Closed Vs. Open Population Models

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

2. 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

3. 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

4. Cormack-Jolly-Seber approachmodels both survival and captures

5. New captures possible each session

6. 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 ;

7. Building CJS capture histories probabilities Survey 1 Survey 2 capture history probability Φ1p2 caught 11 p2 1-p2 Φ1 Alive not caught Φ1(1-p2) 10 Caught, Marked, & Released 1-Φ1 1 - Φ1p2 10 Dead (1-Φ1)

8. 3 session capture history uiis the number of individuals first captured on session i (i=1..K-1)

9. Attributes of capture histories • If ends in 1 all intervening φi are in probability and pi or (1-pi) depending on 1 or 0 in ith position. • If ends in 0 need to include all the ways no observation could be made • φ2 and p3 always occur together. NON-identifiable. • Probabilities conditional because only begin calculating probabilities after individuals first seen.

10. Removal/loss after last capture

11. 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 ;

12. A multinomial likelihood

13. Program Mark Example:Estimation of CJS model for European Dipper Read data Specify format Run basic CJS View Parameter estimates Graph Parameter Estimates

24. 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

25. 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

26. 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

27. Not implemented in Mark • Rcapture (an R package) • Program JOLLY • Program JOLLYAGE

28. POPAN formulation

29. Burnham and Pradel formulation

30. Choosing formulations All formulations include φ and p parameters

31. Considerations for choosing formulations • Match of biology with formulation • Explicit representation of parameters of interest. • Likelihood based inference • Constraints on parameter space.

32. The Robust DesignMerging Open & Closed models • More precise estimates • Less biased estimates • More kinds of estimable parameters • Fewer restrictive assumptions • Greater realism • More complexity

33. Mixing Open and Closed

34. Explosion of capture models

36. Exposes hidden structure which cause bias and uncertainty

37. SECR Density Spatially Explicit Capture Recapture R package and Windows programs by MG Efford