Closed Vs. Open Population Models

Mark L. TaperDepartment 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

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

Cormack-Jolly-Seber approachmodels both survival and captures

New captures possible each session

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 ;

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

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)

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.

Removal/loss after last capture

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

7 110 φ1p2 yes x7

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 ;

A multinomial likelihood

x

K

i iiii x

uupxP

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1

1

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

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

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

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

Not implemented in Mark

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

POPAN formulation

Burnham and Pradel formulation

Choosing formulations

All formulations include φ and p parameters

Considerations for choosing formulations

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

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

The Robust DesignMerging Open & Closed models

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

Mixing Open and Closed

Explosion of capture models

Exposes hidden structure which cause bias and uncertainty

SECR Density

Spatially Explicit Capture RecaptureR package and Windows programs by

MG Efford