Accurately measuring the rate of spread for expanding populations is important for reliably predicting their future spread, as well as for evaluating the effect of different conditions and management activities on that rate of spread.
Although a number of methods have been developed for such measurement, all these are designed only for one- or two-dimensional spread. Species dispersing along rivers, however, require specific methods due to the distinctly branching structure of river networks.
In this study, we analyse data regarding Eurasian beavers' modern recolonization of the Czech Republic. We developed a new methodology for quantifying spread of species dispersing along streams based on representation of the river network by means of a weighted graph.
We defined two different network-based spread rate measures, one estimating the rate of range expansion, with the range defined as the total length of occupied streams, and the second, named range diameter, quantifying the progress along one or several main streams. In addition, we estimated the population growth rates, and, dividing the population size by the range size, we measured the density of beaver records within their overall range. Using linear regression, we compared four beaver populations under different environmental conditions in terms of each of these measures. Finally, we discuss the differences between our method and the classical approaches.
Our method provided substantially higher spread rate values than did the classical methods. Both population growth and range expansion were found to follow logistic growth. In cases of there being no considerable barriers in dispersal routes, the rate of progress along main streams did not differ significantly among populations. In homogeneous environments, population densities remained relatively constant over time even though overall population sizes increased. This indicates that at large spatial scales, the population growth of beavers occurs through progressive space filling rather than increasing population density.
Spatial spread of organisms is an abundantly studied phenomenon, mainly in relation to the negative consequences of invasions by alien species (e.g. Vitousek et al. 1996; Crooks 2002), but also to recolonization by native organisms (e.g. Lubina & Levin 1988). For effective management and/or conservation of an expanding population, it is important for several reasons to accurately measure the rate of its spread. First, the prediction of future spread, which is crucial for management or conservation planning, is usually based either on extrapolating spread rates estimated from historical data or on simple mechanistic models (Hastings et al. 2005; Gilbert & Liebhold 2010). Even in the latter case, observed spread rates are important for model evaluation. Furthermore, as Gilbert & Liebhold (2010) observed, accurate spread rate measurement enables one to evaluate the effect of different conditions and management activities on the rate of spread.
Three different approaches have been developed for measuring the rate of spread. The simplest approach, widely used by plant ecologists, consists of counting the number of records in an entire region then regressing its log-transformed values as a function of time (e.g. Pyšek & Prach 1995; Mihulka & Pyšek 2001; Delisle et al. 2003; Crawford & Hoagland 2009). The second, more spatially explicit approach is based on analysing range maps, either by regressing the square root of the range area as a function of time (Skellam 1951; Andow et al. 1993; Veit & Lewis 1996; Lensink 1998; Hill et al. 2001) or by estimating the mean displacement between range boundaries in two consecutive time steps (Andow et al. 1993; Lonsdale 1993; Tobin, Liebhold & Roberts 2007). The third, and substantially different, approach is based on measuring the distances between point observations rather than estimating the range boundaries. One possibility is to measure the distance from a new record to a certain fixed point which represents the source of invasion and to regress these distances as a function of time (Liebhold, Halverson & Elmes 1992). Alternatively, one can estimate the mean velocity of individuals' dispersal from distances between a new record and its assumed source (or parental) record (Suarez, Holway & Case 2001; Aikio, Duncan & Hulme 2010).
The principal limitation of the classical methods described above is that they are designed only for one- or two-dimensional spread (i.e. along a single line or over a plane) and that they implicitly assume radial continuous spread through a homogeneous environment (see 'Discussion' in Tobin, Liebhold & Roberts 2007 and Gilbert & Liebhold 2010). In river networks, however, neither of these assumptions in fact holds. A waterway's branching structure is neither one- nor two-dimensional, but rather something in between, and, as a river may represent preferential directions of spread, the landscape may be extremely heterogeneous. It appears, therefore, that the classical methods are not appropriate for measuring the rate of spread in river networks.
