## Introduction

There is consensus among ecologists that species-habitat relationships are integrating over several scales (Martinez, Serrano, & Zuberogoitia 2003; Bowyer & Kie 2006; Ma 2008). Different species may perceive the same habitat on different spatial scales in accordance, for example, with the differing activity or locomotory abilities of those species (Wiens 1989). Also, one species may perceive different habitats on different spatial scales, because the ‘ecological neighbourhood’ (Addicott *et al.* 1987) of this animal may differ between the requirements of daily foraging, of the annual life cycle or of the lifetime movements. Definitions of scale apply to space, time and levels of organization and these dimensions are commonly interrelated (Wu 2007). For example, the organizational hierarchy, i.e. species, populations and individuals, has been shown to interact with grain and extent (Turner 1989; Vaughan & Ormerod 2003) when modelling resource selection functions (Meyer & Thuiller 2006). Generally, empirical evidence for the dependence of species-habitat relationships on spatial scale is drawn from the observation that the correlation between habitat cover and species occurrence is strongest on a particular extent (Steffan-Dewenter *et al.* 2002; Gabriel, Thies, & Tscharntke 2005; Holland, Fahrig, & Cappuccino 2005; Kleijn & van Langevelde 2006). Typically, a humped correlation curve is obtained when regression results are plotted over a continuum of scales (Graf *et al.* 2005; Condeso & Meentemeyer 2007; Pinto & Keitt 2008; Schmidt *et al.* 2008). This has led to two fundamental beliefs in current species-habitat modelling: (a) that the scale of maximum correlation indicates the functional scale of habitat influence (Hirao *et al.* 2008; Holzschuh, Steffan-Dewenter, & Tscharntke 2008; Dallimer *et al.* 2010) and (b) that multi-scale analysis is required to understand how species respond to the multiple facets of their environment (Cushman & McGarigal 2002; Boyce 2006; Laliberte *et al.* 2009). Here, we follow the general concept of scale-dependent species-habitat relationships, and we illustrate that habitat influence across scale should indeed be hump-shaped, but we argue that equating the shape of correlation strength across scale with the shape of habitat influence across scale is misleading.

In the spatial dimension, scale-dependence of habitat influence arises from two mechanisms as depicted in Fig. 1. The first mechanism describes how the amount of habitat, which is available at a given distance, increases linearly (i.e. on the perimeter of a circle of radius *r*) with distance *r* from the focal point – assuming a constant proportion of the habitat in the landscape and a homogeneous quality and effectiveness of the habitat with respect to a particular species. Based on this first mechanism, the influence of the habitat on the occurrence of a species will therefore grow with increasing distance (Fig. 1a). Conversely, the second mechanism describes how habitat influence decreases with increasing distance from the focal point because of the degrading forces of dilution and dispersion, or owing to the increasing energy demand of organisms for movement over longer distances. This decay follows a certain distance kernel, for example a Gaussian function (Fig. 1b). The two mechanisms combined result in a *distance function* showing a humped shape of habitat influence over distance (Fig. 1c). These curves represent density functions of the influence of habitat precisely located at a given distance from the focal point, i.e. the influence of an infinitesimally thin circle around the focal point. The next panels illustrate how habitat influence sums up across the surface of a buffer area around the focal point, i.e. how the distance function integrates over the interval from radius zero to the radius of the buffer margin. Again, both mechanisms are effective: Habitat area increases quadratically (proportional to *r*^{2}) with increasing buffer radius (Fig. 1d), and distance decay implies that the accumulation of habitat influence saturates at large buffer radii (Fig. 1e). The two mechanisms combine into a *cumulative distance function* with a sigmoid saturation curve (Fig. 1f). All equations are listed in Table 1a.

