U.S. Department of the Interior

U.S. Geological Survey

Open-File Report 2007–1402

Abundance Trends and Status of the Little Colorado River 

Population of Humpback Chub: An Update Considering 

1989–2006 Data
Abundance Trends and Status of the Little 

Colorado River Population of Humpback 

Chub: An Update Considering 1989–2006 

Data 

By Lewis G. Coggins, Jr.

Open-File Report 2007–1402

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Mark Myers, Director

U.S. Geological Survey, Reston, Virginia: 2007

Revised and reprinted: 2007

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Suggested citation:

Coggins, L.G., Jr., 2007, Abundance trends and status of the Little Colorado River population of humpback chub: an 

update considering 1989–2006 data: U.S. Geological Survey Open-File Report 2007–1402, 53 p.

Cover photograph: Adult humpback chub (Gila cypha). Photograph courtesy of the Arizona Game 

and Fish Department.
iii

Contents

Contents..........................................................................................................................................................iii

Executive Summary.......................................................................................................................................1

Introduction.....................................................................................................................................................2

Methods...........................................................................................................................................................3

Section 1—2006 Humpback Chub Assessment Update with Refinements ................................3

Index-Based Metrics ..................................................................................................................3

Tagging-Based Metrics ..............................................................................................................4

Evaluating Model Fit....................................................................................................................5

Section 2—Estimating the Humpback Chub Growth Function Using Mark-Recapture Data..5

Section 3—Incorporation of Ageing Error in ASMR Assessments..............................................8

Results. ...........................................................................................................................................................

2006 Humpback Chub Assessment Update with Refinements....................................................10

Index-Based Assessments ......................................................................................................10

Tagging-Based Assessments ..................................................................................................10

Closed Population Models .......................................................................................................10

ASMR without Tag-Cohort Specific Data ..............................................................................11

Model Evaluation and Selection ....................................................................................11

ASMR with Tag-Cohort Specific Data....................................................................................11

Model Evaluation and Selection ....................................................................................12

Assessment Update Summary ................................................................................................12

Estimating the Humpback Chub Growth Function Using Mark-Recapture Data .....................12

Incorporation of Ageing Error in ASMR Assessments.................................................................13

Discussion.....................................................................................................................................................13

2006 Humpback Chub Assessment Update with Refinements....................................................13

Estimating the Humpback Chub Growth Function Using Mark-Recapture Data .....................14

Incorporation of Ageing Error in ASMR Assessments.................................................................15

Acknowledgements.....................................................................................................................................15

References....................................................................................................................................................15

Figures

1. Relative abundance indices of sub-adult and adult humpback chub based on hoopnet catch rate (fish/hour) in the lower 1,200-m section of the Little Colorado River and 

trammel-net catch rate of adult humpback chub in the Little Colorado River inflow 

reach of the Colorado River.....................................................................................................19

2. Numbers of humpback chub marked and recaptured by age and year ..........................20

3a. Numbers of humpback chub marked by age in years 1989–92 and subsequently 

recaptured by age and year....................................................................................................21

3b. Numbers of humpback chub marked by age in years 1993–96 and subsequently 

recaptured by age and year....................................................................................................22

3c. Numbers of humpback chub marked by age in years 1997–2000 and subsequently 

recaptured by age and year....................................................................................................23

3d. Numbers of humpback chub marked by age in years 2001–04 and subsequently 
iv

recaptured by age and year....................................................................................................24

3e. Numbers of humpback chub marked by age in years 2005–06 and subsequently 

recaptured by age and year....................................................................................................25

4. Mark-recapture closed population model estimates of humpback chub abundance 

greater than or equal to 150-mm total length in the Little Colorado River.......................26

5. Humpback chub adult abundance (age-4+) estimates from the age-structured mark 

recapture models using data pooled among tag cohorts ..................................................27

6. Humpback chub recruit abundance (age-2) estimates from the age-structured mark 

recapture models using data pooled among tag cohorts ..................................................28

7. Quantile-quantile and Pearson residual plots for age-structured mark recapture 

model 1 (ASMR 1) using data pooled among tag cohorts ..................................................29

8. Quantile-quantile and Pearson residual plots for age-structured mark recapture 

model 2 (ASMR 2) using data pooled among tag cohorts ..................................................30

9. Quantile-quantile and Pearson residual plots for plots for age-structured mark recapture model 3 (ASMR 3) using data pooled among tag cohorts..........................................31

10. Humpback chub capture probability by age and year estimated from age-structured 

mark recapture model 3 (ASMR 3) using data pooled among tag cohorts .....................32

11. Humpback chub adult abundance (age-4+) estimates from the age-structured mark 

recapture (ASMR) models using data stratified by tag cohort..........................................33

12. Humpback chub recruit abundance (age-2) estimates from the age-structured mark 

recapture (ASMR ) models using data stratified by tag cohort.........................................34

13. Coefficient of variation of humpback chub adult abundance estimates (age-4+) for 

data pooled among tag cohorts and stratified by tag cohorts ..........................................35

14. Quantile-quantile and Pearson residual plots for age-structured mark recapture 

model 1 (ASMR 1) using data stratified by tag cohort ........................................................36

15. Quantile-quantile and Pearson residual plots for age-structured mark recapture 

model 2 (ASMR 2) using data stratified by tag cohort ........................................................37

16. Quantile-quantile and Pearson residual plots for age-structured mark recapture 

model 3 (ASMR 3) using data stratified by tag cohort. .......................................................38

17. Humpback chub capture probability by age and year estimated from age-structured 

mark recapture model 3 (ASMR 3) using data stratified by tag cohort............................39

18. Frequency of humpback chub growth intervals used in the growth analysis by timeat-large and total length at the start of the growth interval...............................................40

19. Log-likelihood as a function of the penalty weight ( ) on the standardized von Bertalanffy parameters for the temperature-independent growth model ................................41

20. Fit of the sine curve to average monthly Little Colorado River water temperature .....42

21. Fit of the temperature-independent growth model and the temperature-dependent 

growth model during summer and winter to all observed humpback chub growth-rate 

(dL/dt) data..................................................................................................................................43

22. Fit of the temperature-dependent growth model during summer and winter to 

observed humpback chub growth-rate (dL/dt) data during summer and winter...........44

23. Predicted humpback chub length-at-age from the U.S. Fish and Wildlife Service 

growth curve, the temperature-independent growth model, the temperature-dependent growth model for the Little Colorado River humpback chub population, and the 

temperature-dependent growth model for humpback chub living in the mainstem 

Colorado River under a constant temperature of 10oC .......................................................45

24. Predicted monthly growth rate from the temperature-dependent growth model for 

the Little Colorado River population of humpback chub and for humpback chub living 
v

in the mainstem Colorado River under a constant temperature of 10oC..........................46

25. Seasonal probability surfaces of humpback chub age at a particular length bin........47

26. Estimated adult humpback chub abundance (age-4+) from the age-structured mark 

recapture model 3 (ASMR 3), incorporating uncertainty in assignment of age .............48

27. Coefficient of variation of humpback chub adult abundance (age-4+) estimates 

accounting for uncertainty in age assignment.....................................................................49

28. Estimated recruit abundance (age-1) of humpback chub from age-structured mark 

recapture model 3 (ASMR 3) incorporating uncertainty in assignment of age ..............50

29. Retrospective analysis of adult humpback chub abundance and adult mortality rate 

considering datasets beginning in 1989 and ending in the year indicated in the figure 

legend..........................................................................................................................................51

Tables

1. Akaike information criterion (AIC) model evaluation results among age-structured 

mark-recapture models fit to data pooled among tag cohorts..........................................52

2. Akaike information criterion (AIC) model evaluation results among age-structured 

mark recapture models fit to data stratified by tag cohort ................................................52

3. General growth model results for the humpback chub, temperature-independent 

growth model (TIGM) and the temperature-dependent growth model (TDGM) ............52

4. Parameter correlation matrices for the humpback chub temperature-independent 

growth model and the temperature-dependent growth model.........................................53
vi
Executive Summary

In 1967, the humpback chub (Gila cypha) (HBC) was 

added to the federal list of endangered species and is today 

protected under the Endangered Species Act of 1973. Only six 

populations of humpback chub are currently known to exist, 

five in the Colorado River Basin above Lees Ferry, Arizona, 

and one in Grand Canyon, Arizona. The majority of Grand 

Canyon humpback chub are found in the Little Colorado 

River (LCR)—the largest tributary to the Colorado River in 

Grand Canyon—and the Colorado River near its confluence 

with the Little Colorado River. Monitoring and research of the 

Grand Canyon humpback chub population is overseen by the 

U.S. Geological Survey’s (USGS) Grand Canyon Monitoring 

and Research Center (GCMRC) under the auspices of the Glen 

Canyon Dam Adaptive Management Program (GCDAMP), a 

Federal initiative to protect and improve resources downstream 

of Glen Canyon Dam. 

This report provides updated information on the status 

and trends of the LCR population in light of new information 

and refined assessment methodology. An earlier assessment 

of the LCR population (Coggins and others, 2006a) used data 

collected during 1989–2002; the assessment provided here 

includes that data and additional data collected through 2006. 

Catch-rate indices, closed population mark-recapture model 

abundance estimates, results from the original age-structured 

mark recapture (ASMR) model (Coggins and others, 2006b), 

and a newly refined ASMR model are presented. This report 

also seeks to (1) formally evaluate alternative stock assessment models using Pearson residual analyses and information 

theoretic procedures, (2) use mark-recapture data to estimate 

the relationship between HBC age and length, (3) translate 

uncertainty in the assignment of individual fish age to resulting estimates of recruitment and abundance from the ASMR 

model, and (4) evaluate past and present stock assessments 

considering the available data sources and analyses, recognizing the limitations inherent in both.

A major task of this study was to improve the overall 

methodology used to conduct HBC stock assessment by 

addressing concerns identified in an independent review 

conducted in 2003 (Kitchell and others, 2003). The review 

report identified that the current technique of assigning age 

to individual fish based on length was a potential source of 

bias in ASMR estimates of abundance and recruitment, and 

called for a more complete examination of this potential error 

source. Additionally, the review suggested that further work 

to develop procedures to better arbitrate among alternative 

assessment models (e.g., ASMR 1–3) would be beneficial.

To address the first of the concerns identified by the 

independent review, this study uses mark-recapture data to 

develop a temperature-dependent growth model to characterize the relationship between HBC age and length. This model 

attempts to account for temperature differences resulting from 

both ontogenetic habitat shifts between the Little Colorado and 

the mainstem Colorado Rivers as well as seasonal variation 

in water temperature within the LCR. The resulting growth 

model is then used to characterize the error in assigning age to 

individual fish based on length. Results presented in this study 

suggest that ageing error does not result in large bias in either 

abundance or recruitment estimates from the ASMR model. 

However, incorporating ageing error into the assessment does 

result in less precise estimates, particularly for recruitment.

To address the second concern brought forward in the 

review report related to model selection procedures, this study 

arbitrated among the competing models by both examining 

model fit using Pearson residual analyses and considering 

information theoretic measures. Although adult abundance 

estimates and trend varied little among all models considered, 

these procedures identified ASMR 3 as the model whose 

underlying assumptions were most consistent with the data. 

Because ASMR 3 is also the most complex model, with a 

structure that allows for complex patterns in capture probability across ages and through time, further examination of model 

results suggest a decline in sampling efficacy for middleaged fish since approximately 2001. Although the cause of 

this shift in sampling efficacy is still unknown, it is possible 

that changes in the timing of LCR sampling events or subtle 

changes in sampling gear may be, at least, partly responsible 

for this finding.

Monitoring data and assessment model results reported 

herein continue to support the hypothesis that the adult 

(age-4+) component of the LCR population experienced 

Abundance Trends and Status of the Little Colorado River 

Population of Humpback Chub: An Update Considering 

1989–2006 Data 

By Lewis G. Coggins, Jr.
2 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

approximately a 40%–50% decline between 1989 and 2001. 

More recently, the population appears to have increased, 

reaching between 5,300 and 6,800 individuals in 2006. This 

increase in adult fish abundance since 2001 appears to be a 

result of increased recruitment beginning in the mid- to late1990s and continuing through at least 2002. 

Inclusion of ageing error in the assessment procedures 

has resulted in less precise estimates of adult abundance and 

recruitment. These results suggest that experimental management actions that result in large changes in recruitment 

are much more likely to be detected than actions resulting 

in small changes in recruitment. Therefore, if ASMR results 

continue to be used as the primary measure of HBC recruitment variation, experimental management actions designed to 

induce large changes in HBC recruitment should be preferred 

to those likely to induce only small changes. Adherence to this 

recommendation will help guard against failing to recognize 

beneficial management policies simply because the magnitude 

of the response was not sufficient to be detected by the current 

stock assessment program.