Although the population dynamics in river networks represent an area of increasing interest in recent years (Campbell Grant, Lowe & Fagan 2007), spatial spread in river networks has been investigated to date only by means of theoretical models. Theoretical approaches differ according to whether the models consider species dispersing only along (or in) the streams or also in the rest of the landscape. In the latter case, the spread is regarded as a two-dimensional process in a highly heterogeneous environment (Campbell, Blackwell & Woodward 2002; Jules et al. 2002), whereas in the former the process can be considered as homogenous (Johnson, Hatfield & Milne 1995; Campos, Fort & Méndez 2006; Bertuzzo et al. 2007). However, both of these approaches require some observed spread rate values for model evaluation and application in practice.
Despite that river networks serve as a dispersal vector or dispersal environment for many species, such as riparian plants, aquatic invertebrates, fish or semi-aquatic mammals, no study to our knowledge has addressed empirical measurement of spread rate with respect to their specific geometry. In this article, we suggest a possible methodology for such measurement, and we discuss the analogies and differences relative to the classical approaches developed for one- or two-dimensional spread.
Implementation of our method is based on representing the river network by means of a weighted graph, which enables us to compute the spread rate values efficiently using well-developed graph algorithms. The use of graph theory in ecology has a long tradition (see Bascompte 2007), although utilizing a graph as a direct representation of space first appeared after the pioneering works of Cantwell & Forman (1993) and of Urban & Keitt (2001). Finally, graph representation of river systems as dispersal environments has recently been applied successfully to conservation problems related to landscape connectivity (e.g. Schick & Lindley 2007; Erös et al. 2012).
To demonstrate the effectiveness of our methodology, we analysed the spread of Eurasian beavers (Castor fiber Linnaeus 1758) in four different regions within the Czech Republic. Eurasian beaver is an autochthonous large rodent with substantial impact on the occupied ecosystems and water regime of the cultural landscapes. Recently, beavers are of great conservation and management interest due to their rapid recolonization of the European continent. We estimated the population growths and rates of spread in four distinct populations. Although the rate of beavers' population development and spatial spread has been studied already (Hartman 1994, 1995; Halley & Rosell 2002), use of dense and spatially localized data collected over a long period of roughly 30 years enabled us to investigate it in greater detail. We also focused on the variation in population density during the spread. Using linear regressions, we compared the spread of several distinct populations.
Materials and methods
Eurasian beaver (Castor fiber) is a semi-aquatic herbivorous mammal and the largest European rodent. It is found in the Palaearctic ecozone (Veron 1992), from coastal waterways to high-mountain stream biotopes (Durka et al. 2005), although it prefers larger rivers with stable water level, which it requires for underwater entrance to the lodge. Where no such conditions are available, beavers are famous for their ability to ensure them by building dams.
In general, beavers' dispersion requires river network as a spreading platform (Novak 1987), although beavers are able occasionally to disperse across land from one watershed to another (Hartman 1994). Primary dispersal of single beavers occurs usually in the spring, when the subadult animals are driven out from the parental territory to find a new territory or to conquer an already occupied territory (Sun, Müller-Schwarze & Schulte 2000). In an expanding population, Hartman (1997) described also earlier dispersion of yearlings during the autumn.
The initial population constitution is driven by ideal despotic distribution (Fretwell 1972; Nolet & Rosell 1994) based on hierarchically structured habitat choice. The best biotopes are settled first, and the remaining (i.e. suboptimal) habitats are occupied only as the best are already claimed. This spatial pattern forces dispersal movement so that the dispersers and pioneers will be able to acquire more distant, but superior locations. Another cause of long-distance dispersal events is the need to search for a mate (Svendsen 1989; Sun, Müller-Schwarze & Schulte 2000). Dispersal distance ranges from several kilometres up to more than 80 km (Heidecke 1984; Sun & Müller-Schwarze 1996), but most young beavers (80%) attempt to colonize sites within 5 km from their parental territories (Nolet & Baveco 1996; Sun et al. 2000; Saveljev et al. 2002).
During the 12th to 19th centuries, beavers were massively hunted in Europe (Veron 1992), particularly for their pelts, and their natural range was reduced to eight small and isolated refugia (Nolet & Rosell 1998). At the beginning of the 20th century, beavers were close to extirpation. Nevertheless, following a series of reintroductions from the 1920s to the end of the century (Nolet & Rosell 1998), beavers recently have colonized large parts of their former European ranges (Halley & Rosell 2002), and the process is still ongoing.
