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 1 st captures Unconditional models – Jolly-Seber (JS) models – Calculations model capture process aswell

4. Cormack-Jolly-Seber approach models 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 0 1 ; 1010000 1 0 ; 1010000 0 1 ; 1000000 1 0 ;

7. Building CJS capture histories probabilities Survey 1 Survey 2 capture history probability Caught, Marked, & Released Alive Dead caught not caught 11 10 Φ1 p2Φ1 p2 Φ 1 (1-p 2 ) (1-Φ 1 ) Φ1Φ1 1-Φ 1 p2p2 1-p 2 1 - Φ 1 p 2

8. 3 session capture history Index (ω)historyProbability (π)Count 1111 φ1p2φ2p3 φ1p2φ2p3 X1X1 2110 φ 1 p 2 (1-φ 2 p 3 )X2X2 3101 φ 1 (1-p 2 )φ 2 p 3 X3X3 4100 (1-φ 1 ) + φ 1 (1-p 2 )[1-φ 2 p 3 ]x4x4 5011 φ2p3 φ2p3 x5x5 6010 (1-φ 2 p 3 )x6x6 u i is the number of individuals first captured on session i (i=1..K-1)

9. Attributes of capture histories 1) If ends in 1 all intervening φ i are in probability and p i or (1-p i ) depending on 1 or 0 in i th position. 2) If ends in 0 need to include all the ways no observation could be made 3) φ 2 and p 3 always occur together. NON- identifiable. 4) Probabilities conditional because only begin calculating probabilities after individuals first seen.

10. Removal/loss after last capture Index (ω)historyProbability (π)RemoveCount 2110 φ 1 p 2 (1-φ 2 p 3 )noX2X2 7110 φ1p2 φ1p2 yesx7x7

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 0 1 ; 1010000 1 0 ; 1010000 0 1 ; 1000000 1 0 ;

12. A multinomial likelihood

13. 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

24. Jolly-Seber models CJS approach models recaptures of previously captured individuals – Estimates survival probabilities JS approach models recaptures of previously captured individuals and 1 st 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. M i and U i 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 Design Merging 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