Introduction

The humpback chub is a focal resource of the Glen 

Canyon Dam Adaptive Management Program, a federally 

authorized initiative to protect and mitigate adverse impacts 

to resources downstream of Glen Canyon Dam. The focus on 

HBC is primarily a result of its unique ecological role as one 

of the few remaining endemic aquatic species within Grand 

Canyon and its endangered listing status under the Endangered 

Species Act of 1973 (Gloss and Coggins, 2005). The purpose 

of this report is to provide updated information on the status 

and trends of the Little Colorado River population of HBC in 

light of new information and refined assessment methodology. Such information constitutes the cornerstone of the HBC 

monitoring program within the GCDAMP and is also potentially useful to evaluate the recovery goals for this species 

as specified by the U.S. Fish and Wildlife Service (USFWS; 

U.S. Fish and Wildlife Service, 2002). The U.S. Geological 

Survey’s Grand Canyon Monitoring and Research Center has 

responsibility for the scientific monitoring and research efforts 

for the GCDAMP, including the preparation of reports such as 

this one.

The unique life-history attributes of HBC and the large 

variety of sampling and monitoring programs ongoing since 

the 1980s (Coggins and others, 2006a) prompted the development of a new type of age-structured open population capturerecapture model called the age-structured mark recapture 

model (Coggins and others, 2006b). This model was subsequently used in combination with other capture-recapture and 

index-based assessments to provide a comprehensive assessment of the LCR population of HBC (Coggins and others, 

2006a). The ASMR approach has been subjected to a series 

of independent peer evaluations, both as part of the GCDAMP 

(Kitchell and others, 2003; Otis and Wickham, U.S. Geological Survey, written commun., 2006) and peer-review processes 

required to publish journal articles and USGS products. These 

reviews all provide support for the ASMR as an appropriate modeling approach to evaluate trends in HBC population 

size and recruitment patterns. The ASMR model has undergone continuous refinement since the publication of the last 

assessment to improve the model’s ability to provide insight 

into HBC population dynamics and responses to management actions. Presented herein are improvements to the model 

that include the development of a formal model comparison 

approach using Pearson residuals and information theory to 

evaluate model fit. Comparison of the three main formulations 

of ASMR, detailed below and in Coggins and others (2006b), 

allows more precise examination of model fit and shows how 

incorrect structural assumptions may bias model output. 

A central problem in conducting HBC stock assessment is the assignment of age to individual fish. Though this 

problem is ubiquitous in fish assessment programs (Coggins 

and Quinn, 1998; Sampson and Yin, 1998), it is particularly difficult when working with endangered fish and when 

determination of age is only possible by killing the fish. In the 

case of humpback chub, an endangered species, determining 

the age of an individual requires killing the animal. To avoid 

this problem, individual fish ages must be assigned based on 

fish lengths and assuming some relationship between these 

two metrics. In past HBC assessments (Coggins and others, 

2006a), it was assumed that the age–length relationship was 

adequately described (U.S. Fish and Wildlife Service, 2002); 

however, this age–length relationship is based on an extremely 

small sample size (n ≈ 57) and is therefore suspect. Additionally, when assigning individual age based on this relationship, 

it was originally assumed that fish could be aged without error, 

clearly not a valid assumption. To alleviate these shortcomings, a new method is presented here for estimating the relationship between fish age and length using capture-recapture 

data. The uncertainty in the relationship between fish age and 

length is used to evaluate how uncertainty in the determination of age translates to uncertainty in abundance and recruitment estimates from ASMR using Monte Carlo simulations. 

These analyses offer insight into the HBC assessment and 

other monitoring programs for aquatic and terrestrial species 

where capture-recapture methodologies serve as the core of 

the assessment approach and precise trends in recruitment and 

mortality are difficult to quantify because of uncertainty in 

assigning age. 

Retrospective analyses were employed to evaluate the 

performance of past model predictions of trends in adult HBC 

abundance and recruitment. These retrospective analyses 

consider only a subset of the available data and are structured 

to demonstrate how information collected annually is used to 

“update” estimates from previous years. Mortality and capture 

probability are potentially confounded in open population 

models (Williams and others, 2001). As an example, if an 

animal is not captured in a sample year is it because the animal died, or is it simply because the animal was not captured? 
 3

Though it is possible to estimate capture probability using 

only marked animals known to be alive because of subsequent 

recaptures, as in the Jolly-Seber model (Jolly, 1965; Seber, 

1965), this method limits the sample size available to estimate 

capture probability and does not help with parameter confounding in the terminal year. In ASMR, the virtual population 

analysis structure allows the use of both marked and unmarked 

animals in the calculation of capture probability, but confounding between mortality and capture probability is not explicitly minimized via model structure. Retrospective analyses 

illustrate how perceptions of key population parameters, such 

as mortality rate, have been modified as more information 

becomes available, particularly following publication of Coggins and others (2006a and 2006b). These analyses are also 

useful in understanding how large changes in sampling intensity and protocols (e.g., minimal sampling in the LCR during 

1996–99) may bias or otherwise distort understanding of HBC 

population dynamics based on capture-recapture analyses. 

The primary objective of this report is to provide an 

updated stock assessment for the LCR population of HBC 

using data collected during 1989–2006. This assessment 

includes catch-rate indices, closed population mark-recapture 

model abundance estimates, the original ASMR model (Coggins and others, 2006b), and a newly refined ASMR model 

(detailed below). Coggins and others (2006a) used data 

collected during 1989–2002, and this report includes that 

data and additional data collected through 2006. Supporting 

objectives include (1) formally evaluating alternative stock 

assessment models using Pearson residual statistics and 

information theoretic metrics (Burnham and Anderson, 2002), 

(2) using mark-recapture data to estimate the relationship 

between HBC age and length, (3) translating uncertainty in 

the assignment of individual fish age to resulting estimates of 

recruitment and abundance from the ASMR model, and (4) 

evaluating past and present stock assessments considering the 

available data sources and analyses, recognizing the limitations inherent in both.

The ongoing monitoring program for HBC in Grand Canyon has varied in intensity over the years, but the primary sample locations, techniques, and personnel have remained remarkably consistent (Coggins and others, 2006a). Insight into the 

performance of the model is provided by conducting the annual 

stock assessment and continuously evaluating the performance 

of the assessment with retrospective analyses, independent peer 

evaluations, and tests of the model with simulated data. This 

comprehensive examination may prove useful to other adaptive 

management programs that seek to develop a robust monitoring component. In particular, the model may provide insight 

into (1) the limitations of monitoring alone in assigning cause 

and effect in association with prescriptive management actions, 

(2) the pathologies associated with large changes in monitoring protocols, and (3) a realistic assessment of the considerable 

uncertainty in results for a rare, elusive, long-lived organism, 

even after many years of intensive monitoring.

Methods

The methods employed for this analysis are presented 

in three separate sections. Section 1 describes methods used 

to update the 2002 HBC assessment metrics as presented in 

Coggins and others (2006a) and to refine the ASMR models. 

Additionally, section 1 describes the criteria used to assess 

model fit for each of the ASMR models. Section 2 outlines 

the methods used to estimate the relationship between HBC 

age and length based on mark-recapture information. Finally, 

section 3 describes the Monte Carlo simulations conducted 

to capture the uncertainty in the abundance and recruitment 

estimates that result from uncertainty in age assignment.

Section 1—2006 Humpback Chub Assessment 

Update with Refinements

Monitoring efforts for the humpback chub began in 

1987 when a standardized hoop-net sampling program was 

implemented in the lower reaches of the Little Colorado River. 

During the subsequent 19 years, four sampling periods can 

be generally defined that correspond to different levels of 

sampling effort and protocol (Coggins and others 2006a). The 

initial sampling period (1987–91) consisted mainly of limited hoop netting in the lower 1,200 m of the LCR. Sampling 

period 2 (1991–95) involved an intensive sampling effort in 

both the LCR and the mainstem Colorado River as part of an 

environmental impact statement on the operation of Glen Canyon Dam (U.S. Department of the Interior, 1995). The third 

sampling period (1996–2000) also included both the Colorado River and the LCR but with severely reduced intensities 

compared to period 2. The final sampling period (2000–06) 

involved a higher sampling intensity relative to period 3 but 

decreased relative to period 2. During each of these sampling 

periods, HBC have been collected using multiple types of 

gear, including hoop nets and trammel nets in the LCR, and 

this same gear plus pulsed-DC electrofishing in the mainstem 

Colorado River (Valdez and Ryel, 1995; Douglas and Marsh, 

1996; Gorman and Stone, 1999; Coggins and others, 2006a).

Index-Based Metrics

Although index-based metrics (e.g., catch rate) can be 

unreliable in tracking trends in population size (MacKenzie 

and others, 2006), these indices are frequently examined and 

are potentially useful for comparison with previous assessment efforts. With this caveat in mind and following Coggins 

and others (2006a), two long-term catch-rate time series were 

updated with data from 2003 through 2006, including (1) 

hoop-net catch rate of HBC in the lower 1,200 m of the LCR 

and (2) trammel-net catch rate of HBC in the LCR inflow 

reach of the Colorado River (defined as approximately 9 km 

upstream and 11 km downstream of the confluence; Valdez 
4 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

and Ryel, 1995). Details about these sampling programs are 

provided by Coggins and others (2006a).

Tagging-Based Metrics

The heart of tagging-based assessment is the large number of uniquely tagged sub-adult [150–199-mm total length 

(TL)] and adult (≥200-mm TL) fish that have been captured, 

measured, and implanted with passive integrated transponder 

(PIT) tags. Since 1989, more than 19,000 HBC have been 

captured, tagged, and released with unique identifiers. These 

data are maintained in a central database housed at the USGS 

Grand Canyon Monitoring and Research Center. 

Mark-recapture methods to assess population abundance 

and vital rates have been widely used in fisheries and wildlife 

studies for more than 50 years, and numerous reviews have 

been conducted highlighting the general approaches (e.g., 

Seber, 1982; Williams and others, 2001). Traditional methods 

(e.g., Jolly-Seber-type methods) generally rely on recaptures 

of tagged individuals to estimate abundance, recruitment, and 

survival. Basically, the approach is to create a known population of marked, or tagged, fish that are repeatedly sampled to 

obtain time series estimates of mark rate (i.e., the proportion 

of the overall population that is marked) and the number of 

marked fish alive in the population. These metrics are subsequently used to estimate capture probability, abundance, 

recruitment, and survival. 

The ASMR model differs from the traditional approach, 

because, in general, it contains more structural assumptions 

through the specification of a population accounting structure 

that governs transition of both marked and unmarked animals 

through ages and time. Age-structured stock assessment theory 

(Edwards and Megrey, 1989) is used to annually predict the 

numbers of marked and unmarked fish available for capture 

in a standard fisheries virtual population analysis framework 

(Quinn and Deriso, 1999). The total number of marked fish 

depends on the number of fish recently marked as well as the 

number of previously marked fish decremented by mortality 

rate. The number of unmarked fish depends on the recruitment 

over time, the number of fish marked from a given brood-year 

cohort, and the mortality rate. These annual predictions of the 

abundance of marked and unmarked fish are further segregated 

by age such that age-specific survival and capture probability 

may be modeled. Parameters are estimated by comparing predicted and observed age- and time-specific captures of marked 

and unmarked fish in a Poisson likelihood framework. 

The ASMR model has three different parameterizations (ASMR 1–3) that vary in how the terminal abundance is 

estimated and how age- and time-specific capture probability 

is modeled. Both ASMR 1 and ASMR 2 assume that age- and 

time-specific capture probability can be modeled as the product of an annual overall capture probability multiplied by agespecific vulnerability. This is similar to the common parameterization of fishing mortality in assessment models under the 

“separability assumption” (Megrey, 1989) and diminishes the 

size of the parameter set since it is not necessary to separately 

estimate each age- and time-specific capture probability. 

These models further assume that vulnerability is asymptotic 

with age. As such, vulnerability is assumed to be unity for 

fish age-6 and older and estimated only for the younger fish. 

Finally, annual age-specific vulnerabilities are assumed to 

be equal among each sampling period, as described above. 

Implicit in this assumption is that within a sampling period, 

annual age-specific capture probabilities differ only as a scalar 

value related to the annual overall capture probability.

The primary difference between ASMR 1 and ASMR 

2 is how the terminal abundances are calculated. ASMR 1 

estimates an overall terminal-year capture probability and 

calculates age-specific terminal abundances (both marked 

and unmarked) as the ratio of age-specific catch (both marked 

and unmarked fish) and age-specific capture probability (i.e., 

product of the terminal-year capture probability and sampling period 4 age-specific vulnerability). In contrast, ASMR 

2 treats age-specific terminal abundances up to age-13 as 

individual parameters. Terminal abundances for subsequent 

ages are estimated by applying age-specific survivorship to the 

age-13 abundance. This difference in formulations decreases 

the parameter count for ASMR 1 relative to ASMR 2 at the 

expense of assuming that the vulnerability schedule in the 

terminal year is identical to the rest of period 4. 

