Open population models with SECR 

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Updated 20 May  My thoughts section
added; 15 May  plots for fishers; 13 May

more info in several places; 8 May
 black bears; 7 May 2018  Field vole plots
added. We are thinking about models for multiyear SECR analysis for sea cucumbers and other species. For that the usual closure assumption is implausible. Recall the closure assumption for SECR:
Various implementations of multiyear SECR have relaxed the first part of this, allowing for recruitment and deaths/emigration, with a "robust" design (Pollock 1982). The second part is more difficult. Options include:
See some conclusions below. Examples from the literaturePampas catsGardner, B., Reppucci, J., Lucherini, M., & Royle, J.A. (2010) Spatially explicit inference for open populations: estimating demographic parameters from cameratrap studies. Ecology, 91, 33763383. The real data is only for 2 years, but they provide code for simulations for 3 years. The data are sparse, so assume AC doesn't change, but suggest a random walk for changes in AC with bivariatenormal steps. The pampas cats also get a mention in Royle, J.A. & Gardner, B. (2011). Hierarchical spatial capture–recapture models for estimating density from trapping arrays. In Camera traps in animal ecology: methods and analyses (eds A.F. O'Connell, J.D. Nichols & K.U. Karanth), pp. 163–190. Springer, New York. OvenbirdsRoyle, J.A., Chandler, R.B., Sollmann, R., & Gardner, B.
(2014) Spatial capturerecapture. Elsevier (SCR book),
Chapter 16 give a number of analyses of the ovenbird data
collected by Dawson and Efford from 2005 to 2009 and included in
the They present code for a spatial multiseason model (p.375, this models changes in population density but does not track individual birds) and both nonspatial robust JS (p.408) and spatial JS (p.413) models. For their spatial models they use independent uniform priors for the ACs in each year. The posterior distributions for these for a bird caught in years 1, 3, 4 and 5 but not in year 2 are shown below with a 150m buffer.
Clearly the birds do move around and attempting
to fit a single AC location for all years would be suboptimal.
The parameter
For birds captured, the Having completely independent locations in each year, with no constraint on the location when the bird is not captured, won't work. Royle et al (p.426) discuss modelling movement, suggesting either a movement kernel (eg, bivariate normal) or an autoregressive model. They comment: "At this point, very little work has been done to model movement using SCR models; however, we expect that in the near future this will be one of the most exciting areas of research." Field volesErgon, T. & Gardner, B. (2014) Separating mortality and emigration: modelling space use, dispersal and survival with robustdesign spatialcapturerecapture data. Methods in Ecology and Evolution, 5, 13271336. They model changes in AC as a random walk, with movement from one year to the next occurring in a random direction, then a distance drawn from a distribution, which may be exponential, gamma or lognormal. They also tried zeroinflated movement models, with a certain probability of no movement. They give lots of code for simulation and estimation in JAGS, including goodness of fit procedures. A few unusual features of this analysis (compared with other SCR models):
As a result of that, I'm not sure whether we will be able to combine these ideas with our usual densityestimation methods. Still, I ran the code and got some plots of ACs. Below are plots for a male and a female, both captured in primary occasions 1 and 2 and not captured in 3 and 4:
In session 3, the male is probably dead (mean z is the
posterior probability of being alive), but if alive it has moved
out of the trapping array, and remains there in session 4. The
female is almost certainly dead, but if alive stays at the same
place. This is consistent with the conclusion that males on
average disperse further than females between primary sessions
(6m vs 3m; trap spacing is 7m), have higher survival probability
(0.91 vs 0.75), and higher capture probability if inside the
array (as indicated by These models seem to all have poor mixing and high
crosscorrelations. For the voles, I'm getting Redbacked shrikesSchaub, M. & Royle, J.A. (2014) Estimating true instead of apparent survival using spatial CormackJollySeber models. Methods in Ecology and Evolution, 5, 13161326. As with field voles, this is a modified CJS model, conditioning on first capture. They used area search instead of traps, and if the bird is detected the nest is found and this is the AC. The detection function is categorical: 1 if the AC is inside the search area, 0 if outside. Their random walk is bivariate normal, as with black bears below, instead of direction plus distance (though they comment that the latter may be more biologically meaningful). Code for simulations given, not real data. I didn't explore this in detail as very different to the data we collect. GrizzliesWhittington, J. & Sawaya, M.A. (2015) A comparison of grizzly bear demographic parameters estimated from nonspatial and spatial open population capturerecapture models. PLoS ONE, 10, e0134446. Their data set has > 250,000 hairsnare samples (budget?) for 80 individual bears from 630 traps over 3 years, so likely a long run, which I haven't attempted. Animal activity centres were assumed not to change over the 3 years (= pampas cats). Black bearsRoyle, J.A., Fuller, A.K., & Sutherland, C. (2016) Spatial capture–recapture models allowing Markovian transience or dispersal. Population Ecology, 58, 5362. Single season model, but allowing for changes in occasionspecific AC. For some or all of the population (transients) the ACs move as a random walk controlled by a bivariate normal. A similar model for dispersal involves a single large move to a new location. They did lots of simulations, and the good news is that the bias in the density estimates due to ignoring AC movement is remarkably small (see Tables 13 in the paper). The somewhat bad news is that the estimate of σ increases with time, so be careful if you are using this as a measure of home range size. They analyse the Gardner et al (2010, J Wildlife Mant,
see also the SCR book, chapter 4) Fort Drum bear hair
snare data as their example, where the occasions are 1 week in
length. Mixing is
poor, even with an overnight run with 400k iterations: The first plot is for a bear detected on the first occasion at a location on the edge of the trap array and not thereafter (yellow dot is the location of capture, the black dot the centroid of the posterior estimates of the AC):
It's AC is modelled as outside the array, and it moves slightly further out in subsequent occasions, with less and less certainty about the location. Animals caught only on the last occasion show the reverse pattern, lots of uncertainty at the beginning and gradually homing in on the location of capture. The next animal was caught twice in the northern part of the array and a third time in the south, and is the one with the most movement (the blue dots indicate the previous home range centroids):
Most animals tended to stay in much the same place and the next plots are more typical:
One of my reservations about the usual camera trap analysis that we do is that captures are not actually independent: if an animal was captured in one trap last night, it's not going to appear tonight at the opposite end of its territory. Over a short period (say, 1 week) captures are likely to be clustered, and this may be one way to model that. FishersGreen, D.S., Matthews, S.M., Swiers, R.C., Callas, R.L., Yaeger, J.S., Farber, S.L., Schwartz, M.K., Powell, R.A., & Griffen, B. (2018) Dynamic occupancy modelling reveals a hierarchy of competition among fishers, grey foxes and ringtails. Journal of Animal Ecology, 87, 813824. In
addition to the occupancy modelling for the 3 species, they also
ran SCR analysis for fishers, individually identified from hair
samples, annually from 2006 to 2013. They modelled ACs as
independent for each year (= ovenbirds).
