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Keywords:

  • competition;
  • Dirichlet tessellation;
  • focal point;
  • Voronoi diagram

Abstract

  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Supporting Information

Thiessen polygons are often used to model territory characteristics. However, information about the quality of Thiessen polygon-based estimates is currently lacking. We used published data to investigate the match between Thiessen polygons and mapped bird territories regarding territory size, shape and neighbourhood. Although territory sizes and the number of neighbours were strongly correlated between these two methods, both parameters were overestimated by the Thiessen polygons. Therefore, caution is required when Thiessen polygons are used as a model for absolute values and when the assumptions of Thiessen polygons, such as formation of discrete territories and a contiguous study area, are not met.

Because territory acquisition and tenure are important for individual breeding success and survival, measuring territory parameters is often relevant in studies in behavioural ecology. However, mapping territories directly through observations or radiotracking is difficult and time-consuming, and not possible to do post hoc. Therefore, an alternative to territory mapping to assess territory boundaries becomes important.

Thiessen polygons (also known as Dirichlet tessellations or Voronoi diagrams) define the area of influence of each focal point (e.g. nest-site) by a polygon encompassing the area closer to the target point than to any other point (Aurenhammer 1991). The definition of Thiessen polygons (henceforth TPs) reflects a competitive process by which the available space (suitable habitat) is partitioned among neighbouring individuals (e.g. Morrell & Kokko 2005). This renders TPs a useful model for estimating territory boundaries and areas, as well as the number and identity of neighbours occupying adjacent territories (e.g. Wilkin et al. 2006, Valcu & Kempenaers 2008, Kempenaers et al. 2010). TPs are based entirely on focal points, which can be any set of points around which territories are formed (‘centres of defence’ such as nest-sites: Adams (2001) and references therein). Therefore, territories can be estimated without any further information on the individuals (e.g. their competitive abilities) or the environment (e.g. habitat heterogeneity). This is especially useful for re-analysing existing datasets, where important parameters of the habitat or of the individuals may not be known and cannot be investigated post hoc. Significant relationships between territory size estimated by TPs and measures of breeding success indicate that TPs capture biologically meaningful information (e.g. Valcu & Kempenaers 2008, Grabowska-Zhang et al. 2012). However, how well TPs approximate the different properties of a territory has not been investigated. On the basis of 14 studies presenting mapped territories, this study investigated the quality of the approximation of the TPs, and potential biases. As possible focal points (e.g. nest-sites) were mostly unknown, we investigated the quality of TPs depending on the distance of the focal points to the centres of mass of the respective mapped territories. We review published literature on the use of TPs and present a framework for the practical application of TPs.

Methods

  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Supporting Information

We used the digitized mapped territories from 14 published studies on 12 bird species (from nine families) described in Valcu and Kempenaers (2010) and the TPs that were calculated for these 14 studies. As a minor change, we used only the larger of two study areas (‘cattle creek’) in one study (Wortman-Wunder 1997). We only selected studies in which (1) territories were obtained via detailed observations of territorial behaviour of individually marked animals; (2) territories of more than 10 individuals were mapped; and (3) territories were not obviously constrained by the geography of the study site. All the maps were saved in a raster format and each territory was manually digitized and saved in a vector format (details in Valcu & Kempenaers 2010).

Because studies using mapped territories do not typically provide focal point data (only four studies in our dataset give a focal point position), we used a series of numerical simulations to investigate the influence of the focal points on the construction of the TPs. First, we constructed TPs using the midpoints (centroids, centres of mass) of the mapped territories. Secondly, we defined a series of focal points for TP construction by sampling at specific distances from the centroid in random directions. We used distances between 2% (almost identical to the centroid) and 98% (almost on the border of the territory) at 2% intervals. We repeated the random sampling process 50 times for each distance.

Borders of study areas were defined using a convex Ripley–Rasson estimate (Ripley & Rasson 1977). An example is shown in Supporting Information Figure S1. We calculated the territory size and the number of neighbours for both mapped territories and TPs. The number of neighbours was defined as the number of individuals with adjoining territory borders for the TPs, and the number of territories within a distance defined such that each individual had at least one neighbour for the mapped territories. We used linear mixed-effects models with ‘study’ as the random intercept, the mapped territory measure as the dependent variable and the corresponding TP-based estimate as the predictor. By overlaying mapped territories and TPs we were able to calculate the underestimation and the overestimation of the polygons for both territory location and neighbour identity. We defined ‘underestimation’ as the percentage of the mapped area (or neighbours) that was not assigned to the corresponding TPs, and ‘overestimation’ as the percentage of the TP area (or neighbours) that exceeded (or projected beyond) the corresponding mapped area (or neighbours).

