We selected the eight best estimators from the 24 evaluated by 1 to test using seventeen fully enumerated field data sets. In the discussion that follows the closest individual (CI) is the individual that is closest to the random sample point and this individual can have a nearest neighbor (NN). The closest individual to the NN is referred to as the second nearest neighbor (2NN). One or more of the following distances need to be measured depending on the estimator: from the i^th random point to the first, second or third closest individual; from the closest individual to the first or second nearest neighbor and; the distance from a transect baseline of width w, to the g^th event such that all g events are within the transect. Estimators used in this study (Table 1) comprise four general types: basic distance; Kendall-Moran; ordered distance and angle order; and variable area transect. The quadrat method was done to check that the simulation routines were working correctly (see Additional file 1) and not as an explicit test of this method as this has been done elsewhere 13. No attempt was made to optimize the dimensions of the quadrat or the VAT. The latter has been dealt with explicitly elsewhere 4.

Basic distance estimators

Kendall-Moran estimators

Ordered distance and angle order methods

The

variable area transect method

Eight plotless density estimators were examined in the present study using 5000 Monte Carlo simulations, Table 1. The simulation program was written in Fortran 77, and each simulation was a specific combination of a spatial data set and sample size (10, 25, 50 and 100 samples per simulation were undertaken). The uniform random number generator, UNIF 11, was used to locate sampling points and, where required, the VNORM routine 11 was used to convert uniform random numbers to normal random numbers to generate the synthetic data sets used for comparison with natural data sets (see below). The input for each simulation included: the name of the data file containing the location of all events as X-Y coordinates on a Cartesian plane; selects the number of samples to be taken; the sizes of the VAT width and quadrat; an output file specification and; the number of simulations to be performed. These inputs were provided within in a batch-processing environment and could be left to run unattended. The output file, one for each data set comprised the estimated density, relative bias and relative root mean square error for each estimator.

Seventeen data sets (Table 2) were obtained from unpublished studies by the authors and colleagues that included animal damage to rice and corn, bird nest locations, active rodent burrows and distribution of plants. Densities ranged from 0.06 m^-2 (bee-eater nest sites) to 19.3 m^-2 (damaged sugar). A boundary strip of 10% of the length and width of the extent of the population of points was used to remove the bias associated with sampling close to the edge of the study area.

For ground or cliff nesting birds the density of nest sites provide important information on the number of breeding females or pairs. Two data sets were used with densities of 0.06 (bee eater) and 3.2 m^-2 (Alaskan waterfowl nests).

Burrowing species such as gophers and rabbits can be monitored through the presence of active burrows. Two data sets of a population of pocket gophers measured in two successive years were used to demonstrate the application of PDE as a suitable method for monitoring populations.

The use of PDEs for monitoring damage to crops was done using corn and rice in the Philippines, and sugar cane in Hawaii.

The remaining data set is from a coastal sand island, north of Brisbane, Australia. Grass trees, Xanthorrhoea sp., grow in heath communities inland from the foredunes. Unlike the crop data sets these are naturally occurring communities.

Five data sets whose spatial characteristics were predetermined were also included for comparison. The artificial data sets (where n is the number of individuals, λ is the density m^-2) had distributions that were Poisson (n = 100, λ = 1), uniform – regular lattice (n = 100, λ = 1), hexagonal – regular triangular (n = 100, λ = 0.9), first-order clumped (n = 100, λ = 1.1, number of offspring per parent (nop) = 10, clump radius (cr) = 0.5 m) and second order clumped (n = 100, λ = 2.1, nop = 10, cr = 0.5 m). The Poisson or random pattern was created by generating the required number of random coordinates within the designated area. The uniform data set was generated by first dividing the area into a grid of rectangles, the same number as the population size. One population member was randomly located within each grid cell. The hexagonal pattern was generated so that population members were located at the vertices of a lattice of equilateral triangles. For the clumped data sets, the required number of clump centers was randomly created within the designated area. In addition to the clump center point, offspring for the clumps were located within a designated radius from the parent. These offspring were located within the clump about the parent using coordinates randomly generated using a standard bivariate normal distribution. For the second order clumping, the individuals in the clump are used for parent points. The two individuals of the sub-clumps include the parent plus offspring points, which are randomly generated from the standard bivariate normal distribution. The radius for the sub-clump is limited to half that for the clump. The second order clumping approximates the situation that can occur with rodent damage in field crops.

