Maintenance Cost, Toppling Risk and Size of Trees in a Self-thinning Stand

Keywords: allometric, allometry, biomass, self-thinning law, wind drag

Wind routinely topples trees during storms, and the likelihood that a tree is toppled depends critically on its allometry. Yet none of the existing theories to explain tree allometry consider wind drag on tree canopies. Since leaf area index in crowded, selfthinning stands is independent of stand density, the drag force per unit land can also be assumed to be independent of stand density, with only canopy height influencing the total toppling moment. Tree stem dimensions and the self-thinning biomass can then be computed by further assuming that the risk of toppling over and stem maintenance per unit land area are independent of stand density, and that stem maintenance cost is a linear function of stem surface area and sapwood volume. These assumptions provide a novel way to understand tree allometry and lead to a self-thinning line relating tree biomass and stand density with a power between -3/2 and -2/3 depending on the ratio of maintenance of sapwood and stem surface.

Introduction

Interest in tree allometry has increased in recent years as forests have been identified as a potential sink for atmospheric carbon. Studies on the proportions of tree organs indicate that much of the biomass in a forest is in stems and that this proportion increases with forest age. Another generalization states that when there are more individuals per unit land area (e.g., per hectare), the total biomass of the individuals decreases [START_REF] Yoda | Self-thinning in overcrowded pure stands under cultivated and natural conditions[END_REF]. This relationship, called the "self-thinning law", is so consistent that it has been called "the only generalization worthy of the name of a law in plant ecology" [START_REF] Hutchings | ECOLOGYS LAW IN SEARCH OF A THEORY[END_REF]. Numerous empirical papers have shown that for even-aged monocultures, the maximum biomass of an individual is a power function of stand density (individuals per unit land area), with a slope of approximately -3/2 when both axes are plotted on logarithmic scales [START_REF] Osawa | ALLOMETRIC THEORY EXPLAINS SELF-THINNING RELATIONSHIPS OF MOUNTAIN BEECH AND RED PINE[END_REF]. Mathematically, this can be described as

2 3 v D w t
, Equation 1where w t is the average total biomass (all symbols listed in Table 1) of a plant and D is stand density and the sign in between stands for proportionality (with equal sign a constant would need to be added). Many theories have been proposed to explain the selfthinning law (e.g. [START_REF] Adler | A model of self-thinning through local competition[END_REF][START_REF] Givnish | BIOMECHANICAL CONSTRAINTS ON SELF-THINNING IN PLANT-POPULATIONS[END_REF]), but none has gained wide support. [START_REF] Schiel | EFFECTS OF DENSITY ON MONOSPECIFIC STANDS OF MARINE-ALGAE[END_REF] showed that the self-thinning law does not apply to marine algae, where the size of individuals can even increase with increasing density. This

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4 suggests that self-thinning might be related to structural requirements for staying erect on terrestrial environments. Seven years later [START_REF] Weller | A REEVALUATION OF THE -3/2 POWER RULE OF PLANT SELF-THINNING[END_REF] demonstrated that many datasets on terrestrial vascular plants published earlier to support the self-thinning law in fact lead to exponents that diverge from -3/2 (in Equation 1) when the correct statistical approach is applied. His paper [START_REF] Weller | A REEVALUATION OF THE -3/2 POWER RULE OF PLANT SELF-THINNING[END_REF] lead to a decrease in papers on selfthinning even though he concluded that the variable exponent "may provide a valuable measure of the ecological differences among species and stands, and a powerful stimulus for further research".

The objective of this paper is to present a model to explain self-thinning law from a new perspective, which will probably be the first attempt to explain the self-thinning law based on wind friction in canopies. Because of the novelty of the approach, the presented model is simple in order to introduce the new assumptions and implications, which will hopefully encourage more complex modelling based on the same principles. In the next three sections I present ideas necessary for understanding the thinking behind the assumptions of this new model.

Stagnation and self-thinning

A central element of forest management is the use of thinning to manipulate stand structure and maximise growth [START_REF] Davis | Forest Management[END_REF]. In a sparse stand, much light penetrates the canopy without being used by the trees and therefore the gross primary

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production of trees is low. On the other hand, growth in an overly dense stand suffers from high maintenance cost relative to the gross primary production of trees. The maintenance cost is composed of both maintenance respiration and replacement of leaves and fine roots. In extreme cases when the stand is dense and nearly all energy (or carbohydrates) is allocated into maintenance, growth ceases and the stand reaches "stagnative equilibrium". Theoretically it is obvious that with a given stand density and closed canopy, gross primary production is relatively constant and that larger stems require more energy for maintenance. Therefore, growth ceases at a certain stem size if individuals are identical (or all individuals die).

In practice, stagnative equilibrium has been documented in surprisingly few studies.

Ecological papers typically focus on stands with individuals of varying size, age and normally even species and therefore certain individuals start dying (self-thinning occurs)

when the most competitive individuals are still growing significantly (e.g. [START_REF] Osawa | ALLOMETRIC THEORY EXPLAINS SELF-THINNING RELATIONSHIPS OF MOUNTAIN BEECH AND RED PINE[END_REF]). Forestry literature describes how stand density influences stem growth of often nearly identical individuals but unfortunately the interest has been mainly in wood production and therefore in stand densities leading to high or moderate stem growth and not those close to stagnative equilibrium [START_REF] Davis | Forest Management[END_REF]. Significant exceptions are the well studied very dense Pinus contorta stand that have been naturally regenerated after fire from serotinous cones and which grow very slowly and are close to stagnative equilibrium and can be released from it by thinning (e.g. [START_REF] Farnden | Severely repressed lodgepole pine responds to thinning and fertilization: 19-year results[END_REF]). At the individual level many understory trees are at a similar situation that nearly all energy produced is required for maintenance and growth is extremely slow.

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6 Although at stand level stagnative equilibrium is rare in nature it is ecologically and physiologically nearly identical to self-thinning if the variability between individuals is small and is the simplest basis to understand tree stem dimensions and allometry in general.

What determines the stem dimensions of trees?

Several theories have been put forward to explain tree stem dimensions. The "pipe model" assumes that the quantity of leaves above a given level is proportional with stem cross-sectional area at that level [START_REF] Shinozaki | A quantitative analysis on plant form -The pipe model theory. I -Basic analyses[END_REF], but does not explain stem height, heart wood and inter-specific variability. The three main approaches to understand the height-radius ratio are: geometric similitude, elastic buckling and stress similarity [START_REF] Niklas | Plant Biomechanics -An Engineering Approach to Plant Form and Function[END_REF]. The theory of geometric similitude simply assumes that ratios of dimensions remain the same when trees grow. However, geometric similitude is not based on understanding of evolutionary drivers of allometry, i.e. survival or reproduction.

