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Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

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Page 1: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Closed Vs. Open Population Models

Mark L. TaperDepartment of Ecology

Montana State University

Page 2: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Fundamental Assumption of Closed Population Models

• Births, Immigration, Deaths, & Emmigration do not occur

• Ecologists are deeply interested in these processes

• Open population models relax this assumption in various ways

Page 3: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Two Classes of Open Models

• Conditional models – Cormack-Jolly-Seber (CJS) models– Calculations conditional on 1st captures

• Unconditional models– Jolly-Seber (JS) models– Calculations model capture process aswell

Page 4: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Cormack-Jolly-Seber approachmodels both survival and captures

Page 5: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

New captures possible each session

Page 6: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Capture Histories/* European Dipper Data, Live Recaptures, 7 occasions, 2 groups Group 1=Males Group 2=Females */1111110 1 0 ;1111100 0 1 ;1111000 1 0 ;1111000 0 1 ;1101110 0 1 ;1100000 1 0 ;1100000 1 0 ;1100000 1 0 ;1100000 1 0 ;1100000 0 1 ;1100000 0 1 ;1010000 1 0 ;1010000 0 1 ;1000000 1 0 ;1000000 1 0 ;1000000 1 0 ;

Page 7: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Building CJS capture histories probabilities

Survey 1 Survey 2 capture history probability

Caught,Marked, &Released

Alive

Dead

caught

notcaught

11

10

10

Φ1 p2

Φ1(1-p2)

(1-Φ1)

Φ 1

1-Φ1

p 2

1-p2

1 - Φ1 p2

Page 8: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

3 session capture historyIndex (ω) history Probability (π) Count

1 111 φ1p2φ2p3 X1

2 110 φ1p2(1-φ2p3) X2

3 101 φ1(1-p2)φ2p3 X3

4 100 (1-φ1) + φ1(1-p2)[1-φ2p3] x4

5 011 φ2p3 x5

6 010 (1-φ2p3) x6

ui is the number of individuals first captured on session i (i=1..K-1)

Page 9: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Attributes of capture histories

1) If ends in 1 all intervening φi are in probability and pi or (1-pi) depending on 1 or 0 in ith position.

2) If ends in 0 need to include all the ways no observation could be made

3) φ2 and p3 always occur together. NON-identifiable.

4) Probabilities conditional because only begin calculating probabilities after individuals first seen.

Page 10: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Removal/loss after last capture

Index (ω) history Probability (π) Remove Count2 110 φ1p2(1-φ2p3) no X2

7 110 φ1p2 yes x7

Page 11: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Capture Histories/* European Dipper Data, Live Recaptures, 7 occasions, 2 groups Group 1=Males Group 2=Females */1111110 1 0 ;1111100 0 1 ;1111000 1 0 ;1111000 0 -1 ;1101110 0 1 ;1100000 -1 0 ;1100000 1 0 ;1100000 1 0 ;1100000 1 0 ;1100000 0 1 ;1100000 0 1 ;1010000 1 0 ;1010000 0 1 ;1000000 1 0 ;1000000 1 0 ;1000000 1 0 ;

Page 12: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

A multinomial likelihood

x

K

i iiii x

uupxP

!

!,,|

1

1

Page 13: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Program Mark Example:Estimation of CJS model for

European Dipper

1) Read data2) Specify format3) Run basic CJS

4) View Parameter estimates5) Graph Parameter Estimates

Page 14: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 15: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 16: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 17: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 18: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 19: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 20: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 21: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 22: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 23: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 24: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Jolly-Seber models

• CJS approach models recaptures of previously captured individuals– Estimates survival probabilities

• JS approach models recaptures of previously captured individuals and 1st capture process.– Estimates “population sizes” and recruitment

Page 25: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

General Jolly-Seber assumptions

• Equal catchability of marked and unmarked animals

• Equal survival of marked and unmarked animals

• Tag retention• Accurate identification• Constant study area

Page 26: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Jolly-Seber original formulation

-The number of marked and unmarked individual in population i.e. Mi and Ui Are now parameters to be estimated.-Builds on previous likelihood by adding binomial components

Page 27: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Not implemented in Mark

• Rcapture (an R package)• Program JOLLY• Program JOLLYAGE

Page 28: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

POPAN formulation

Page 29: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Burnham and Pradel formulation

Page 30: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Choosing formulations

All formulations include φ and p parameters

Page 31: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Considerations for choosing formulations

• Match of biology with formulation• Explicit representation of parameters of

interest.– Likelihood based inference– Constraints on parameter space.

Page 32: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

The Robust DesignMerging Open & Closed models

• More precise estimates• Less biased estimates• More kinds of estimable parameters• Fewer restrictive assumptions• Greater realism• More complexity

Page 33: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Mixing Open and Closed

Page 34: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Explosion of capture models

Page 35: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University
Page 36: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

Exposes hidden structure which cause bias and uncertainty

Page 37: Closed Vs. Open Population Models Mark L. Taper Department of Ecology Montana State University

SECR Density

Spatially Explicit Capture RecaptureR package and Windows programs by

MG Efford