Abundance Trends and Status of the Little Colorado River

Population of Humpback Chub

LEWIS G. COGGINS JR.*

U.S. Geological Survey, Southwest Biological Science Center, Grand Canyon Monitoring and Research

Center, 2255 North Gemini Drive, Flagstaff, Arizona 86001, USA

WILLIAM E. PINE III

Department of Fisheries and Aquatic Sciences, University of Florida, 7922 Northwest 71st Street,

Gainesville, Florida 32653, USA

CARL J. WALTERS

Fisheries Centre, University of British Columbia, 2259 Lower Mall,

Vancouver, British Columbia V6T 1Z4, Canada

DAVID R. VAN HAVERBEKE

U.S. Fish and Wildlife Service, Arizona Fishery Resources Office,

323 North Leroux, Suite 401, Flagstaff, Arizona 86001, USA

DAVID WARD

Arizona Game and Fish Department, Research Branch,

2221 West Greenway Road, Phoenix, Arizona 85023, USA

HELENE C. JOHNSTONE

SWCA Environmental Consultants,

114 North San Francisco Street, Suite 100, Flagstaff, Arizona 86001, USA

Abstract.—The abundance of the Little Colorado River population of federally listed humpback chub Gila

cypha in Grand Canyon has been monitored since the late 1980s by means of catch rate indices and capture–

recapture-based abundance estimators. Analyses of data from all sources using various methods are consistent

and indicate that the adult population has declined since monitoring began. Intensive tagging led to a high

proportion (.80%) of the adult population being marked by the mid-1990s. Analysis of these data using both

closed and open abundance estimation models yields results that agree with catch rate indices about the extent

of the decline. Survival rates for age-2 and older fish are age dependent but apparently not time dependent.

Back-calculation of recruitment using the apparent 1990s population age structure implies periods of higher

recruitment in the late 1970s to early 1980s than is now the case. Our analyses indicate that the U.S. Fish and

Wildlife Service recovery criterion of stable abundance is not being met for this population. Also, there is

a critical need to develop new abundance indexing and tagging methods so that early, reliable, and rapid

estimates of humpback chub recruitment can be obtained to evaluate population responses to management

actions designed to facilitate the restoration of Colorado River native fish communities.

The humpback chub Gila cypha, a cyprinid endemic

to the Colorado River basin, was described in 1945

from a specimen captured near the mouth of Bright

Angel Creek in Grand Canyon, Arizona (Miller 1946).

This morphologically unique cyprinid has been

characterized as the most specialized member of the

genus Gila and is highly adapted to exist in turbulent

canyon-bound reaches of the Colorado River basin

(Minckley 1973). This species was included on the

federal list of endangered species in 1967 (USOFR

1967) and is protected under the Endangered Species

Act (ESA) of 1973. Presently, only six populations are

known, five in the upper basin of the Colorado River

(i.e., above Lees Ferry) and one in Grand Canyon

(Valdez and Ryel 1995).

The humpback chub population in Grand Canyon is

centered near the confluence of the Colorado and Little

Colorado rivers (LCR) in Grand Canyon (36.198N,

111.798W; Kaeding and Zimmerman 1983; Douglas

and Marsh 1996; Gorman and Stone 1999). Valdez and

Ryel (1995) describe the humpback chub distribution

as consisting of nine aggregations throughout Marble

* Corresponding author: lcoggins@usgs.gov

Received May 6, 2005; accepted August 15, 2005

Published online February 3, 2006

233

North American Journal of Fisheries Management 26:233–245, 2006

 Copyright by the American Fisheries Society 2006

DOI 10.1577/M05-075.1

[Article]
and Grand canyons. However, only the aggregation

near the confluence of the LCR and Colorado River

(hereafter referred to as the LCR population) is known

to successfully reproduce. Because of abiotic and biotic

changes in the Colorado River after the construction of

Glen Canyon Dam in 1963, the LCR population relies

on the LCR as its primary spawning and juvenile

rearing habitat (Gorman and Stone 1999).

Humpback chub demonstrate potadromous spawning migrations between the Colorado River and the

LCR (Gorman and Stone 1999). Recaptures of tagged

animals suggest that the geographic extent of this

population is the lower 15 km of the LCR and the Little

Colorado River inflow reach of the Colorado River

(LCR inflow reach, defined as approximately 9 km

upstream and 11 km downstream of the confluence;

Valdez and Ryel 1995). The spawning migration

creates both difficulties and opportunities for monitoring the population. Typically, adult humpback chub

stage near the mouth of the LCR in March and April,

ascend into the LCR in April and May, and return to

the LCR inflow reach over a protracted time period

from June to September (Valdez and Ryel 1995;

Gorman and Stone 1999). Though the existence of

a spawning migration between the LCR inflow reach

and the LCR is accepted, a number of uncertainties

remain. The most important of these are the size or age

at which fish begin to exhibit migratory behavior and

whether a proportion of the adult population fails to

migrate every year (i.e., skip spawn). Douglas and

Marsh (1996) suggested that two populations exist: one

resident population in the LCR and one that migrates

between the LCR and LCR inflow reach. However,

Gorman and Stone (1999) suggested that the majority

of adult humpback chub larger than 300 mm total

length (TL) live in the LCR inflow reach except during

the spawning migration.

Because of ongoing management disputes related to

water use issues within the Colorado River basin and

the potential effect of these management actions on

humpback chub population viability, a monitoring

program coordinated by the U.S. Geological Survey,

Grand Canyon Monitoring and Research Center

(GCMRC), was developed to track changes in the

abundance and recruitment of the LCR population.

Information from this program is also potentially useful

in evaluating the ESA recovery goals for humpback

chub, which specify capture–recapture studies to

estimate absolute population size and trend (USFWS

2002). The GCMRC program includes analyses of

historical data, ongoing sampling, and external peer

review (Kitchell et al. 2003). Here we describe

a synthesis of data to characterize the dynamics of

the population. We assessed humpback chub population status and trend using catch rate indices and

capture–recapture-based abundance estimates from

open and closed models. While each of these methods

requires various assumptions, the results of the

synthesis are clear: the LCR humpback chub population has steadily declined since at least 1989. We

present these results and suggest strengths and

weaknesses in the various approaches as related to

stock assessment of the LCR population.