ASMR 3 is the most general model; it makes no assumption as to the age- or time-specific pattern in capture probability. The conditional maximum likelihood estimates of 

age- and time-specific capture probability are used to predict 

the age- and time-specific catch of marked and unmarked fish. 

Full details of each of the models are provided by Coggins and 

others (2006b).

In addition to the ASMR assessments, the time series of 

the annual spring abundance estimates in the LCR are updated. 

Abundance of HBC in the LCR greater than or equal to 150 

mm TL was estimated during the early 1990s and 2001–06, 

using closed population models. These models included 

the CAPTURE suite of models (Otis and others, 1978) and 

Chapman-modified, Lincoln-Petersen, length-stratified models 

(Seber, 1982). The recent estimators use data collected annually during two sampling occasions in the spring. Full details 

of the sampling and estimation methods are provided by 

Douglas and Marsh (1996) and Coggins and others (2006a). 

Coggins and others (2006b) recommended exploring the 

use of individual capture histories within the ASMR framework to reduce confounding between capture probability and 

mortality. Though the updated ASMR models presented in 

this report do not yet incorporate individual capture histories, 

they do model recaptured fish by annual-tagging cohort with 

the intent of reducing parameter confounding by increasing 

the number of observations available for parameter estimation. 

In the non-tag cohort, or pooled, version of ASMR described 

above and by Coggins and others (2006b), age- and time-specific predictions of recaptured fish are not separated by year of 

tagging. As an example, assume that ASMR 3 predicts that 50 

marked age-6 HBC should be captured in 2002. These 50 fish 

could be comprised of fish tagged as age-5 in 2001, age-4 in 
 5

2000, age-3 in 1999, or age-2 in 1998. However, as the model 

is currently formulated, all age-6 fish recaptured in 2002 are 

pooled for a single observation. Assuming that the age- and 

time-specific captures of marked and unmarked fish are Poisson distributed, the log-likelihood, ignoring terms involving 

only the data, is computed as:

where ma,t is the observed number of age-a, unmarked fish 

captured in year t, ma,t ˆ is the predicted number of unmarked 

fish captured, ra,t is the observed number of marked fish captured (i.e., recaptures), a t r , ˆ is the predicted number of marked 

fish captured, and is the parameter vector to be estimated. 

Notice in the second term that the individual log-likelihood 

terms are summed over age and time. However, it may be 

more informative to stratify the recapture data by tagging 

cohort. The proposed log-likelihood is then:

where c is the tag cohort (i.e., all fish marked in year t). In 

principle, this modified log-likelihood should provide additional information on time-specific capture probability and 

may improve parameter estimation.

Evaluating Model Fit

Following Baillargeon and Rivest (2007), standardized 

Pearson residuals of observed and predicted age composition for both unmarked and marked fish were used to evaluate model fit among the three different ASMR models. The 

standardized Pearson residual is the difference between the 

observed and predicted values scaled by an estimate of the 

standard deviation as:

where nt is the number of observations (e.g., the number of 

marked fish recaptured each year) and oa,t and pa,t are the 

proportions of fish in each year and age class observed and 

predicted, respectively. The individual Pearson residuals for 

each combination of age and time were plotted to look for 

consistent bias for individual brood-year cohorts. In addition, 

quantile-quantile (Q-Q) plots were used to compare the distribution of the Pearson residuals to a theoretical normal distribution. The intercept of the theoretical curve is approximately 

the standard deviation of the distribution of Pearson residuals, 

where a small value of the intercept indicates a narrow distribution of the residuals. Deviations from the theoretical curve 

indicate a non-normal distribution of the Pearson residuals and 

imply that the model error is not well distributed (e.g., tending 

to more often either over- or under-predict age proportions) 

and possibly inducing bias in parameter estimates.

In addition to examination of model fit using Pearson 

residuals, information theory was also used to aid in model 

evaluation. The use of this approach is increasingly common 

in ecological studies to arbitrate among competing models 

and is primarily concerned with estimating the KullbackLeibler (K-L) distance between the model and the “truth” as 

a measure of model support (Burnham and Andersen, 2002). 

The Akaike information criterion (AIC; Akaike, 1973) is the 

standard estimator for the relative K-L distance and is computed as a function of model likelihood and number of model 

parameters. Following review of the ASMR method in 2003 

(Kitchell and others, 2003), it was pointed put that although 

ASMR uses a quasi-likelihood structure of estimating equations and true likelihood, estimates of relative K-L distance 

using AIC, though not strictly appropriate, would be useful for 

model arbitration (C. Schwartz, Simon Fraser University, written commun., 2003). Therefore, in addition to the evaluation 

based on Pearson residuals, an AIC evaluation was conducted.

Section 2—Estimating the Humpback Chub 

Growth Function Using Mark-Recapture Data

Capture-recapture data have long been used by biologists in an attempt to characterize growth rates of fish. The 

basic technique for estimating growth model parameters from 

capture-recapture data is to predict the amount of growth in 

the elapsed time between capture and recapture. Assuming 

the standard von Bertalanffy growth curve (Bertalanffy, 1938) 

predictions of length at time t and at time t+Δt, Fabens (1965) 

developed the most basic model where the predicted growth 

increment is given as:

where L∞ and k are the asymptotic length and the rate at 

which length approaches L∞ , respectively (Quinn and Deriso, 

1999). Parameter estimates are found by minimizing the difference between predicted and observed growth increments.

Though this technique has been widely applied, numerous authors have pointed out that resulting parameter estimates 

will be biased if individual fish exhibit growth variability (e.g., 

(2)

(1)

(3)

(4)

,
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 positively 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). Von 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 demonstrating 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 recruitment occurs when fish are spawned 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 potadromous migration between the Colorado 

River and the LCR (Gorman and Stone, 1999). This ontogenetic 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.; 2007) (hereafter Walters and Essington, 

written commun., 2007) to fit growth increment data to a general growth model (Paloheimo and Dickie, 1965) were used to 

describe the rate of change in weight as:

where the first term describes anabolic (i.e., mass-acquisition) 

processes and is governed by a term representing the massnormalized rate at which the animal acquires mass (H), the 

mass of the animal (W), and a parameter (d) describing the 

scaling of the anabolic process with mass. The second term 

represents catabolic (i.e., mass loss through basal metabolism, 

activity, and gonad production) processes where m is the massnormalized rate at which the animal looses mass and n is the 

scaling factor of catabolic processes with mass. Assuming a 

constant relationship between length and weight over time as:

where L is length and a and b are constant, simple algebra 

provides an analogous relationship for the rate of change in 

length as:

Constants in this relationship are related to those in equations (5) and (6) as:

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 2/3 mass, and results in the standard von 

Bertalanffy growth model:

where t0 is the theoretical age where body length is equal to zero.

Walters and Essington (written commun., 2007) then 

define a protocol to estimate the parameters of equation (5). 

To restate this definition, first assume that measurement errors 

in the length of fish are normal with variance , and that all 

fish follow a standard von Bertalanffy growth curve (equation 12) 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:

Assuming that individual L∞ is normally distributed with variance , the variance of each L(t + ∆t) given L(t), mean L∞ , 

, and will be: 

(6) (14)

(7)

(8)

(11)

(10)

(9)

and

(12)

,

.

(5)

,

(13)

.
 7

Deviations between observed and predicted growth increment 

for individual fish (i) are given as:

Walters and Essington (written commun., 2007) note that 

it is then possible to estimate the parameter vector = { L∞ , k, 

, } by maximizing the log-likelihood function:

where s is the number of growth increments. This is essentially 

an inverse-variance weighting strategy. However, they further 

note that the variance components of the measurement error 

and L∞ are typically not separately estimable, such that, in 

practice, it is usually necessary to specify one of them a priori. 

Though this procedure is applicable assuming fish growth 

is described by equation (12), if fish growth is described by 

equation (7), there is no analytical solution of , as in 

equation (14). However, Walters and Essington (written commun., 2007) note that if one first estimates , and in particular 

k, using equations (14–16), and assuming that the individual 

variances computed using equation (14) are adequate, then 

deviations from the general model (equation 7) can be used in 

the log-likelihood. These deviations are computed as:

After specifying the parameters a and b from equation (6), 

estimation proceeds as above with the parameter vector = 

{H, d, m, n, σ2

L

, σ2

m}.

The above procedure was implemented using both Excel 

Solver tool (Lasdon and Allan, 2002) and AD Model Builder 

(ADMB; Fournier, 2000) to obtain estimates of . The parameter set was reduced by specifying σ2

m = 31.8 mm2

, based on 

an analysis of the observed error between consecutive measurements of identical fish within 10 days. The a and b parameters 

for equation (6) were specified as 0.01 and 3, respectively. 

To calculate the conditional variance of each L(t + ∆t), k was 

specified as 0.145, based on previous analyses. Additionally, 

penalty terms were included in the log-likelihood equation (16) 

to constrain d and n so that they did not deviate too far from the 

theoretical values, assuming standard von Bertalanffy growth 

of 2/3 and 1, respectively. Alternative weight values on these 

penalty terms were evaluated to find an appropriate tradeoff 

between minimum weights and decreased log-likelihood.

Because all the information contained in the markrecapture data are for fish larger than 150 mm TL, extrapolating results to the growth rate of smaller fish could be 

problematic. Though this is not necessarily a concern related 

to assignment of age to fish greater than or equal to 150 

mm TL, as required by ASMR, an accurate growth curve 

across all sizes would be desirable. Fortunately, Robinson 

and Childs (2001) conducted rigorous monthly sampling of 

juvenile HBC in the LCR during 1991–94. They used these 

data to estimate average monthly length from age-0 months 

to age-32 months. These estimates were used in an additional 

log-likelihood term to constrain the predicted lengths from 

the general model to be similar to those reported by Robinson and Childs (2001). Using these auxiliary data and assuming normal deviations allowed the incorporation of information on the growth rate of fish before they are large enough to 

be implanted with PIT tags. With these constraints in place, 

the full log-likelihood was:

where is the weighting value for the penalty terms, L(i) is 

the predicted length in month i from the general model, and 

(i) is the predicted length over mos = 32 months, as reported 

by Robinson and Childs (2001). The weighting term can be 

interpreted as the prior variance on the standard von Bertalanffy parameters (d = 2/3 and n = 1).

A logical extension of the general model is to assume 

temperature dependence in growth rate. Accounting for 

changes in growth rate as a function of temperature is likely 

to be very important for the analysis of this dataset for two 

reasons. The first is to account for the growth rate differences 

associated with occupancy in either the LCR or the mainstem 

Colorado River. The second is to account for seasonal changes 

in water temperature within the LCR. The importance of this 

second consideration is further magnified by the temporal 

distribution of sampling within the LCR. Sampling in the LCR 

typically occurs in the spring and fall. Therefore, much of the 

observed growth increment data corresponds to either summer 

growth (i.e., observations of fish captured in spring and again 

in fall) or winter growth (i.e., observations of fish captured 

in fall and the following spring). Because growth varies with 

temperature (Paloheimo and Dickie, 1966), one would expect 

slower growth rates during winter than during summer. This 

general prediction is also consistent with both field (Robinson 

and Childs, 2001) and laboratory (Clarkson and Childs, 2000) 

observations of HBC.

Walters and Essington (written commun., 2007) present a method to allow temperature dependence in α and κ in 

(15)

(16)

(17)

(18)

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

equation (7). This is accomplished by defining temperaturedependent multipliers of α and κ as:

where f (T ) c is the temperature-dependent multiplier of α,

and f (T ) m is the temperature-dependent multiplier of κ. The 

consumption and metabolism coefficients (Qc and Qm ) of a 

Q10 relationship allow these multipliers ( f (T ) c and f (T ) m ) 

to increase or decrease with temperature (T). One can think 

of these constants as the amount that the anabolic or catabolic 

processes will change with an increase in temperature from 

10°C to 20°C. Inclusion of these temperature-dependent multipliers into equation (7) yields:

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 of LCR occupancy is assumed to be:

where L is fish total length and Lt 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 yearround LCR residency approaches unity at lengths much less 

than Lt and decreases to 0.2 at lengths much larger than Lt. 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 observation that adult HBC use the LCR for spawning (Gorman 

and Stone, 1999).

