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 longdistance movement (Nowicki et al. 2005c).
Study area
The study area (3·8 × 3·5 km^{2}) 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 landscapeuse, 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 m^{2}, 50% of patches were smaller than 2500 m^{2}, 90% smaller than 10 000 m^{2} and the largest covered 25 600 m^{2}. 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 markrecapture 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.
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 predefined 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 recaptured 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:
 ( eqn 1)
Here, L(i,j) specifies the probability of a transition from start patch i to target patch j, with d_{i,j} being the Euclidian distance between the two patches, α the distance dependence of dispersal, A_{j} 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 densitypdf (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 distancepdf (the probability of a certain distance travelled) is thus proportional to 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 1p, respectively – of two negative exponentials with different values for dispersal distance (α_{1}, α_{2}). To achieve this, we modified S_{i,j} in the following way:
 ( 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 timedependent adjustment of α would significantly improve the fit of the model, i.e.:
 ( eqn 3)
with α’∋ being the timeadjusted 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 , 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 (α_{1}, α_{2}) in proportions of p and 1p), ‘Mixed + Area’ (mixed kernel with immigration scaling) and finally ‘Mixed + Area + Time’ (NEF with immigration scaling and time dependence of α_{2}, i.e. the longdistance 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 comparisonModel  BIC  Dispersal parameters 

α_{(1)}  ζ  τ  p  α_{2}  τ_{2} 

Distance dependence (shortdistance component)  Immigration scaling with target patch area  Time dependence of α_{(1)}  Fraction of shortdistance dispersers  Distance dependence (longdistance component)  Time dependence of α_{2} 


Simple  930·4  0·00218  n.f.  n.f.  n.f.  n.f.  n.f. 
0·00187 
0·00255 
Simple + Area  891·1  0·00229  0·5013  n.f.  n.f.  n.f.  n.f. 
0·00260  0·3462 
0·00198  0·6564 
Simple + Area + Time  891·4  0·00313  0·5021  0·1996  n.f.  n.f.  n.f. 
0·00387  0·3468  0·0528 
0·00239  0·6575  0·3464 
Mixed  919·7  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 + Area  871·8  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 + Time  873·4  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).