Methods

Field methods.—Repeated sampling for long-term

population trends in native fish abundance began in

1987 with the initiation of standardized hoop net

sampling in the lower LCR. During 1991–1995,

intensive sampling was conducted both in the Colorado

River and the LCR in conjunction with an environmental impact study of the operation of Glen Canyon

Dam (USBOR 1995). Sampling in both the Colorado

River and the LCR has continued from 1996 to the

present, but at reduced intensity. Although sampling

effort and research groups have varied across the study

period, the methodologies and sampling personnel

have remained fairly consistent (Table 1). Fish were

predominately collected using hoop nets (0.5–1.0 m in

diameter, 1.0–5.0 m long, 6-mm mesh, single or double

10-cm throat) and trammel nets (7.6–45.7 m long, 1.8

m deep, 1.3–3.8-cm inner mesh and 30-cm outer mesh)

in the LCR (Douglas and Marsh 1996; Gorman and

Stone 1999) and hoop nets, trammel nets, and pulsed

DC electrofishing (Coeffelt Mark XXII CPS) in the

Colorado River (Valdez and Ryel 1995).

Index-based assessments.—We analyzed two longterm catch rate data sets for trends in the abundance of

the LCR population. The first data set consists of the

hoop net catch rate in the lower LCR, which was

collected annually for 20–30 consecutive days in April

and May during the years 1987–1999 and 2002–2003.

Hoop nets were deployed at 13 standardized locations

in the LCR within 1,200 m of the confluence with the

Colorado River. The nets were fished daily for about

24 h throughout the monitoring period. The second

data set consists of the monthly trammel net catch rate

in the LCR inflow reach. In various months between

1990 and 2003, trammel nets were deployed during

crepuscular and night periods. Sample locations were

chosen at slow-water and current separation points. We

use the results of these two analyses as gross indices of

long-term population trends with which to compare the

estimated population sizes from the capture–recapture

models described below.

Tagging-based assessments.—Between 1989 and

2002, over 14,500 unique humpback chub (TL .

150 mm) were measured and implanted with passive

234 COGGINS ET AL.
integrated transponder (PIT) tags to provide a unique,

long-term identification mark. Capture–recapture data

were used in a simple, two-sample closed-population

model to estimate population size (Nˆ ) and two Jolly–

Seber-type open-population models to generate estimates of capture probability (pˆ), rate of population

change (^

k), and population size (Seber 1982; Williams

et al. 2002). We used an extension of the Jolly–Seber

population model to incorporate information on age at

first capture (Pollock 1981) to estimate mortality,

capture probability, and LCR population size with the

program MARK (White and Burnham 1999). We also

used methods presented by Pradel (1996) to estimate

the rate of population change directly without having to

estimate population size.

In addition, we developed a new age-structured,

open-population capture–recapture model (ASMR)

similar to the Jolly–Seber models but developed

specifically to address analysis needs related to this

robust capture–recapture data set. The ASMR model

predicts age at first capture from length, the numbers of

marked and unmarked fish at risk of capture from the

age-specific survival rate, and age- and time-specific

capture probabilities. Details of the ASMR model are

provided in the companion paper in this issue (Coggins

et al. 2006, this issue).

Model background.—Capture–recapture methods

have rigorous assumptions and are broadly defined as

‘‘closed’’ or ‘‘open’’ depending on whether the

population is allowed to change over the sample

interval (open models) or not (closed models). Shortterm capture–recapture studies are usually closedpopulation estimates and are based on the assumption

that all members of the population are vulnerable to

sampling and that no animals leave or enter the

population between sampling occasions. Open-population models allow for changes in the population due to

births, deaths, and movement into and out of the area

and are generally more suited for longer-term studies

(Pine et al. 2003).

Most multiyear capture–recapture studies use some

variation of the classic open Jolly–Seber model to

estimate population size, recruitment, and survival

(Jolly 1965; Seber 1965; Pollock et al. 1990). A key

assumption of open-population models is that every

animal (both marked and unmarked) present in the

population during a sampling period has the same

probability of being captured. Parameter estimates from

open-population models can be biased if capture

probabilities are heterogeneous (Pollock et al. 1990).

In many fisheries applications, much of this heterogeneity in capture probability is related to animal age.

Incorporating age information into the Jolly–Seber

model reduces estimation bias by accounting for this

heterogeneity in capture probability (Pollock 1981;

Pollock et al. 1990).

Closed-population models.—Between 2001 and

2003, we made a series of two-pass, closed-population

abundance estimates of humpback chub (TL . 150

mm) in the lower 14.2 km of the LCR during two 12–

14-d annual spring sampling trips. Population size and

variance were estimated using a Chapman-modified,

TABLE 1.—Sample sizes of humpback chub by year, gear type, and location. Values for nets are the numbers of net sets;

values for electrofishing are the numbers of shoreline sections (typically 100–400 m in length) sampled. The Colorado River

samples were collected in the Little Colorado River inflow reach (approximately 9 km upstream and 11 km downstream of the

confluence with the Little Colorado River). The Little Colorado River samples were collected within 15 km of the confluence

with the Colorado River.

Colorado River Little Colorado River

Year Electrofishing Trammel netting Hoop netting Trammel netting Hoop netting

1987 85 142

1988 179 399

1989 177 454

1990 20 142 73 356

1991 231 1,076 4 319 2,826

1992 407 847 198 3,712

1993 558 1,104 15 146 4,602

1994 37 19 2 103 3,885

1995 137 71 2,371

1996 89 122 4 1,114

1997 89 163 20 936

1998 70 42 372 27 1,435

1999 106 54 550 12 1,046

2000 225 344 284 7 2,700

2001 47 1,098 605 64 2,651

2002 168 120 199 1 2,998

2003 119 100 108 8 2,786

Total 2,303 5,231 2,139 1,494 34,413

LITTLE COLORADO RIVER HUMPBACK CHUB 235
length-stratified, LincolnPetersen abundance estimator (Seber 1982). Sampling for each abundance

estimate consisted of a marking event (i.e., the first

trip during each spring) and a recapture event (i.e., the

second trip). We compare these recent estimates with

earlier estimates presented by Douglas and Marsh

(1996) and other closed-population estimates of

humpback chub abundance (TL . 200 mm) in the

LCR inflow reach (Valdez and Ryel 1995; Trammell

and Valdez 2003).