A weighted temperature function experienced by fish of a 

particular length can then be defined as:

where T (t) LCR is the time-dependent water temperature in the 

LCR and is the time-dependent water temperature in the 

mainstem Colorado River. This overall temperature experienced 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 fit to these data as:

where t is time in fraction of a year starting April 1, peak t is 

a phase shift allowing predicted peak temperature to align 

temporally with the observed peak temperature, Tave is the 

1⁄2-amplitude temperature and roughly corresponds to the 

average annual temperature, and Tmax is the maximum annual 

temperature. Values for peak t , Tave , and Tmax 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 10°C 

was assumed. This value corresponds 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 = {H, d, m, n, Qc , Lt } was estimated 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 (17) to compute the 

deviations between observed and predicted growth. Walters 

and Essington (written commun., 2007) recommend constraining 1.8 < Qm < 2.4 following guidance from a meta-analysis 

by Clark and Johnson (1999). Therefore, 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, σ2

L

 was specified 

as 2,000 to correspond with a coefficient of variation of about 

10%, as is typically observed in fish populations (S. Martell 

and C. Walters, University of British Columbia, oral commun., 

2006). The relative fit to the data for the temperature-independent growth model (TIGM) and the temperature-dependent 

growth mode (TDGM) were evaluated using AIC techniques 

(Burnham and Andersen, 2002).

Section 3—Incorporation of Ageing Error in 

ASMR Assessments

As mentioned above, Coggins and others (2006a) 

assigned age to individual fish strictly as an inverse von 

(23)

(22)

(24)

(21)

(20)

,

(19)

and

,
 9

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 particular length interval was estimated following methods 

reported by Taylor and others (2005). First, the probability of 

an age-a fish having length within length bin l is specified as:

where length bin l has mid-point length l, minimum length l-d, 

and maximum length l+d. These probabilities can be thought 

of as a matrix with rows corresponding to length bins and columns 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 density with mean al and variance . The mean length-at-age is 

computed from the temperature-dependent growth model and 

procedures described previously. The variance in length-at-age 

is , according to the assumption that coefficient of 

variation in length ( ) at age is constant ( =0.10).

With available, one option to compute 

would be to normalize each matrix cell by the sum of its 

row as:

However, Taylor and others (2005) suggest that this type 

of procedure will induce bias if the population has experienced size-dependent mortality (e.g., size-selective fishing 

mortality). This result is because within a particular age class, 

fast-growing individuals (i.e., large L∞ ) may experience either 

a higher or lower mortality rate than their cohorts and therefore 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 exploitation 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 of fish in 

each age and length bin were computed as:

where Na is the abundance of fish at each age. If the agespecific mortality rate ( M a ) is available and recruitment (R) is 

assumed constant, abundance-at-age is given by:

With abundance at each age and length bin thus available, 

the proportion in each age and length bin can then be calculated as:

where 

The probability of age given length is then calculated as:

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 natural 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:

where M ∞ is the mortality rate suffered by an adult fish of size

L∞ . This mortality schedule was calculated with M ∞ specified as 0.148, as estimated by ASMR 3 considering tag-cohort 

specific data (see results below).

Four seasonal P(a | l) matrices were computed and used 

to assign age to fish captured at different times of year. Growth 

during the year could thus be accounted for by recalculating , such that length-at-age for a particular age-a was 

computed as either l(a), l(a+.25), l(a+0.50), or l(a+0.75). The 

resulting seasonal P(a | l) 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 based on the seasonal P(a | l) matrices. To understand this procedure it is first helpful to recognize that given a 

fish with length in bin l, the resulting probabilities of belonging to each age is a multinomial probability distribution with 

(26)

(27)

(28)

(29)

(30)

(25)

(31)
10 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

number of categories equal to the number of ages. Assigned 

age for a single fish having length in bin l is therefore a 

multinomial random variable with probabilities P(a | l) with 

a single draw. The multinomial random number generator 

within program R (R Development Core Team, 2006) was 

used to randomly assign age to tagged fish. Age-at-recapture 

was calculated as the sum of age-at-tagging and time-at-large. 

For each resulting dataset of captures- and recaptures-at-age, 

adult (age-4+) abundance and recruitment was estimated using 

ASMR 3. Additionally, ADMB was used to compute the 95% 

profile confidence interval for adult abundance and recruitment. This procedure was repeated to generate and analyze 

1,000 datasets (i.e., Monte Carlo trials).

Results

2006 Humpback Chub Assessment Update with 

Refinements

Index-Based Assessments

Between 1987–99 and 2002–present, the Arizona Game 

and Fish Department sampled HBC using hoop nets in the 

lower 1,200-m section of the LCR. Examination of this 

catch-rate index suggests that abundance of both subadult 

(150–199-mm TL) and adult (≥200-mm TL) HBC declined 

between 1987 and 1992 and remained relatively constant 

through much of the 1990s (fig. 1a). Since 2003, this index 

suggests a slight upward trend in the abundance of subadult 

fish. Note that several data points in the index are shifted 

slightly relative to those reported by Coggins and others 

(2006a). This adjustment resulted from additional standardization of the data used to construct this index (David Ward, 

Arizona Game and Fish Department, oral commun., 2007). 

The index of adult abundance in the LCR inflow reach of the 

Colorado River indicates a similar trend in adult fish (≥200 

mm, fig. 1b). In general, this index shows a stable to declining trend in relative abundance through the 1990s with a slight 

indication of increased abundance in most recent years. All 

monthly trammel-net samples from the LCR inflow reach for 

1990–2006 are presented in figure 1b. However, only samples 

from 1990–93, 2001, and 2005–06 (i.e., dark circles in figure 

1b) represent robust sampling coverage throughout the entire 

reach. Annual sample sizes in 1994–2000 and 2002–03 (i.e., 

hollow circles in figure 1b) were between 2% and 50% of the 

1990–93 average sample size, and in some years effort was 

focused near the LCR confluence where HBC density may 

have been highest. Thus, the 1990–93, 2001, and 2005–06 data 

are likely to best depict the overall trend of relative abundance 

within this reach. Simple linear regression analyses provide 

estimated slopes that are not significantly different from zero 

(p = 0.16 for all data and p = 0.26 for the preferred data; fig. 1b).

Tagging-Based Assessments

As described above, the data required for the ASMR 

models are numbers of fish marked and recaptured each year 

and for each age. For the results contained in this section, all 

ages were assigned based on the standard von Bertalanffy 

growth curve as described in Coggins and others (2006b). 

With that in mind, examination of the age distribution of fish 

marked and recaptured since 1989 provides insight into the 

trends in sampling effort and also provides important information related to HBC survival (fig. 2). The top panel of figure 2 

shows the numbers of newly tagged fish by age and year. The 

total number of fish marked annually is influenced by both 

trends in sampling effort and numbers of unmarked fish alive. 

The most consistent period of sampling has been 2001–present; approximately 1,100 fish have been tagged annually. 

Because a large fraction of the population was tagged in 

sampling periods 1 and 2, the majority of fish tagged in recent 

years are young fish and the number of new fish tagged each 

year declined with fish age. The bottom panel of figure 2 

represents the numbers of fish of each age recaptured each 

year. The same patterns related to sampling effort are evident, 

but there are some very interesting patterns that result from 

the high sampling effort in the early to mid-1990s (fig. 2). For 

example, in 1995 a total of 1,244 HBC were collected and 902 

of these fish had been marked in previous years. This pattern 

is evident for several years of data, indicating that the high 

sampling effort in the early 1990s resulted in marking upwards 

of 70% of the HBC population. The sampling program since 

2000 (period 4) has been plagued by the low sampling effort 

in the mid- to late-1990s. For example, figure 2 shows that 

because few age-3 to age-5 fish were tagged during period 

3, there were few age-8 to age-10 fish recaptured in the early 

2000s. This contributes to the “spoon” shapes in the lower 

panel of figure 2, where there were relatively large numbers of 

fish younger than age-10 recaptured in period 4, lower catches 

of age-10 to age-15 fish, and relatively stable numbers of older 

than age-15 fish.

Another finding is the extreme longevity of HBC. This 

is most evident by examining the number of HBC of each age 

marked in each year and recaptured in subsequent years (fig. 

3). Figure 3a shows the number of HBC of each age marked 

in 1989–92 and recaptured in subsequent years. This figure 

shows that a remarkable number of old fish (>age-15), first 

tagged in the early 1990s, continued to be recaptured into 

2006. This slow decay pattern of tagged fish demonstrates the 

low mortality rate suffered by older HBC.

Closed Population Models

The time series of abundance estimates for HBC greater 

than or equal to 150 mm TL in the LCR during spring indicate 
 11

a decline in abundance from the early 1990s to the present 

(fig. 4). However, as is apparent in the data, these estimators 

are very imprecise with corresponding poor ability to detect 

significant trends. Additionally, preliminary analyses of data 

collected in this program suggest that the 2007 estimate may 

be up to twice as large as the 2006 estimate (R. Van Haverbeke, U.S. Fish and Wildlife Service, oral commun., 2007).

ASMR without Tag-Cohort Specific Data

Overall, the three ASMR formulations generally agree 

that adult (age-4+) HBC abundance has gradually increased 

since about 2001 (fig. 5). Among these three models, the 

2006 adult abundance estimate is 6,690 (95% credible interval 

(CI) 6,403–6,994), 6,768 (95% CI 6,397–7,131), and 6,648 

(95% CI 6,222–7,102) for models ASMR 1, 2, and 3, respectively. These results suggest that this population has increased 

35%–40% from an estimated low abundance of approximately 

4,800–5,000 during 2000–01. Estimated recruitment (age-2) 

among models is also in agreement (fig. 6). Following low 

recruitment for brood years during the early 1990s, all the 

models suggest that recruitment increased through the latter 

part of the 1990s. The biggest discrepancy among the three 

models is that ASMR 1 suggests a decline in recruitment 

following the 2000 brood year, while the other two models 

suggest stability. The structural assumptions of model ASMR 

3 (see Coggins and others, 2006b) do not permit a reliable 

recruitment estimate for brood year 2003. An additional difference in the model results is in the estimation of instantaneous 

adult mortality ( M ∞ ), where adult mortality ranges from 

0.119 (ASMR 1) to 0.133 (ASMR 3).

Model Evaluation and Selection

With these results in hand, the question becomes which 

model is best? Stated another way, which model produces 

results most consistent with or best supported by the data? In 

this case, the discrepancies among model results related to 

adult abundance are not large; as a result, selecting the best 

model is probably not critical from a management or conservation perspective. However, the models do reflect rather 

different hypotheses about recruitment trends. Model ASMR 1 

supports the hypothesis that recruitment has declined following the 2000 brood year, while the other two models suggest 

relative stability. Therefore, selecting the model (hypothesis) 

that is the most consistent with the data is desirable.

Model fit to the data was assessed by plotting the Pearson residuals of the observed and predicted numbers of fish 

marked and recaptured for each year and age (figs. 7–9). The 

patterns in Pearson residuals for both ASMR 1 (fig. 7) and 

ASMR 2 (fig. 8) demonstrate systematic lack of fit for particular sets of cohorts. This is best seen in the recapture residuals 

where it is apparent that there are more fish observed than 

predicted for about eight pre-1990 cohorts, particularly for 

observations after 2000. Additionally, there are fewer recaptures associated with the 1992 cohort than are predicted. These 

systematic trends are likely imposing bias in the model results 

for ASMR 1 and ASMR 2. In contrast, there is much less 

systematic lack of fit in the residual patterns for ASMR 3 (fig. 

9). Additionally, among the three models the Pearson residual 

standard deviation is smallest for ASMR 3.

The finding that ASMR 3 has the best fit among the three 

models is not surprising, since it has the largest parameter set. 

Although ASMR 3 only varies 13 parameters in the direct 

numeric search, the conditional maximum likelihood estimates 

are used for each age- and time-specific capture probability 

(Coggins and others, 2006b). Therefore, assuming a liberal 

maximum longevity of 50 years, ASMR 3 has 895 parameters. The question then becomes whether these additional 

parameters are justified. To provide insight into this question, 

the relative K-L distance was estimated using AIC (table 1). 

These results strongly indicate that model ASMR 3 is superior 

to ASMR 1 and 2; a finding congruent with the evaluation of 

model fit using Pearson residuals.

Although it is comforting to find agreement between 

these two evaluations, one should ask: why is the more 

complicated structure of ASMR 3 needed to adequately fit 

the data? Since the fundamental difference between ASMR 

1 and 2 and ASMR 3 is the amount of flexibility in age- and 

time-specific capture probabilities, the pattern in ASMR 3 

estimated capture probabilities is of interest (fig. 10). The 

discrepancy in capture probabilities between sampling period 

2 (i.e., 1991–95; heavy gray lines) and sampling period 4 (i.e., 

2000–06; heavy black lines) suggests a major shift in the gear 

selectivity. Sampling since 2000 appears to be much less effective at capturing fish between ages 9–20 years old than was 

sampling during the second period. Since structural assumptions in ASMR 1 and ASMR 2 require that vulnerability is 

asymptotically related to age, it is not surprising that these 

models are not able to account for this unexpected pattern, and 

thus display poor model fit.