The model with all the covariates runs
very slowly (25 secs per iteration!) but with 20 cores and 3
days I managed to get
Looking at the AC locations in successive years, it seems that a random walk would be a good model. In that case, the location in year 6 would be not far from the array and detection probability would be higher and probability of survival lower. ACs in years 7 and 8 would be less constrained but not scattered over the entire state space. TigersWang, T., Royle, J.A., Smith, J.L.D., Zou, L., Lü, X., Li, T., Yang, H., Li, Z., Feng, R., Bian, Y., Feng, L., & Ge, J. (2018) Living on the edge: Opportunities for Amur tiger recovery in China. Biological Conservation, 217, 269279. This was a 2year study, appears they used the Gardner et al 2010 method (= pampas cats). JaguarsGlennie, R., Borchers, D.L., Murchie, M., Harmsen, B.J., & Foster, R.J. Open population maximum likelihood spatial capturerecapture. (submitted). They have a 12year data set for male jaguars in Belize and (like it says) want to do an ML estimation. They do not allow for change in AC, which seems bold given the length of the period, but they do use the CMSA parameterisation for recruitment (see below). My thoughts and experimentsCJS vs JSCormackJollySeber (CJS) methods only consider animals captured and only their survival after first capture, as with field voles and shrikes above. That literature has interesting ideas about modeling survival and movement, but is generally not what we want. JollySeber (JS) methods, in particular the robust versions, fit best with our over approach, where we are interested in abundance and recruitment as well as survival. Modelling AC movement over timeFixed ACs are in general implausible, as animals are almost certain to 'drift' as the years go by. So we need to be able to model changes in AC over time; that does not mean that movement models will be the best in every case, and I have worked with some sparse data where AC drift and the usual SCR scale parameter were not separable. Independent ACs each year are easy to implement but hard to estimate. In particular, survival estimates depend on the area of the state space, as seen with the ovenbirds. A random walk, where this year's AC depends on last year's AC, seems the way to go; no one has chosen the "hyperAC' idea. Movement is modelled independently as drawn from a bivariate normal (or similar) between primary sessions.
Modeling the walk as a random direction plus a distance drawn
from a halfnormal (or similar) distribution, seems biologically
sensible, as used for field voles and
suggested for shrikes. Unfortunately,
JAGS has no sampler to deal with circular
distributions. (The Egon & Gardner (2014) prior for direction is coded
as Recruitment parameterisationRecruitment gets tangled with data augmentation. With our SECR models, we use data augmentation to be able to estimate abundance, and an inclusion parameter (usually \(\psi\) or \(\Omega\)) controls the probability that individuals in the augmented population are in the population available for detection. The parameter has no biological significance and depends on the degree of data augmentation. An early approach, used for all the above except jaguars, is to add recruits from the augmented population, with a parameter \(\gamma_t\) controlling the probability that an individual in the augmented population which has not yet been added to the real population will be added in year \(t\). Again, the parameter has no biological meaning, and it would be difficult to put any hierarchical structure on it or even to enforce constant recruitment. More efficient is to borrow the idea of a superpopulation from the CrosbieManlySchwarzArneson (CMSA) literature (see Link & Barker 2010 p.255). The superpopulation consists of all the individuals in the population  and available for detection  at any primary session. Membership of the superpopulation is determined by \(\psi\) or \(\Omega\) as usual. For these individuals, recruitment is a question of when, not if. The timing of recruitment is a categorical variable determined by a vector of probabilities, where \(\alpha_t\) is the probability that an individual will be recruited just before primary session \(t\), with \(\alpha_1\) the probability of recruitment at some time before the first survey, and all the \(\alpha_t\) summing to 1. Now it's clear how to model recruitment as a function of covariates or how to reduce the number of parameters, for example with:
Survival parameterisationThe usual apparent survival parameter (\(\phi_t\)) works fine here, giving the probability that an individual will leave the population through death or permanent emigration before primary session \(t\). Slices for sessionsThe standard singleseason SCR study assumes the population is closed (no births/deaths/immigration/emigration). For many rare and elusive species, passive detectors need to be deployed for long periods to get enough data We often discard data outside a restricted period to try to meet closure assumptions and still hope that infringements will not cause too big a bias. Some projects have moved to continuous monitoring of populations. Slicing these extended periods short sessions and using a robust analysis method may be the best way to utilise the data. 