^{1}

^{*}

^{1}

^{2}

^{3}

The understanding of habitat demands of single species needs an explicit habitat element approach which includes both the effect of intensity of the habitat element on the species population and the spatial effect of that habitat element in a given matrix (

Carabid beetles are often used as indicator species to assess the effects of environmental changes on insect populations arising from human impact (

The link of species to defined environmental conditions is at the heart of the habitat concept (

Species are linked to essential parts of a habitat, which may include both abiotic as well as biotic components (

For both aforementioned interspecific spatial relationships (

Spatial point patterns of any kind of specified subjects (

Up till now, the analysis of spatial association by point pattern analysis in ecology mostly has been applied to plants (

In this paper we describe (i) an approach to transform random field data of mobile organisms into point patterns, and (ii) present a method to overcome the problem caused by non-rectangular windows in the toroidal shift test. We replace the random field values by point patterns in the cells and (re)construct larger point patterns in a rectangle containing the original window. The result is a bivariate point pattern in the polygon and a univariate point pattern in the surrounding rectangle. This permits application of the toroidal shift test. We applied this approach to the case of carabid beetles (random field data) and oak trees (point pattern). The aim of this study was to test the hypothesis of independence of oak and beetle distribution to gain further insight to the beetle’s habitat demands. The null hypotheses was the independence of oak and beetle distribution. By this, the study is a case study and the system of beetles and oaks in pine stands serves as a model.

The females of

The study was established in a homogeneous Scots pine (

The sessile oak trees (hereafter referred to as oaks) in the stand were considered important habitat elements as they were the only deciduous trees in the otherwise pure coniferous stand. The oaks were stem-mapped and their dbh (diameter at breast height) measured. As the stand was predominantly a pure stand of Scots pine, this species, though considered ecologically important as well, was regarded as noise in this study. The shape of the stand was irregular (

To collect the carabid beetles (

To combine the spatial distribution of carabid beetles with their species-specific properties, we recorded the gender (male

For analyzing the relationship between beetle density and oak positions we used the approach described in

After the transformation of the trap data to point data, we had two point patterns in the same window, _{12}

If the cross-pair correlation function obviously deviates from the line of “complete spatial randomness”, we should test the hypothesis of independence or “no association”.

If the hypothesis of independence is rejected, a “proof” of spatial correlation between the two point patterns and, consequently, between the random field and point process is obtained and the function _{12}_{12} function with _{12} functions belonging to newly-simulated bivariate patterns. While in the reference publications the test is based on the _{12} function, it is here carried out using the _{12} function. Using the _{12} function does not change the principle of the test.

The newly simulated patterns are obtained as follows: the 1-points are fixed, while the new 2-points (

In this way, a number _{12} functions determined. The extremes of these associated _{12} functions lead to envelopes that include the strip of deviation from the empirical _{12} function which are acceptable if the independence hypothesis is true. If the empirical _{12} function extends beyond this strip and crosses one of the envelopes, the independence hypothesis is rejected. Clearly, the number of simulations (

All computations were performed using the software package “spatstat” in R version 3.0.1 (

For the toroidal shift null model a rectangular window is necessary. However, our window (^{c}.

In the toroidal shift test only the pattern of 2-points (

The parameters of the algorithm are described in

The aim of the reconstruction algorithm was to obtain random points in ^{c} that followed statistical rules similar to those of the points in

The number _{2} of 2-points in ^{c} is fixed as (

and rounded to integers, where _{2} is the density of ^{c}) the area of ^{c}. The algorithm starts with _{2} points uniformly and independently distributed in ^{c}. It uses a “deviation-energy” which is minimized during the algorithm. The energy of the 2-point pattern _{k} in the

where ^{(k)}(_{k} and _{s}(_{s}^{(k)}(_{k}, and (

and (

The integrals in

In the case of reconstruction we carried out the toroidal shift test as follows. The 1-points in _{12}(

_{2} = 0.000885 oaks m^{-2}.

The numbers of female beetles caught in the traps is displayed in _{1} = 0.015745 beetles m^{-2}.

The true beetle density in the forest is of course higher than the empirical density of 45 beetles per 2.8263 hectare, since not all beetle females were caught. We estimated the true beetle density using information from the literature. To this purpose we used data from ^{2}). However, the results of the toroidal shift test were similar when 5 or 20 were chosen as scaling factors.

A classical, non-spatial, statistical approach yielded a coefficient of correlation of 0.1397 (Spearman’s rank correlation) between the number of oak trees in the 15 × 15 m grid squares and the number of beetles in the corresponding traps. This low value indicated no significant correlation. However, this result was questionable since the spatial aspect was completely ignored. The assignment of oaks to particular grid squares may be too rigorous, and effects between neighboring grid squares should be taken into account.

