closed vs. open population models mark l. taper department of ecology montana state university
TRANSCRIPT
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Closed Vs. Open Population Models
Mark L. TaperDepartment of Ecology
Montana State University
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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
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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
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Cormack-Jolly-Seber approachmodels both survival and captures
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New captures possible each session
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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 ;
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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
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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)
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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.
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Removal/loss after last capture
Index (ω) history Probability (π) Remove Count2 110 φ1p2(1-φ2p3) no X2
7 110 φ1p2 yes x7
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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 ;
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A multinomial likelihood
x
K
i iiii x
uupxP
!
!,,|
1
1
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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
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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
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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
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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
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Not implemented in Mark
• Rcapture (an R package)• Program JOLLY• Program JOLLYAGE
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POPAN formulation
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Burnham and Pradel formulation
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Choosing formulations
All formulations include φ and p parameters
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Considerations for choosing formulations
• Match of biology with formulation• Explicit representation of parameters of
interest.– Likelihood based inference– Constraints on parameter space.
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The Robust DesignMerging Open & Closed models
• More precise estimates• Less biased estimates• More kinds of estimable parameters• Fewer restrictive assumptions• Greater realism• More complexity
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Mixing Open and Closed
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Explosion of capture models
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Exposes hidden structure which cause bias and uncertainty
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SECR Density
Spatially Explicit Capture RecaptureR package and Windows programs by
MG Efford