Density Estimation with Closed CR Models 5.10 UF-2015

Density Estimation Based on CR Estimates: Basic Problem Animals from outside the grid may be captured Animals from outside the grid may be captured Sample area not known Sample area not known –except in discrete habitats (e.g., islands, woodlots, isolated parks) Area physically covered by traps, A, is too small, so that is too large Area physically covered by traps, A, is too small, so that is too large

Density Estimation Based on CR Estimates: Solution Estimate boundary strip width, Estimate boundary strip width, –Account for animals with ranges that partially overlap the grid Combine boundary area and trap area to obtain Combine boundary area and trap area to obtain Estimate density as Estimate density as

Approaches to Estimation of Boundary Strip Width ½ home range diameter ½ home range diameter –Estimate diameter as mean maximum distance moved between trap captures –Ad hoc approach –½ MMDM supported by simulations (Wilson and Anderson 1985)

Approaches to Estimation of Boundary Strip Width ½ MMDM will underestimate ½ home range ½ MMDM will underestimate ½ home range –Use 10+ trapping occasions –Use MMDM (Parmenter et al. 2003) – (Jett and Nichols 1987) –Radio telemetry

Nested Grid Approach Basic idea: naïve density estimates based on the area covered by traps, A, is most biased for small areas and least biased by large areas Basic idea: naïve density estimates based on the area covered by traps, A, is most biased for small areas and least biased by large areas

Nested Grid Approach Key is the ratio of perimeter/area Key is the ratio of perimeter/area –Greatest bias when this ratio is big If we estimate density over increasing A, we expect estimates to be progressively less biased If we estimate density over increasing A, we expect estimates to be progressively less biased This progression is predictable and can be used to estimate w This progression is predictable and can be used to estimate w

Nested Grid Approach Estimate abundance and naïve density for each grid Estimate abundance and naïve density for each grid Estimate w using the relationship between naïve density and area Estimate w using the relationship between naïve density and area Mathematical development by Otis et al. (1978) Mathematical development by Otis et al. (1978) Estimation for grid systems available in CAPTURE (Rexstad and Burnham 1991) Estimation for grid systems available in CAPTURE (Rexstad and Burnham 1991)

Nested Grid Approach Assumes constant density Assumes constant density –A gradient would affect the naïve density to area relationship Need large sample size Need large sample size Jett and Nichols 1987 used 16x16 grid Jett and Nichols 1987 used 16x16 grid

Trapping Web Place traps on radial spokes of web Place traps on radial spokes of web Use distance sampling to estimate density based on captures occurring at different distances from a random point Use distance sampling to estimate density based on captures occurring at different distances from a random point Anderson et al. (1983) Anderson et al. (1983)

Trapping Web Density Estimation: Assumptions All animals at center of web captured (p=1) All animals at center of web captured (p=1) Distances moved by animals on the web are small relative to web size Distances moved by animals on the web are small relative to web size Distances from web center to traps are measured exactly Distances from web center to traps are measured exactly Estimation in program DISTANCE (Buckland et al. 2001) Estimation in program DISTANCE (Buckland et al. 2001)

Trapping Web: Geometric Analysis Based on “closest trap assumption” that number of captures at a trap is proportional to region of web for which that trap is the closest trap Based on “closest trap assumption” that number of captures at a trap is proportional to region of web for which that trap is the closest trap Estimation is based on number of captures at each trap and computation of the size of the region for which each trap is closest Estimation is based on number of captures at each trap and computation of the size of the region for which each trap is closest Barker and Link (1994) Barker and Link (1994)

Gradient Designs Trapping web creates a gradient in capture probability by varying density of traps with distance Trapping web creates a gradient in capture probability by varying density of traps with distance Other ways to create this gradient (e.g., linear transects of traps, with trap density decreasing with distance from center line) Other ways to create this gradient (e.g., linear transects of traps, with trap density decreasing with distance from center line)

Concluding Comments In studies requiring density (as opposed to abundance) estimates, the estimation of sampled area is a weak link In studies requiring density (as opposed to abundance) estimates, the estimation of sampled area is a weak link –Camera-trap studies of carnivores Use ½ MMDM or MMDM, but no theoretical basis for selecting an approach Use ½ MMDM or MMDM, but no theoretical basis for selecting an approach Use Distance methods Use Distance methods Spatially explicit CR models hold great promise Spatially explicit CR models hold great promise