Open-population models.—We used MARK (White

and Burnham 1999) to estimate the mortality and

capture probability for humpback chub (TL . 150

mm) from both age-structured (i.e., Jollyage models;

Pollock 1981) and non-age-structured models (i.e.,

traditional Jolly–Seber models; Seber 1982). Models

were developed based on humpback chub life history

(i.e., that they are long-lived fish with type III survival)

and variability in sampling effort over time. We used

the corrected Akaike information criterion (AICc

;

Burnham and Anderson 1998) as a guide in evaluating

the fit of models built in MARK (Table 2). Each model

was further evaluated to be the most biologically

reasonable, parsimonious model. We compared ageindependent models with the age-structured models

(Table 2) to evaluate differences in the mortality

estimates and capture probability trends resulting from

the two model types. We estimated the population size

of adult humpback chub as defined by the ESA

recovery goals (i.e., TL . 200 mm, age-4 and older;

USFWS 2002) using an age-structured Jolly–Seber

model within MARK. Variance estimates were calculated using the Delta method, and confidence intervals

were constructed following methods from Chao (1989)

outlined in Williams et al. (2002). Because humpback

chub are long lived and capture–recapture data were

sparse for individuals greater than age 15, capture

probabilities and mortality for fish older than age 15

were assumed to be equal to those of 15-year-old fish.

We estimated population change (^

k) across all ages

(i.e., independent of age) in MARK using temporal

symmetry models (TSMs) described by Pradel (1996;

Table 2). Estimates of population growth are more

robust to heterogeneity in capture probability than

traditional Jolly–Seber models (Schwarz 2001; Hines

and Nichols 2002). However, these models do assume

that capture probability does not change radically over

time. This approach to estimating ^

k is also dependent

on the sampled area’s remaining constant (e.g., if the

sampling area expands during the study, biased

population growth rate estimates are likely). To meet

this assumption, the TSMs were fit with a subset of the

data, namely, collections made only within the LCR.

The population change estimates do not depend on the

geographic closure of the sample area but do assume

that the trends in the sampled population are indicative

of the population as a whole. Previous investigations

suggest extensive annual movement of fish between the

LCR inflow reach and the LCR (see Introduction), so

that trends generated from data collected in the LCR

should be representative of the overall LCR humpback

chub population.

We chose candidate TSMs to estimate population

change based on the life history characteristics of

humpback chub and sampling methodologies used

throughout the study. The models had three parameters

(survival, /ˆ ; capture probability, pˆ ; and rate of

population change, ^

k), and each parameter was either

time dependent or time independent. A value of k in

excess of one indicates a population increase, a value

less than one indicates a population decrease, and

a value equal to one indicates that the population is

stable. When constraints (such as fixed time effects) are

placed on either /ˆ or pˆ, part of the variation around

each of these parameters is shifted to the model

parameters without constraints. We followed the

guidelines in Franklin (2001) to only place constraints

TABLE 2.—Jolly–Seber and Pradel population trend models fit to humpback chub data, 1989–2002. The best-fitting model is

the one with the lowest corrected Akaike information criterion (AICc) value.

Model AICc DAICc

AICc

weights

Model

likelihood

Number of

parameters

Jolly–Seber models

Age-dependent survival; time-dependent capture probability 41,303.2 0.0 1.0 1.0 27

Time-dependent survival and capture probability 42,478.1 1,174.9 0.0 0.0 26

Fixed survival; time-dependent capture probability 42,718.8 1,415.6 0.0 0.0 14

Age-dependent survival and capture probability 45,072.5 3,769.3 28

Fixed survival and capture probability 47,188.8 5,885.6 2

Pradel models

Time-dependent survival, capture probability, and rate of population change 99,755.7 0.0 1.0 1.0 38

Fixed survival; time-dependent capture probability and rate of population change 99,916.8 161.1 0.0 0.0 27

Time-dependent survival and capture probability; fixed rate of population change 100,014.6 258.9 0.0 0.0 28

Fixed survival; time-dependent capture probability; fixed rate of population change 100,441.1 685.3 0.0 0.0 16

236 COGGINS ET AL.
on the model parameter of interest. We again evaluated

model fit using AICc (Table 2).

The annual ASMR model.—We used the ASMR

model (Coggins et al. 2006) to estimate the age-specific

mortality rate, the age- and time-specific capture

probabilities, adult abundance, and recruit (age-2)

abundance. Like the traditional Jolly–Seber type

models, the annual ASMR model uses data aggregated

within years and ignores multiple within-year recapture

information. Three formulations (ASMR 1–3) of the

overall model structure were used to explore alternatives in estimating time- and age-specific capture

probabilities. We used a Markov chainMonte Carlo

algorithm to assess parameter uncertainty for each

model formulation (Coggins et al. 2006).

Multistate movement models.—The ASMR and

traditional Jolly–Seber models do not explicitly

represent seasonal spawning movements between the

LCR and the LCR inflow reach. Because sampling has

historically been concentrated in the LCR (there were

over 12,500 recapture events in the LCR versus about

1,100 in the LCR inflow reach), it is possible that the

capture probabilities calculated from spatially aggregated data could lead to misleading population

estimates (e.g., underestimates of the numbers of adult

fish that are seasonally resident in the LCR inflow

reach). To evaluate this possibility, we developed

a spatially explicit model for individual recapture

histories (Appendix), treating fish as being in two

possible location states (the LCR or the LCR inflow

reach) in each sampling month. The binomial likelihood function for this model was combined with the

Poisson likelihood function for unmarked fish from the

ASMR model (Coggins et al. 2006) to provide an

overall model for the dynamics of both marked and

unmarked fish. Our goal was to estimate age-specific

mortality rates, monthly movement rates to and from

the LCR, monthly capture probabilities in both

locations, and the abundance of unmarked fish over

time by maximizing this combined likelihood function.

We also evaluated multistate movement models

(Brownie et al. 1993; Hestbeck et al. 1991) in MARK

that allow animals to move between strata and provide

a probability of transitioning between strata.

Results

High mark rates (.80%) were achieved in the mid1990s (Figure 1). Examination of the monthly mark

rates among recapture locations (LCR versus LCR

inflow reach) indicates that fish mix and move between

the two areas, as there are similar patterns in overall

mark rate with large changes in sampling intensity.

Index-Based Assessments

Long-term catch rate indices from the standardized

hoop and trammel net sampling show 2–3-fold declines

in catch rate from the late 1980s to the present (Figure

2). Although all monthly trammel net samples from the

LCR inflow reach for 1990–2003 are presented, only

samples from 19901993 and 2001 represent robust

FIGURE 1.—Observed monthly mark rates of adult (age-4

and older) humpback chub captured in the Little Colorado

River (diamonds) and the Little Colorado River inflow reach

of the Colorado River (squares).