It is evident that extra height is beneficial in exposing trees to more light and that the stem radius should be as small as possible in order not to waste energy. The theories of elastic buckling and stress similarity are the main options to explain how small is possible. McMahon (1973) showed that if the fresh biomass of the above-ground parts of a plant are above the limit of "elastic buckling" the stem will bend irreversibly. This theory is based on thinking that crucial in plant allometry is the avoidance of elastic McMahon, 1973). This mathematically simple relation has been used in dozens of journal articles (e.g. correctly [START_REF] King | The role of wood density and stem support costs in the growth and mortality of tropical trees[END_REF] and incorrectly [START_REF] West | A general model for the structure and allometry of plant vascular systems[END_REF] as they do not take into account the influence of subsequent branching). However, empirical data has shown that most plants are far from the maximal height leading to elastic buckling. For example Niklas (1994a) measured plants belonging to 111 species and concluded that the "safety factor", which is the ratio of the height leading to elastic buckling and actual height was very variable but on average approximately 4. Because of non-linear relationships, for a tree with a safety factor of 4, gravity causes just 1.6 % of the necessary force needed for elastic buckling. However, it is probable that the safety factor is lower in general in self-thinning stand [START_REF] King | TREE DIMENSIONS -MAXIMIZING THE RATE OF HEIGHT GROWTH IN DENSE STANDS[END_REF].

The need for a safety factor has been correctly explained to be necessary as in addition to gravity acting on the fresh biomass of the plants temporary forces such as wind and snow load could potentially cause toppling over and death. However, the approach to understand tree allometry based on elastic buckling is misleading as the additional forces required to cause toppling are so weakly linked and important relative to gravity acting on tree. This causes the wide range of safety factors found in trees. For example swaying in the wind has been shown to decrease the height-radius ratio [START_REF] Coutand | Mechanical stimuli regulate the allocation of biomass in trees: Demonstration with young Prunus avium trees[END_REF] and therefore increasing the safety factor. In forests in snow free climates wind is basically the only significant force toppling over trees in addition to their own fresh biomass (with the main exception of trees pulled down by other toppling over trees). I therefore argue

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8 that in simple modelling of mature trees when only one toppling over force can be taken into account it is best to focus on wind and the theory of stress similarity that focuses on breaking strengths of stems [START_REF] Niklas | Plant Biomechanics -An Engineering Approach to Plant Form and Function[END_REF] , as I do in this paper. Plants such as shrubs which can be bent down to the ground without breaking need another approach.

The drag caused by wind friction is difficult to model as bending of the branches and leaves and variable wind speeds for each leaf complicate the phenomenon. The moments resisting uprooting are also very difficult to model as they depend on both the roots and the soil. However, beam theory in basic engineering science suggests that stems resist breaking (bending stress of a beam) simply with a moment (m)

3 r m v , Equation 2
where r is stem radius inside the bark (in this equation at any height but in subsequent equations at any given height relative to stem height), when the wood is homogenous and shear forces are not taken into account [START_REF] Niklas | Plant Biomechanics -An Engineering Approach to Plant Form and Function[END_REF]. The third power for r comes from the fact that increasing radius not only increases the number of vertical fibres responsible for the strength of the wood, but also increases their average distance from the neutral axis (where neither elongation or shortening occurs when the stem is bending), thus increasing the average moment of the fibres resisting bending [START_REF] Larjavaara | Rethinking the value of wood density[END_REF]. The moment (m) is equal to the product of force and length of the lever arm (e.g. height of a crown). This theory of stress similarity has been successfully used to

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9 explain stem tapering [START_REF] Dean | An evaluation of the uniform stress hypothesis based on stem geometry in selected North American conifers[END_REF], trunk and branch wood density [START_REF] Anten | The Role of Wood Mass Density and Mechanical Constraints in the Economy of Tree Architecture[END_REF] and life-history variation [START_REF] Sterck | Mechanical branch constraints contribute to life-history variation across tree species in a Bolivian forest[END_REF].

A strong stem is useless for trees that uproot easily. Therefore evolutionary pressure has strengthened the root system in allometric balance to resist approximately the same moment as the stem in conditions in which the root system would otherwise be weaker (shallow, moist or nutrient-rich soils). This is supported by studies showing that both uprooting and stem breakage occur in many species in a given stand [START_REF] Niklas | Plant Biomechanics -An Engineering Approach to Plant Form and Function[END_REF][START_REF] Peltola | Mechanical stability of Scots pine, Norway spruce and birch: an analysis of tree-pulling experiments in Finland[END_REF][START_REF] Putz | UPROOTING AND SNAPPING OF TREES -STRUCTURAL DETERMINANTS AND ECOLOGICAL CONSEQUENCES[END_REF]. I therefore argue that, Equation 2 can be used as the basis for theories on tree allometry as is demonstrated in section "Structure of the new model".

How to model wind drag on the canopy?

Equation 2 has probably never been used in modelling height-radius ratio, presumably because the drag force in canopies is difficult to model at the tree level (Coutts and Grace, 1995) compared to elastic buckling. The self-thinning law has been developed to describe density-size relations in a monocultural even-aged stand. Independent of the stand density, leaves need a certain level of light for a positive energy balance. Therefore the leaf area index (total leaf area per unit area) is independent of the stand density once the stand has recovered from disturbances as in the case of self-thinning stands [START_REF] Osawa | ALLOMETRIC THEORY EXPLAINS SELF-THINNING RELATIONSHIPS OF MOUNTAIN BEECH AND RED PINE[END_REF]. Therefore where a is the leaf area of one tree and D is stand density and "const" refers to a constant.

This implies that the drag force caused by the wind on leaves per unit land area is also independent of stand density assuming that the average wind force on an individual leaf is independent of stand density. The moment per unit area caused by this force can then be simply computed from the height of the leaves.

Structure of the new model

I now focus on even-aged stands at the self-thinning limit and in stagnative equilibrium to understand the self-thinning law better. I assume that wood density, structure and size of individual roots and branches as well as the total number of roots and branches per unit land area are independent of the stand density. I also assume that the leaf area index is independent of stand density. As growth is very slow, its variation depending on stand density is insignificant and therefore the energy usable for the maintenance per unit land area is also independent of stand density (same assumption justified if growth is significant but the same fraction of energy is always allocated in growth). As the energetic maintenance cost of leaves, branches and roots per unit land area are independent of stand density the energy available for maintenance of stems per unit land area also needs to be independent of stand density at the self-thinning limit. In addition, I

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11 assume that gravity does not act on bending trees (see section "What determines the stem dimensions of trees?").

Wood provides support and transports sap to the canopy. As dead heartwood provides support but does not require maintenance it is justifiable to assume that the sap flow function of the wood causes the total wood maintenance cost and that this cost is proportional with the product of the distance and amount of sap that needs to be transported. By assuming that the height of branches relative to height of the tree (h) is independent of the stand density in fully stocked stands and that the leaf area of one tree (a) is proportional with the amount of sap transported the wood maintenance cost (e w ) is ah e w v , Equation 4based on the function of tree stems. Alternatively, based on the tree structure, the same relation can be derived assuming the total wood maintenance (e w ) is proportional with sapwood volume and that the cross-sectional area of sapwood is proportional with the leaf area above it as in the pipe model [START_REF] Shinozaki | A quantitative analysis on plant form -The pipe model theory. I -Basic analyses[END_REF].