(a) Distance effect assumed in the simulations | ||
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Available habitat | Distance decay | Distance function |

where Erf(x) is the integral of the Gaussian distribution, given by *b*= regression parameter,*c*_{1}–*c*_{3}= constants,*d*= scaling factor of distance,*r*= radius (distance).
| ||

cπ_{1}·2 | y(r) = c_{2}· | Y(r) = c_{3}·r· |

Cumulative habitat | Cumulative decay | Cumulative distance function |

c_{1}·πr^{2} | y(r) = (c_{2}·d·√π)/2·Erf(r/d) | Y(r) = (c_{3}·d^{2})/2·(1 – ) |

(b) Distance weightings used in the regressions | ||

Name | Distance kernel | Distance function |

Gaussian decay | y(r) = c_{2}· | Y(r) = b·r· |

Habitat area | y(r) = c_{2} | Y(r) = b·r |

Proximity index | y(r) = c_{2}·r^{−1} | Y = b |

Root index | y(r) = c_{2}·r^{−1} + c_{3}·r^{−1/2} | Y(r) = b_{1} + b_{2}·r^{1/2} |

Gaussian index | y(r) = c_{2}· | Y(r) = b·r· |

This cumulative distance function has an important interpretation. It shows the influence of increasingly larger areas on the focal point. The influence originating from a small buffer will be small, because the area is small. With increasing buffer size, habitat influence will grow, until it levels off at buffers large enough to capture the relevant part of the habitat. This concept applies to the spatial scale-dependence of any ecological mechanism, be it effective at a resolution of a few millimetres or several kilometres. Clearly, the extent, i.e. the buffer size where saturation of habitat influence occurs, varies depending on the organism and hierarchy of the selected habitat under observation.

Taking the concept of a hump-shaped distance function of habitat influence as a basis, it is obvious why a conventional regression cannot establish a good match between habitat and habitat influence at all scales. By ‘conventional’ regression, we imply a regression model where habitat is not explicitly weighted according to its distance from the focal point. In such a regression, habitat area within a buffer of given radius is specified as predictor variable, and habitat influence given as the numerical response, i.e. occurrence probability or abundance of an animal species, is specified as response variable. This applies irrespective of whether habitat area is expressed in absolute (m², ha) or relative (density, per cent cover) terms because linear transformation does not affect explained variance in a regression. Therefore, if a conventional regression is performed over increasing buffer sizes, the predictor variable of habitat influence is modified by the first mechanism of increasing habitat area, but is not corrected for the second mechanism, namely the decay of habitat influence by distance. Thus, such a modification of the predictor variable implies a very unrealistic distance function of habitat influence (identical to Fig. 1a). Using a small buffer size, this unrealistic distance function of the predictor variable might match the true distance function (from Fig. 1c) quite well. However, the larger part of the habitat’s influence on the focal point is not represented in the predictor variable and cannot be explained in a regression using this predictor. It therefore seems obvious that regressions conducted on too small extents will generally exhibit low explained variance even if the resolution of the data is appropriate to the scale at which the ecological mechanism operates.

If the predictor variable is measured within larger buffers, the proportion of habitat influence inside the buffer increases gradually, and the correlation between habitat area and total habitat influence possibly improves. However, with increasing buffer radius, a mismatch between the curve of the predictor variable and the curve of the dependent variable within the buffer arises. Because all habitat is equally weighted within the buffer, central parts of the habitat are underrated and peripheral parts of the habitat are overrated by the predictor variable, thus compromising the correlation with the dependent variable.

Thus, there is a conflict inherent to the conventional regression technique, between either choosing a small buffer radius and thereby neglecting the influence of distant habitat, or choosing a large buffer radius and thereby overestimating the influence of distant habitat. Our main hypothesis is that introducing a realistic distance weighting into the regression will resolve this conflict and will lead to sigmoid correlation curves across scale. We suspect that the humped shape of a conventional correlation curve across scale (Steffan-Dewenter *et al.* 2002; Holland, Fahrig, & Cappuccino 2005) results erroneously from a lack of information at small extents and model mismatch at large extents and that humped correlation curves are improper means to elucidate the characteristic scale.

To examine our hypothesis, we devised four levels of distance weighting with increasing capability to match habitat influence at large radii. Using a landscape simulation model in which we had control over the assumed distance function of habitat influence, we compared regressions with the different distance weightings for the scale-dependence of correlation strength. We then applied regression models with the same distance weightings to a data set of Eurasian lynx (*Lynx lynx*) covering Central Europe to examine whether theoretical findings are of relevance in empirical species-habitat relationships. We conclude by discussing a linear and a nonlinear approach for integrating distance weighting into regression models.