HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Trends and Status of the

Little Colorado River

population of Humpback

Chub: 1989-2011

Scott Vanderkooi on behalf of Steven Martell

University of British Columbia

s.martell@mail.ubc.ca

April 6, 2012
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Outline

Introduction to ASMR

Input data summary

Results

Abundance Recruitment

Residuals

Retrospecitve Analysis

Uncertainty

Sensitivity to M

Summary
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Introduction to ASMR

Mark-recapture information between 1989 and 2011 is used

to estimate the abundance and recruitment of Humpback

Chub (HBC).

Catch-at-length data are transformed into catch-at-age data

using length-age relationship developed from a bioenergetics

model.

The unmarked population is reconstructed using Virtual

Population Analysis (VPA) with and assumed value of

natural mortality rate.

The fate of marked individuals is tracked using an

age-structured model, and the capture-recapture probability

is assumed to be a Poisson sampling process.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

ASMR-3 Analytical details

Age Structure Mark Recapture (ASMR-3)

Data: Marks & Recaptures mt,a,rt,a (1)

Estimate unmarked numbers ⇥ = Uˆt=2012,a=2:14 (2)

Survival rate Sa = exp(Ml1/la) (3)

Unmarked animals Uˆt,a = Uˆt+1,a+1

Sa

+ mt,a (4)

Marked animals Mˆ a+1,t+1 = Sa(Mˆ t,a + mt,a) (5)

Predicted new marks ˆmt,a = ˆpt,aUˆt,a (6)

Predicted recaps ˆrt,a = ˆpt,aMˆ t,a (7)

Capture probability ˆpt,a = mt,a + rt,a

Uˆt,a + Mˆ t,a

(8)

Negative log likelihood `t,a = (mˆ t,a + mt,a ln( ˆmt,a))

+(ˆrt,a + rt,a ln(ˆrt,a)) (9)
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

ASMR-3 Analytical details

Age Structure Mark Recapture (ASMR-3)

Assumptions:

• Natural mortality is a function of length.

• Natural mortality is constant over time.

• Marked & unmarked fish have same capture probablility

& survival.

• Observation error only.

• Growth is time invariant.

• Age is a function of length (slicing).

• Walters & Martell are never wrong!
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Marks released

Year

Age

1990 1994 1998 2002 2006 2010

5

10

15

20

25

30

35

40

45

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Figure: Number of marks released by age-year. Area of bubble is

proportional to the number of marks released.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Marks recaptured

Year

Age

1990 1994 1998 2002 2006 2010

5

10

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Figure: Number of marks recaptured by age-year. Area of bubble

is proportional to the number of marks released.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Marks recaptured

Recapture year

1990 1994 1998 2002 2006 2010

2010

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Figure: Tags released each year (red circles) and number

recaptured by tag-year (row of black circles).
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Age-4+ abundance

1990 1995 2000 2005 2010

0

2

4

6

8

10

12

14

Year

Abundance (1000 age−4+)

Figure: Maximum likelihood estimates of age-4+ abundance.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Age-2 Recruits

1990 1995 2000 2005

0

1

2

3

4

5

6

Brood year

Age−2 recruits

Figure: Maximum likelihood estimates of age-2 recruits.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Capture probability

1990 1995 2000 2005 2010

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Year

Capture probability

Figure: Maximum likelihood estimates of annual capture

probability.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Residuals (released marks)

Year

Age

1990 1994 1998 2002 2006 2010

5

10

15

20

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Figure: Pearson residuals (observed - expected) for new marks

released (black=positive).
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Residuals (recaptured marks)

Year

Age

1990 1994 1998 2002 2006 2010

5

10

15

20

25

30

35

40

45

50

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Figure: Pearson residuals (observed - expected) for recaptured

marks (black=positive).
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Retrospective age-4+ abundance

1990 1995 2000 2005 2010

0

2

4

6

8

10

12

14

Year

Abundance (1000 age−4+)

●

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Figure: Retrospective estimates of age-4+ abundance.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Retrospective age-2 recruits