Empirical studies indicate that the energetic maintenance cost of inner bark is high [START_REF] Pruyn | Respiratory potential in sapwood of old versus young ponderosa pine trees in the Pacific Northwest[END_REF] and that stem surface area better explains total stem maintenance cost than sapwood biomass or volume [START_REF] Bosc | Variability of stem and branch maintenance respiration in a Pinus pinaster tree[END_REF]. This can be caused by both the respiration of the inner bark and increased wood respiration due to closeness of stem surface. The importance of stem surface area in determining stem maintenance cost is not

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12 biochemically well understood but could be linked to protection against pathogens which often enter through the stem surface. It is clear that sapwood of a living tree is actively protected against pathogens as sapwood decays less likely in the living tree but more likely in a dead tree than heart wood (David A. King, unpublished). In the new model the maintenance cost of stem surface (e s ) is proportional with the surface area of the stem, which can be calculated assuming that the stem taper is independent of the stand density as rh e s v . Equation 5Combining Equations 4 and 5 and multiplying by the number of individuals per unit area (D) or stand density leads to the total stem maintenance cost per unit land area (E)

krh jah D E
, Equation 6where j and k are parameters dependent on the ratio of maintenance cost of the wood and of the stem surface. These parameters need to be added as the relatively importance of these two costs is unknown and probably climate and species specific. According to the assumptions E and the product of D and a are independent of stand density (Equation 3) and can be replaced by a constant. Taking this into account when solving h, Equation 6converts to kDr j h const const . Equation 7A c c e p t e d m a n u s c r i p t 13 Based on another assumptions (Equation 2), the height of trees leading to breaking at a given wind speed is a r h 3 v , Equation 8as the moment caused by the drag of wind on the base of the stem is proportional to both the height (h) and leaf area (a). Evolutionary pressure forces trees to maximize their height (h), within the constraints of not taking excessive risks of toppling over (Equation 8) and not exceeding the highest possible maintenance cost (Equation 7) and have dimensions close to the point corresponding to the highest possible value of height (h).

Because Equation 8 is rising (when h is shown in function of r for a given D and therefore a) and Equation 7is descending (except when k is zero) the intersection of these two functions reveals the dimensions of tree stems (see Fig. 1). The intersection can be solved by combining Equations 7 and 8 as shown in the appendices.

Allometry and self-thinning law based on the new model

In the theoretical situation when the value parameter j of Equations 6 and 7 is zero and sapwood is assumed to make no contribution to stem maintenance

v D w s

Equation 9

A c c e p t e d m a n u s c r i p t 14 where w s is biomass of the stem of a tree individual, as shown in Appendix A. This is equal to Equation 1 with the exception that Equation 9 is for stem biomass and Equation 1for total biomass (or total above ground biomass). In the reverse theoretical case when the parameter k is zero and sapwood causes all the maintenance of a stem In reality, since both stem surface and sapwood always cause maintenance, both extremities are unrealistic. For example [START_REF] Bosc | Variability of stem and branch maintenance respiration in a Pinus pinaster tree[END_REF] showed that for the Pinus pinaster that they studied stem surface maintenance cost is approximately half of the total stem maintenance cost.

The classic self-thinning law focuses on the total or above-ground biomass of an individual and not just stem biomass. The new model assumes that the biomass excluding stems per unit land area is independent of stand density and the slope of the self-thinning line excluding stems is therefore -1. The slopes of self-thinning lines for total tree biomass are intermediate between -1 and the self-thinning line for stem biomass, which depends on the values of j and k. The higher the proportion of stem biomass is, and the higher the proportion of stem surface maintenance of total stem maintenance, the steeper the slope of the self-thinning line is (except when sapwood causes a very large proportion of stem maintenance).

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Increasing complexity of the new model

The model described in the previous sections is simple and presented in this short article to encourage development of more realistic and complex models following the same principles. However, most of the dozens of possible adjustments making the model more realistic are species-specific.

Perhaps the most obvious areas in which to make the model more realistic are related to the structure of the trees. Most trees have a main stem as assumed in the new model but they also have varying branch lengths depending on tree size, which is not taken into account in the new model. Furthermore, the model assumes that the biomass and energetic cost per unit land area of roots is independent of stand density. [START_REF] Ryan | The hydraulic limitation hypothesis revisited[END_REF] review numerous studies indicating that the assumption of invariable gross primary productivity with invariable leaf area index and variable tree height is incorrect. Increasing height may decrease the efficiency of photosynthesis because of hydraulic limitations [START_REF] Ryan | The hydraulic limitation hypothesis revisited[END_REF]. This could be taken into account in the model.

Also the assumption of equal size of all tree individuals and stagnative equilibrium would not be needed if a game-theoretical approach including realistic ranges of tree radii, heights and growth rates could be chosen.

A c c e p t e d m a n u s c r i p t 16

The drag force created by wind acting on an individual leaf is central to the new model.

However, wind also acts on stem and branches, causing an additional drag and the average wind speeds acting on leaves might increase with size as the canopy roughness changes when leaves are arranged more like vertical clusters than a horizontal layer (Coutts and Grace, 1995). As explained earlier, in most falls of healthy trees the force of wind acting on the canopy is probably the most significant single factor. However, gravity acting on the fresh biomass of the tree is always after some bending an additional factor that could be included in the model.

Conclusions

Focusing on the allometry of an individual without paying enough attention to its biotic surroundings has been the dominant approach in developing theories on tree structure (Niklas, 1994b). However, understanding the allometry of trees in an overcrowded stand that is stagnating and is at the limit of self-thinning is the simplest and therefore in many ways the best approach to model tree structure. This enables the use of Equation 3for modelling the toppling moment caused by wind drag and together with assumptions on stem maintenance not only explains the height-radius ratio as some earlier models (McMahon, 1973) but explains both the height and radius relative to stand density.

Thanks to this it can be applied to self-thinning law and lead to realistic self-thinning line slopes ranging upwards from -3/2. Increasing stand density decreases biomass per unit land area (i.e. self-thinning law) as the trees in them need to be shorter for stability and More sophisticated species-specific models including some of the aspects presented in the previous section could be developed and tested with simple self-thinning data.

Alternatively ecophysiological data on maintenance or diameter, height and leaf area data for tree individuals in stagnating stands could be used to test some of the assumptions presented in this article. The model could be also applied to non-stagnating stands when growth or leaf area information is available or on estimating biomass based on remote sensing data on tree heights when data on stand density is not available. A 10 and B 10 are computed from a tenfold stand density compared to A 1 and B 1 .

The arrows show how a 90% decrease in stand density changes stem dimensions from the intersection between A 10 and B 10 to the intersection between A 1 and B 1 .

  , when only stem biomass (branches and leaves excluded) is taken into account the height leading to elastic buckling scales to stem radius to the power of 2/3 (

  Appendix B. These two theoretical extremities are shown graphically in Fig 1A and Fig 1B.

  a large stem surface area requiring maintenance. Numerous species and site-specific complicating factors influence the precise slope of the self-thinning line.