The modern colonization of Europe by beavers is relatively well documented, especially due to the visible marks of occupancy in the form of freshly cut trees. In Central Europe, the process originates from several sources (Šafář 2002). In 1977, animals from an Austrian reintroduction programme (Kollar & Seiter 1990) had spontaneously colonized the extensive and widely spread low-lying floodplain forests along the rivers Morava (‘March’ in German) and Dyje (‘Thaya’ in German) of the South Moravia region. In 1979, dispersers from a Bavarian reintroduction programme (Schwab & Schmidbauer 2000) started to cross a watershed divide by means of a broad colonization front to reach the River Berounka watershed in Western Bohemia. Conditions are considerably poorer for inhabitation in this region, as riparian stands are lower in quantity and diversity and more highly fragmented in comparison with other regions. In 1991, the North Bohemian region was settled by free-ranging animals originating from a German autochthonous population (Babik, Durka & Radwan 2005). In this region, the quality of critical resources (woody vegetation and water environments) rather corresponds to those of floodplain forests, but with a high level of fragmentation. Spatial spread is constrained by steep valley sides and a large weir at Střekov, which creates an insurmountable obstacle for beavers' dispersal. Finally, in a reintroduction programme, 20 animals were translocated from Lithuania and Poland and released into the Central Morava basin during 1990–1991 (Kostkan & Lehký 1997). Local habitat conditions are close to those of the South Moravian low-lying floodplain forests, with no restrictive circumstances.
To analyse the beavers' spatial spread in the Czech Republic, we used historical records summarized by Šafář (2002) and by Anděra & Červený (2004), as well as our own unpublished data bases developed under a long-term monitoring programme underway at the Czech University of Life Sciences Prague since 1998. These records consist of all evidenced marks of beaver occupancy in the Czech Republic during the study period, including beaver buildings, observed animals, road-killed animals, felled or nibbled trees, footprints, etc. Only those records including overwinter occupancy were considered, i.e. recording of shelters, dams, observed breeding, isolated large clumps of felled trees. We removed all uncertain dispersal data or unclear evidence of year-to-year occupancy (e.g. single treefalls or nibbles, single animals, cadavers, etc.). We also included only records with sufficient spatial and temporal accuracy, i.e. those for which we were able to determine a year of observation and a concrete place near water bodies. We combined information from these sources to develop a data base consisting of historical evidence of beaver settlement in the area.
Thus, we developed a spatially oriented sequential primary data base of beaver's spatio-temporal spread. We used the application SurveyPro running within freeware Janitor 1·0 (CENIA GIS Laboratory, http://janitor.cenia.cz, Žďár nad Sázavou, Czech Republic) to create a point layer from the beaver data base. Subsequently, we used ArcGIS 9·2 (ESRI, Redlands, CA, USA) for placing the point data onto the river network layer provided by the T.G. Masaryk Water Research Institute, Prague (the DIBAVOD data base, available for free at http://heis.vuv.cz).
Our data consisted of the following four populations (see Fig. 1): (1) Morava-south, originating in an Austrian population on the River Danube; (2) Morava-central, initiated by human-controlled reintroduction; (3) Berounka and (4) Elbe populations originating from two different populations in Germany. Although several other populations existed in the Czech Republic, we chose only those for which sufficiently long and accurate time-series records were available.
As can be seen in Fig. 1, the two Morava populations share the same watershed and overlay. There was no possibility to distinguish between these two populations directly in the data, inasmuch as we did not know to which population a particular record would belong. We nevertheless decided to treat them as distinct, because of their quite different colonization histories (spontaneous dispersal for Morava-south vs. reintroduction for Morava-central). To achieve this, we assigned each record to one of these populations by tracing the series of dispersal trajectories connecting the record with its predecessors. These dispersal trajectories have been estimated in the same way as is described below (see section 'Spread rate measures').
