Do all inter-patch movements represent dispersal? A mixed kernel study of butterfly mobility in fragmented landscapes


  • Thomas Hovestadt,

    Corresponding author
    1. University of Würzburg, Biozentrum, Field Station Fabrikschleichach, Glashüttenstraße 5, 96181 Rauhenebrach, Germany
    2. Muséum National d’Histoire Naturelle, CNRS UMR 7179, 1 Avenue du Petit Château, 91800 Brunoy, France
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  • Birgit Binzenhöfer,

    1. Landschaftsökologische Gutachten & Kartierungen, Friedhofstr. 1, 97475 Zeil am Main, Germany
    2. UFZ, Helmholtz Centre for Environmental Research, Department of Conservation Biology, Permoserstr. 15, 04318 Leipzig, Germany
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  • Piotr Nowicki,

    1. Institute of Environmental Sciences, Jagiellonian University, Gronostajowa 7, 30–387 Kraków, Poland
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  • Josef Settele

    1. UFZ, Helmholtz Centre for Environmental Research, Department of Community Ecology, Theodor-Lieser-Str. 4, 06120 Halle, Germany
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1. In times of ongoing habitat fragmentation, the persistence of many species is determined by their dispersal abilities. Consequently, understanding the rules underlying movement between habitat patches is a key issue in conservation ecology.

2. We have analysed mark-release-recapture (MRR) data on inter-patches movements of the Dusky Large Blue butterfly Maculinea nausithous in a fragmented landscape in northern Bavaria, Germany. The aim of the analysis was to quantify distance dependence of dispersal as well as to evaluate the effect of target patch area on immigration probability. For statistical evaluation, we apply a ‘reduced version’ of the virtual migration model (VM), only fitting parameters for dispersal distance and immigration. In contrast to other analyses, we fit a mixed dispersal kernel to the MRR data.

3. A large fraction of recaptures happened in other habitat patches than those where individuals were initially caught. Further, we found significant evidence for the presence of a mixed dispersal kernel. The results indicate that individuals follow different strategies in their movements. Most movements are performed over small distances, nonetheless involving travelling between nearby habitat patches (median distance c. 480 m). A small fraction (c. 0·025) of the population has a tendency to move over larger distances (median distance c. 3800 m). Further, immigration was positively affected by patch area (I∼Aζ), with the scaling parameter ζ = 0·5.

4. Our findings should help to resolve the long-lasting dispute over the suitability of the negative exponential function vs. inverse-power one for modelling dispersal. Previous studies on various organisms found that the former typically gives better overall fit to empirical distance distributions, but that the latter better represents long-distance movement probabilities. As long-distance movements are more important for landscape-level effects and thus, e.g. for conservation-oriented analyses like PVAs, fitting inverse-power kernels has often been preferred.

5. We conclude that the above discrepancy may simply stem from the fact that recorded inter-patch movements are an outcome of two different processes: daily routine movements and genuine dispersal. Consequently, applying mixed dispersal kernels to disentangle the two processes is recommended.


In the face of ongoing habitat fragmentation and global change, the persistence of species will primarily depend on their ability to survive as metapopulations (Pullin 2002) and to colonize new habitats as they become available (Hill, Thomas & Blakeley 1999; Hill et al. 2001; Thomas et al. 2001; Simmons & Thomas 2004). Within metapopulation theory (Levins 1970; Hanski 1994, 1999), dispersal is consequently one of the key mechanisms, and knowledge about dispersal is of great relevance for making predictions of the future fate of populations (Thomas 2000; Hovestadt & Poethke 2006; Driscoll 2007; Hawkes 2009). Understanding the rules driving species dispersal is thus of primary importance to develop reasonable conservation strategies and foresee current or future threats to species persistence (Brook et al. 2000).

There is accumulating evidence that individuals may differ in their dispersal propensity as well as abilities, and that the probability of undertaking dispersal is context dependent (Bowler & Benton 2005; Clobert et al. 2009). This empirical evidence supports a growing body of theoretical studies that show that context-dependent strategies should consistently result in higher fitness benefits for emigrating individuals (e.g. Clobert et al. 2009; Enfjäll & Leimar 2009; Hovestadt, Kubisch & Poethke 2010); consequently, the evolution of such strategies is clearly predicted. One outcome of many of the theoretical models is that the fraction of individuals dispersing should become small as soon as dispersal is risky and populations are reasonably large and stable (Travis & Dytham 1998; Gandon & Rousset 1999; Poethke, Hovestadt & Mitesser 2003). Although animals move around all the time, most of their movements may thus not be motivated by a ‘desire’ to disperse but simply by their need to satisfy daily needs like foraging (Van Dyck & Baguette 2005).