ASMR with Tag-Cohort Specific Data

In addition to repeating the analyses by Coggins and 

others (2006a) above, the ASMR models were also fit to the 

tag-cohort specific data using the log-likelihood in equation 

(2). The trends in adult abundance and recruitment are quite 

similar to those found using the simpler log-likelihood (figs. 

11 and 12). In general, adult abundance estimates are slightly 

higher at the beginning of the time-series and slightly lower at 

the end. Adult abundance estimates for 2006 are 6,057 (95% 

CI 5,797–6,308), 6,138 (95% CI 5,842–6,458), and 5,893 

(95% CI 5,554–6,242) for the ASMR 1, 2, and 3 models, 

respectively. Adult mortality ( M ∞ ) estimates from the models 

fit to the stratified data indicate slightly higher adult mortality 

(0.128, 0.137, and 0.148 for ASMR 1–3, respectively) than 

when fit to the pooled data. This finding is consistent with the 

more rapid decay observed in the time-series of adult abundance. Another difference is a slight increase in the precision 

of the estimates using the stratified data (fig. 13). This observation provides marginal confirmation of the assertion that it 
12 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

might be possible to extract additional information considering 

the stratified, rather than pooled, data.

Model Evaluation and Selection

Examination of Pearson residuals for the tag-cohort 

specific models suggests similar patterns in model misspecification for ASMR 1 and ASMR 2 (figs. 14 and 15). However, 

there appears to be even less residual pattern for ASMR 3 (fig. 

16), suggesting slightly better agreement with the data than the 

pooled data model fit. This, again, suggests that segregation of 

the data is permitting greater extraction of information. Model 

evaluation using AIC methods strongly indicates that ASMR 

3 is preferable (table 2), which is generally in agreement 

with the residual evaluation. Finally, a very similar pattern in 

estimated capture probability from ASMR 3 emerges, suggesting a similar mechanism to explain the poor performance of 

models ASMR1 and 2 (fig. 17).

Assessment Update Summary

The adult portion of the LCR HBC population appears to 

have increased in recent years as a result of increased recruitment, particularly associated with brood years 1999 and 

later. In addition, model evaluation procedures indicate that 

the results from model ASMR 3 are most consistent with the 

available data. Using data stratified by tagging cohort added 

additional information to the assessment, as indicated by the 

slightly higher precision of adult abundance estimates. However, adult abundance estimates, as reported, are extremely 

precise. This level of precision is questionable, since the assessment doesn’t incorporate uncertainty in the assignment of age.

Estimating the Humpback Chub Growth Function 

Using Mark-Recapture Data

Both the temperature-independent (TIGM) and temperature-dependent (TDGM) growth models described in the 

methods section were fit to 14,971 observed growth intervals 

extracted from the HBC mark-recapture database. All fish 

were larger than 150 mm TL and the time interval between 

capture and recapture exceeded 30 days. Though greater than 

60% of the fish were at large for 1 year or less, a small fraction 

of the observations were for much longer time intervals (fig. 

18). The longest time interval in the dataset was 5,538 days 

(about 15 years).

The measurement error contained in the dataset was 

estimated by computing the observed difference in measured 

lengths of fish captured and recaptured within 10 days. This 

resulted in a measurement error variance of 31.8 mm2 across 

all sizes of fish, implying that most TL measurements were 

within 11 mm of the true TL. This amount of measurement 

error is fairly high but not unexpected, considering the difficulty in measuring live fish. However, this error rate contributes substantially to the variability of observed growth rate.

The TIGM was fit with prior variance weighting terms 

on the d and n parameters ={0.00001, 0.0001, 0.001, 0.01, 

0.1,0.5,1,10,100,1,000, and 10,000} to explore the effect of 

constraining these parameters to values near standard von Bertalanffy values. The log-likelihood is nearly identical for all 

values of = 0.01 and greater, but reducing below = 0.01 

caused large changes in the log-likelihood (fig. 19). Therefore, 

= 0.01 was specified as the optimal weighting value for both 

the TIGM and TDGM.

To estimate the parameters of the TDGM it was necessary 

to first fit the time-dependent LCR water temperature model. 

Fortunately, the sine curve function with parameters peak t = 

-0.011, Tave = 17.9, and Tmax = 23.2 fit the observed average 

monthly temperatures very well (fig. 20).

The estimated parameters, log-likelihood, and AIC 

statistics for the TIGM and TDGM are presented in table 3. 

The parameter values for the TIGM suggest an extremely low 

value for the catabolic constant (m) and a catabolic scaling 

parameter value (n) greater than unity. This is a rather unlikely 

situation from a biological perspective and indicates that this 

model may not be well supported by the data. In contrast, the 

estimated scaling parameters for the TDGM are not much 

different than what would be expected under the standard von 

Bertalanffy model, where the anabolic scaling parameter (d) 

should be close to 2/3 and the catabolic scaling parameter (n) 

should be close to unity. Results from AIC indicate strong 

support for the TDGM over the TIGM. However, the parameter correlation matrices for each of these models show very 

high correlation, indicating that all of the parameters are 

not separately estimable (table 4). In situations such as this, 

where the model is not full rank, it has been suggested that 

the K-L distance is undefined (Viallefont and others, 1998; 

Bozdogan, 2000).

An alternative way to arbitrate among these two models 

is simply to examine the model fit to the data. The observed 

growth rate as a function TL at the start of the interval is 

extremely variable, particularly at smaller sizes (fig. 21). It 

is also apparent that all three lines (the fit of the TIGM and 

the fit of the TDGM corresponding to LCR water temperatures during summer and winter) differ from the strict linear 

relationship implied by a standard von Bertalanffy model. The 

temperature-independent model is somewhat of a compromise between the temperature-dependent summer fit and the 

temperature-dependent winter fit. It is also clear that observed 

summer growth is generally greater than observed winter 

growth, suggesting that growth rate is oscillating with temperature (fig. 22).

Each of the models was used to predict length as a function of age. In addition to the two models fit above, length-atage was also predicted using the growth function reported in 

the USFWS recovery goals document (U.S. Fish and Wildlife 

Service, 2002) and using the TDGM for a constant temperature of 10oC (fig. 23). This last curve is equivalent to a fish 

experiencing a constant 10oC temperature and is a prediction of length-at-age for a fish spending its entire life in the 

mainstem Colorado River. Examination of these curves show 
 13

that the USFWS growth curve tends to predict somewhat 

smaller sizes at young ages and larger sizes at older ages than 

is implied by the mark-recapture data. The TIGM and TDGM 

predict very similar length-at-age with the exception of 

10–25-year-old fish. Two features are apparent in the TDGM: 

(1) temperature-dependent periodic change in growth rate at 

ages younger than about age-5, and (2) an apparent “bend” 

in relationship at approximately age-4. This age corresponds 

to the length-at-transition (Lt) where HBC are rapidly shifting from primarily LCR occupancy to primarily mainstem 

Colorado River occupancy. A Lt length of 236 mm TL is most 

strongly supported by the data and the TDGM (table 3).

Finally, it is informative to use the TDGM to predict 

monthly growth increments as a function of TL. These predictions, based solely on field data, can then be compared to 

laboratory observations of the same or similar species. Growth 

rate predictions from the LCR population are much higher 

than a population experiencing constant 10oC temperature 

(fig. 24). This latter growth rate is presented as a prediction of 

monthly growth rates that would be observed in the mainstem 

Colorado River.

Incorporation of Ageing Error in ASMR 

Assessments

The TDGM and the procedures identified in the methods 

sections were used to construct seasonal P(a | l) matrices. The 

resulting probability distributions were subsequently plotted as 

surfaces to allow examination of the uncertainty in predicting 

age given length (fig. 25). The most obvious feature of these 

probability surfaces is the increasing uncertainty in age assignment with increasing length. For instance and considering the 

April–June P(a | l) surface (fig. 27, top left panel), one can see 

that a 150-mm-TL fish is age-2 with highest probability, but 

there is some chance that it could be any age between age-1 

and age-4. In contrast, a 300 mm fish is approximately age-7 

with highest probability, but could be as young as age-4 or as 

old as age-18. It is such uncertainty that this assessment was 

intended to incorporate.

As described in the methods section, a stochastic assignment of age to each fish was made using the appropriate 

P(a | l) matrix, depending on the time of year the fish was first 

captured. Using this procedure, a total of 1,000 input datasets 

were generated and the ASMR 3 model was fit to each. For 

each model fit, the estimated annual adult abundance and 

95% profile likelihood confidence bounds were retained. The 

estimated brood year recruitment and 95% profile likelihood 

confidence bounds were also retained. Note that because of 

the uncertainty in assigning age to even the smallest fish in 

the dataset, newly tagged fish now have a possibility of being 

age-1. As a result, it was necessary to expand the age range of 

the model such that recruitment estimates were for age-1 fish.

Estimated adult abundance (age-4+) ranged from 9,322 

(95% CI 8,867–9,799) in 1989 to 6,017 (95% CI 5,369–6,747) 

in 2006 (fig. 26). The coefficient of variation for these 

estimates ranges from approximately 1%–7%, in contrast 

to 0.5%–3% if uncertainty in assignment of age is ignored 

(figs. 13 and 27). The trend in recruitment considering the 

new growth function and ageing error contains much greater 

uncertainty than when ageing error is ignored (figs. 12 and 

28). Although the point estimates from the two models are in 

agreement that recruitment has been increasing since the mid1990s, the uncertainty in the recruitment estimates from the 

latter assessment makes statements about differences among 

years tenuous.

Discussion

2006 Humpback Chub Assessment Update with 

Refinements

The overall result of the mark-recapture-based open 

population model assessment is that the adult portion of the 

LCR HBC population appears to have increased in abundance 

since 2001. The assessment model best supported by the data 

is ASMR 3 with ageing error. This model produces a 2006 

adult abundance estimate of approximately 6,000 fish. In addition, this analysis suggests that there has been an increase of 

approximately 20%–25% in adult abundance since 2001. This 

increase is likely related to an increasing recruitment trend 

beginning perhaps as early as 1996, but likely no later than 

1999. Recruitment of juvenile HBC since 2000 appears stable, 

but the precision of these estimates is low when ageing error is 

included in the assessment.

The LCR hoop-net abundance index suggests a modest 

increase in the abundance of juvenile fish and stability in the 

abundance of adult fish. The LCR inflow reach trammel-net 

abundance index indicates stability with a slight indication of 

increased abundance in 2005 and 2006. Although confidence 

in the mark-recapture-based open population model results 

might be higher if the catch-rate metrics indicated similar 

trends, it is not surprising that the catch-rate metrics are not 

able to detect a 25% increase in abundance. The basic, and 

frequently violated, assumption that must be made when 

evaluating a catch-rate time series is that capture probability 

must remain constant for the metric to be well correlated with 

abundance (MacKenzie and Kendall, 2002). There is good 

reason to suspect that this assumption is violated for the index 

data series presented in this update because of the influence 

of abiotic factors on catchability (Arreguin-Sanchez, 1996). 

As an example, a likely significant driver of catchability in 

the LCR is turbidity (Dennis Stone, U.S. Fish and Wildlife 

Service, oral commun., 2007; U.S. Fish and Wildlife Service, 

unpub. data, 2007), and turbidity varies greatly in the mainstem Colorado River and the LCR as a function of tributary 

freshets and dam operations.

A more significant concern is the lack of correlation 

between ASMR 3 results and the closed population model 
14 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

mark-recapture estimates in the LCR. However, given the low 

precision in the LCR estimates and the possible influence of 

migration magnitude and timing on the results of this program, 

it is, again, not too surprising that these assessments are not 

strongly correlated with the ASMR 3 results. The low precision of these estimates may not permit detection of a 25% 

increase in adult abundance. Additionally, preliminary analyses of data collected during 2007 suggests that the abundance 

estimate for 2007 may be twice as large as the 2006 estimate 

(R. Van Haverbeke, U.S. Fish and Wildlife Service, oral commun., 2007). Though this result would provide support for the 

ASMR 3 results, it would also call into question the ability 

of the LCR program to provide a consistent metric of overall 

population size. One would have to reconcile whether that 

level of change was related to a very large age class entering 

the sampled population, a larger than normal fraction of the 

population entering the LCR during the sampling period, or 

some other factor.

