Appendix A. Estimating the probability of detection of birch seedlings within the treeline ecotone.

Within line transect sampling, the likelihood of detecting an individual decreases the further the individual is from the transect line. This can be problematic in the estimation of population density of a species. Statistical techniques have been developed to account for decreasing probability of detection within line transect sampling, allowing density or abundance to be estimated based upon the decrease in observed individuals with distance from the transect line (Thomas et al*.* 2010). Such analyses require the perpendicular distance from the transect line to the observed objects to be measured. Distance analysis techniques are most commonly applied to surveys of wild animal populations, however, they are also particularly well suited to modelling the detection probability of plants, particularly as two of the underlying assumptions of distance analyses are that objects are stationary and distance measurements from the transect line to object location are exact (Thomas et al. 2010). These assumptions are clearly less likely to be violated in the case of plants, than for mobile animals.

During line transect sampling of birch within the experimental enclosures, birch recruits were recorded within 5 m either side of the transect line, thus transects were 10 m wide. The perpendicular distance from the transect line to the individual was recorded. It was assumed that detection probability of birch recruits on the transect line was 1 and that detection probability decreases with distance from the sampling line. Detection functions were developed to account for this decrease using the software Distance 6.0 (Thomas et al. 2009). The area under the detection function curve corresponds to the detection probability across the transect width (Fig. A1). As detection probability was predicted to vary with vegetation type and grazing treatment, detection functions were fitted for each treatment vegetation combination. The detection function for each combination of grazing treatment and vegetation type was modelled with the half normal key function, using cosine, simple polynomial and Hermite polynomial series expansions, and the hazard rate key function, using cosine and simple polynomial series expansions (Buckland et al. 2001). The optimal model was selected based on minimum Akaike’s Information Criterion (AIC) for each treatment – vegetation combination.

Each transect was fragmented into 10 m segments, equal in length to the transect width (Hedley and Buckland 2004) and the number of birch saplings and birch recruits within each 10 × 10 m segment was presence–absence transformed. The observed presence–absence data was fitted to the explanatory variables at each transect segment within a general linear model with binomial error distribution. The estimated detection probability from the selected model for the particular vegetation type and treatment was used as an offset in the models within the Design library (Harrell 2009) in the R environment (R Development Core Team 2009).

Example detection functions within the dwarf shrub heath, the most common vegetation community at the study site, are shown in Fig. 1. The detection probabilities (area under the curve) within this vegetation type were 0.63, 0.45, and 0.15 for ungrazed, low, and high sheep density treatments respectively. This indicates that there is a 63% chance of detecting a birch recruit within the dwarf shrub heath in the ungrazed treatment and a 15% chance at high sheep density. Such treatment differences may be attributable to the impact of sheep herbivory upon vegetation openness or birch size and apparency.

FIG. A1. Example detection functions for the most common vegetation community within the study site, dwarf shrub heath, for each grazing treatment. |

LITERATURE CITED

Buckland, S. T., D. R. Anderson, K. P. Burnham, J. L. Laake, D. L. Borchers, and L. Thomas. 2001. Introduction to distance sampling: estimating abundance of biological populations*. *Oxford University Press, Oxford, UK.

Harrell, E. J. 2009. Design: Design Package R package version 2.3-0.

Hedley, S. L., and S. T. Buckland. 2004. Spatial models for line transect sampling. Journal of Agricultural, Biological, and Environmental Statistics 9:181.

R Development Core Team. 2009. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

Thomas, L., S. T. Buckland, E. A. Rexstad, J. L. Laake, S. Strindberg, S. L. Hedley, J. R. B. Bishop, T. A. Marques, and K. P. Burnham. 2010. Distance software: design and analysis of distance sampling surveys for estimating population size. Journal of Applied Ecology 47:5–14.

Thomas, L., J. L. Laake, E. Rexstad, S. Strindberg, F. F. C. Marques, S. T. Buckland, D. L. Borchers, D. R. Anderson, K. P. Burnham, M. L. Burt, S.L. Hedley, J. H. Pollard, J. R. B. Bishop, and T. A. Marques. 2009. Distance 6.0. Release 2. Research Unit for Wildlife Population Assessment, University of St. Andrews, UK.