Measuring dispersal is hence not a straightforward task. First, empirical investigations of dispersal need to cover an area of an appropriate spatial extend, and because of logistic difficulties, many studies apparently fail to meet this requirement (Schneider 2003). Second, depending on the structure (‘grain’) of the landscape, dispersal can come about by daily routine movement (Baguette & Van Dyck 2007), but specific ‘dispersal episodes’ may be governed by very different movement rules (Van Dyck & Baguette 2005; Schtickzelle et al. 2007; Getz & Saltz 2008; Nathan et al. 2008). Data collected, for example, in mark-release-recapture (MRR) may thus contain a mixed bag of movement observations generated by collecting data from individuals following completely different motivations and moving according to different rules. If we use all the data as evidence for estimating the properties of a dispersal kernel, we may grossly underestimate the relevant parameters (e.g. mean dispersal distance) because the data set is dominated by values taken from individuals that were just involved in the standard daily movement activities; the latter are typically restricted to rather small ranges. Consequently, while estimating parameters of dispersal, one should account for the possible double nature of individual movements.

In the literature on seed dispersal, the problem of different dispersal patterns generated by different dispersal processes has long been recognized (Higgins & Richardson 1999; Clark et al. 2005). A number of studies indeed indicate that mixed models can fit dispersal data considerably better than models assuming a single-kernel models (that implicitly assume a single underlying movement mechanism). Some examples include secondary dispersal of seeds by kestrels that pick up seeds from lizards they predate on (Padilla & Nogales 2009), seed dispersal patterns generated by monkeys (Russo, Portnoy & Augspurger 2006), or data on pollen dispersal (Slavov et al. 2009).

Surprisingly, mixed models have rarely been fitted to the mark-recapture-release (MRR) data frequently collected in animal studies. In this paper, we investigate the benefit of fitting a mixed kernel model in the analysis of MRR data collected in a study of the Dusky Large Blue butterfly, Maculinea nausithous (Lycaenidae, Bergsträsser 1779).

Materials and methods

We reanalyse data collected in 1994 and presented in greater detail already in Binzenhöfer & Settele (2000). That study includes data on population size, mortality, and habitat preferences, but here, we focus on data about the movements of M. nausithous butterflies between habitat patches. The study also covered the less abundant sympatric Maculinea teleius, but the number of observations of the latter species was too small to reasonably analyse them in the way outlined below. In their tendency, analyses of M. teleius data, however, lead to similar conclusions than those reported here. Butterflies of the genus Maculinea are assumed to be rather sedentary showing little tendencies for long-distance movement (Nowicki et al. 2005c).

Study area

The study area (3·8 × 3·5 km2) is located in northern Bavaria in the Steigerwald region, an area characterized by its large forests and open valleys along the small rivers and creeks running trough the hilly landscape. The study took place in the valley of the Aurach, a small creek running SE in this area. With a mean annual temperature of 8 °C, and mean annual precipitation of 750–800 mm, the study area is cooler and more humid than the larger surrounding region. However, the summer of 1994 was one of the warmest recorded since the onset of regular weather reporting in 1879.

Study sites were located in the open (agricultural) sections surrounding the villages of Oberschleichach, Neuschleichach and Unterschleichach. Habitat patches suitable for M. nausithous were defined on the basis of landscape-use, vegetation composition and the density of Sanguisorba officinalis, which is the only larval foodplant and at the same time the principal adult nectar source for this butterfly. Based on vegetation attributes that reflect gradual differences in habitat suitability, 211 patches were initially identified, covering a total area of 81 ha. The smallest patch had only 180 m2, 50% of patches were smaller than 2500 m2, 90% smaller than 10 000 m2 and the largest covered 25 600 m2. Many of the 211 patches, however, shared borders with other patches, i.e. the movement from one patch to another would not necessarily imply that the butterfly traversed any unsuitable habitat. We thus carried out the analyses for data sets with all patches pooled together that shared a common border, reducing the number of patches to 69 (Fig. 1). The median distance between these patches was >2000 m, but the median distance to the nearest patch was only 156 m. As the position of butterflies was not exactly noted, we assume that individual butterflies were caught in the centre of the patches. Dispersal distances given in the analyses are thus always from the centre of the start patch to the centre of the target patch. More details of the mark-recapture procedure have been described by Binzenhöfer & Settele (2000). All sites were searched with equal (in proportion to area) intensity and visited in intervals of c. 3 days.