FIGURE 2.—Annual mean humpback chub catch rate using

hoop nets (top panel; catch/h, with 95% confidence intervals)

in the lower 1,200 m of the Little Colorado River (LCR) and

mean monthly humpback chub (TL .200 mm) catch rate

using trammel nets (bottom panel; catch h

1

100 m1) in the

Little Colorado River inflow reach (LCR inflow reach) of the

Colorado River. In the bottom panel, diamonds represent the

extensive sampling that took place in 1990–1993 and 2001

and that was well distributed throughout the LCR inflow

reach. Squares represent the limited sampling that took place

in 1994–2000 and 2002–2003 and that, in some years, was

concentrated near the confluence of the LCR. Though both

regression lines (dashed, 1990–2003; solid, 1990–1993 and

2001) suggest a declining trend in abundance, the slope of

neither regression is significantly different from zero (P1⁄40.14

for both regressions).

LITTLE COLORADO RIVER HUMPBACK CHUB 237
sampling coverage throughout the entire reach. Annual

sample sizes in 1994–2000 and 2002–2003 were

between 2% and 50% of the 1990–1993 average

sample size, and in some years effort was focused near

the LCR confluence. The lack of robust coverage and

limited sampling in the mid to late 1990s suggest that

comparing the 1990–1993 and 2001 data best depicts

the overall trend of relative abundance. However,

though both data sets suggest an overall decline in

adult humpback chub abundance, simple linear regression analyses provide estimated slopes that are not

significantly different from zero (P 1⁄4 0.14 for both

regressions; Figure 2).

Closed-Population Models

Population size estimates for fish (TL . 150 mm)

from the Chapman-modified, LincolnPetersen closedpopulation model ranged from about 2,000 in 2001 to

about 3,400 in 2003, a decline from the 4,300–5,400

individuals estimated in the 1990s (Figure 3). Recent

closed-population abundance estimates in both the

LCR and the LCR inflow reach (Figure 3) suggest

smaller population sizes in the early 2000s than in the

early 1990s, although the precision for all LincolnPetersen estimates is low.

Open-Population Models

The bootstrap goodness-of-fit test in MARK was

able to calculate a cˆ to adjust for the overdispersion of

the data. This estimate was used to evaluate the effects

of overdispersion on AICc model selection. No changes

in model selection were found across the range of cˆ

values evaluated. The best (lowest AICc

) Jolly–Seber

model for estimating mortality and capture probability

was the age-dependent mortality and time-dependent

capture probability model (Table 2). Annual mortality

decreased with age, from about 68% for age-2 fish to

about 16% for fish over age 15 (Figure 4). Mortality

estimates from each of the ASMR model formulations

were similar to those of the Jollyage methods; these

estimates were age dependent and ranged from about

65% for age-2 fish to about 18% for fish over age 15

(Figure 4).

Annual capture probabilities were consistent across

all models. Annual capture probability for the Jolly

age model ranged from 0.04 to 0.51 (Figure 4). Capture

probability from the three ASMR model formulations

was similar, ranging from about 0.04 to 0.47 (Figure

4). In both the ASMR and Jollyage models capture

probability was highest in the early 1990s, when

FIGURE 3.—Abundance estimates for humpback chub from

closed-population capture–recapture models in the Little

Colorado River (top panel; all estimates for fish larger than

150 mm total length) and the Little Colorado River inflow

reach of the Colorado River (bottom panel; all estimates for

fish larger than 200 mm total length). In the top panel, the

estimates depicted as diamonds are from the studies of

Douglas and Marsh (1996) and those depicted as squares are

from this article. In the bottom panel, the estimates depicted as

diamonds are from the studies of Valdez and Ryel (1995) and

that depicted as a square are from the studies of Trammell and

Valdez (2003). The error bars in both panels indicate 95%

confidence intervals.

FIGURE 4.—Humpback chub annual mortality rates across

ages (top panel) and capture probabilities across years (bottom

panel) from the three formulations of the annual agestructured, open-population capture–recapture (ASMR) model

and the age-structured Jolly–Seber model. The error bars on

the ASMR estimates are 95% credible intervals from Markov

chainMonte Carlo sampling of posterior parameter distributions; those on the Jolly–Seber estimates are 95% confidence

intervals.

238 COGGINS ET AL.
sampling effort was high, and lowest in the late 1990s,

when sampling effort was limited. More intensive

sampling effort since 2000 has resulted in increased

capture probability from about 0.10 to 0.25 (Figure 4).

Population Change

The best TSM model (Table 2) was a fully timedependent model that did not place time constraints on

any model parameter. The estimated annual rates of

population decline across all years were variable

because of poor model fit and imprecise estimates in

1996–1999, when sampling effort was low. Excluding

these years, the geometric mean annual population

change declined 14% annually (^

k 1⁄4 0.86). In addition

to using the fully time-dependent TSM model, we

followed Franklin’s (2001) suggestion to allow mortality and capture probability to remain time dependent

and make population change time independent. The ^

k

estimate for this model also indicates a negative trend

in population growth of about 1% annually (SE 1⁄4

0.006). Because humpback chub are a relatively large,

long-lived species with few natural predators as adults,

natural mortality should be relatively constant. To

examine this, we fit a model with fixed survival and

time-dependent capture probability and population

change. The geometric mean of population change

from this model was also negative, indicating a decline

of about 4% annually. When the low sampling years of

1996–1999 are excluded, the overall findings for

population growth indicate a decline of up to 14%

annually.

Recruitment

As described in our companion paper (Coggins et al.

2006), the ASMR method allows for estimation of

recruitment preceding the onset of sampling activities

by utilizing back-calculation methods from virtual

population analysis. The recruitment reconstructions

among the three ASMR formulations suggest a peak in

recruitment in the late 1970s to early 1980s of 13,500–

18,500 age-2 fish (Figure 5). After that peak, an overall

decline was evident to the early 1990s, when annual

recruitment stabilized at about 2,000 age-2 fish.

Recruitment may have stabilized or increased in the

late 1980s before resuming its decline and again

stabilizing through the 1990s.

Population Size

The best (i.e., biologically reasonable with a low

AICc value) Jollyage model for estimating the

population size of age-4 and older humpback chub

allowed mortality to vary with age and capture

probability to vary with time. This model gave a better

fit than non-age-structured models, demonstrating the

strong affect of age on mortality (Table 2). Adult

population estimates for this Jolly–Seber model ranged

from about 14,500 in 1989 to about 2,400 in 2001

(Figure 5). Adult population estimates from the ASMR

methods were similar across methods and ranged from

10,00011,000 in 1989 to 3,1004,400 in 2001

(Figure 5). Population estimates from both the

Jollyage and ASMR models were similar in magnitude and trend from the early 1990s through 2001.