River network representation
We represented a river network by means of a weighted graph. For the sake of clarity, we summarize here some basic definitions from graph theory used in this article. A graph G consists of a set of vertices V(G), set of edges E(G) and mapping assigning to each edge a pair of vertices that it connects. A weighted graph is a graph followed up with a mapping that assigns to each edge e a certain real number w(e) called its weight. A path between vertices v0 and vn in a graph is a sequence of vertices and edges of the form v0,e1,v1,e2,v2,…,en,vn, where the edge ei connects the vertices vi and vi−1 for i = 1,2,…,n, and where no vertex occurs more than once. In a weighted graph, the length of a path is the sum of the weights of its edges, and therefore the shortest path between two vertices is the path between these vertices with the least length. The so-called graph diameter is defined as the longest of all the shortest paths in the graph. A subgraph G* of a given graph G is defined as a graph for which V(G*) ⊆ V(G) and E(G*) ⊆ E(G).
In our case, the vertices are of two types, representing both individuals (i.e. beaver records) and stream ends (i.e. the confluences, sources and estuaries), and the edges represent river reaches (i.e. the stream segments between any types of vertices) with weights defined as the reaches' lengths (see Fig. 2).
We created the graph and performed all further computations using our own computer program written in C# programming language and built in Microsoft Visual Studio 2008 (Microsoft, Redmond, WA, USA). Both the program and the source codes are available at http:/fzp.czu.cz/~bartakv/spreadanalyst.
Spread rate measures
First, we estimated the population size as the cumulative number of records. Although it reflects primarily the population growth rather than its spatial spread, this provides information about the number of possible dispersers. We plotted log-transformed values of cumulative number of records against time and performed linear regression in the linear part of the plot, which represents a phase of exponential population growth. We used the regression slope from the linear part as a quantification of population growth rate. The linear parts were identified visually, because the more objective piecewise regression method (Toms & Lesperance 2003) could not be used in this instance. The reason for this is that, because the break points are located close to the edges of the series, there are too few data points to fit a separate regression line before (or after) the break point. The same applies to the other linear regressions with break points in the rest of the article.
The second measure was the range expansion, i.e. the temporal variation in the range size. As we consider species dispersing along streams and occupying their close vicinity, we define the range as the union of all ‘occupied streams’ and range size as the length of such a union. We consider a stream to be ‘occupied’ when it has been used for dispersal or is a part of a territory. This requires the estimation of dispersal trajectories as well as locations of territories. As we had no information about where a particular individual came from, we approximated its dispersal trajectory by the shortest path connecting that individual with the nearest individual already settled down. We identified the shortest paths by means of Dijkstra's algorithm, which is designed for finding shortest paths in a weighted graph. We considered the territory of any individual to consist of a 2 km part of a stream around the individual, 1 km on each side.
We also needed to identify individuals that were ‘source’ or ‘introduced’ (i.e. that had no parents in the data set). We did not estimate dispersal trajectories for such individuals. In the Elbe and Morava-south data sets, we considered the first year of the series as consisting of the ‘source’ individuals. That is to say, we assumed that these individuals had come from the source populations in Germany and Austria respectively. In the Morava-central data set, we denoted as ‘introduced’ all eight and one records originating from human-controlled reintroduction during 1990–1991 and 1996 respectively. Finally, in the Berounka data set, where apparently were several separate infiltrations from the source German population, we considered as ‘source’ individuals those records for which the estimated dispersal trajectories seemed to look highly unlikely.
As the temporal growth of the range size appeared in most cases to be roughly exponential (or, in fact, logistic; see Fig. 4), we estimated a relative measure of range expansion as the regression slope in linear parts of the log-transformed values of range sizes plotted against time (i.e. the same procedure we used for the population growth analysis).
As the temporal growth of the range size reflects the progress along the main stream as well as the colonizing of small tributaries, we also were interested in measuring the progress only along one or several main directions. Such a measure would reflect the velocity of the ‘dispersal front’ rather than the rate of range expansion. We suggest that an appropriate measure can be the range diameter, defined as the diameter of a subgraph that defines the range. Note that this measure is analogous to the range radius used in classical, two-dimensional methods (see 'Introduction').
In the time vs. range diameter plot (see Fig. 5), we identified visually the break point between the initial lag phase and successive phase of rapid and roughly linear growth, as well as a possible break point between such linear growth and successive slowdown, and then we performed linear regression on that linear part. As we expected a terraced range diameter growth rather than one purely linear, and because such a series would suffer from dependent errors, we used only those points in which a change in diameter value (in comparison with the previous year) appeared. Then we used the regression slopes as the spread rate values.