Figure 1.

 Schematic map of the study area in the Aurach valley, Steigerwald, Northern Bavaria, Germany (c. N49°56′, E10°38′). The map shows the resulting 69 patches of Maculinea nausithous habitat after merging those of the originally mapped 211 patches of Binzenhöfer & Settele (2000) that shared a common border. The remaining matrix includes forest, agricultural areas, and human settlements.

Statistical analysis

In the last years, several advanced statistical methods for the dispersal analysis of MRR data have been developed (Grosbois & Tavecchia 2003; Ovaskainen 2004; Ovaskainen et al. 2008), of which the best example is the Virtual Migration (VM) model introduced by Hanski, Alho & Moilanen (2000). Yet, the application of these new methods requires extensive data collected according to pre-defined protocols. The data used in the present paper were collected in 1994, i.e. before the aforementioned models became available, and thus the fieldwork did not always conform to the data collection protocol demanded. Apart from this, in the current study, we are primarily interested in estimating parameters of the dispersal kernel and not in other parameters, e.g. mortality rates within habitat and during dispersal that can be fitted by the full VM model. Consequently, we modified the VM approach, limiting the analyses to individuals that have been re-captured in other sites, i.e. individuals known to have survived, to have emigrated, and to have been found in a new location. Neither emigration probability nor survival parameters can be extracted from such data (cf. Hanski, Alho & Moilanen 2000). However, they allow estimation of the parameters for dispersal distance and immigration:

image( eqn 1)

Here, L(i,j) specifies the probability of a transition from start patch i to target patch j, with di,j being the Euclidian distance between the two patches, α the distance dependence of dispersal, Aj the target patch area and ζ a (possible) dependence of immigration on the target patch area. Unlike in the standard VM model (Hanski, Alho & Moilanen), 1/α does not represent the mean distance decay for a single day but remains undefined in relation to the time lag between capture and recapture. Provisional analysis suggests, however, that the distance covered between the marking and recapture of an individual is not correlated with the time lag between the two events (see below and results). Note that the negative exponential function (NEF) fitted is the density-pdf (probability density function) distribution of the dispersal process, i.e. the density of immigrants on the circle with radius r around the starting point. The distance-pdf (the probability of a certain distance travelled) is thus proportional to inline image with mean dispersal distance 2/α (see Cousens, Dytham & Law 2008).

In addition, we were interested to test for the presence of a mixed dispersal kernel, i.e. one composed – with proportions p and 1-p, respectively – of two negative exponentials with different values for dispersal distance (α1, α2). To achieve this, we modified Si,j in the following way:

image( eqn 2)

If dispersal of butterflies across the landscape at least to some degree resembles a diffusion process, i.e. a random walk, observed dispersal distance should increase with the time lag between capture and recapture, i.e. α should decline with time. We thus tested whether a time-dependent adjustment of α would significantly improve the fit of the model, i.e.:

image( eqn 3)

with α’∋ being the time-adjusted value of α, Δt the time interval between marking and recapture and τ a scaling parameter. Under a diffusion model, mean distance should increase in proportion to inline image, i.e. τ should be approximately 0·5.

Overall, we thus compare six different statistical models (see Table 1: ‘Simple’ (only NEF) ‘Simple + Area’ (NEF with immigration scaling), ‘Simple + Area + Time’ (NEF with immigration scaling and time dependence of α), ‘Mixed’ (mixed kernel of two separate NEFs (α1, α2) in proportions of p and 1-p), ‘Mixed + Area’ (mixed kernel with immigration scaling) and finally ‘Mixed + Area + Time’ (NEF with immigration scaling and time dependence of α2, i.e. the long-distance component of dispersal). As preliminary analyses did not suggest differences between the two sexes, we did not incorporate gender as a factor into the analysis (which would have inflated parameter numbers by a factor of two).