Movement Model Results

Unfortunately, the combined movement and ASMR

model is overparameterized and unable to provide

reliable estimates of movement rates. For example,

seasonal changes in the capture rates in the LCR can be

explained equally well as the result of seasonal

movement to the LCR inflow reach or seasonal

changes in capture probability for fish still in the

LCR. To complicate matters further, seasonal movement rates probably vary with age, younger fish being

much less likely to migrate downstream after the

spawning season (Gorman and Stone 1999).

To avoid the overparameterization problem, we used

capture probabilities and population size from the

ASMR models. We used this information to examine

a series of seasonal movement rate scenarios designed

FIGURE 5.—Age-2 humpback chub recruitment by brood

year estimated using the three formulations of the annual agestructured, open-population capture–recapture (ASMR) model

(top panel) and adult (age-4 and older) humpback chub

abundance estimates using the three formulations of the

ASMR model and the age-structured Jolly–Seber model

(bottom panel). The error bars on the ASMR estimates are

95% credible intervals from Markov chainMonte Carlo

sampling of posterior parameter distributions; those on the

Jolly–Seber estimates are 95% confidence intervals.

LITTLE COLORADO RIVER HUMPBACK CHUB 239
to represent a range of possibilities for explaining the

observed changes in catch rate in the LCR as well as

the ontogeny of downstream migration. This is

unsatisfactory as a long-term solution to the problem

of disentangling movement and sampling intensity

effects, but it allowed us to determine whether

recognition of seasonal movement results in different

abundance trend estimates than those obtained from the

spatially aggregated methods. The results of these

scenario tests were essentially the same as those of the

annual ASMR and Jolly–Seber models shown in Figure 5. We were not able to fit multistate models in

MARK for reasons similar to those described above for

ASMR: low capture probabilities between states and

the associated failure to estimate a transition probability.

Discussion

Population Trend and Abundance Assessments

All of our evidence implies that the adult humpback

chub population in the LCR has experienced a large

decline in abundance since 1989. Low sampling effort

in the late 1990s (1996–1999), heterogeneity in capture

probability related to age, and a large number of ageclasses make complicated models with large numbers

of parameters necessary to realistically estimate

population parameters. However, population trend

and annual population size estimates from all catch

indices and methods indicate declines in population

size of 30–60% since the early 1990s.

A comparison of the recent closed-population

(LincolnPetersen) estimates with other closed-population estimates from the early 1990s indicates large

declines in population size. The only published

population estimates for humpback chub in the LCR

are closed-population estimates of fish exceeding 150

mm TL by Douglas and Marsh (1996). Their estimates

for the spring of 1992 ranged from about 4,300 to

5,500 (Figure 3). Our closed-population estimates from

spring 2000–2003 ranged from about 2,000 to 3,400

fish (TL .150 mm; Figure 3). A comparison of

closed-population estimates for fish occupying the

inflow reach suggests a decline in adult abundance of

about 60% between the early 1990s and 2001 (Valdez

and Ryel 1995; Trammell and Valdez 2003). Although

the assumptions of closed-population models (particularly the lack of animals moving into and out of the

sampling area) may not have been met, LincolnPetersen estimates do allow some relaxation of

this closure assumption (Pollock et al. 1990). For

example, if unmarked animals immigrate into the

sampling area, then the LincolnPetersen population

estimate is not biased for the second sample. If both

marked and unmarked animals emigrate between the

samples, then the LincolnPetersen estimate is unbiased by emigration for the first sample. If only

untagged animals emigrate between samples, then

population size estimates would be negatively biased.

However, it is unlikely that tagging effects only

occurred in the early 2000s, and any bias caused by

tagging probably occurred across all closed-population

estimators in the 1990s and 2000s. As such, these

estimates still serve as a relative indicator of an

apparently large decline in the humpback chub

population.

Using open-population models (Jolly–Seber and

ASMR), the estimated LCR humpback chub population of adult fish (i.e., TL .200 mm, age 4 and older)

is currently between 2,400 and 4,400 individuals

(Figure 5). No other open-population size estimates

have been published for this endangered species. The

TSM open-population trend estimates provide an

alternative to the age-dependent Jolly–Seber and

ASMR model annual population estimates. Population

growth estimates using TSM methods are robust to

heterogeneity in capture probability under the assumption that the heterogeneity of the population does not

change over time (e.g., capture rates for smaller fish are

always lower than those for larger fish; Williams et al.

2002). Estimates of population growth also do not

depend on geographic closure of the sample area as

long as the measured population change of the sampled

area matches that of the population as a whole

(Schwarz 2001; Hines and Nichols 2002). All biologically reasonable TSM models, including constrained models that allow survival and capture

probability to remain time dependent and population

change to be time independent (Franklin 2001),

indicate annual declines in population size of up to

14%. If the trends in annual estimated population size

from open-, closed-, or TSM population models are

examined, all approaches agree that the abundance of

the LCR humpback chub population has declined

significantly since at least the early 1990s.

Model Performance and Potential Assessment Errors

Abundance trends from the Jollyage and ASMR

models are similar, though the absolute abundance

estimates differ between the two approaches, particularly in the early years of the study. These differences

arise because the ASMR model predicts the number of

fish available for capture by using the existing age

structure of the population at the beginning of the study

while the Jollyage approach only uses information

gained from recapture of tagged fish. Thus, as the

tagged-fish population increases in the middle to late

years of the study, the estimates from the two

approaches converge (Figure 4).

240 COGGINS ET AL.
The modeling approach for the age-structured Jolly–

Seber and ASMR methods assumes that the initial age

is correctly assigned based on size at first capture.

Errors in initial age assignment could lead to a negative

bias in population point estimates and to the ‘‘smearing’’ of estimated recruitment among cohorts in the

ASMR model (Coggins et al. 2006). However, the

population trend estimates from the TSM models are

not age dependent, so that incorrect age assignment

does not bias the population trend estimates from these

methods.

The most critical assumption in the open-population

models is that mortality rates of fish have been stable

over time. Without this assumption, cumulative

survivors of fish tagged in previous years cannot be

safely used in calculating capture probabilities (and

population estimates) for any one year. If, for example,

there had been a high mortality rate of all fish in any

one year, we would have seen a subsequent drop in the

recapture rates of tagged fish from all previous years.