Though the GCDAMP is fortunate to have such a large 

mark-recapture database for these high-profile endangered 

animals, significant changes in sampling protocol over time 

continue to cause ambiguity. As identified by Melis and others 

(2006), retrospective analyses of the data suggest a continual 

updating of the adult mortality rate estimates as additional 

information has been collected since 2000. Following addition 

of the 2006 data, this updating is again apparent (fig. 29). It 

appears that adult mortality rate estimates may be stabilizing 

as more data are collected, but it is difficult to be certain. The 

likely cause of this updating is the sampling program essentially having to “catch-up” following the low sampling effort 

during period 3. When focused analyses of this dataset began 

with open population models in 2000 (U.S. Geological Survey, 

unpub. data, 2000), there had been so little sampling in the 

mid- to late-1990s that the models interpreted the lack of old 

fish captures as a relatively high adult mortality rate. As additional data were collected through a more rigorous sampling 

program during 2000–06, mortality rate and capture probability estimates were adjusted. The hope is that if the GCMRC 

continues with a fairly uniform sampling program over time, 

adult mortality rate will stabilize and only abundance estimates in the last few years of the dataset will be subject to 

significant updating.

An additional finding, identified by Martell (2006) and 

in this assessment, is the major change in gear selectivity 

between sampling period 2 and sampling period 4. Though 

it is presently unknown what may have caused this change, 

several possibilities have been suggested. First, it is possible 

that elimination of the extensive trammel-netting effort during 

sampling period 4 may have reduced the capture probability 

of middle-aged fish. If this is true, fitting the ASMR 3 model 

only to data collected in the LCR should indicate similar 

patterns in capture probability during both sampling periods. 

However, when this fit was conducted, the patterns were 

essentially unchanged from those predicted from the entire 

dataset (U.S. Geological Survey, unpub. data, 2007). This 

is not too surprising, since fish captured in the LCR inflow 

reach of the Colorado River represent only about 11% of the 

entire HBC mark-recapture database. Second, it is possible 

that reducing the use of large hoop nets in the LCR during 

sampling period 4 reduced the catch rate of larger fish. Though 

this is possible, the ASMR 3 results indicate a reduction in 

capture probability of mid-sized fish, rather than in the largest 

individuals. Finally, it is possible that sampling in the LCR 

only 4 months of the year during period 4, as opposed to 

10–12 months of the year during period 2, may be the cause. 

This is certainly possible, particularly if there is some differential migration timing for the middle-aged fish relative the 

oldest individuals.

One obvious result of all this confusion is that large 

changes in sampling protocol should be carefully considered 

in light of how those changes may affect the ability to infer 

population change. This is particularly true for populations 

that are in low abundance and exhibit low capture probability. A careful simulation of considered changes may help to 

expose potential problems or, at the very least, help to clarify 

thinking related to proposed changes in sampling protocol. 

Finally, those considering implementing a mark-recapturebased monitoring program should plan to expend considerable 

sampling effort using similar protocols for the duration of the 

monitoring program. The results herein support the recommendations of Williams and others (2001) that the objectives 

of the monitoring program, with regard to issues such as 

precision of measured quantities, should not only be clearly 

identified, but that the measured quantities should be directly 

linked to the management objectives.

Estimating the Humpback Chub Growth Function 

Using Mark-Recapture Data

Understanding the relationship between fish age and fish 

length is necessary to address a host of fundamental issues 

across a broad spectrum of fisheries management. Though this 

relationship is typically estimated using paired observations of 

individual fish age and length (Quinn and Deriso, 1999), this 

often requires sacrificing the animal to obtain the age information. The TIGM and TDGM seek to obtain this information 

through non-lethal sampling and using information that is 

frequently collected in routine mark-recapture studies. Particularly for endangered species such as the HBC, a non-lethal 

method to obtain information on growth rate is mandatory.

In the postdam Colorado River, temperature is thought 

to be a limiting factor affecting native fish spawning, rearing, 

growth, and survival (Kaeding and Zimmerman, 1983; Valdez 

and Ryel, 1995; Gorman and Stone, 1999). As a result, much 

effort has been expended attempting to better understand how 

temperature affects basic functions such as growth (Clarkson 

and Childs, 2000; Robinson and Childs, 2001; Petersen and 

Paukert, 2005), swimming ability (Ward and others, 2002), 

and predation risk (Ward and Bonar, 2003). However, because 

of the sensitivity of HBC at larval and juvenile stages, much 

of the emphasis on understanding the effects of temperature 
 15

on growth has been focused on these smaller individuals. The 

analyses described in this report address the effect of temperature on growth rate of HBC and attempt to estimate the length 

at which fish transition from primarily LCR occupancy to 

primarily mainstem occupancy. The general implication from 

the findings reported herein is that growth rate will increase 

substantially with a temperature increase from 10oC to 20oC, as 

is indicated by the values of Qc = 4.6 and Qm = 2.0. These coefficients suggest that anabolic processes will more than double 

relative to catabolic processes across this temperature range.

As an additional evaluation of the results of the analysis, monthly growth rates from the TDGM were compared to 

laboratory observations of juvenile HBC growth. Clarkson and 

Childs (2000) conducted laboratory experiments to evaluate 

the growth rate of larval HBC at 10oC, 14oC, and 20oC. They 

report monthly growth rates of 1 mm/month, 13 mm/month, 

and 17 mm/month for these temperatures, respectively. Considering the estimated monthly growth rates from the TDGM 

in figure 24, the TDGM tends to overestimate the growth rates 

reported by Clarkson and Childs (2000) at 10oC and underestimate the growth rate at 20oC. However, the results reported 

herein are in overall agreement with the laboratory study.

The TDGM and related age–length function should be of 

considerable use to researchers studying HBC throughout the 

Colorado River Basin. Additionally, this case history should 

also be useful to anyone wishing to recover temperaturedependent bioenergetic parameters for fish using capturerecapture data, or to estimate the relationship and associated 

uncertainty between fish age and length using non-lethal 

techniques. This technique shows considerable promise in 

extracting useful information on fish growth from field data, 

rather than from laboratory studies, where such information is 

typically obtained.

Incorporation of Ageing Error in ASMR 

Assessments

A major criticism of the ASMR technique, as previously 

applied, is that it does not explicitly account for uncertainty 

in the assignment of age to individual fish (Kitchell and others, 2003). As a result, abundance, recruitment, and mortality 

estimates may contain excessive bias. Additionally, estimates 

of precision are likely overstated by not incorporating this 

important source of uncertainty. The analyses presented in 

this report attempt to address these concerns by constructing a 

more rigorous model to predict length as a function of age and 

to incorporate uncertainty from age assignments into estimates 

of abundance and recruitment. Coggins and others (2006b) 

conducted sensitivity analyses on the effect of random ageing 

error and found little systematic bias in reconstructed recruitment trends. However, the current analysis is a more rigorous 

treatment of the problem and has two major implications.

First, model results of estimated adult abundance are still 

very precise even when uncertainty in the assignment of age is 

explicitly accounted for in the assessment. Following reviews 

by Kitchell and others (2003) and Otis and Wickham (U.S. 

Geological Survey, written commun., 2006), this assessment 

lends additional credibility to results from ASMR, indicating 

that it provides a rigorous measure of the state of the adult 

portion of the LCR HBC population. It is recommended that 

this assessment be considered “best available science” for use 

in contemplating management decisions both within the Glen 

Canyon Dam Adaptive Management Program and the U.S. 

Fish and Wildlife Service.

Second, this analysis points out the difficulty that open 

population models generally have in the precise estimation 

of recruitment (Williams and others, 2001; Pine and others, 

2003). Because many of the most critical management questions for HBC center around how best to improve recruitment, 

particularly considering improved rearing conditions in the 

mainstem Colorado River, it will be difficult for ASMR to 

detect statistically significant changes in recruitment, unless 

those changes are quite large. As a result, experimental 

adaptive management actions designed to increase recruitment should consider first and foremost how to achieve large 

changes in recruitment. Small-scale experimental treatments 

of short duration, or so-called “mini-experiments,” should be 

summarily discounted recognizing that the monitoring program is unlikely to detect small recruitment change even if it 

occurs. Additionally, multiyear experiments should be strongly 

favored in order to help offset not only unexpected and uncontrollable effects, but the low precision in recruitment estimates.

Acknowledgements

For their thorough review of the GCMRC assessment 

methodology, I would like to thank members of the Humpback Chub Assessment Review Panel: James F. Kitchell, 

Churchill Grimes, Steven T. Lindley, David Otis, and Carl 

Schwarz. Steve Martell is acknowledged for his help and 

council in developing ADMB programs and conceptualizing 

how to incorporate ageing error into the current assessment 

program. Carl Walters, Bill Pine, Mike Allen, and Tom Frazer 

are deserving of thanks for early reviews and suggestions 

that much improved this manuscript. I would also like to 

thank Tina Kister and Lara Schmit for publication assistance. 

Finally, and perhaps most deservedly, my thanks to field crews 

past and present whose tireless efforts provide the data upon 

which this assessment relies completely.

References

Akaike, H.X., 1973, Information theory as an extension of the 

maximum likelihood principle, in Petrov, B.N., and Csaki, 

F., eds., International symposium on information theory, 

2nd, Akademiai Kiado, Budapest, 1973 [Proceedings], p. 

267–281.
16 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Arreguin-Sanchez, F.X., 1996, Catchability: a key parameter 

for fish stock assessment: Reviews in Fish Biology and 

Fisheries, v. 6, no. 2, 221–242.

Baillargeon, Sophie, and Rivest, L.P., 2007, Recapture: loglinear models for capture-recapture in R: Journal of Statistical 

Software, v. 19 no. 5, 1–31.

Bertalanffy, L.von, 1938, A quantitative theory of organic 

growth (Inquiries on growth laws II): Human Biology, v. 10, 

p. 181–213. 

Bozdogan, H., 2000, Akaike’s information criterion and recent 

developments in information complexity: Journal of Mathematical Psychology, v. 44, p. 62–91. 

Burnham, K.P., and Andersen, D.R., 2002, Model selection 

and multimodel inference: A practical information-theoretic 

approach (2nd ed.): New York, Springer, 488 p.

Clarke, Andrew, and Johnston, N.M., 1999, Scaling of metabolic rate with body mass and temperature in teleost fish: 

Journal of Animal Ecology, v. 68 no.5, p. 893–905.

Clarkson, R.W., and Childs, M.R., 2000, Temperature effects 

of hypolimnial-release dams on early life stages of Colorado 

River big-river fishes: Copeia, v. 2000, no. 2, p. 402–412.

Coggins, L.G., Jr., Pine, W.E., III, Walters, C.J., Martell, 

S.J.D., 2006b, Age-structured mark-recapture analysis: a 

virtual-population-analysis-based model for analyzing agestructured capture-recapture data: North American Journal 

of Fisheries Management, v. 26, no. 1, p. 201–205.

Coggins, L.G., Jr., Pine, W.E., III, Walters, C.J., Van Haverbeke, D.R., Ward, D., and Johnstone, H.C., 2006a, 

Abundance trends and status of the Little Colorado River 

population of humpback chub: North American Journal of 

Fisheries Management, v. 26 no. 1, p. 233–245.

Coggins, L.G., Jr., and Quinn, T.J., II, 1998, A simulation 

study of the effects of aging error and sample size on sustained yield estimates, in Funk, F., Quinn, T.J., II, Heifetz, 

J., Ianelli, J.N., Powers, J.E., Schweigert, J.F., Sullivan, P.J., 

and Zhang, C.I. Fishery Stock Assessment Models: Alaska 

Sea Grant College Program Report No. AK-SG-98-01, Fairbanks, Alaska, University of Alaska, p. 955–973.

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

population movements of Gila cypha, an endangered 

cyprinid fish in the Grand Canyon region of Arizona: 

Copeia, v. 1996, no. 1, p.15–28.

Edwards, E.F., and Megrey, B.A., eds., 1989, Mathematical 

analysis of fish stock dynamics. American Fisheries Society 

Symposium 6, Bethesda, Md., 214 p.

Essington, T.E., Kitchell, J.F., and Walter, C.J., 2001, The von 

Bertalanffy growth function, bioenergetics, and the consumption rates of fish: Canadian Journal of Fisheries and 

Aquatic Sciences, v. 58, no. 11, p. 2129–2138.

Fabens, A.J., 1965, Properties and fitting of the von Bertalanffy growth curve: Growth, v. 29, no. 3, p. 265–289.

Fournier, D., 2000, AD Model Builder: Sidney, British Columbia, Canada, Otter Research Ltd. [http://otter-rsch.com/

admodel.htm]. 

Francis, R.I.C.C., 1988, Are growth parameters estimated 

from tagging and age-length data comparable?: Canadian 

Journal of Fisheries and Aquatic Sciences, v. 45, p. 936–

942.