Figure caption Figure 1 .

 caption1 Figure caption

Fig

  Fig.1Ademonstrates the situation when stem surface and Fig.1Bwhen sapwood is assumed to cause all the maintenance. The functions B 10 and B 1 form an identical horizontal line in Fig.1B. Functions A 10 and A 1 are based on Equation8(where a is constant for a given function), functions B 10 and B 1 in Fig.1Aon Equation A.1 (where D is constant for a given function) and function B 10&1 in Fig. 1B on Equation B.1. The assumption that stem dimensions are determined by the intersection between functions A 10 or A 1 and B 10&1 to give the maximum possible stem height is irrelevant in the theoretical case of Fig. 1B, as equal heights but larger radii are equally possible, leading to greater strength.

Table 1 .

 1 Symbols in equations. 1

	Symbol Definition	Possible
			unit
	a	leaf area of a tree	m 2
	D	stand density	m -2
	const	constant (changes from one equation to	-
		another)
	E	stem maintenance cost per unit land area	W/m 2
	e s	stem surface maintenance cost of a tree	W
	e w	wood maintenance cost of a tree	W
	h	height of the tree	m
	j	parameter on maintenance of sapwood	-
	k	parameter on maintenance of stem surface	-
	m	moment that a stem of a tree can resist	Nm
	r	stem radius of a tree	m
	w s	stem biomass of a tree	kg
	w t	total biomass of a tree	kg
	2

Acknowledgements

I want to thank David A. King, Helene C. Muller-Landau, Annikki Mäkelä, Juho Pennanen and anonymous reviewers for valuable comments, my wife Katja Sidoroff for general support and HSBC Climate Partnership for funding.

A c c e p t e d m a n u s c r i p t

Appendix A When sapwood is assumed not to make contribution to stem maintenance and the parameter j is therefore zero Equation 7 simplifies to


Grit Greiser Oliver Keuling
email: oliver.keuling@tiho-hannover.de

Andreas Grauer

Egbert Strauss

Martina Bartel-Steinbach

Roland Klein

Ludger Wenzelides

Armin Winter

Grit Greiser

Egbert Strauß

G Greiser Lfe

The German wildlife information system (WILD): population densities and den use of red foxes (Vulpes vulpes) and badgers (Meles meles) during 2003-2007 in Germany

Keywords: Monitoring, Reference areas, Population density, Small game predators, Management, Sett usage

The German wildlife information system (WILD): population densities and den use of red foxes () and badgers () during 2003-2007 in Germany

Introduction

The German wildlife information system "WILD" (Wildtier-Informationssystem der Länder Deutschlands) was founded in 2001 and is the first monitoring programme recording population sizes of huntable game species throughout Germany. The project was initiated on behalf of the German hunting association (Deutscher Jagdschutz-Verband e.V. (DJV)) as a permanent integral part of environmental assessment with the aim of developing strategies for conservation and sustainable use of game populations (see [START_REF] Strauß | The German wildlife information system: population densities and development of European Hare (Lepus europaeus PALLAS) during 2002-2005 in Germany[END_REF]) by monitoring live animals instead of estimating population sizes from hunting bags. Especially for cryptic or rare species, many detection errors persist within any survey method [START_REF] Elphick | How you count counts: the importance of methods research in applied ecology[END_REF]. Therefore, within every monitoring programme, the most practicable method has to be established, as different methods have different levels of cost-benefit ratios, i.e. accuracy and precision versus cost efficiency (Gaidet-Drapier et al. 2006;[START_REF] Mc | Comparing methods for sampling large-and medium-sized mammals: camera traps and track plots[END_REF][START_REF] Vine | Comparison of methods to detect rare and cryptic species: a case study using the red fox (Vulpes vulpes)[END_REF]. Hunting bag statistics are most practical for large area monitoring, as they deliver area-wide data casually. Hunting bags give a good view on long-term population tendencies; however, they depend very much on the willingness and ability of hunters as well as on weather conditions [START_REF] Gaillard | Effects of hurricane Lothar on the population dynamics of roe deer[END_REF][START_REF] Grauer | Management of chamois in Bavaria (Germany): the importance of game activities in scabies control[END_REF]. Therefore, WILD established a live survey network of socalled reference areas, in which the densities of the European hare (Lepus europaeus, first results presented in [START_REF] Strauß | The German wildlife information system: population densities and development of European Hare (Lepus europaeus PALLAS) during 2002-2005 in Germany[END_REF], the red fox (Vulpes vulpes), the badger (Meles meles), the carrion crow (Corvus corone) and the hooded crow (Corvus cornix) are recorded using standardised methods [START_REF] Diezel | Biologie et écologie du blaireau Meles meles (Mustelidae, Carnivora) dans la Broye: résultats préliminaires[END_REF]. As the accuracy of such live surveys depends a lot on accessibility to area and interaction with local people (Gaidet-Drapier et al. 2006), the reference areas are monitored by the local hunters. At periodical intervals, other small game species are estimated by querying hunters areawide all over Germany.

Red fox populations, as indicated by annual hunting bags, decreased in Germany owing to rabies and the gassing of dens in the 1960s. Since gassing stopped in the mid (FRG) and later (GDR) 1970s, the population increased a little but remained stable until the start of vaccination against rabies in 1986 [START_REF] Diezel | Biologie et écologie du blaireau Meles meles (Mustelidae, Carnivora) dans la Broye: résultats préliminaires[END_REF][START_REF] Bellebaum | Bestandsentwicklung des Fuchses in Ostdeutschland vor und nach der Tollwutimpfung[END_REF][START_REF] Pegel | Fuchsbejagung -Sinn oder Unsinn?[END_REF]). Since the late 1980s, the population has increased very rapidly and has varied since 1995 at a high level. The annual hunting bag in Germany increased from about 180,000 in the late 1960s to around 630,000 recently, with a tendency for a slight decrease [START_REF] Diezel | Biologie et écologie du blaireau Meles meles (Mustelidae, Carnivora) dans la Broye: résultats préliminaires[END_REF][START_REF] Diezel | Biologie et écologie du blaireau Meles meles (Mustelidae, Carnivora) dans la Broye: résultats préliminaires[END_REF][START_REF] Bellebaum | Bestandsentwicklung des Fuchses in Ostdeutschland vor und nach der Tollwutimpfung[END_REF]. Monitoring fox populations is important for the conservation of endangered prey species, management of small game [START_REF] Reynolds | The ecology of the red fox Vulpes vulpes in relation to small game in rural southern England[END_REF][START_REF] Baker | The potential impact of red fox Vulpes vulpes predation in agricultural landscapes in lowland Britain[END_REF]) and for disease control (e.g. rabies or echinococcosis, [START_REF] Bellebaum | Bestandsentwicklung des Fuchses in Ostdeutschland vor und nach der Tollwutimpfung[END_REF][START_REF] König | Integrated-baiting concept against Echinococcus multilocularis in foxes is successful in southern Bavaria, Germany[END_REF][START_REF] Vos | Feasibility of oral rabies vaccination campaigns foxes (Vulpes vulpes) against rabies in summer[END_REF].

