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
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 approach
models 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
11
Φ1 p 2
not
caught
10
Φ1(1-p2)
Alive
Caught,
Marked, &
Released
1 - Φ1 p 2
Dead
10
(1-Φ1)
3 session capture history
Index (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. NONidentifiable.
4) Probabilities conditional because only begin
calculating probabilities after individuals first
seen.
Removal/loss after last capture
Index (ω) history Probability (π) Remove
2
110
φ1p2(1-φ2p3) no
Count
X2
7
x7
110
φ1p2
yes
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
 K 1 
 i 1 ui ! 
x
Px | i 
, pi 
, ui   
  
  x !  
 

Program Mark Example:
Estimation of CJS model for
European Dipper
1) Read data
2) Specify format
3) Run basic CJS
4) View Parameter estimates
5) 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 Design
Merging 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 Recapture
R package and Windows programs by
MG Efford