Capture-recapture Models for Open Populations “Single-age Models” 6.13 UF-2015

Capture-recapture Models for Open Populations Open: gains and losses of animals between sampling periods (mortality, movements,…) Studies typically conducted over longer time periods Estimation focus is on survival, recruitment and population growth rate

Cormack-Jolly-Seber (CJS) Cormack (1964) –Estimated survival rates of a population of fulmars (color rings) –Captures and resightings of banded birds at the breeding colony each year –Estimated annual survival: 0.94 Jolly (1965) and Seber (1965): similar approach (focus on abundance) CJS model = focus on survival !

Capture History Data Row vector of 1’s (indicating capture/resighting) and 0’s (indicating no capture/resighting) Examples: Caught in periods 1,3,4,7 of an 8-period study Caught in periods 2,3,5,7,8 of an 8-period study

Conditional Capture History

Modeling of Capture History Data  i = survival probability, from i to i+1 p i = capture probability at time I (no parameter N here) Model parameters under the Cormack-Jolly-Seber (CJS) single-age model:

Capture History Modeling CJS: Conditional on first capture Confounded at last occasion First capture at occ. 2 => Start modeling at occ. 2

Parameter Estimation Data: numbers of new releases each period and number of animals with each capture history Model: probability structure for each capture history Likelihood: Multinomial likelihood, conditional on new releases in each sample period Maximum likelihood (e.g., program MARK)

CJS Model Assumptions Homogeneity: All marked animals (within stratum) have similar capture and survival probabilities at time I (stratum = age, sex,…) No mark loss: Marks are not lost, overlooked or incorrectly recorded Mortality during the sampling period is negligible Independence of fates (affects variance estimates only)

Program MARK Computes parameter estimates and measures of uncertainty (variances, confidence intervals) Computes model selection statistics (AIC) Computes likelihood ratio tests between competing models

To go further More single-age models

More complicated models Trap-dependence models: Capture-history dependence (trap responses in survival and capture probabilities) Time-specific covariates (e.g., environmental) Multiple groups (e.g., sexes, treatments, habitats) Individual covariates (for static covariates such as mass at fledging)

1. Modeling Trap Dependence

Capture-history Dependence Trap response in survival probability –Marked and unmarked animals have different  Trap response in capture probability –Animals have different p depending on whether or not they were caught the previous period

Capture-history Dependence in  : Initial Marking Effect

Capture-history Dependence in p p i = capture probability for animals caught at i-1 p’ i = capture probability for animals not caught at i-1 P(1011|Release in 1) = P(1001|Release in 1) = Can be model by decomposing capture histories of individuals and adding a 2-year age effect. Trap-dependence model (see Pradel 1993)

2. Covariate Modeling Time-specific covariates and group effects

Covariate Modeling: Time- and Group-Specific Older “add-hoc” approach: –Estimate parameters, e.g.  t –Use regression approach to estimate relationship between  t and time-specific covariate, x t Current approach: build model directly into likelihood and estimate relationship directly in 1 step (easily implemented in MARK)

Covariate Modeling: Time- and Group-Specific Frequently use linear-logistic model: Estimate  ’s directly via ML

Linear-logistic Modeling: Parallel Group Effects Additive models with time and group effects but no time-group interaction Hypothesis is that parameter may differ by group (g) and over time (t), but temporal variation occurs in parallel

Linear-logistic Modeling: Different Group Effects Models with time and group effects that include a time-group interaction Hypothesis is that parameter may differ over time (t) in different ways for the groups (g)

Time-varying  for pop. models Annual variability in survival affects predictions in population models Time-constant underestimates variability (we know survival is not constant) Time-varying models will overestimate actual variability (sampling error included) Can we separate variation due to sampling error and process variability?

Time-varying  for pop. models

Hierarchical model High-survival year Low-survival year Average - survival year 0

3. Modeling individual variation in vital rates

Individual heterogeneity in vital rates Three ways of modeling it: 1.Individual covariates are available (e.g. size of individual) 2.No covariate available: latent heterogeneity can be modeled as a mixture of fixed effects 3.No covariate available: latent heterogeneity can be modeled with individual random effects

1. Individual Covariates Each captured animal has 1 or more associated covariates Ask whether covariate is related to survival or capture probabilities Restriction: covariate applies to animal for duration of study (i.e., individual covariate cannot vary over time) Time varying covariates –Joint modeling of covariate change and survival needed –Multistate models or hierarchical models

2. Mixture models

3. Individual random effects Assumes each individual has its ‘own value’ of survival (normally distributed) STRONG Individual WEAK Individual Average Individual 0

3. Individual random effects

Concluding remarks… about model assumptions

CJS Model Assumptions All marked animals within stratum have similar capture and survival probabilities at time i Marks are not lost, overlooked or incorrectly recorded Mortality during the sampling period is negligible Independence of fates (affects variance estimates only)

Robustness to Assumption Violations Variable capture probabilities –Carothers (1979) used simulations to show that bias generally small (less problem than abundance) –Permanent trap happy not a problem (condition on first capture Variable Survival –Individual variation biases estimates high (Pollock et al not large for levels considered) –But now, we can easily use the heterogeneity models (individual covariates, individual random effects)

Robustness to Assumption Violations Tag loss –Indistinguishable from mortality –Estimates of tag loss or joint modeling can be used to account for this Non-instantaneous Sampling –Leads to heterogeneous survival probabilities –Bias depends on how large variation is

Robustness to Assumption Violations