Bayesian open models allow abundance (N) and density (D) to vary over longer time occasions, so that these states are now indexed over time, N[t] and D[t]. There are many forms of open models, here we'll consider types that include both survival over [t,t+1] and recruitment (entry) into the population over [t,t+1]. These models typically include (at least implicitly) a simple equation for population change
N[t+1]<- N[t]S[t]+N[t]G[t]
where S[t] and G[t] are survival rate (probability) and per-capita recruitment over [t,t+1]. Actually, in the JAGS formulation, since we are getting at abundance by data augmentation, we use a relationship such as
mu[i,t]<-(S[t-1]*z[i,t-1])+(G[t-1]*A[i,t-1]), t=2, T
where mu[i,t] is the Bernoulli inclusion probability at time t, and A[i,t-1] is an indicator variable signifying "has not yet entered the population at t-1".
The inclusion states of the individuals i at times t are then modeled as
z[i,1]~dbern(psi)
z[i,t]<-dbern(mu[i,t]), t=2, ...T
Spatial captures are then used as they are with closed models at each of the open periods to estimate psi (initial inclusion), mu[i,t], t>1, N[t] and D[t]. The model is fairly easily generalized to a Robust Design, in which sampling at trap stations occurs at closed secondary sessions (J= 1, ....K) nested within the T primary occasions.
Here I applied a Robust spatial CMR analysis to collections of scats from coyotes (Canis latrans) in South Carolina with 3 primary occasions each containing 3-4 secondary occasions. Scats were collected along transect intervals through a study area, and identified via genetic markers to individual; thus, re-encounters of scat with the same genetic ID constitute a recapture. The JAGS model was quite slow, but did provide reasonable convergence within 1K or so iterations of 3 chains:
At least as currently configured, secr or other ML procedures that I am aware of do not provide open CMR analogs. Thus, Bayesian modeling provides us a powerful tool to apply these methods to studying population dynamics.
Script for coyote example
R data objects for coyote example
Summary results for coyotes example
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