Table 1.  Model comparison
ModelBICDispersal parameters
α(1)ζτ p α2τ2
Distance dependence (short-distance component)Immigration scaling with target patch areaTime dependence of α(1)Fraction of short-distance dispersersDistance dependence (long-distance component)Time dependence of α2
  1. Performance of different dispersal models tested, based on BIC criterion and their parameter estimates (in bold) with the lower and upper 95% confidence limits (in italics). The best-fitting model ‘Mixed + Area’ is highlighted in grey. Parameters not fitted in a model are marked ‘n.f.’. BIC, Bayesian Information Criterion.

Simple930·4 0·00218 n.f.n.f.n.f.n.f.n.f.
Simple + Area891·1 0·00229 0·5013 n.f.n.f.n.f.n.f.
0·00260 0·3462
0·00198 0·6564
Simple + Area + Time891·4 0·00313 0·5021 0·1996 n.f.n.f.n.f.
0·00387 0·3468 0·0528
0·00239 0·6575 0·3464
Mixed919·7 0·03387 n.f.n.f. 0·9740 0·00188 n.f.
0·01160 0·9264 0·00157
0·05614 1·0000 0·00219
Mixed + Area871·8 0·00413 0·5350 n.f. 0·9745 0·00053 n.f.
0·00298 0·3781 0·9412 0·00033
0·00528 0·6919 1·0000 0·00102
Mixed + Area + Time873·4 0·00416 0·5350 n.f. 0·9681 0·00129 0·4490
0·00597 0·3778 0·8616 0·00000 −0·5449
0·00234 0·6923 1·0000 0·00274 1·4439

For all model variants, parameter estimation was carried out by specifying the appropriate likelihood functions and then minimizing -2LL by optimizing choice of parameter values using R functions ‘optimize()’ for single parameter models, and ‘optim()’ (R Development Core Team 2009) for models with two or more parameters. We applied the default Nelder and Mead algorithm, but checked robustness of estimates by also applying simulated annealing; in all the cases, parameter estimates generated by the two methods differed only marginally. Model selection was then based on the Bayesian Information Criterion (BIC), but using AIC would have led to the same outcome. Estimates of confidence intervals for single parameter models were based on the likelihood ratio (see Hilborn & Mangel 1997), while (approximated) estimates of confidence intervals for more complex models were based on the Hessian matrices calculated by function ‘optim()’ (Bolker 2008).


The total number of M. nausithous individuals caught was 1634, with 1330 individuals captured only once, 273 captured twice, 29 thrice and 2 four times, that is 337 recaptures in total. The distribution of recapture events per individual very closely matches a Poisson distribution (χ2 = 0·01, = 0·99, d.f. = 3). This suggests that capture probability was not affected by capture and/or handling. Including further recaptures collected in addition to the standard capture routine, 157 of 388 recaptures (40%) involved individuals that changed patches. Movements of M. nausithous between habitat patches were therefore relatively frequent events in the landscape studied. In the following, we only analyse the data on 145 first recaptures that involve a transfer between patches. The average movement distance recorded was 600 m, with the median being 390 m, and the maximum nearly 3800 m (Fig. 3a). For a comparison, the maximum distance between any two patches included in the study is slightly above 5800 m. The scale of the study area and/or the distribution of suitable habitats could thus limit especially the largest distances observed. In that case, we should find similar estimates for the dispersal parameter if we would randomize (randomly draw from the list of targets without replacement) the sequence of target patches across observations (see Hovestadt & Nowicki 2008 for the rationale of the approach). We performed such a randomization 10 000 times, fitting the ‘Simple’ model to each generated data set. We found that mean dispersal distance generated in such a way was substantially larger than that obtained through fitting the model to the empirical data (3780 vs. 920 m).

There was weak indication that movement distances were declining over the course of the season (Spearman ρ = –0·18, < 0·05). More surprising, however, there was no correlation of the time interval between captures and the distance travelled (Fig. 2; Spearman ρ = 0·02, > 0·8), a conclusion also supported by the likelihood models where models including parameter τ (time dependence of α) performed worse than those not including τ (Table 1). A significant influence of patch area on immigration probability was in turn quite clear (Table 1). Estimates for ζ (immigration scaling with target patch area A) are close to 0·5 in all the models, in which this parameter was fitted.

Figure 2.