But because sampling has changed over time (so that

capture probabilities cannot be assumed to have been

stable), we would attribute that drop in recapture rates

to changes in capture probability rather than to the

change in survival and we would therefore overestimate the number of surviving fish. We do not see

trends in calculated monthly capture probabilities that

would indicate such an error. However, the limited

sampling effort in the late 1990s and the associated low

catch rates make it difficult to evaluate the mortality

and contribution of these year-classes to the overall

population (new recruits to the population in the late

1990s were unlikely to be collected given the low

sampling effort).

Is it possible that the adult population was in fact

stable from the early 1990s to the early 2000s and that

the analyses indicating large declines over this period

have been misleading? All of the tagging-based

estimates, using both open- and closed-population

estimates, indicate such a decline, as does the trammel

net catch rate index for the LCR inflow reach. Only the

hoop net catch rate index of larger (TL .200 mm) fish

does not show a decline over the key decade of

intensive monitoring (the 1990s). However, the hoop

net index suggests a decline from the beginning of data

collection (i.e., the late 1980s) to the present.

For the tagging-based methods, there is no assumption that sampling methodology was consistent over

time (i.e., capture probabilities are not assumed to have

been stable). For the newer tagging estimates to be

biased downward by 50% or more, these estimates

would have to involve a doubling in recapture

probability for marked fish relative to the probability

of capturing an unmarked fish. In capture–recapture

studies, such bias is typically due to restricting

sampling to a reduced portion of the area occupied

by the population (Seber 1982). There has been no

such systematic reduction in area sampled within the

LCR or the LCR inflow reach. There has been

a restriction in the seasonal time coverage of sampling

(from monthly sampling in the early 1990s to spring

and fall events in the 2000s) that could have the same

effect as a spatial restriction of sampling. This seasonality could cause downward bias in closed-population

estimates based on monthly aggregated data, but it is

difficult to see how it would affect the tagging methods

based on annually aggregated data (annual ASMR,

Jolly–Seber) where there is ample time (1 year) for

marked and unmarked fish to mix before recaptures are

used in estimation.

Another possible cause of differentially high capture

probabilities for marked fish is the use of baited hoop

nets in the LCR in 2001–2002. If that practice caused

marked fish to be more likely to reenter nets than

unmarked fish were to enter them in the first place,

then capture probabilities were overestimated for both

the closed- and open-population estimation methods.

This would have caused underestimates of the numbers

of unmarked fish at the end of 2002, and in the ASMR

methods this would in turn have caused the backcalculated (virtual population) numbers of unmarked

fish to be too low for at least the 1998–2001 time

period. However, when we artificially inflated the 2002

unmarked numbers by a factor of two, we found that

the back-calculated overall population trend from

ASMR methods still indicated a decline of at least

40% over the last decade, even though the calculated

2001 population size was increased by only 500–800

adult fish. When we simply excluded the likelihood

terms for catches of unmarked fish during the baiting

period from the calculation of maximum likelihood

estimates, there was no noticeable effect on the

parameter estimates. When we attempted to directly

estimate the relative capture probability for marked

versus unmarked fish during the baiting period, we

obtained estimates of 0.96–0.98 for all the ASMR

(annual, monthly, and with or without movement)

formulations, suggesting that there is no differential

capture probability between marked and unmarked

fish. These statistical checks do not prove that there

was no strong differential baiting effect, but they do

indicate that it is not parsimonious to assume such an

effect.

Finally, the effects of random sampling variation on

the ASMR estimates can be evaluated by comparing

estimates of recruitment or adult population trends

LITTLE COLORADO RIVER HUMPBACK CHUB 241
based on different total numbers of years of sample

data. This retrospective analysis (Figure 6) indicates

that the recruitment estimates for the early years (brood

years 1987–1992) of the sampling program are quite

stable (i.e., are not altered by including more years of

data), but that the estimates for the later sampling years

are more variable. Additionally, the adult abundance

estimates display more variability in both trend and

magnitude and generally suggest lower initial abundance (1989) and higher terminal abundance (2001)

when more years of data are considered. This trend in

adult abundance is a result of changes in sampling

intensity over time that essentially cause the model to

trade off higher mortality estimates with lower capture

probability during the years of low sampling intensity

(1996–1999). As the analysis is supplied with more

recapture data during the 2000–2002 time period,

mortality estimates fall and the trend in adult

abundance becomes less severe. Although the retrospective analysis does not suggest any severe structural

problem in model formulation, as there is a general

convergence in recruitment and abundance estimates

during the early years of data collection, it does further

reinforce the need to minimize the variability in

sampling effort over time to minimize assessment

errors resulting from parameter confounding.

All of the evidence suggests that the LCR humpback

chub population suffered a major decline in adult

abundance between the late 1980s and the present. All

of the evidence also points to major declines in

recruitment sometime in the 1980s. Though it is

difficult to believe that the adult population size

remained stable over the 1990s despite these recruitment declines, changes in sampling intensity

between the early 1990s and early 2000s make it

impossible to categorically deny that the adult

population size has been stable since 1990. Efforts to

improve monitoring since 2000 (e.g., baited hoop nets)

FIGURE 6.—Retrospective analysis of age-2 humpback chub recruitment by brood year (top panel) and adult (age-4 and older)

humpback chub abundance (bottom panel) using the first of the three annual age-structured, open-population capture–recapture

models (ASMR 1). The trend lines reflect analyses using annually truncated data sets beginning in 1989 and ending as specified

in the legend.

242 COGGINS ET AL.
have contributed to uncertainty in each of the assessments, as have reductions in overall sampling effort

since 1995.

Management Implications

U.S. Fish and Wildlife Service recovery goals for

humpback chub state that down-listing can proceed if

the following developments occur over a 5-year period

(USFWS 2002):

1. The trend in adult (age-4 and older) point estimates

for each of the populations does not decline

significantly;

2. Mean estimated recruitment of age-3 naturally

produced fish equals or exceeds mean annual adult

mortality;

3. Two genetically and demographically viable, selfsustaining core populations are maintained such that

the point estimate for each core population exceeds

2,100 adults; and

4. Certain site-specific management tasks to minimize

or remove the threats to the population are

performed.

The results of the present assessment suggest that

items 1–2 are not being met for the LCR population of

humpback chub. Although our analysis suggests that

there are currently more than 2,100 adults (item 3), at

the present rate of population decline the abundance

will fall below the minimum levels listed within 10–15

years.