1990 1995 2000 2005

0

1

2

3

4

5

6

Brood year

Age−2 recruits

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Figure: Retrospective estimates of age-2 recruits.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Uncertainty: Age-4+ & Age-2

1989 1992 1995 1998 2001 2004 2007 2010

0

2

4

6

8

10

Year

Abundance (1000)

0

2

4

6

8

10

Figure: Marginal distributions for age-4+ (blue) abundance and

age-2 recruits (orange).
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Age-4+ abundance in 2011

Age−4+ abundance in 2011

Posterior density

8600 8700 8800 8900 9000 9100 9200 9300

Figure: Marginal posterior density for age-4+ abundance in 2011.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Sensitivity to natural mortality (M)

1989 1992 1995 1998 2001 2004 2007 2010

0

2

4

6

8

10

12

Year

Abundance (1000)

0

2

4

6

8

10

12

0

2

4

6

8

10

12

0

2

4

6

8

10

12

Nt4 M=0.130

Rt M=0.130

Nt4 M=0.094

Rt M=0.094

Figure: Freely estimating natural mortality results in increased

estimates of age-4 abundance and fewer age-2 recruits.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Sensitivity to natural mortality (M)

Age−4+ abundance in 2011

Posterior density

9000 10000 11000 12000 13000

M=0.130

M=0.094

Truth lies somewhere in

between these distributions

Figure: Marginal posterior density for age-4+ abundance in 2011

with M=0.13 (tan) and M=0.094 (light blue).
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Summary

Strong residual pattern arising age-assignment of newly

marked fish (max age=30).
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Summary

Strong residual pattern arising age-assignment of newly

marked fish (max age=30).

Recapture residuals suggest lower natural mortality rate.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Summary

Strong residual pattern arising age-assignment of newly

marked fish (max age=30).

Recapture residuals suggest lower natural mortality rate.

Assuming the asymptotic natural mortality rate of 0.13:

• Median age-4+ in 2011: 8912 (8736, 9095)–95%CI

• Median age-2 in 2011: 3998 (3814, 4195)–95%CI
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Summary

Strong residual pattern arising age-assignment of newly

marked fish (max age=30).

Recapture residuals suggest lower natural mortality rate.

Assuming the asymptotic natural mortality rate of 0.13:

• Median age-4+ in 2011: 8912 (8736, 9095)–95%CI

• Median age-2 in 2011: 3998 (3814, 4195)–95%CI

Freely estimating natural mortality rate (M = 0.094):

• Median age-4+ in 2011: 12274 (11773, 12812)–95%CI

• Median age-2 in 2011: 3635 (3463, 3838)–95%CI
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Summary

Strong residual pattern arising age-assignment of newly

marked fish (max age=30).

Recapture residuals suggest lower natural mortality rate.

Assuming the asymptotic natural mortality rate of 0.13:

• Median age-4+ in 2011: 8912 (8736, 9095)–95%CI

• Median age-2 in 2011: 3998 (3814, 4195)–95%CI

Freely estimating natural mortality rate (M = 0.094):

• Median age-4+ in 2011: 12274 (11773, 12812)–95%CI

• Median age-2 in 2011: 3635 (3463, 3838)–95%CI

Uncertainty is grossly under-estimated due to observation

error only, and assignment of age from length.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Future Work

Developing a length-based version of ASMR where there is

no age-assignment from length.

Key Features:

• Mixed-error model (better uncertainty estimates)

• Growth based on length-transition matrix (no aging

info)

• Can incorporate information on individuals < 150 mm.
HBC Status

Scott Vanderkooi

on behalf of

Steven Martell

Introduction

Input data

Results

Abundance

Residuals

Retrospective

Uncertainty

Sensitivity to M

Summary

Acknowledgments

Lew Coggins, Carl Walters, Josh Korman and Bill Pine for

technical assistance and development of ASMR.

GCMRC Sta↵: Scott Vanderkooi, Bill Persons, Ted Melis,

Glenn Bennett, Mike Yard & the many other volunteers

(some even busting their ass, literally!) who worked in the

canyon collecting all of these data.