Badger populations, as indicated by annual hunting bags, were reduced much more by rabies and gassing [START_REF] Schwierz | Studie über die Ursachen des starken Rückganges der Dachspopulation in Hessen im Zeitraum 1952-1977[END_REF] than was fox density. Since the end of gassing of setts, the annual hunting bag has increased from less than 5,000 in the 1960s and 1970s and now remains stable at a high level of about 50,000 since 2003 [START_REF] Diezel | Biologie et écologie du blaireau Meles meles (Mustelidae, Carnivora) dans la Broye: résultats préliminaires[END_REF][START_REF] Diezel | Biologie et écologie du blaireau Meles meles (Mustelidae, Carnivora) dans la Broye: résultats préliminaires[END_REF][START_REF] Pegel | Zur Bestandssituation des Dachses in Baden-Württemberg[END_REF]. However, reliable large-scale population data are sparse today in most parts of Europe [START_REF] Griffiths | The conservation and management of the European badger Meles meles[END_REF][START_REF] Kowalczyk | Badger density and distribution of setts in Białowieża Primeval Forest (Poland and Belarus) compared to other Eurasian populations[END_REF][START_REF] Sleeman | How many Eurasian badgers Meles meles L. are there in the Republic of Ireland?[END_REF]). Monitoring badgers is important for wildlife management [START_REF] Eichstädt | Auswirkungen der Verkehrsmortalität auf populationsökologische Parameter des Dachses (Meles meles L. 1758)[END_REF]Revilla et al. 2001;[START_REF] Sleeman | How many Eurasian badgers Meles meles L. are there in the Republic of Ireland?[END_REF]), agricultural damage [START_REF] Roper | Damage by badgers Meles meles to wheat Triticum vulgare and barley Hordeum sativum crops[END_REF][START_REF] Moore | Survey of badger Meles meles damage to agriculture in England and Wales[END_REF][START_REF] Schley | The badger Meles meles and the wild boar Sus scrofa: distribution and damage to agricultural crops in Luxembourg[END_REF][START_REF] Delahay | Managing conflict between humans and wildlife: trends in licensed operations to resolve problems with badgers Meles meles in England[END_REF]) and disease control (e.g. bovine tuberculosis in Britain and Ireland, Griffin et al. 2005;[START_REF] Woodroffe | Effects of culling on badger abundance: implications for tuberculosis control[END_REF].

Our aims were to assess densities and density changes of foxes and badgers in Germany between the years 2003 and 2007 as well as to monitor differences of densities between geographical regions. We discuss the reliability of the methods and data. Additionally, we queried the distribution of badgers from every German hunting ground. Besides population data, also habitat parameters for breeding dens were evaluated from the mapping data. The data from WILD may serve as a basis for further research.

Material and methods

Since 2002, the project has established more than 800 reference areas, randomly distributed over all German agricultural regions. The mean size of these reference areas is 738 ha (minimum 95, maximum 4,500 ha). For several reasons (e.g. organisational matters and weather), the number of reference areas providing data differs between years. In total, 360 reference areas delivered density data for fox and badger populations.

Hunters monitoring the reference areas were instructed and trained according to the WILD manual, which includes detailed descriptions of the methods used [START_REF] Diezel | Biologie et écologie du blaireau Meles meles (Mustelidae, Carnivora) dans la Broye: résultats préliminaires[END_REF]. The local hunters familiar with their hunting grounds map all fox dens and badger setts every winter. Therefore, the hunters visit every known den and sett. Additionally, they search for dens and setts systematically. Unknown dens might be found by following fox and badger tracks in winter (Briedermann and Dittrich 1982;[START_REF] Stubbe | Die ökologischen Grundlagen zur Bewirtschaftung des Dachses Meles meles (L. 1758) in der DDR[END_REF], especially during fox mating and in snow (January and February). In spring, during rearing of puppies (April-June), the hunters control the mapped dens and setts and noted several parameters: type, location and habitat of den, number of entrances, inhabited or not, species inhabiting, breeding den or not, litter size and additional notes.

The sex ratio of red foxes is male biassed [START_REF] Goretzki | Zur Biologie des Rotfuchses Vulpes vulpes (L., 1758) in einem intensiv landwirtschaftlich genutzten Gebiet[END_REF][START_REF] Tryjanowski | The relationship between hunting methods and sex, age and body weight in a non-trophy animal, the red fox[END_REF]). Thus, for calculating the minimum spring fox density, we multiplied the number of litters by the factor 2.5, adding nonreproductive females and surplus males to the litters' parents [START_REF] Wandeler | Rotfuchs[END_REF][START_REF] Stubbe | Das Populationsdynamogramm eines Fuchsbestandes[END_REF]. Potential minimum summer densities were calculated by adding the spring adult density and cub density. The cub density was the observed mean number of cubs per litter at the breeding den multiplied by the number of observed litters.

For calculating badger densities, every sett was observed carefully during spring and early summer, to record the litter density and size. The sex ratio of badgers was assumed to be 1:1 [START_REF] Macdonald | Population dynamics of badgers (Meles meles) in Oxfordshire, U.K.: numbers, density and cohort life histories, and a possible role of climate change in population growth[END_REF], although populations are often slightly female biassed (Do Linh [START_REF] Diezel | Biologie et écologie du blaireau Meles meles (Mustelidae, Carnivora) dans la Broye: résultats préliminaires[END_REF]. As badgers live in their setts all the year, the potential minimum spring density was calculated by multiplying the litter density by the factor 2 (number of parents) and adding non-reproductive animals represented by the number of inhabited outlier setts without litters [START_REF] Stubbe | Die ökologischen Grundlagen zur Bewirtschaftung des Dachses Meles meles (L. 1758) in der DDR[END_REF][START_REF] Noack | Kartierung von Rotfuchs-und Dachsbauen als Grundlage für die Bestandsschätzung von Rotfuchs und Dachs im Nationalpark "Unteres Odertal[END_REF], although the relationship between the number of setts and the number of group members still remains unclear [START_REF] Tuyttens | Estimating group size and population density of Eurasian badgers Meles meles by quantifying latrine use[END_REF][START_REF] Rogers | The role of setts in badger (Meles meles) group size, breeding success and status of TB (Mycobacterium bovis)[END_REF][START_REF] Wilson | Quantification of badger (Meles meles) sett activity as a method of predicting badger numbers[END_REF][START_REF] Sadlier | Methods of monitoring red foxes Vulpes vulpes and badgers Meles meles: are field signs the answer?[END_REF]). Admittedly, non-reproducing badgers use more than one sett [START_REF] Brøseth | Function of multiple badger Meles meles setts: distribution and utilisation[END_REF][START_REF] Loureiro | Use of multiple den sites by Eurasian badgers, Meles meles, in a Mediterranean habitat[END_REF]). The minimum summer density was the spring density plus the number of litters multiplied by the mean number of cubs per litter. All densities were calculated per 100 ha hunting ground.