An example of a simulated beetle distribution map is given in ^{c}. The rectangle area was 40253.6 m².

The partial pair correlation function _{12}

_{12}_{12}

In this study we found a positive association between the two point patterns of female

The results obtained in this study may support forest management activities aimed at increasing the population density of beetles by increasing the oak tree proportion and setting a more regular spatial distribution of single admixed oak trees. Likewise,

Our findings may also help a better understanding of the reasons underlying the attractiveness of oaks to female beetles. Indeed, previous research carried out in the same Scots pine forest reported that the leaf litter of oak trees does not extend beyond a distance of 15 meters from the oak stem (

Our study was based on three simulation steps: (i) simulation in the transition from random-field data to point pattern data; (ii) simulation in the reconstruction algorithm; and (iii) simulation in the toroidal shift test. The first two steps could find wide application in ecology. Indeed, ecologists are often confronted with random-field data in the context of mobile objects (as in our study) or coverage data (as in

In many applications in ecology, the windows of observation are not rectangular, since there are constrains due to natural edges that define such windows. The restriction of statistical analyses to rectangles inside the original irregular window often results in a considerable loss of information. While the estimation of summary characteristics can conveniently be carried out with point-process software, rectangular windows are essential for the important toroidal shift test. The reconstruction approach presented here may overcome this problem. However, the approach used in this study (

Some prerequisites are necessary in the application of point pattern statistics to data from traps such as in this investigation, regarding (i) trap density, (ii) trap spacing and (iii) trap-to-tree ratio. The same issues are also of concern in the “inverse modeling”

(i) A total trap number of 123 may be regarded as high. However, 91 of these traps did not catch any

(ii) The spacing of the traps should be regular to some extent, as the simulation was built on squares of equal size for each trap; different quadrat sizes would need some justification. Likewise, for inverse modeling of wind-dispersed seeds,

(iii) Finally, the number of traps in relation to the number of trees should be as high as to obtain an appropriate spatial resolution of beetle positions in relation to different oaks. Accuracy of the estimates will suffer severely when this ratio shifts due to much higher oak numbers as the allocation of beetles to particular oaks will be more difficult.

A habitat element considered positive, relevant or essential to a species population will cause a significantly increasing effect on the species intensity as compared with a habitat where this element is missing. The species-intensity effect of the habitat element should be spatially correlated to the element, either to be observed in the element itself, as in sunspots in the forest and the speckled wood butterfly (

Additionally, we may be interested in the gradient of the effect beginning at the habitat element and decreasing or increasing with distance (

Forest science increasingly attempts to develop strategies which integrate the conservation of biodiversity in forest management concepts (

In general, the available information on species’ demands in terms of amount and pattern of habitat elements is often insufficient. The method proposed here enables both to establish an intensity effect of a habitat element on a mobile species population and to describe a gradient of this effect into the surrounding matrix of Scots pine forest.

We thank the Michael-Jahr-Foundation, which financially supported the study through a scholarship to Alexandra Wehnert. The data set used was part of a more complex study concerned with the single-tree approach to forest ecology research. We thank Thorsten Wiegand for the idea to only reconstruct the pattern of 2-points; in an earlier version of the paper we had reconstructed the bivariate pattern. We finally thank two anonymous reviewers whose comments helped to substantially improve the manuscript.

Map showing the positions of 25 sessile oaks (blue circles) and 123 traps (red triangles) within the Scots pine stand. The width of the circles is proportional to the diameter of the oaks, the largest and smallest diameter being 80.2 and 14.2 cm, respectively. The black polygon represents the window of observation, with an area of 28263.53 m². The scale is given in meters.

Map showing the positions of 25 sessile oaks (blue circles) and 32 traps (red triangles) where at least one female

Example of simulated beetle positions inside the window (homogeneous Poisson process, scaling factor = 10, compare to

Results of bivariate (_{12}

Result of the toroidal shift test for one of the simulated beetle patterns (compare to _{12}

Stand characteristics of the Scots pine forest where the investigation took place. The entire stand area is 28263.53 m². The mean annual precipitation amounts to 546 mm, elevation is 64 m a.s.l., and mean annual temperature is 9.4 °C.

Parameter | Scots pine | Sessile oak |
---|---|---|

age [years] | 61 | 70-150 |

number | 2455 | 25 |

mean dbh [cm] | 24.2 | 27.3 |

mean height [m] | 19.5 | 14.8 |

mean height of lowest living branch [m] | 13.0 | 3.2 |

mean crown length [m] | 6.5 | 11.6 |

mean crown radius [m] | 1.3 | 3.8 |