 Plots of Maculinea nausithous displacement distance over (a) the day of first capture (the reference date is 1 July 1994) and (b) the time interval between capture and recapture (measured in days).

Most importantly, our analyses provided strong evidence for the existences of two types of movements performed by the investigated butterflies. The model ‘Mixed + Area’, including a mixed dispersal kernel and immigration scaling, was clearly the best-fitting for our empirical data (Table 1). According to this model, only about 2·5% of movements fall into the long-distance category. The rest constitute short-distance dispersal. The estimated average dispersal distances for the two kernels were c. 480 and 3800 m (Fig. 3).

Figure 3.

 (a) Histogram of empirical displacement distances of Maculinea nausithous as well as dispersal kernels of the best-fitting model ‘Mixed + Area’ (see Table 1). The continuous line shows the overall mixed kernel, and the hatched line indicates the ‘long-distance component’ of the kernel (α2), which contributes to only c. 2·5% of the overall mixed kernel. Note that the kernels drawn assume a constant size of the target patch (1 ha), while the distribution of empirical data is also affected by the significant influence of target patch area. (b) Pair-wise confidence ellipses (continuous line 95%, hatched line 90%) for the two distance-dependence parameters α1 and α2 of model ‘Mixed + Area’ as derived from the Hessian matrix (see Bolker 2008).


Our observations reveal that almost half of M. nausithous individuals switched habitat patches in their adult lifetime. The respective proportion found by other studies on this species was considerably lower, typically around 10% (Nowicki et al. 2005a, 2007). Also, the distances covered by the butterflies in the Steigerwald appear relatively long when compared with those previously reported for Maculinea (see review in Nowicki et al. 2005b). This indicates that mobility is a landscape-specific trait, showing strong intra-specific variability (cf. Stevens, Pavoine & Baguette 2010a; Stevens, Turlure & Baguette 2010b). Fundamentally, our findings of relatively high mobility of the investigated butterflies may not come as a surprise as most of the habitat patches in the study area were smaller than 1 ha. Such patches may be too small to satisfy all the requirements of an adult butterfly over the course of its lifetime.

The more important of our findings is the considerable benefit of fitting a mixed kernel model over a simple negative exponential kernel in spite of the fact that the ‘contribution’ of the long-distance kernel is estimated at only about 2·5%. Correspondingly, while the median movement distance reported is 390 m, 5% of the individuals travelled further than 2100 m. It is certainly tempting to assume that this is about the proportion of individuals performing long-distance dispersal. These results and the fact that net distance travelled is not time dependent and does not conform to a diffusion process are in agreement with the idea that movements of a small fraction of dispersing individuals may follow different rules than those governing ‘daily routine’ movements (Van Dyck & Baguette 2005). The latter may nonetheless include moving between different habitat patches in a fragmented landscape.

The good performance of a mixed kernel that we documented should help to resolve the long-lasting dispute over the suitability of the negative exponential vs. inverse-power functions for modelling dispersal (Okubo & Levin 1989; Turchin 1998). The negative exponential kernel has, e.g. when compared to the power model, the benefit that it is rooted in the theory of a correlated random walk (with settlement; see Turchin 1998). However, a single exponential kernel cannot be ‘fat-tailed’, an attribute often encountered with empirical data. Not surprisingly, the past studies across different taxa found that the former typically gives better overall fit to empirical distance distributions, but the latter better represents long-distance movement probabilities (Fitt & McCartney 1986; Schwartz 1993; Baguette 2003). Accounting for a small fraction of individuals dispersing over long distance can considerably change predictions concerning the spread of species (Clark et al. 2005) or their persistence in fragmented landscapes (Baguette 2003). Consequently, despite the need to compromise model fit for short- to medium-distance movements, the inverse-power kernel is often preferred in ecological applications for which long-distance movement probability is of primary importance, e.g. in Population Viability Analyses (cf. Schtickzelle et al. 2005).

Compared to using the inverse-power function or some kernel that just fits the ‘fat tail’ of the movement data well, applying a mixed kernel model has the benefit that its components could possibly correspond with (depending on landscape attributes) data emerging because of different movement rules, i.e. small-scale daily routine movements of residents on the one hand and to ‘true’ dispersal on the other. One could even consider mixing kernels of different functional types, e.g. a normal with a negative exponential density. This should promote statistical models that can more directly be related to underlying mechanisms (as recommended by Turchin 1998) and thus reveal more about the processes underlying dispersal. Our study suggests that fitting mixed dispersal kernels offers a valuable alternative in such cases. Obviously, it should be left to statistical evaluation whether increasing the parameter number as necessary for constructing such mixed models is justified.