Estimating density by mapping and counting active setts, number and size of litters, and numbers of reproductive and non-reproductive animals may be more precise than calculating densities from hunting bags, line transects and other methods, especially in low densities [START_REF] Stiebling | Zur Habitatnutzung des Rotfuchses Vulpes vulpes (L. 1758) in der uckermärkischen Agrarlandschaft -Ergebnisse zur Populationsdichte und -dynamik[END_REF]Heydon et al. 2000;[START_REF] Tuyttens | Estimating group size and population density of Eurasian badgers Meles meles by quantifying latrine use[END_REF]). However, with these calculations, only minimum densities are estimated, as unknown numbers of undetected dens, litters and non-reproducing animals exist depending on habitat and landscape structure [START_REF] Stiebling | Zur Habitatnutzung des Rotfuchses Vulpes vulpes (L. 1758) in der uckermärkischen Agrarlandschaft -Ergebnisse zur Populationsdichte und -dynamik[END_REF][START_REF] Good | A habitat analysis of badger (Meles meles) setts in a semi-natural forest[END_REF]. Especially for badgers in large setts, two litters of two females might be possible [START_REF] Diezel | Biologie et écologie du blaireau Meles meles (Mustelidae, Carnivora) dans la Broye: résultats préliminaires[END_REF], and several non-reproducing animals might live within large families, which utilise outlier setts in different proportions (e.g. [START_REF] Roper | Sett use in European badgers Meles meles[END_REF][START_REF] Rogers | The role of setts in badger (Meles meles) group size, breeding success and status of TB (Mycobacterium bovis)[END_REF][START_REF] Sadlier | Methods of monitoring red foxes Vulpes vulpes and badgers Meles meles: are field signs the answer?[END_REF]. For these reasons, spring and summer densities have to be figured more as indices.

Additionally, in 2006, the distribution of badgers was surveyed by a nationwide questionnaire, querying badger occurrence in hunting grounds all over Germany. The questionnaire was sent to every owner, tenant or manager of private or state forest hunting grounds by the regional hunting associations and forestry offices and vice versa.

Habitat preferences for breeding dens and setts were calculated from their location compared to the availability of forest and agricultural areas within reference areas for the year 2006 (maximum participation) with Jacobs' selection index [START_REF] Jacobs | Quantitative measurement of food selection. A modification of the forage ratio and Ivlev's electivity index[END_REF]. Dens and setts were divided into three categories: (1) natural dens, (2) artificial dens specifically constructed mostly for foxes and (3) others dens of mainly anthropogenic origin, such as culverts, bunkers, stacks of wood or straw bales.

We divided Germany into six geographical regions (GR, see Fig. 1) according to the habitat and land use mapping of the German Federal Nature Conservation Agency (BfN 2008): North-Western Lowlands (NWL), North-Eastern Lowlands (NEL), Western Low Mountain Range (WLMR), Eastern Low Mountain Range (ELMR), South-Western Low Mountain Range (SWLMR) and the German Alpine Foothills (AF). These regions and the observed years were analysed for differences in litter densities and litter size with the analysis of variance (ANOVA) function in SPSS 11 (© SPSS Inc.).

Results

The participation of reference areas and, thus, the number of surveyed dens and setts as well as observed litters changed during the years due to organisational matters (Table 1). The fox and badger are distributed all over Germany with the exception of some small gaps and subject to regionally differing densities.

Foxes occurred in every reference area; however, not every reference area reported fox litters. (median) and 1.0 (mean) adult foxes per 100 ha, and the potential minimum summer density was 2.1 (median) and 2.7 (mean) foxes per 100 ha for Germany in 2007. The highest density within a single reference area was 7.1 adult foxes per 100 ha in spring and 20 foxes per 100 ha in summer. Fox litter densities differed slightly within the GR (Fig. 1), remaining stable in NEL and ELMR, and decreasing slightly in NWL and WLMR (n.s., p=0.175 for NWL and p=0.518 for WLMR; Fig. 1). Differences between the GR were significant every year (ANOVA, 2003, F=3.252, p=0.007;2004, F=3.726, p=0.003;2005, F=7.719, p≤0.001;2006, F=6,513, p≤0.001;2007, F= 4.829, p≤0.001). Especially in NWL, litter densities were comparatively low (Fig. 2). Fox cubs averaged 4.4 per litter (mean; median=4, N= 1,273 litters, Fig. 3). At 9% (N=141) of the breeding dens, cubs were reported to have been killed. On average, 3.6 (median 3) cubs were shot per litter. The mean numbers of cubs per litter fluctuated during 2003-2007 around this overall average (mean, 4.21-4.49) and were not significantly different between years (ANOVA, F=1.238, p= 0.293), but within the GR (ANOVA, F=4.996, p≤0.001). Litter size was higher in NWL (mean, 4.6) and ELMR (mean, 4.7) than in the other GR.

In total, 880 specified breeding dens were reported in 2006, 88.2% of which were natural dens, 6.9% artificial dens and 4.9% other dens. Analysis of locations of 610 natural breeding dens out of 220 reference areas revealed a preference for forest and habitats offering shelter (Fig. 4).

In the nationwide questionnaire of 2006, the badger was present in 93.6% of the participating (62.5%) municipal areas (Fig. 5) and in 74.7% of participating (N=31,403) hunting grounds (Fig. 5). However, in NWL, the badger occurred in far fewer hunting grounds per municipal area than in other GR (Fig. 5).

Densities of badger litters within the reference areas remained stable since 2003 (Fig. 1, ANOVA, F=0.552, p= 0.698). The potential minimum spring density was 0.2 (median) and 0.4 (mean) adult badgers per 100 ha, and the minimum summer density was 0.5 (median) and 0.8 (mean) badgers per 100 ha for Germany in 2007. The highest density within one reference area was 2.4 adult badgers per 100 ha in spring and 5.9 badgers per 100 ha in summer.

Litter densities differed significantly between the GR (Fig. 1, ANOVA, F=36,974, p≤0.001); this outcome is mainly based on the distribution of badger densities (Fig. 5). Especially in NWL, litter density was low, based on a high proportion of reference areas without litters. A mean of 3.0 (median 3, Fig. 3) badger cubs were observed per litter (N=319 litters). The mean numbers of cubs per litter was similar for all GR and all years.

In 2006, 413 breeding setts were recorded with specification, 96.4% of which were natural setts, 1.9% artificial setts and 1.7% other setts of anthropogenic origin such as bunkers. The locations of 297 of the natural breeding setts out of 145 reference areas were analysed. A strong preference of forest and habitats offering shelter was clear (Fig. 4) for the natural breeding setts.