We used R 2.14.1 for all statistical analyses (R Development Core Team 2011). Specifically, we used the packages ‘lme4’ for linear mixed-effect models (Bates et al. 2011), ‘effects’ for graphical representation of the regressions (Fox 2003), ‘spatstat’, ‘maptools’, ‘spdep’, ‘rgeos’ and ‘rgdal’ for spatial calculations, and ‘stats’ for fitting smooth lines (Baddeley & Turner 2005, Bivand et al. 2008, R Development Core Team 2011, Bivand & Rundel 2012, Keitt et al. 2012, Lewin-Koh & Bivand 2012, respectively). TPs were calculated following the method described in Valcu and Kempenaers (2010). Estimates of territory size were log-transformed to achieve normality and scaled (standardized and centralized) to achieve valid intercepts and comparability among studies. The number of neighbours was centralized to achieve valid intercepts. All transformations were performed for each study using all data points for the respective area (values of both TPs and mapped territories) to maintain comparability of the absolute values of the two methods. Note that t-values and P-values reflect effects caused by both the true correlations and the spatial autocorrelations among the TPs (Valcu & Kempenaers 2010).

Results

  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Supporting Information

The mean (± se) number of mapped territories per study was 30.2 ± 6.4 (range 13–107). Territory size as estimated by the TPs was strongly correlated with mapped territory size, although it was on average overestimated (Fig. 1a, slope = 0.87 ± 0.06, = 15.30, < 0.001, = 423; intercept = −0.07 ± 0.01, = −6.05, < 0.01). Similarly, the number of neighbours estimated by the TPs was strongly correlated with the number of neighbours defined by the mapped territories, although the number of neighbours was overestimated on average (Fig. 1b, slope = 0.56 ± 0.04, = 13.02, < 0.001, = 423; intercept = −1.31 ± 0.15, = −8.95, < 0.01). All parameters decreased in fit as the distance of the focal points used for TP construction to the centroid of the mapped territory increased. However, both regressions (territory size and number of neighbours) remained significant throughout (all P-values ≤ 0.01).

image

Figure 1. Regression of mapped territories against Thiessen polygons with respect to territory size (a, c) and the number of neighbours (b, d). Panels (a) and (b) show model estimates and confidence intervals (black), regressions for individual studies (grey) and the data frequency (parallel to the axes). The dotted line indicates a perfect fit. Axes in (a) are given as the percentage of the largest (log-transformed) value per study. Note that in contrast to the statistical model, the data shown here are non-centralized. Panels (c) and (d) show the slope of the respective regression depending on the distance of the focal point used for the construction of Thiessen polygons to the centroid of the mapped territory. Grey circles are individual estimates from the randomizations, and the black line is a smooth curve through the randomized points.

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The mapped territories of the 14 studies showed little overlap (range 0–0.8%). The mean percentage of the mapped area that was not assigned to the corresponding TP ranged from 2% to 25% (11 ± 2%, Supporting Information Fig. S2a, ‘underestimation’), and the mean percentage of the area of the TP that exceeded the corresponding mapped area ranged from 26% to 72% (49 ± 4%, Fig. S2a, ‘overestimation’). The mean percentage of neighbours that were not identified by the TP ranged from 0% to 16% (4 ± 1%, Fig. S2b, ‘underestimation’), and the mean percentage of neighbours identified by the TP that did not correspond to neighbours based on the mapped area varied between 16 and 44% (30 ± 2%, Fig. S2b, ‘overestimation’). Neighbour identity and territory location showed a decreasing fit as the distance of the focal point used for Thiessen polygon construction to the centroid of the mapped territory increased (Fig. S2).

We defined a two-step framework for the use of TPs in a specific study where focal points could be assessed (Fig. 2). At step 1, the habitat or the population boundary is selected assuming that all space within that boundary is partitioned among individuals. In some cases, clear boundaries can be defined a priori (e.g. an island in a lake, a forest patch surrounded by agricultural fields). When the area that encompasses a population is not clear a priori but relatively well defined, then the boundary can be estimated as a convex polygon. If any boundary can be established, TPs can be constrained to lie within the boundary. Otherwise edge territories should be removed and/or polygon boundaries should be constrained by intersection with a circle of a given radius to avoid unrealistically large territories. These methods can be combined in cases in which boundaries can only be established for some parts of the study area (e.g. a lake at one side of the study forest). At this point, TPs completely cover the area within which they are calculated. At step 2, if a priori knowledge of covariation of individual traits with territory parameters exists, TPs can be further improved by scaling the polygon through a homothetic transformation (e.g. younger males' territories are smaller than those of older males, and are therefore down-scaled) or by altering its shape in accordance with a known behavioural process (e.g. the competitive ability of two experienced breeders may be more similar than that of an experienced and a first-time breeder, so that in the latter case the territory border can be moved in favour of the experienced breeder).

image

Figure 2. Flow chart for the application of Thiessen polygons to estimate territory characteristics.