The relative root mean square error (RRMSE) was used as the basis of comparisons between the different PDEs 112, where I is the number of simulations (5000), D_est is the estimated density and λ is the true density in the population, such that:

R R M S E = ∑ ( D e s t − λ ) 2 / λ 2 I MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOuaiLaemOuaiLaemyta0Kaem4uamLaemyrauKaeyypa0ZaaOaaaKqbagaadaWcaaqaamaaqaeabaGaeiikaGIaemiraq0aaSbaaeaacqWGLbqzcqWGZbWCcqWG0baDaeqaaiabgkHiTiabeU7aSjabcMcaPmaaCaaabeqaaiabikdaYaaacqGGVaWlcqaH7oaBdaahaaqabeaacqaIYaGmaaaabeqabiabggHiLdaabaGaemysaKeaaaWcbeaaaaa@4532@

In addition, relative bias (RBIAS) shows the bias relative to the true density and the direction of that bias such that:

R B I A S = ( ∑ D e s t / I ) − λ λ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOuaiLaemOqaiKaemysaKKaemyqaeKaem4uamLaeyypa0tcfa4aaSaaaeaacqGGOaakdaaeabqaaiabdseaenaaBaaabaGaemyzauMaem4CamNaemiDaqhabeaacqGGVaWlcqWGjbqscqGGPaqkcqGHsislcqaH7oaBaeqabeGaeyyeIuoaaeaacqaH7oaBaaaaaa@42BF@

The R index, 13, was calculated for all data sets (Table 3) including examples of simulated distributions such that:

R O = ∑ i = 1 n r i n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOuai1aaSbaaSqaaiabd+eapbqabaGccqGH9aqpjuaGdaWcaaqaamaaqahabaGaemOCai3aaSbaaeaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6gaUbGaeyyeIuoaaeaacqWGUbGBaaaaaa@3B7F@

where R_O is the average observed nearest neighbor distance, r_i is the nearest neighbor distance to the i^th sample point and n is number of nearest neighbor distances measured;

R E = 1 2 λ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOuai1aaSbaaSqaaiabdweafbqabaGccqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabikdaYmaakaaabaGaeq4UdWgabeaaaaaaaa@33E1@

where R_E is the expected nearest neighbor distance for a random pattern of events;

R = R O R E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeOuaiLaeyypa0tcfa4aaSaaaeaacqWGsbGudaWgaaqaaiabd+eapbqabaaabaGaemOuai1aaSbaaeaacqWGfbqraeqaaaaaaaa@33C8@

R was calculated for the complete data set less a 10% buffer. When the pattern is entirely random R = 1, if the events are uniform then R > 1 (R = 2.149 for a perfect hexagonal uniform distribution) and conversely when the population of events is clumped R < 1 (R approaches 0 for maximally clumped distributions). The z test statistic was calculated that measured the difference between the observed and expected values of R, i.e. it considers a null hypothesis that the spatial distribution is random.

z = | R O − R E | s e MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOEaONaeyypa0tcfa4aaSaaaeaadaabdaqaaiabdkfasnaaBaaabaGaem4ta8eabeaacqGHsislcqWGsbGudaWgaaqaaiabdweafbqabaaacaGLhWUaayjcSdaabaGaem4CamNaemyzaugaaaaa@3AEB@

where se is the standard error of R_E

s e = 0.26136 n λ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4CamNaemyzauMaeyypa0tcfa4aaSaaaeaacqaIWaamcqGGUaGlcqaIYaGmcqaI2aGncqaIXaqmcqaIZaWmcqaI2aGnaeaadaGcaaqaaiabd6gaUjabeU7aSbqabaaaaaaa@3A4E@

A Spearman (rank) correlation coefficient was calculated between the log of (λ) and the log of D_est for AO3Q, BDAV3, KM2P and VAT across all natural data sets.