In the model we fitted, because of the limitations of the data set, we were not able to estimate the daily distance dependence of dispersal, as should ideally be done (Hanski, Alho & Moilanen 2000). The fact that we could not find a significant relationship between distance travelled and the time lag between capture and recapture as well as the reduction in the likelihood when fitting time dependence of the dispersal parameter in the models suggests that the time intervals between capture and recapture (median = 4 days) were typically long enough for the butterflies to travel anywhere in the study area. This raises the question whether the spatial scale of the study area was large enough to investigate the dispersal abilities of M. nausithous at all; the movement distances observed could rather be determined by the limited distances between habitat patches in the study area (see Schneider 2003) than by the ability or willingness of butterflies to travel even larger distances. However, this is excluded by the results of our randomization test. Thus, we conclude that the movement distances observed were indeed limited by the butterflies’ decisions or limitations rather than opportunity. The lacking time dependence of movement may thus indicate the presence of a home-range (possibly spanning several habitat patches) in M. nausithous, which has been proposed before, based on different evidence (Hovestadt & Nowicki 2008; Korosi et al. 2008).

However, at some spatial scale, or when time intervals between capture and recapture become too short, the possible distances travelled must become time dependent. We thus suggest fitting models including time dependence of α whenever a reasonably large data set is available and includes travel distances that appear large in relation to the movement abilities of the organisms under investigation. Whether to maintain this effect in the model or not can then be decided based on the BIC or some other criterion.

In this study, we also found strong evidence for significant area dependence in immigration as proven by the substantial decrease in the BIC value when the parameter for immigration scaling was included in both the simple and the mixed dispersal kernel models. The estimates for parameter ζ were close to 0·5 in all of the models where it was included. This result is in good agreement with those of other butterfly studies, and it conforms to the value predicted for the immigration probability proportional to patch diameter (Englund & Hamback 2007). Diameter scaling of immigration can be expected when butterflies moving though the matrix would either fly more or less linearly (Dusenbery 1989); rather straight inter-patch movement of butterflies has, for example, been reported by Delattre et al. (2010). Diameter scaling would also be expected if butterflies could detect a possible habitat site from a considerable distance (Englund & Hamback 2007). It is noteworthy that the latter ability would also reduce the risk for an emigrating butterfly to become lost in the matrix. However, detection distances reported for butterflies seem to be limited and are often considerably smaller than 100 m (Conradt et al. 2000; Conradt, Roper & Thomas 2001; Cant et al. 2005; Schtickzelle et al. 2007). For the Fender Blue (Icaricia icarioides fenderi), the perceptional range is 10–22 m (Schultz & Crone 2001). Provided that M. nausithous adults have similar abilities they could to a certain degree follow a rather risk-poor strategy of ‘island hopping’ in the study area even if they do not settle in the first patch they encounter, but clearly many of the patches are further away from their nearest neighbour than 25 m.

In conclusion, we have found that fitting a mixed kernel model to animal movement data collected with the help of MRR studies may be particularly useful. It enables us to detect a small but important fraction of long-distance dispersal events, usually hidden in the multitude of non-dispersive movement witnessed in such studies. These long-distance movements are the key to understand how species might survive under different levels of fragmentation and how they might be able to adapt to many threats derived from global change. These data give us particularly precious information on the probability that species might be able to track climatic changes and disperse in parallel with newly established geographical settings of their (not only climatic) niches (Schweiger et al. 2008, 2011; Settele et al. 2008; Settele & Kühn 2009). In the instance of M. nausithous, for example, the species may indeed have a better chance and ability to bridge even large habitat gaps than previously thought.


The authors gratefully acknowledge support by the following EU funded projects: MacMan (FP 5: EVK2-CT-2001-00126), SCALES (FP7 grant agreement no. 226852) and CLIMIT, the latter funded through the FP6 BiodivERsA Eranet by the German Federal Ministry of Education and Research (JS), the French ANR (TH). PN was funded by the Polish Ministry of Science and Higher Education (grant N N304 064139). We thank Michel Baguette and an anonymous reviewer for constructive comments.