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Discussion

  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Supporting Information

We assessed how well TPs estimated the four most commonly used parameters of territories (size, location, number of neighbours and neighbour identity). There was a strong correlation between mapped territories and the estimates from TPs regarding territory size and number of neighbours (Fig. 1). Both territory size and the number of neighbours were overestimated by the TPs. This overestimation is also obvious in Figure S2, and has to be kept in mind when estimating absolute territory sizes from TPs (see below). Note that Figure 1(b) suggests a stronger overestimation when more neighbours are present. With regard to territory size and the number of neighbours, the strength of the correlation between TPs and mapped territories decreased when the focal points used for the construction of TPs were further away from the centroids of the mapped territories, but remained strongly significant at all distances.

Although the mean percentage of the mapped area that was not assigned to the corresponding TP (‘underestimation’) was small, the mean percentage of the area of the TP that did not overlay the corresponding mapped area (‘overestimation’) was large. For example, for more than half of the territories, 46% of the area that was predicted to be defended as a territory by the TPs did not overlay the corresponding mapped territory. Similarly, the identities of almost all mapped neighbours were correctly predicted, whereas for half of the territories more than 25% of the TP neighbours were mis-assigned (Fig. S2). TPs are therefore expected to be a ‘noisy’ measurement for studies on, for example, environmental attributes of territories. They should not be used when an overestimation of territory location or neighbour identity can lead to spurious results. Both overestimation and underestimation increased if focal points were fixed to be further away from the centroid of the mapped territory (Fig. S2).

TPs are an appropriate model for classically territorial species (Adams 2001). As expected for territorial species, the mapped territories overlapped little in the focal 14 studies (range 0–0.8%). This means that discrete territories were defended. However, some studies had low-density settlement. Additionally, boundaries of study areas and unusable habitat patches were rarely defined. This is in part responsible for both the overestimations of territory size and the number of neighbours. To avoid such biases and increase the fit of TPs, it is important to consider carefully how to set biologically meaningful conditions for a specific context (Fig. 2). The construction of TPs resembles the biological process of territory formation only if the available habitat is partitioned into distinct territories through a competitive process between adjacent individuals. If territories are formed by fundamentally different mechanisms (e.g. habitat patchiness, distance of movement from the nesting site), there is no theoretical basis for the use of TPs. In low-density populations, where neighbour encounters may be only partly responsible for territory formation, unrealistically large territories can be avoided by assuming that an individual will not move further than a certain distance from its nesting site (McLeod et al. 2002). However, in many cases the radius of the respective circle will not be clear a priori (e.g. Wilkin et al. 2006). The information that will be included to achieve a valid fit of the TPs largely depends on the available information on habitat structure, study area boundaries and individual parameters (Fig. 2), as well as on the precision of the parameter estimate needed for addressing the respective research question.

In conclusion, our results suggest that TPs without any biological or ecological refinements are a useful tool for estimating territory areas, locations and neighbourhoods of territorial bird species. However, they generally tend to overestimate both the territory size and the number of neighbours. They may be fine-tuned by including ecological and biological information in the construction process, which in turn may decrease overestimations. If the main assumptions underlying the use of TPs are not met, or if the overestimation of parameters is problematic for the specific study, other models should be used that are less general and carefully adjusted to the specific context.

We thank two anonymous reviewers and Stephan J. Schoech, whose sophisticated comments greatly improved the manuscript. This work was funded by the Max Planck Society and the International Max Planck Research School for Organismal Biology.