Discussion

Fox populations seem to be steady, since the litter densities were mainly stable within German WILD reference areas. However, the annual hunting bags varied much more than we would predict from our results for reproduction and fox density indices. Hence, hunting bags may depend more on hunting intensities, which are influenced by short-term weather and other local conditions as well as individual hunter activity and fashion than is the reproductive rate (see also [START_REF] Cavallini | Reproduction of the red fox Vulpes vulpes in Central Italy[END_REF]Heydon and Reynolds 2000b;[START_REF] Selås | Possible impact of snow depth and ungulate carcasses on red fox (Vulpes vulpes) populations in Norway, 1897-1976[END_REF][START_REF] Tryjanowski | The relationship between hunting methods and sex, age and body weight in a non-trophy animal, the red fox[END_REF].

Litter size and density are supposed to belong on habitat quality [START_REF] Funk | Zur Dichteabhängigkeit der räumlichen und sozialen Organisation und der Reproduktion beim Rotfuchs (Vulpes vulpes L.): Eine Studie bei zeitlich und räumlich durch Jagd und Tollwut variierenden Populationsdichten in Südwest-Deutschland und Ost-Frankreich[END_REF][START_REF] Cavallini | Reproduction of the red fox Vulpes vulpes in Central Italy[END_REF]Heydon and Reynolds 2000a). The differences in fox litter density indices between geographical regions presumably reflect the biological needs of the species. The most important resource limiting the fox population in Germany seems to be the availability of suitable territories containing convenient breeding dens [START_REF] Stiebling | Zur Habitatnutzung des Rotfuchses Vulpes vulpes (L., 1758) in der uckermärkischen Agrarlandschaft: Anlage der Baue[END_REF][START_REF] Dekker | A spatial analysis of a population of red fox (Vulpes vulpes) in the Dutch coastal dune area[END_REF]) and food availability [START_REF] Jankowiak | Habitat use, food and the importance of poultry in the diet of the red fox Vulpes vulpes in extensive farmland in Poland[END_REF]. The red fox shows a clear preference for forest and sheltering structures for breeding dens (compare also [START_REF] Behrendt | Beiträge zur Ökologie des Rotfuchses (Vulpes vulpes L.)[END_REF][START_REF] Weber | Habiat use by the red foy Vulpes vulpes in a mountainous area[END_REF]. As the fox is also able to use artificial dens for parturition (in our study, about 12% of breeding dens existed of anthropogenic structures), especially in urban areas where food and den sites are much more plentiful than in forested or agricultural regions, very high densities can occur [START_REF] Baker | Flexible spatial organization of urban foxes, Vulpes vulpes, before and during an outbreak of sarcoptic mange[END_REF][START_REF] Gloor | The rise of urban fox populations in Switzerland[END_REF].

The results of the nationwide badger survey are coincident with those of regional studies (e.g. [START_REF] Pegel | Zur Bestandssituation des Dachses in Baden-Württemberg[END_REF]). Some authors have assumed that lack of forest, inappropriate soil properties and shallow groundwater are the main causes of local absence of badgers [START_REF] Noack | Kartierung von Rotfuchs-und Dachsbauen als Grundlage für die Bestandsschätzung von Rotfuchs und Dachs im Nationalpark "Unteres Odertal[END_REF][START_REF] Sleeman | How many Eurasian badgers Meles meles L. are there in the Republic of Ireland?[END_REF]). This hypothesis is supported by the observation of lower densities and gaps in the distribution of badgers within NWL, where these three factors, often simultaneously, occur frequently. Analysis of the locations of breeding setts confirms previous evidence of a close bond between badgers and forest or forest-like habitat [START_REF] Walliser | Einfluß der Landschaftszerschnittenheit und des Landnutzungsmusters auf die Raum-Zeitstruktur des Dachses (Meles meles L., 1758)[END_REF]. In our study, the badgers strongly preferred natural habitats and natural setts for breeding and very seldom used anthropogenic structures. The minimum habitat requirements of badgers are for grassland and shelter [START_REF] Good | A habitat analysis of badger (Meles meles) setts in a semi-natural forest[END_REF][START_REF] Delahay | Habitat correlates of group size, bodyweight and reproductive performance in a high-density Eurasian badger (Meles meles) population[END_REF][START_REF] Palphramand | Spatial organization and behaviour of badgers (Meles meles) in a moderate-density population[END_REF], plus adequate food [START_REF] Da Silva | Habitat, food availability and group territoriality in the European badger, Meles meles[END_REF]. If those key habitats and resources are sparse, fewer territories may be occupied, leading to lower population density [START_REF] Doncaster | What determines territory configurations of badgers?[END_REF][START_REF] Rosalino | Resource dispersion and badger population density in Mediterranean woodlands: is food, water or geology the limiting factor?[END_REF][START_REF] Delahay | Distribution of badger latrines in a high-density population: habitat selection and implications for the transmission of bovine tuberculosis to cattle[END_REF]) in continental Europe.

The badgers litter densities and indices of abundance have remained steady since 2003, corresponding to the stable hunting bags (DJV 2009) during this period. The annual hunting bag within our reference areas equals the annual German hunting bag (about 0.16 badgers per 100 ha) and, thus, amounts to only one third of the calculated reproduction. As hunting bags, as well as calculated density indices and reproduction, have been stable since 2003, we assume a high natural mortality and a high proportion of unreported traffic mortality (compare [START_REF] Broekhuizen | Durchlässe für Dachse und ihre Effektivität[END_REF].

To fit the needs of a monitoring programme, the most practicable method has to be selected, as different methods have different levels of cost-benefit ratios, i.e. accuracy and precision versus cost efficiency (Gaidet-Drapier et al. 2006;[START_REF] Elphick | How you count counts: the importance of methods research in applied ecology[END_REF][START_REF] Mc | Comparing methods for sampling large-and medium-sized mammals: camera traps and track plots[END_REF][START_REF] Vine | Comparison of methods to detect rare and cryptic species: a case study using the red fox (Vulpes vulpes)[END_REF].

Hunting bag statistics in total might be correlated with population data [START_REF] Grauer | Management of chamois in Bavaria (Germany): the importance of game activities in scabies control[END_REF] and show relative population trends, especially when verified or exemplarily proved. However, hunting bags depend a lot on ability (weather conditions, changing hunting restrictions etc.) and willingness of hunters [START_REF] Gaillard | Effects of hurricane Lothar on the population dynamics of roe deer[END_REF][START_REF] Kalchreuter | Sauen als Beutegreifer -Welchen Einfluss kann Schwarzwild auf andere Tierarten ausüben? in: 13[END_REF]Keuling 2007). Additionally, hunting bags might show completely opposite or wrong trends or dimensions, especially for migratory birds [START_REF] Frederiksen | The dynamics of hunted Icelandic goose populations: a reassessment of the evidence[END_REF] or due to insufficient harvest rates [START_REF] Grauer | Management of chamois in Bavaria (Germany): the importance of game activities in scabies control[END_REF]. Therefore, live monitoring seems to be necessary to develop strategies for conservation and sustainable use of game populations (compare also [START_REF] Strauß | The German wildlife information system: population densities and development of European Hare (Lepus europaeus PALLAS) during 2002-2005 in Germany[END_REF].