Gloss, S.P., and Coggins Jr., L.G., 2005, Fishes of Grand Canyon, in Gloss, S.P., Lovich, J.L., and Melis, T.S., eds., The 

state of the Colorado River ecosystem in Grand Canyon: 

U.S. Geological Survey Circular 1282, 220 p.

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

humpback chub (Gila cypha), in the Little Colorado River 

near Grand Canyon, Arizona: Environmental Biology of 

Fishes, v. 55, no. 1-2, p. 115–133.

James, I.R., 1991, Estimation of von Bertalanffy growth curve 

parameters from recapture data: Biometrics, v. 47, no. 4, p. 

1519–1530.

Jolly, G.M., 1965, Explicit estimates from capture-recapture 

data with both death and immigration-stochastic model: 

Biometrika, v. 52 no. 1-2, p. 225–247.

Kaeding L.R., and Zimmerman, M.A., 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, v. 112, no. 5, p. 577–594.

Kirkwood, G.P., and Somers, I.F., 1984, Growth of two species of tiger prawn, Penaeus esculentus and P. semisulcatus, 

in the Western Gulf of Carpentaria: Australian Journal of 

Marine and Freshwater Research, v. 35 no. 6, p. 703–7 12.

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

C., 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, Ariz., www.gcmrc.gov/library/reports/biological/Fish_studies/Kitchell2003.pdf, accessed May 2005.

Lasdon, L., and Allan, W., 2002, Solver: Incline Village, NV, 

Frontline Systems Inc., Incline Village, Nev. [http://www.

solver.com].
 17

Laslett, G.M., Eveson, J.P., and Polacheck, T., 2002, A flexible 

maximum likelihood approach for fitting growth curves 

to tag-recapture data: Canadian Journal of Fisheries and 

Aquatic Sciences, v. 59, no. 6, p. 976–986.

Lorenzen, Kai, 2000, Allometry of natural mortality as a 

basis for assessing optimal release size in fish-stocking 

programmes: Canadian Journal of Fisheries and Aquatic 

Sciences, v. 57, no. 12, p. 2374–2381.

MacKenzie, D.I., and Kendall, W.L., 2002, How should detection probability be incorporated into estimates of relative 

abundance?: Ecology, v. 83, no. 9, p. 2387–2393.

MacKenzie, D.I., Nichols, J.D., Royle, J.A., Pollock, K.H., 

Bailey, L.L., and Hines, J.E., 2006, Occupancy estimation 

and modeling: inferring patterns and dynamics of species 

occurrence: Burlington, Mass., Elsevier, 324 p.

Martell, S.J.D., 2006, Review of the status and trends of 

humpback chub (Gila cypha) in the Grand Canyon: 1989–

2005: Report submitted to Grand Canyon Monitoring and 

Research Center, Flagstaff, Ariz., 19 p.

Megrey, B.A., 1989, Review and comparison of age-structured 

stock assessment models from theoretical and applied points 

of view, in Edwards, E. F., Megrey, B. A., eds., American 

Fisheries Society Symposium 6, Bethesda, Md., p. 8–48.

Melis, T.S., Martell, S.J.D., Coggins, L.G., Jr., Pine, W.E., 

III, and Andersen, M.E., 2006, Adaptive management of 

the Colorado River ecosystem below Glen Canyon Dam, 

Arizona: using science and modeling to resolve uncertainty 

in river management, in Specialty Summer Conference 

on Adaptive Management of Water Resources, Missoula, 

Mont., CD-ROM Proceedings (ISBN 1-882132-71-8): 

Middleburg, Va., American Water Resources Association.

Otis, D.L., Burnham, K.P., White, G.C., and Anderson, D.R., 

1978, Statistical inference from capture data on closed animal populations: Wildlife Monograph 62, p. 1–135.

Paloheimo, J.E., and Dickie, L.M., 1965, Food and growth of 

fishes. I. A growth curve derived from experimental data: 

Journal of the Fisheries Research Board of Canada, v. 22, p. 

521–542.

Paloheimo, J.E., and Dickie, L.M., 1966, Food and growth of 

fishes. II. Effects of food and temperature on the relation 

between metabolism and body weight: Journal of the Fisheries Research Board of Canada, v. 23, p. 869–908.

Petersen, J.H., and Paukert, C.P., 2005, Development of a 

bioenergetics model for humpback chub and evaluation of 

water temperature changes in the Grand Canyon, Colorado 

River: Transactions of the American Fisheries Society v. 

134, no. 4, p. 960–974.

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

Rice, J.A., 2003, A review of tagging methods for estimating fish population size and components of mortality: 

Fisheries, v. 28, p. 10–23.

Quinn, T.J., II, and Deriso, R.B., 1999, Quantitative fish 

dynamics: New York, Oxford University Press, 566 p.

R Development Core Team, 2006, R: A language and environment for statistical computing: Vienna, Austria, R Foundation for Statistical Computing (ISBN 3-900051-07-0) 

[http://www.R-project.org].

Robinson, A.T., and Childs, M.R., 2001, Juvenile growth of 

native fishes in the Little Colorado River and in a thermally 

modified portion of the Colorado River: North American 

Journal of Fisheries Management, v. 21, p. 809–815.

Sainsbury, K.J., 1980, Effect of individual variability on the 

von Bertalanffy growth equation: Canadian Journal of Fisheries and Aquatic Sciences, v. 37, no. 2, p. 241–247.

Sampson, D.B., and Yin, Y., 1998. A Monte Carlo evaluation of the stock synthesis assessment program, in Funk, 

F., Quinn, T.J., II, Heifetz, J., Ianelli, J.N., Powers, J.E., 

Schweigert, J.F., Sullivan, P.J., and Zhang, C.I., eds., Fishery Stock Assessment Models: Alaska Sea Grant College 

Program Report No. AK-SG-98-01, Fairbanks, Alaska, 

University of Alaska, p. 315–338.

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

Biometrika, v. 52, p. 249–259. 

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

related parameters (2d ed.): Griffin, London, Blackburn 

Press, 654 p.

Taylor, N.G., Walters, C.J., and Martell, S.J.D., 2005, A new 

likelihood for simultaneously estimating von Bertalanffy 

growth parameters, gear selectivity, and natural and fishing 

mortality: Canadian Journal of Fisheries and Aquatic Sciences, v. 62, no. 1, p. 215–223.

U.S. Department of the Interior, 1995, Operation of Glen Canyon Dam Final Environmental Impact Statement: Salt Lake 

City, Utah, Bureau of Reclamation, 337 p., http://www.

gcmrc.gov/library/reports/LawoftheRiver/GCDEIS_Final.

pdf, accessed May 2005.

U.S. Fish and Wildlife Service, 2002, Humpback chub (Gila 

cypha) recovery goals: amendment and supplementation 

to the Humpback Chub Recovery Plan: Denver, Colorado, 

U.S. Fish and Wildlife Service, Mountain-Prairie Region 

(6), http://www.gcmrc.gov/library/reports/biological/Fish_

studies/USFWS/Humpbackchub_recoverygoals2002.pdf, 

accessed May 2005.
18 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

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

of the humpback chub (Gila cypha) in the Colorado River, 

Arizona: Final report, contract No. 0-CS-40-09110, Salt 

Lake City, Utah [http://www.gcmrc.gov/library/reports/biological/Fish_studies/GCES/Valdez1995f.pdf].

Viallefont, A., Lebreton, J.D., and Reboulet, A.M., 1998, 

Parameter identifiability and model selection in capture-recapture models: a numerical approach: Biometrical Journal, 

v. 40, p. 313–325. 

Voichick, N., and Wright, S.A., 2007, Water-temperature data 

for the Colorado River and tributaries between Glen Canyon 

Dam and Spencer Canyon, Northern Arizona 1988–2005: 

U.S. Geological Data Series 251, 28 p.

Wang, Y-G., Thomas, M.R., and Somers, I.F., 1995, A maximum likelihood approach for estimating growth from tagrecapture data: Canadian Journal of Fisheries and Aquatic 

Science, v. 52, p. 52–259.

Ward, D.L., and Bonar, S.A., 2003, Effects of cold water on 

susceptibility of age-0 flannelmouth sucker to predation by 

rainbow trout: The Southwestern Naturalist, v. 48, no. 1, p. 

43–46.

Ward, D.L., Maughan, O.E., Bonar, S.A., and Matter, W.J., 

2002, Effects of temperature, fish length, and exercise on 

swimming performance of age-0 flannelmouth sucker: 

Transactions of the American Fisheries Society, v. 113, no. 

3, p. 492–497.

Williams, B.K., Nichols, J.D., and Conroy, M.J., 2001, 

Analysis and management of animal populations: London, 

Academic Press, 817 p. 
 19

Figures Hoop−net Catch Rate

19871988198919901991199219931994199519961997199819992000200120022003200420052006

0.00 0.10 0.20 0.30

150−199 mm

>200 mm

A.

Year

Trammel−net Catch Rate

19911992199319941995199619971998199920002001200220032004200520062007

0.0 0.2 0.4 0.6 0.8 1.0

B.

Figure 1. Relative abundance indices of sub-adult (150–199 mm total length (TL)) and adult (>200 mm TL) humpback chub based on 

hoop-net catch rate (fish/hour) in the lower 1,200-m section of the Little Colorado River (A) and trammel-net catch rate (fish/hour/100m) 

of adult (>200 mm TL) humpback chub in the Little Colorado River inflow reach of the Colorado River (B). Error bars in the top panel 

are 95% confidence intervals. In the lower panel, the solid line represents a regression model fit to the subset of data (solid circles) 

representing robust sampling and the dashed line represents a regression model fit to the entire dataset (all circles).
20 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 2. Numbers of humpback chub marked and recaptured by age and year. The annual sample size is indicated by the number at 

the top of each bubble column and the distribution among ages is indicated by relative size of bubbles within each column.

1990 1995 2000 2005

10

20

30

40

Year

Age

451709

374423602832

885342

6061

213217696

113111031001

827

13871245

Number of Fish Marked

1990 1995 2000 2005

10

20

30

40

Year

Age

0

19

218

154417371604

902108133102115132469647561500490556

Number of Fish Recaptured
 21

Figure 3a. Numbers of humpback chub marked by age in years 1989–92 (dark circles) and subsequently recaptured (light circles) by 

age and year. The annual sample size is indicated by the number at the top of each bubble column and the distribution among ages is 

indicated by the relative size of the bubbles within each column.

1990 1995 2000 2005

10

20

30

40

Year

Age

451

194854302410

32121757520

1990 1995 2000 2005

10

20

30

40

Year

Age

709170151126

6049

58786

12222313

59

1990 1995 2000 2005

10

20

30

40

Year

Age

37441339

859501326

3843283840

128137

97978555

1990 1995 2000 2005

10

20

30

40

Year

Age

2360

722411258

2944403537

100131

78657549
22 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 3b. Numbers of humpback chub marked by age in years 1993–96 (dark circles) and subsequently recaptured (light circles) by 

age and year. The annual sample size is indicated by the number at the top of each bubble column and the distribution among ages is 

indicated by the relative size of the bubbles within each column.

1990 1995 2000 2005

10

20

30

40

Year

Age

2832

608185

2122151411454740173218

1990 1995 2000 2005

10

20

30

40

Year

Age

885

74

67569

12251210

79

1990 1995 2000 2005

10

20

30

40

Year

Age

342

66125

1213

5644

1990 1995 2000 2005

10

20

30

40

Year

Age

60

1210010000
 23

Figure 3c. Numbers of humpback chub marked by age in years 1997–2000 (dark circles) and subsequently recaptured (light circles) by 

age and year. The annual sample size is indicated by the number at the top of each bubble column and the distribution among ages is 

indicated by the relative size of the bubbles within each column.

1990 1995 2000 2005

10

20

30

40

Year

Age

61

321431433

1990 1995 2000 2005

10

20

30

40

Year

Age

213

70963451

1990 1995 2000 2005

10

20

30

40

Year

Age

217

221510

4425

1990 1995 2000 2005

10

20

30

40

Year

Age

696125

29

9954
24 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 3d. Numbers of humpback chub marked by age in years 2001–04 (dark circles) and subsequently recaptured (light circles) by 

age and year. The annual sample size is indicated by the number at the top of each bubble column and the distribution among ages is 

indicated by the relative size of the bubbles within each column.

1990 1995 2000 2005

10

20

30

40

Year

Age

1131

218

77291714

1990 1995 2000 2005

10

20

30

40

Year

Age

1103

205

843224

1990 1995 2000 2005

10

20

30

40

Year

Age

1001

153

6526

1990 1995 2000 2005

10

20

30

40

Year

Age

827151101
 25

Figure 3e. Numbers of humpback chub marked by age in years 2005–06 (dark circles) and subsequently recaptured (light circles) by 

age and year. The annual sample size is indicated by the number at the top of each bubble column and the distribution among ages is 

indicated by the relative size of the bubbles within each column.