In addition to the concerns they raise relative to ESA

listing status, humpback chub are a central concern in

the design of adaptive, experimental water management plans for the Colorado River in Grand Canyon.

Planned and ongoing experimental treatments range

from mechanical removal of nonnative fishes to

warming of the Colorado River through temperature

control devices on Glen Canyon Dam. Timely

estimates of the responses in recruitment and abundance to these treatments are critical to the experimental program. None of the methods we have used to date

give quick results, as all ongoing experiments require

several years of monitoring. Our analyses of the

historical data indicate that no existing monitoring

method can provide reliable estimates of such

responses for at least 3 years after the response has

begun. It takes 23 years until the larger number of

humpback chub recruits caused by an experimental

treatment have reached the ages at which PIT tagging

can begin to give estimates of abundance, and it may

take several more years until a reliable estimate of the

population trend caused by this recruitment change

becomes evident. Earlier response indications may be

obtained from index hoop netting in the lower LCR,

but we do not consider such indices reliable enough to

use as a guide for experimental treatment planning.

There is a critical need to develop new abundance

indexing and tagging methods that allow detection of

abundance changes earlier in the life cycle.

Acknowledgments

This assessment of the Little Colorado River

population of humpback chub is based on an

assimilation of data collected between 1987 and

present. Many individuals belonging to governmental,

academic, and private institutions contributed to this

database. We acknowledge all the individuals who

have been involved in sampling humpback chub in the

Grand Canyon. In particular, Arizona State University,

Arizona Game and Fish Department, U.S. Fish and

Wildlife Service, the Navajo Nation, and BioWest

conducted field sampling that contributed importantly

to the database and thus to current understanding of

humpback chub. Permits allowing humpback chub

research have been provided through time by Grand

Canyon National Park, Arizona Game and Fish

Department, U.S. Fish and Wildlife Service, and the

Navajo Nation. We also thank J. Kitchell, C. Grimes,

S. Lindley, D. Otis, J. Rice, C. Schwarz, M. Alldredge,

S. Martell, and K. Pollock for providing an earlier

review and much discussion on the ASMR method and

humpback chub stock assessment in the Grand

Canyon.

References

Brownie, C., J. E. Hines, J. E. Nichols, K. H. Pollock, and J.

B. Hestbeck. 1993. Capturerecapture studies for

multiple strata including non-Markovian transition probabilities. Biometrics 49:1173–1187.

Burnham, K. P., and D. R. Anderson. 1998. Model selection

and inference: a practical information-theoretic approach.

Spring-Verlag, New York.

Chao, A. 1989. Estimating population size for sparse data in

capture–recapture experiments. Biometrics 45:427–438.

Coggins, L. G., Jr., W. E. Pine III, C. J. Walters, and S. J. D.

Martell. 2006. Age-structured markrecapture analysis:

a virtual-population-analysis-based model for analyzing

age-structured capturerecapture data. North American

Journal of Fisheries Management. 26:89–93.

DeValpine, P., and A. Hastings. 2002. Fitting population

models incorporating process noise and observation

error. Ecological Monographs 72:57–76.

Douglas, M. E., and P. C. Marsh. 1996. Population estimates/

population movements of Gila cypha, an endangered

cyprinid fish in the Grand Canyon region of Arizona.

Copeia 1996:15–28.

Franklin, A. B. 2001. Exploring ecological relationships in

survival and estimating rates of population change using

program MARK. Pages 290–296 in R. Field, R. J.

LITTLE COLORADO RIVER HUMPBACK CHUB 243
Warren, H. Okarma, and P. R. Sievert, editors. Wildlife,

land, and people: priorities for the 21st century:

proceedings of the Second International Wildlife Management Congress. The Wildlife Society, Bethesda,

Maryland.

Gorman, O. T., and D. M. Stone. 1999. Ecology of spawning

humpback chub (Gila cypha), in the Little Colorado

River near Grand Canyon, Arizona. Environmental

Biology of Fishes 55:115–133.

Hestbeck, J. B., A. Dzubin, J. B. Gollop, and J. D. Nichols.

1991. Estimates of movement and site fidelity using

markresight data of wintering Canada geese. Ecology

72:523–533.

Hines, J. E., and J. D. Nichols. 2002. Investigation of potential

bias in the estimation of k using Pradel’s (1996) model

for capturerecapture data. Journal of Applied Statistics

29:573–588.

Jolly, G. M. 1965. Explicit estimates from capturerecapture

data with both death and immigration-stochastic model.

Biometrika 52:225–247.

Kaeding, L. R., and M. A. Zimmerman. 1983. Life history and

ecology of the humpback chub in the Little Colorado and

Colorado rivers of the Grand Canyon. Transactions of the

American Fisheries Society 112:577–594.

Kitchell, J. F., C. Grimes, S. T. Lindley, D. Otis, and C.

Schwarz. 2003. Report to the Adaptive Management

Work Group, Glen Canyon Dam Adaptive Management

Program: an independent review of ongoing and proposed scientific methods to assess the status and trends of

the Grand Canyon population of the humpback chub

(Gila cypha). Final report submitted to the Grand Canyon

Monitoring and Research Center, Flagstaff, Arizona.

Available: www.gcmrc.gov/library/reports/biological/

Fish_studies/Kitchell2003.pdf. (May 2005).

Lorenzen, K. 2000. Allometry of natural mortality as a basis

for assessing optimal release size in fish-stocking

programmes. Canadian Journal of Fisheries and Aquatic

Sciences 57:2374–2381.

Miller, R. R. 1946. Gila cypha: a remarkable new species of

cyprinid fish from the Colorado River in Grand Canyon,

Arizona. Journal of the Washington Academy of Science

36:409–415.

Minckley, W. L. 1973. Fishes of Arizona. Arizona Game and

Fish Department, Phoenix.

Pine, W. E. III, K. H. Pollock, J. E. Hightower, T. J. Kwak,

and J. A. Rice. 2003. A review of tagging methods for

estimating fish population size and components of

mortality. Fisheries 28(10):10–23.

Pollock, K. H. 1981. Capturerecapture models allowing for

age-dependent survival and capture rates. Biometrics

37:521–529.

Pollock, K. H., J. D. Nichols, C. Brownie, and J. E. Hines.

1990. Statistical inference for capturerecapture experiments. Wildlife Monographs 107:1–97.

Pradel, R. 1996. Utilization of capturemarkrecapture for

the study of recruitment and population growth rate.