Indirect counts by searching for field signs are more practicable at a large scale. However, pure field signs might not deliver accurate data for density estimates as often the correlations between number of signs, activity and number of individuals are not clear [START_REF] Sadlier | Methods of monitoring red foxes Vulpes vulpes and badgers Meles meles: are field signs the answer?[END_REF]). Thus, indirect counts by field signs are only useful for the detection of the presence of one species but not for density estimates or reliable indices [START_REF] Sadlier | Methods of monitoring red foxes Vulpes vulpes and badgers Meles meles: are field signs the answer?[END_REF][START_REF] Vine | Comparison of methods to detect rare and cryptic species: a case study using the red fox (Vulpes vulpes)[END_REF].

Direct census methods such as capture-mark-recapture, spotlight counts, camera trapping and radio tracking are predominantly of high quality and deliver the most accurate results, nevertheless are also very expensive and costly [START_REF] Delahay | Managing conflict between humans and wildlife: trends in licensed operations to resolve problems with badgers Meles meles in England[END_REF][START_REF] Grilo | Spatial-temporal patterns in Mediterranean carnivore road casualties: consequences for mitigation[END_REF]. Additionally, direct counts usually deliver insufficient data for rare or cryptic species (Heydon et al. 2000;[START_REF] Sadlier | Methods of monitoring red foxes Vulpes vulpes and badgers Meles meles: are field signs the answer?[END_REF][START_REF] Vine | Comparison of methods to detect rare and cryptic species: a case study using the red fox (Vulpes vulpes)[END_REF]. Direct methods might be very useful in special cases [START_REF] Delahay | Managing conflict between humans and wildlife: trends in licensed operations to resolve problems with badgers Meles meles in England[END_REF][START_REF] Grilo | Spatial-temporal patterns in Mediterranean carnivore road casualties: consequences for mitigation[END_REF]) like local management programmes for cryptic, rare or endangered species even with low densities (e.g. genotyping, [START_REF] Frantz | Estimating population size by genotyping remotely plucked hair: the Eurasian badger[END_REF]) as well as for populations causing economic problems at a regional scale. The live monitoring programme used in this study was carried out with comparably low effort and cost, while achieving its aim of providing indices of abundance (or "estimates of density") for cryptic species. Furthermore, this method is independent from factors which influence hunting bags [START_REF] Gaillard | Effects of hurricane Lothar on the population dynamics of roe deer[END_REF][START_REF] Kalchreuter | Sauen als Beutegreifer -Welchen Einfluss kann Schwarzwild auf andere Tierarten ausüben? in: 13[END_REF][START_REF] Grauer | Management of chamois in Bavaria (Germany): the importance of game activities in scabies control[END_REF] and supplies minimum density numbers instead of detecting presence or absence of a cryptic species solely.

Of course, there are some disadvantages of a survey performed with local volunteers in sampling areas exclusively, e.g. there is the problem of missing data [START_REF] Elphick | How you count counts: the importance of methods research in applied ecology[END_REF]: the hunters might miss dens [START_REF] Stiebling | Der Rotfuchs, Vulpes vulpes (L., 1758), im Biosphärenreservat Schorfheide-Chorin[END_REF]. Nevertheless, the hunters, well trained in using standardised methods, deliver most reliable data as they have the knowledge of the area (compare Gaidet-Drapier et al. 2006). Although the reference areas represent only small sampling plots, the number of reference areas and the consistency of the results with other specific small scale studies (e.g. fox: [START_REF] Goretzki | Zur Biologie des Rotfuchses Vulpes vulpes (L., 1758) in einem intensiv landwirtschaftlich genutzten Gebiet[END_REF]Heydon et al. 2000;[START_REF] Bartón | Winter severty limits red fox populations in Eurasia[END_REF]Zalewski 2007, badger: Walliser andRoth 1997;[START_REF] Kowalczyk | Badger density and distribution of setts in Białowieża Primeval Forest (Poland and Belarus) compared to other Eurasian populations[END_REF]Do Linh San et al. 2007) strengthen the use of litter and animal densities giving reliable trends for Germany's badger and fox populations.

As the hunters collect data voluntarily, this method is indeed restricted to countries with area-based hunting systems, where hunters hold (owned or rented) private hunting grounds. This method would not deliver reliable data without a high effort of non-local assistants in a system based on hunting licences on public game management units. The high level of variation between years within the GR might also be influenced by differing participation of reference areas (see "Materials and methods") which did not deliver data continuously every year.

The assumptions on which spring and summer densities were based contain a high degree of uncertainty [START_REF] Elphick | How you count counts: the importance of methods research in applied ecology[END_REF], as the relationships between field signs, litter densities and individual densities are not clear. For these reasons, spring and summer densities have to be used as relative indices. To achieve more reliable individual density estimates, such large-scale indices would have to be calibrated with more accurate and precise methods. However, this is a common issue for many survey methods, such as hunting bags and field signs (compare also [START_REF] Palomares | Die Dichte des Rotfuchses und die Beute an Niederwild während der Periode der Jungenaufzucht im Südosten Spaniens[END_REF]Heydon et al. 2000;Grauer et al. 2009), and it would be worthwhile validating survey methods.

Concurrent to the hunting bag statistics (DJV 2009), the populations of badgers and foxes stayed stable at a high level. Two main key resources seem to be necessary for both species: suitable setts or dens for breeding and adequate nutrition (e.g. [START_REF] Baker | Flexible spatial organization of urban foxes, Vulpes vulpes, before and during an outbreak of sarcoptic mange[END_REF]Revilla and Palomares 2001;[START_REF] Rosalino | Resource dispersion and badger population density in Mediterranean woodlands: is food, water or geology the limiting factor?[END_REF].

To summarise, den mapping and observation of litters by local hunters is an appropriate method for surveying fox and badger populations in Germany. At a large scale, this method has less effort and costs than direct counts; however, it provides live population estimates of more accuracy than hunting bags do. Additionally, direct mappings might deliver additional data on the species biology like habitat utilisation.

  Fig. 1 Litter densities of foxes and badgers within WILD reference areas in Germany and its geographical regions 2003-2007 (for description of regions, see text). Box and whisker plots show median (horizontal line within box), 25% and 75% percentiles (box) and range (whiskers); statistical outliers and extreme values are excluded

Fig. 2

 2 Fig. 2 Fox litter densities within the German reference areas for 2006. Geographical regions: North-Western Lowlands (NWL), North-Eastern Lowlands (NEL), Western Low Mountain Range (WLMR),

Fig. 3

 3 Fig. 3 Percentage (%) of occurrence of litter sizes within all observed litters for fox (N=1,273) and badger (N=319)

Acknowledgements We would like to thank all involved hunters for the work they did. We thank the hunting associations of the federal states for their support. Additional thanks go to several colleagues all over Europe who helped with literature and comments, especially to T. J. Roper for discussing all the badger questions and an early draft of this manuscript. The four anonymous reviewers helped a lot advancing this manuscript. In memoriam Dr. Heike Nösel, we will keep you in our hearts.