References

  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Supporting Information
  • Adams, E. 2001. Approaches to the study of territory size and shape. Annu. Rev. Ecol. Syst. 32: 277303.
  • Aurenhammer, F. 1991. Voronoi diagrams – a survey of a fundamental geometric data structure. Comput. Surv. 23: 345405.
  • Baddeley, A. & Turner, R. 2005. Spatstat: an R package for analysing spatial point patterns. J. Stat. Softw. 12: 142.
  • Bates, D., Maechler, M. & Bolker, B. 2011. lme4: Linear Mixed-Effects Models Using S4 Classes. R package version 0.999375–42 Available at http://CRAN.R-project.org/package=lme4. (accessed 2 February 2012).
  • Bivand, R. & Rundel, C. 2012. rgeos: Interface to Geometric Enginge – Open Source (GEOS). R package version 0.2–5 Available at http://CRAN.R-project.org/package=rgeos (accessed 21 March 2012).
  • Bivand, R., Pebesma, E. & Gómez-Rubio, V. 2008. Applied Spatial Data Analysis with R. New York: Springer Science+Business Media.
  • Davies, N.B. & Hartley, I.R. 1996. Food patchiness, territory overlap and social systems: an experiment with dunnocks Prunella modularis. J. Anim. Ecol. 65: 837846.
  • Fox, J. 2003. Effect displays in R for generalized linear models. J. Stat. Softw. 8: 127.
  • Grabowska-Zhang, A., Sheldon, B. & Hinde, C. 2012. Long-term familiarity promotes joining in neighbour nest defence. Biol. Lett. 8: 544546.
  • Halls, P.J., Bulling, M., White, P.C.L., Garland, L. & Harris, S. 2001. Dirichlet neighbours: revisiting Dirichlet tessellation for neighbourhood analysis. Comput. Environ. Urban Syst. 25: 105117.
  • Keitt, T.H., Bivand, R., Pebesma, E. & Rowlingson, B. 2012. rgdal: Bindings for the Geospatial Data Abstaction Library. R package version 0.7–8 Available at http://CRAN.R-project.org/package=rgdal (accessed 15 March 2012).
  • Kempenaers, B., Borgström, P., Loës, P., Schlicht, E. & Valcu, M. 2010. Artificial night lighting affects dawn song, extra-pair success, and lay date in song birds. Curr. Biol. 20: 17351739.
  • Kenkel, N.C., Hoskins, J.A. & Hoskins, W.D. 1989. Edge effects in the use of area polygons to study competition. Ecology 70: 272274.
  • Lewin-Koh, N.J. & Bivand, R. 2012. maptools: Tools for Reading and Handling Spatial Objects. R package version 0.8–14 Available at http://CRAN.R-project.org/package=maptools (accessed 20 February 2012).
  • McLeod, D.R.A., Whitfield, D.P., Fielding, A.H., Haworth, P.F. & McGrady, M.J. 2002. Predicting home range use by Golden Eagles Aquila chrysaetos in western Scotland. Avian Sci. 2: 117.
  • Morrell, L.J. & Kokko, H. 2005. Bridging the gap between mechanistic and adaptive explanations of territory formation. Behav. Ecol. Sociobiol. 57: 381390.
  • Pedersen, H.C. 1984. Territory size, mating status, and individual survival of males in a fluctuating population of Willow Ptarmigan. Ornis Scand. 15: 197203.
  • R Development Core Team. 2011. R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing. ISBN 3-900051-07-0, Available at http://www.R-project.org/ (accessed 30 January 2012).
  • Ripley, B. & Rasson, J. 1977. Finding the edge of a Poisson forest. J. Appl. Probab. 14: 483491.
  • Valcu, M. & Kempenaers, B. 2008. Causes and consequences of breeding dispersal and divorce in a Blue Tit, Cyanistes caeruleus, population. Anim. Behav. 75: 19491963.
  • Valcu, M. & Kempenaers, B. 2010. Is spatial autocorrelation an intrinsic property of territory size? Oecologia 162: 609615.
  • Wilkin, T.A., Garant, D., Gosler, A.G. & Sheldon, B.C. 2006. Density effects on life-history traits in a wild population of the great tit Parus major: analyses of long-term data with GIS technique. J. Anim. Ecol. 75: 604615.
  • Wortman-Wunder, E. 1997. Territory size in Lincoln's Sparrows (Melospiza lincolnii). Southwest. Nat. 42: 446453.

Supporting Information

  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Supporting Information
FilenameFormatSizeDescription
ibi12105-sup-0001-FigureS1.epsimage/eps902K 
ibi12105-sup-0002-FigureS2.epsimage/eps1563K 
ibi12105-sup-0003-FigS1-S2.docWord document649K

Figure S1. Visualization of mapped territories (grey) vs. Thiessen polygons (black).

Figure S2. Underestimation and overestimation of the Thiessen polygons regarding (a) territory area and (b) neighbour identity in relation to the distance of the focal point to the respective centroid.

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