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

Bertalanffy function. This procedure ignores variability in the
age of fish of a particular length and tacitly assumes that age
assignments can be made much more precisely than is true. To
account for uncertainty in the assignment of age using length,
the probability of age for fish having length within a particu-
lar length interval P(a I) was estimated following methods
reported by Taylor and others (2005), First. the probability of
an age-a fish having length within length bin [ is specified as:
(25)

I /+<1
PVla)~ J

(]'" ..[2; I-d

(I - (, Y ]dl
2 2 '
(J'"

where length bin [ has mid-point length [, minimum length [-d,
and maximum length [+d. These probabilities can be thought
of a~ a mau"ix with rows cOITesponding to length bins and col-
umns as ages. As is obvious from equation (25), entries within
a particular column (age) can be thought of as resulting from
the integral over each length bin of a normal probability den-
sity with mean Ia and vari,mce (1,,2. The mean length-at-age is
computed from the temperature-dependent growth model and
proce,dures described previously. The variance in length-at-age
is 0",," c::;'l.c\', , according to the assumption that coefficient of
variation in length (CV{ ) at age is constant (ev{ =().I 0).
With P{fia) available, one option to compute P(all}
would be to normalize each matrix cell by the sum of its
row as:

( I ) _ P,,' Via)
paJ - 1 .
L p(lli)

,~I

(26)

However, Taylor and others (20()5) suggest that this type
of procedure will induce bias if the population has experi-
enced size-dependent mortality (e.g., size-selective fishing
mortality). This result is because within a particular age class,
fast-growing individuals (i.e., largeL~.) may experience either
a higher or lower mortality rate than their cohorts and there-
fore be either over- or under-represented in the population.
This "sorting" by growth rate can either favor slow-growing
individuals, as in the case of increasing vulnerability to exploi-
tation with size, or fast-growing individuals, as in the case
of reduced natural mortality with size. Therefore, Taylor and
others (2005) suggest that an adjustment for mortality must
be made to accurately predict the proportion of individuals in
each age and length bin. Accordingly, the numbers offish in
each age and length bin were computed as:

(27)

Nf,a == N'"PVla),

where Na is the abundance of fish at each age. If the age-
specific mortality rate (Ma) is available and recruitment (R) is
assumed constant, abundance-at-age is given by:

9

(28)

'~:\f.
;;::Re 71

With abundance at each age and length bin thus available,
the proportion in each age and length bin can then be calcu-
lated as:

N
p. -~
f." - lv'
, t

(29)

where

l\r T = z:: z:: N"jJ .
I a

The probability of age given length is then calculated as:
(30)

p
P(al/);;::~.
LJ~.,

1=1

Taylor and others (2005) focus on age-specific mortality
driven by vulnerability to exploitation. For the unexploited
HBC, age-specific mortality as a function of changes in natu-
ral mortality was included. Lorenzen (2000) demonstrated that
much of the variation in natural mortality was explained by the
size of fish. Thus, Lorenzen's allometric relationship between
natural mortality and length was used to calculate a declining
mortality rate with age as:

(31)

\1 _"'1",. L"
~ rJ -- I f
a

where M ~ is the mortality rate suffered by an adult tish of size
L.~. This mortality schedule was calculated with M~, speci-
lied as 0.148, as estimated by ASMR 3 considering tag-cohort
speciflc data (see results below).
Foul' seasonal P(a II) matrices were computed and used
to assign age to fish captured at diflerent times of year. Growth
during the year could thus be accounted for by recalculat-
ing P(ila), such that length-at-age for a particular age-a was
computed as either I(a), l(a+.25), l(a+O.50), or l(a+0.75). The
resulting seasonal P(a II) matrices were then used to assign
age to a fish depending on the quarter of the year in which it
was captured.
To incorporate the uncertainty in assigning age based on
length into the overall assessment, a Monte Carlo procedure
was employed in which age was stochastically assigned to
each fish ba..<;ed on the seasonal P(a 11) matrices. To under-
stand this procedure it is first helpful to recognize that given a
fish witb length in bin I, the resulting probabilities of belong-
ing to each age is a multinomial probability distribution with