Biometrics 52:703–709.

Schwarz, C. J. 2001. The JollySeber model: more than just

abundance. Journal of Agricultural, Biological, and

Environmental Statistics 6:195–205.

Seber, G. A. F. 1965. A note on the multiple-recapture census.

Biometrika 52:249–259.

Seber, G. A. F. 1982. The estimation of animal abundance and

related parameters, 2nd edition. Griffin, London.

Trammell, M., and R. Valdez. 2003. Native fish monitoring

activities in the Colorado River within Grand Canyon

during 2001. Final report submitted to the Grand Canyon

Monitoring and Research Center, Flagstaff, Arizona.

Available: www.gcmrc.gov/library/reports/biological/

Fish_studies/swca/01WRAG0046/trammell2003.pdf.

(May 2005).

USBOR (U.S. Bureau of Reclamation). 1995. Operation of

Glen Canyon Dam, Colorado River storage project,

Arizona: a final environmental impact statement. USBOR, Salt Lake City, Utah. Available: www.gcmrc.gov/

library/reports/LawoftheRiver/GCDEIS_Final.pdf. (May

2005).

USFWS (U.S. Fish and Wildlife Service). 2002. Humpback

chub (Gila cypha) recovery goals: amendment and

supplementation to the Humpback Chub Recovery Plan.

USFWS, Denver, Colorado. Available: www.gcmrc.gov/

library/reports/biological/Fish_studies/USFWS/

Humpbackchub_recoverygoals2002.pdf. (May 2005)

USOFR (U.S. Office of the Federal Register). 1967. Native

fish and wildlife: endangered species. Federal Register

32:48(11 March 1967):4001.

Valdez, R. A., and R. J. Ryel. 1995. Life history and ecology

of the humpback chub (Gila cypha) in the Colorado

River, Grand Canyon, Arizona. Final report to Bureau of

Reclamation, Salt Lake City, Utah. Available: www.

gcmrc.gov/library/reports/biological/Fish_studies/

Biowest/Ryel1995.pdf. (May 2005).

White, G. C., and K. P. Burnham. 1999. Program MARK:

survival estimation from populations of marked animals.

Bird Study 46(Supplement):120–138.

Williams, B. K., J. D. Nichols, and M. J. Conroy. 2002.

Analysis and management of animal populations:

modeling, estimation, and decision making. Academic

Press, San Diego, California.

Appendix Follows

244 COGGINS ET AL.
Appendix: Likelihood Function for Tag Recaptures Including Movement

For each tagged fish i, we have a recapture history of the form

Yt 1⁄4 f0, k, 0, 0, . . ., k, 0, . . .g, where for each possible

sampling date an observation of yt 1⁄4 0 denotes no recapture

and an observation of yt 1⁄4 k denotes at least one recapture in

location k (in this case either the Little Colorado River or the

Little Colorado River inflow reach). We calculate the

likelihood P(Yt

) of each history using the recursive method

reviewed in DeValpine and Hastings (2002), where P(Yt

) is

represented by the equation

PðYtÞ 1⁄4 PðYt1ÞPðytjYt1Þ: ðA:1Þ

To use this representation, we note that the probability

P(yt

jYt1) can be written as

PðytjYt1Þ 1⁄4 X

s

PðstjYt1ÞPðytjst; Yt1Þ; ðA:2Þ

where st represents possible fish states (dead, alive in location

1, alive in location 2, etc.) and P(yt

jst, Yt1) is the probability

of the observation yt given that the fish is in state st (i.e., the

capture probability for fish in state st at time t if yt . 0, 1

minus this capture probability if yt 1⁄4 0). Representing

P(yt

jYt1) in terms of states st expresses the problem of

calculating it as two simpler problems, namely, calculating

location state probabilities P(st

jYt1) over time and the capture

probabilities.

The location state probabilities are updated over time using

Bayes’ theorem. We first calculate the ‘‘posterior’’ probabilities as

PðstjYtÞ 1⁄4 pðytjstÞPðstjYt1Þ=PðytÞ; ðA:3Þ

where the total probability of the yt data is given by P(yt

) 1⁄4

Rs p(yt

jst)P(stjYt1) and p(ytjst)P(stjYt1) is simply 1 or 0 (0 if

the fish is either dead or not recaptured in location st

, 1 if the

fish is recaptured in location st

). Calculation of P(stjYt) is

nontrivial only for the case yt

1⁄4 0. We then calculate P(stþ1jYt)

(which is P(st

jYt1) for the next time step) as follows:

Pðstþ1jYtÞ 1⁄4 S

X

s

PðstjYtÞMðs;s0Þ; ðA:4Þ

where S is a (possibly age- and time-varying) survival rate to

be estimated from the data and M(s, s0

) is a movement

probability matrix representing the probability of a live fish’s

moving from location state s at time t to state s0 at time t þ 1

(the elements are set to 0 for s 1⁄4 dead; M(s, s) is the

probability of a fish’s staying in location s).

In calculating the likelihood function, the capture probabilities

P(kt

jst, Yt1) are set to their conditional maximum likelihood

estimates given by

Pðktjst; Yt1Þ 1⁄4 nkt=

X

i

PiðstjYt1Þ; ðA:5Þ

where nkt is the number of fish captured at location k at time t,

and the sum over fish i of the probabilities of being alive and

at location k represents the expected number of fish at risk of

capture in location k at time t.

Seasonal and age dependencies are incorporated in the

estimation by (1) assigning each fish i an apparent age at

first capture based on length, then varying S for subsequent

recapture times using apparent age and the Lorenzen survival

function (Lorenzen 2000); (2) calculating the capture

probabilities in equation (A.5) separately by apparent fish

ages up to age 10; and (3) using a different movement rate

matrix Mm(s, s0

), m 1⁄4 1, . . ., 8, for at least the first eight

calendar months of each year (we believe that there is

relatively little movement in late summer and fall; Valdez and

Ryel 1995). These assumptions require estimating at least 2 3

8 þ 1 leading parameters (assuming capture probabilities to be

given by equation A.5), 16 parameters for movements to and

from the LCR and 1 parameter for the asymptotic mortality

rate, Madult. In some estimation trials we have multiplied the

M(s, s0

) movement rates by a logistic age factor representing

lower probabilities of movement out of the LCR for younger

fish, thereby adding two parameters.

LITTLE COLORADO RIVER HUMPBACK CHUB 245