Spatially explicit capture-recapture models for relative density
Abstract
Spatially explicit capture-recapture (SECR) methods are used widely to estimate animal population density and related parameters. Maximum likelihood has been applied to two flavors of SECR model - a full model that includes absolute density as a spatially varying parameter, and a model conditional on the number of detected individuals that assumes uniform density. We show how the conditional model may be extended to estimate relative density as a function of habitat and other spatial variables. Unlike absolute density, coefficients of relative density are robust to bias from heterogeneous individual detection. The conditional model has the further advantage of seamlessly including individual covariates of detection, and it is faster to fit than the full model. The spatial absolute density surface may be derived from the conditional model fit. Population growth rate (relative density with respect to time) may also be estimated from a conditional model, given temporal data. These properties are demonstrated by examples and simulation. It is suggested that conditional-likelihood models, including those for relative density, play a central role in SECR.
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In-depth summary
Claude Opus 5.5 · xhigh effort · 5 Nov 2026 · from the full text of version 2 (24 Oct 2025) · CC BY-NC-ND 4.0
Maximizing the conditional likelihood in spatially explicit capture-recapture models estimates spatial relative density and population growth rates rapidly while remaining robust to unmodeled individual detection heterogeneity.
In spatially explicit capture-recapture (SECR), full-likelihood models of spatially varying density are biased by unmodeled individual detection heterogeneity and cannot easily incorporate continuous individual covariates. Conditioning on the number of detected individuals removes the density intercept to estimate relative density and individual detection covariates within a single framework, leaving absolute density as a derived parameter.
Key results
ShownRelative density coefficients estimated from the conditional SECR likelihood are unaffected by unmodeled individual variation in detection hazard and match full-likelihood estimates at lower computational cost.
Across 2000 simulated replicates per level of log-normal detection hazard heterogeneity (mean lambda0 = 0.2, CV from 0 to 0.8; average n declining from 265 with SD 71.6 to 232 with SD 60.7), the absolute density intercept beta0 was strongly biased while the spatial covariate slope beta1 was unbiased, differed from the full-likelihood estimate by < 0.0003, and ran in 59% to 70% of the execution time. · Conditional likelihood for relative density; Simulations
ShownConditional SECR models jointly accommodate spatial habitat effects on relative density and continuous individual covariates on detection in pitfall-trapped skinks.
In 253 trapped Oligosoma infrapunctatum skinks across two 231-trap grids, density in high-cover habitat exceeded low-cover habitat by a factor of exp(beta1) = 43.2 whether or not snout-vent length (SVL) spline smooths were included on baseline detection hazard lambda0, and the conditional habitat model without individual covariates fit in 58% of the full-likelihood runtime. · Example. Spatial distribution of skinks; Discussion; Figure legends
ShownMulti-session closed-population conditional SECR directly estimates annual population growth rates that match ratios of sequential full-likelihood density estimates.
In a 10-year (1996-2005) trapping dataset of 698 brushtail possums (Trichosurus vulpecula) captured 4408 times across 167 traps, session-specific growth rates lambda from the multinomial conditional likelihood overlaid successive density ratios from the full likelihood while fitting in 78% of the time. · Multi-session conditioning; Example. Brushtail possum population dynamics; Figure legends
Methods, data and reproducibility
Factorizes the SECR likelihood to condition on total detected individuals n, cancelling the log-linear density intercept exp(beta0) to estimate spatial relative density coefficients and individual detection covariates simultaneously. Recovers absolute density via a Horvitz-Thompson-like estimator and models multi-session counts as multinomial to estimate finite population growth rates in the R package secr.
- Simulated spatial detection histories on a 10 x 10 grid of binary proximity detectors with inhomogeneous Poisson activity centers · n = 2000 replicates per each of 5 CV(lambda0) levels (mean n = 232-265 individuals)
- Pitfall-trapping capture-recapture dataset of diurnal skinks over 3 days in November 1995 in Upper Buller Valley, South Island, New Zealand · Oligosoma infrapunctatum (New Zealand skink) · n = n = 253 trapped individuals across 2 grids of 231 traps each
- Multi-year June live-trapping dataset of brushtail possums in mixed forest, New Zealand (1996-2005) · Trichosurus vulpecula (brushtail possum) · n = 698 individuals captured 4408 times across 167 traps over 10 annual sessions
- Methods & tools
- secr (R package), secr.fit, derivedDcoef, derivedDsurface
- Reproducibility gaps
- Simulation code and full skink data are cited as external references (Efford 2025b, 2025c) and Appendices S1-S3 without direct repository URLs or software version numbers in the text.
Limitations (4)
- Variance estimates for derived absolute density rely on the delta method with numerical gradients rather than profile likelihood intervals, and adjusting for individual detection covariates can substantially reduce the precision of absolute density estimates. · stated by the authors
- The conditional model cannot treat absolute density as a derived parameter when another likelihood component depends on it, such as when the spatial scale of detection sigma is modeled as a function of absolute density. · stated by the authors
- Multi-session closed-population variance estimates for population growth rate assume independent Poisson samples of activity centers in each session, which is violated when individuals survive and retain their activity centers across sessions. · stated by the authors
- Exact equivalence between conditional-likelihood derived density and full-likelihood density requires activity centers to follow a Poisson distribution and does not hold exactly when population size in a region is treated as fixed (binomial n). · stated by the authors
Who should read:Read this if you fit spatially explicit capture-recapture models in ecology and need to model spatial density covariates, temporal population growth, or individual detection heterogeneity.
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