1990 1995 2000 2005

10

20

30

40

Year

Age

1387

234

1990 1995 2000 2005

10

20

30

40

Year

Age

1245
26 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 4. Mark-recapture closed population model estimates of humpback chub abundance greater than or equal to 150-mm total 

length in the Little Colorado River. Error bars represent 95% confidence intervals.

Year

Abundance (x1,000)

199219931994199519961997199819992000200120022003200420052006

0 1 2 3 4 5 6 7 8
 27

Figure 5. Humpback chub adult abundance (age-4+) estimates from the age-structured mark recapture (ASMR) models using data 

pooled among tag cohorts. Error bars are 95% credible intervals from 200,000 Markov-Chain Monte Carlo trials.

4

5

6

7

8

9

10

ASMR 1

4

5

6

7

8

9

10

Adult Abundance (x1000)

ASMR 2

19891990199119921993199419951996199719981999200020012002200320042005

2006

4

5

6

7

8

9

10

ASMR 3

Year
28 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 6. Humpback chub recruit abundance (age-2) estimates from the age-structured mark recapture (ASMR) models using data 

pooled among tag cohorts. Error bars are 95% credible intervals from 200,000 Markov-Chain Monte Carlo trials.

0

1

2

3

4

5

ASMR 1

0

1

2

3

4

5

Recruit Abundance (x1000)

ASMR 2

198919901991199219931994199519961997199819992000200120022003

0

1

2

3

4

5

ASMR 3

Brood Year
 29

Figure 7. Quantile-quantile (Q-Q) and Pearson residual plots for age-structured mark recapture model 1 (ASMR 1) using data pooled 

among tag cohorts.

−3 −2 −1 0 123

−3

−2

−1

0

1

2

3

Q−Q Plot for Marks

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Marks

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

Q−Q Plot for Recaptures

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Recaptures
30 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 8. Quantile-quantile (Q-Q) and Pearson residual plots for age-structured mark recapture model 2 (ASMR 2) using data pooled 

among tag cohorts.

−3 −2 −1 0 123

−3

−2

−1

0

1

2

3

Q−Q Plot for Marks

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Marks

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

Q−Q Plot for Recaptures

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Recaptures
 31

Figure 9. Quantile-quantile (Q-Q) and Pearson residual plots for plots for age-structured mark recapture model 3 (ASMR 3) using 

data pooled among tag cohorts.

−3 −2 −1 0 123

−3

−2

−1

0

1

2

3

Q−Q Plot for Marks

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Marks

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

Q−Q Plot for Recaptures

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Recaptures
32 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 10. Humpback chub capture probability by age and year estimated from age-structured mark recapture model 3 (ASMR 3) using 

data pooled among tag cohorts.

Age

Capture Probability

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

0.0

0.2

0.4

0.6

0.8

1.0

2468 10 12 14 16 18 20 22 24 26 28 30
 33

Figure 11. Humpback chub adult abundance (age-4+) estimates from the age-structured mark recapture (ASMR) models using data 

stratified by tag cohort. Error bars are 95% credible intervals from 200,000 Markov-Chain Monte Carlo trials.

4

5

6

7

8

9

10

11

ASMR 1

4

5

6

7

8

9

10

11

Adult Abundance (x1000)

ASMR 2

198919901991199219931994199519961997199819992000200120022003200420052006

4

5

6

7

8

9

10

11

ASMR 3

Year
34 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 12. Humpback chub recruit abundance (age-2) estimates from the age-structured mark recapture (ASMR) models using data 

stratified by tag cohort. Error bars are 95% credible intervals from 200,000 Markov-Chain Monte Carlo trials.

0

1

2

3

4

5

ASMR 1

0

1

2

3

4

5

Recruit Abundance (x1000)

ASMR 2

198919901991199219931994199519961997199819992000200120022003

0

1

2

3

4

5

ASMR 3

Brood Year
 35

Figure 13. Coefficient of variation (CV) of humpback chub adult abundance estimates (age-4+) for data pooled among tag cohorts and 

stratified by tag cohorts.

Year

CV (%)

198919901991199219931994199519961997199819992000200120022003200420052006

0

1

2

3

4 Pooled data

Stratified by tag cohort
36 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 14. Quantile-quantile (Q-Q) and Pearson residual plots for age-structured mark recapture model 1 (ASMR 1) using data 

stratified by tag cohort.

−3 −2 −1 0 123

−3

−2

−1

0

1

2

3

Q−Q Plot for Marks

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Marks

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

Q−Q Plot for Recaptures

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Recaptures
 37

Figure 15. Quantile-quantile (Q-Q) and Pearson residual plots for age-structured mark recapture model 2 (ASMR 2) using data 

stratified by tag cohort.

−3 −2 −1 0 123

−3

−2

−1

0

1

2

3

Q−Q Plot for Marks

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Marks

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

Q−Q Plot for Recaptures

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Recaptures
38 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 16. Quantile-quantile (Q-Q) and Pearson residual plots for age-structured mark recapture model 3 (ASMR 3) using data 

stratified by tag cohort.

−3 −2 −1 0 123

−3

−2

−1

0

1

2

3

Q−Q Plot for Marks

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Marks

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

Q−Q Plot for Recaptures

Theoretical Quantiles

Sample Quantiles

1990 1995 2000 2005

5

10

15

20

25

30

Year

Age

Pearson Residuals for Recaptures
 39

Figure 17. Humpback chub capture probability by age and year estimated from age-structured mark recapture model 3 (ASMR 3) using 

data stratified by tag cohort.

Age

Capture Probability

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

0.0

0.2

0.4

0.6

0.8

1.0

2468 10 12 14 16 18 20 22 24 26 28 30
40 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 18. Frequency of humpback chub growth intervals used in the growth analysis by time-at-large (top panel) and total length at 

the start of the growth interval (bottom panel).

Time-at-Large (years)

Frequency

0 2000 4000 6000

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

11

11.5

12

12.5

13

13.5

14

14.5

15

Total Length at Start of Interval (mm)

Frequency

0 200 400 600 800 1000

150165180195210225240255270285300315330345360375390405420435450465480
 41

Figure 19. Log-likelihood as a function of the penalty weight ( ) on the standardized von Bertalanffy parameters for the temperatureindependent growth model. Log−Likelihood −68200 −68000 −67800 −67600 −67400 −67200 −67000 −66800 1e−05 1e−0

4

0.001

0.01

0.1

1

10

10

0

1000

10000

λ
42 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 20. Fit of the sine curve to average monthly Little Colorado River water temperature. The points are the average observed 

monthly temperature and the line is the predicted monthly temperature.

Month

Temperature (C)

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

12

14

16

18

20

22

24
 43

Figure 21. Fit of the temperature-independent growth model and the temperature-dependent growth model during summer and winter 

to all observed humpback chub growth-rate (dL/dt) data.

0 100 200 300 400 500

−200 −10

0

0 100 200

Total Length at Start of Interval (mm)

dL/dt

Temperature Independent

Temperature Dependent Summer

Temperature Dependent Winter
44 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 22. Fit of the temperature-dependent growth model during summer and winter to observed humpback chub growth-rate (dL/dt) 

data during summer and winter.

0 100 200 300 400 500

−200 −10

0

0 100 200

Total Length at Start of Interval (mm)

dL/dt

Summer

Winter
 45

Figure 23. Predicted humpback chub length-at-age from the U.S. Fish and Wildlife Service (USFWS) growth curve, the temperatureindependent growth model, the temperature-dependent growth model for the Little Colorado River (LCR) humpback chub population, 

and the temperature-dependent growth model for humpback chub living in the mainstem Colorado River under a constant temperature 

of 10oC.

0 5 10 15 20 25 30

0 100 200 300 400

Age

Total Length (mm)

 USFWS 

 Temperature Independent LCR 

 Temperature Dependent LCR 

 Temperature Dependent Mainstem Colorado River 
46 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 24. Predicted monthly growth rate from the temperature-dependent growth model for the Little Colorado River (LCR) population 

of humpback chub and for humpback chub living in the mainstem Colorado River under a constant temperature of 10oC.

Total Length (mm)

Monthly Growth Increment (mm/mo)

Colorado River Growth 

LCR Growth 

0

20406080

10

0

12

0

14

0

16

0

18

0

20

0

22

0

24

0

26

0

28

0

30

0

32

0

34

0

36

0

38

0

40

0

0 2 4 6 8 10 12 1

4
 47

Figure 25. Seasonal probability surfaces of humpback chub age at a particular length bin. These surfaces sum to unity in the vertical 

dimension (i.e., for each length bin) with the height of the surface indicating the probability of a particular age given a particular length 

bin.

April−June

Length Bin

Age

150 200 250 300 350 400 450 500

0 5 10 15 20 25 30

July−September

Length Bin

Age

150 200 250 300 350 400 450 500

0 5 10 15 20 25 30

October−December

Length Bin

Age

150 200 250 300 350 400 450 500

0 5 10 15 20 25 30

January−March

Length Bin

Age

150 200 250 300 350 400 450 500

0 5 10 15 20 25 30
48 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Figure 26. Estimated adult humpback chub abundance (age-4+) from the age-structured mark recapture model 3 (ASMR 3), 

incorporating uncertainty in assignment of age. Point estimates are mean values among 1,000 Monte Carlo trials, and error bars 

represent maximum and minimum 95% profile confidence intervals among 1,000 Monte Carlo trials.

Year

Adult Abundance (x1000)

198919901991199219931994199519961997199819992000200120022003200420052006

4

5

6

7

8

9

10
 49

Figure 27. Coefficient of variation (CV) of humpback chub adult abundance (age-4+) estimates accounting for uncertainty in age 

assignment.

Year

CV(%)

198919901991199219931994199519961997199819992000200120022003200420052006

1

2

3

4

5

6

7

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

Figure 28. Estimated recruit abundance (age-1) of humpback chub from age-structured mark recapture model 3 (ASMR 3) 

incorporating uncertainty in assignment of age. Point estimates are mean values among 1,000 Monte Carlo trials and error bars 

represent maximum and minimum 95% profile confidence intervals among 1,000 Monte Carlo trials.

Brood Year

Recruit Abundance (x1000)

198819891990199119921993199419951996199719981999200020012002

0

2

4

6

8
 51

Figure 29. Retrospective analysis of adult humpback chub abundance and adult mortality rate considering datasets beginning in 1989 

and ending in the year indicated in the figure legend. Adult Abundance (x 1,000)

Data through 2006 

Data through 2005 

Data through 2004 

Data through 2003 

Data through 2002 

Data through 2001 

0 2 4 6 8 10 12

Year

Adult Mortality

Data through 2006 

Data through 2005 

Data through 2004 

Data through 2003 

Data through 2002 

Data through 2001 

198919901991199219931994199519961997199819992000200120022003200420052006

0.00 0.05 0.10 0.15
52 Abundance Trends and the Status of the Little Colorado River Population of Humpback Chub 1989–2006

Tables

Table 1. Akaike information criterion (AIC) model evaluation results among age-structured mark-recapture models fit to data pooled 

among tag cohorts.

Model AIC # Parameters Rank ∆AIC

ASMR 1 -216274 18 3 2492

ASMR 2 -217132 30 2 1634

ASMR 3 -218766 895 1 0

Table 2. Akaike information criterion (AIC) model evaluation results among age-structured mark recapture models fit to data stratified 

by tag cohort.

Model AIC # Parameters Rank ∆AIC

ASMR 1 -196278 18 3 2577

ASMR 2 -197183 30 2 1672

ASMR 3 -198856 895 1 0

Table 3. General growth model results for the humpback chub, temperature-independent growth model (TIGM) and the temperaturedependent growth model (TDGM).

Model H d m n L∞ Qc Lt AIC

#

 ParametersRank ∆AIC

TIGM 163 0.52 .0007 1.15 391 961 -- -- 133,658 6 2 38,493

TDGM 21 0.61 4.6 .89 434 2000 4.59 236 95,165 8 1 0
 53

Table 4. Parameter correlation matrices for the humpback chub temperature-independent growth model (top table) and the 

temperature-dependent growth model (bottom table).

 Temperature-independent growth model:

H d m n

H 1

d -0.99 1

m -0.66 0.73 1

n 0.62 -0.72 -0.99 1

0.14 -0.19 -0.38 0.38 1

Temperature-dependent growth model:

H d m n Qc Lt

H 1

d 0.74 1

m 0.88 0.94 1

n -0.86 -0.934 -0.99 1

Qc -0.98 -0.824 -0.89 0.88 1

Lt 0.55 0.16 0.35 -0.34 -0.46 1