Closed Vs Open Population Models Mark L Taper

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

Building CJS capture histories probabilities Survey 1 Survey 2 p 2 Alive 1 - p Φ 1 Caught, Marked, & Released 1 - 2 capture history probability caught 11 Φ 1 p 2 not caught 10 Φ 1(1 -p 2) 1 - Φ 1 p 2 Φ 1 Dead 10 (1 -Φ 1)

3 session capture history Index (ω) history Probability (π) Count 1 111 φ 1 p 2 φ 2 p 3 X 1 2 110 φ1 p 2(1 -φ2 p 3) X 2 3 101 φ1(1 -p 2)φ2 p 3 X 3 4 100 (1 -φ1) + φ1(1 -p 2)[1 -φ2 p 3] x 4 5 011 φ 2 p 3 x 5 6 010 (1 -φ2 p 3) x 6 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 p 3 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 φ1 p 2(1 -φ2 p 3) no Count X 2 7 x 7 110 φ 1 p 2 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 0 1 ; 1100000 0 1 ; 1010000 1 0 ; 1010000 0 1 ; 1000000 1 0 ;

A multinomial likelihood

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 1 st 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 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