e., mass-acquisition)
processes and is governed by a teml representing the mass-
normalized rate at which the animal acquires mass (ll), the
mass of the animal (W), and a parameter (d) describing the
scaling of the anabolic process with mass. The second term
represents catabolic (Le., mass loss through basal metabolism,
activity, and gonad production) processes where m is the mass-
normalized rate at which the animal looses mass ane!ll is the
scaling factor of catabolic processes with mass. Assuming a
constant relationship between length and weight over time as:

l'V:::::at',

where L is length and a and b are constant, simple algebra
provides an analogous relationship for the rate of change in
length as:

0)

dL =a[," [f!
_ - K._ .
dt

Constants in this relationship are related to those in equa-
tions (5) and (6) as:

a"-JH
a:::::~
b '

(8)

a"-'m
K-
--;-'

(9)

(10)

o:::::bd-b+t,and

(11)

17 ::::: bn - b + t .

Essington and others (2001) review these relationships
and describe the derivation of the standard von Bertalanffy
growth function as the integral of equation (7) when n = 1, b
= 3. and d = 2/3. This is the situation where catabolism scales
linearly with mass, the length-weight relationship is isometric,
anabolism scales as 213 mass, and results in the standard von
Bertalantfy growth model:

(12)

L(/)::::: Ljt_e-k(H"J),

(5)

where lo is the theoretical age where body length is equal to zero.
Walters ane! Essington (written commun., 2007) then
define a protocol to estimate the parameters of equation (5).
To restate this definition, tlrst assume that measurement errors
in the length offish are normal with variance a,:, and that all
fish follow a standard von Bertalanffy growth curve (equa-
tion (2) with shared k and individual L~.. The predicted length
of fish at time of recapture can be found by rearranging the
Fabens equation (4) as:

(13)

L(t + ~t)::::: L(t) + (Lx - L(t)XI- e-k<J.I).

Assuming that individual L~ is nomlaUy distributed with vari-
ance ai ' the variance of eachL(t + M) given L(t), mean L7.'
(1 Z. and (1,~, will be:

(6)

(14)

cr1(r+&)1 ::::: cr,;, (1 + e2kAr. )+ crt (1

J.