We estimated the population density as the cumulative number of records divided by the range size, and we plotted it against time. In this plot, we removed from each series several first data points equalling or very close to the value of 0·5 territories per km, as this value reflects the small initial ranges consisting of pioneers' territories rather than the expanding range between them. In such a shortened series, we performed linear regressions to assess the overall temporal trends in densities.
Whenever we used linear regression, we compared slopes of different populations using analysis of covariance together with an incremental F-test for comparing the full model (i.e. with slopes and intercepts differing among populations) with the model without interactions (i.e. with common slope for all populations). When the differences between slopes were significant, in the next step we used analysis of covariance for each pair of populations separately and tested the significance of an interaction term. In these post-hoc tests, we applied Bonferroni significance level adjustments for multiple comparisons. (Hence, if we compared four regression slopes at overall significance level 0·05, this corresponded to six individual tests and so the minimum significance level required for rejecting the null hypothesis in each of these tests was 0·05/6 = 0·008. Similarly, when comparing three regressions, the required significance level was 0·017.)
Comparison with classical methods
We compared the network-based range diameter growth defined above with a two-dimensional range diameter growth. The latter has been computed as the classical range radius, defined as the radius of a circle with an area equalling the range area, multiplied by two. The range area has been estimated as the number of quadrats occupied, multiplied by the quadrat size. We used the quadrats from the widely utilized KFME scheme (Niklfeld 1971). The two-dimensional range diameter growth has been estimated as the slope of the regression line in the (visually identified) linear part of the time vs. range diameter plot.
The exponential growth periods were determined visually as follows (see Fig. 3): 1977–2002, 1991–2001 and 1980–2011 (i.e. the whole data set) for the Morava-south, Morava-central and Berounka populations respectively. As there was no exponential phase determined in the Elbe population growth, this population was removed from the linear regression.
Regression slopes of the log-transformed population sizes ranged roughly from 0·05 to 0·09 and all differences between them were statistically significant (see Tables 1-3). The Morava-central population showed the fastest population growth (regression slope 0·088), the slowest Berounka population had slope value only half that of the fastest (0·048) and the Morava-south population was in between these (0·068).
Table 1. Results of linear regressions. Significant slopes are bold. The Elbe population has been omitted from the linear regression of population growth and range expansion due to its lack of exponential growth
Table 2. Results of incremental F-tests. The models 1, 2 and 3 refer to the model with a single regression for all the populations, model with differing intercepts, and model with both intercepts and slopes differing respectively. The residual degrees of freedom (DF) of individual models and the F statistics of incremental F-tests are shown. Asterisks beside an F statistic refer to significant improvement of the model's fit in comparison with the previous one, with ** and *** corresponding to P < 0·001 and P << 0·0001 respectively. For the various measures, the acronyms MS, MC, BE and EL in the brackets refer to those populations being compared
(MS, MC, BE)
(MS, MC, BE)
(MS, MC, BE, EL)
(MS, MC, BE)
(MS, MC, BE, EL)
Table 3. Differences between regression slopes. The populations are coded as MC, MS, BE and EL for Morava-south, Morava-central, Berounka and Elbe respectively. The Diff. columns contain absolute differences between regression slopes from analysis of covariance. The P columns contain corresponding P values (i.e. the P values for the interactions among time and populations). Significant values are bold (after Bonferroni correction)
MS x MC
MS x BE
MS x EL
MC x BE
MC x EL
BE x EL
In the plots of log-transformed range size values (Fig. 4), we determined the linear parts as follows: 1997–1999, 1993–1999 and 1980–2011 (again, the whole data set) for the Morava-south, Morava-central and Berounka populations respectively. Again, there was no exponential growth distinguishable in the Elbe range expansion curve, so we did not perform linear regression for this population.
The regression slopes (and therefore the range expansion rates) of both Morava populations were almost the same (0·095 and 0·094 for the Morava-south and Morava-central populations respectively), both being considerably greater than the slope of the Berounka population (0·063). The only significant difference, however, was found between the Morava-south and Berounka populations (that difference being 0·032).
In the range diameter plot, we identified the following linear growth periods (Fig. 5): 1985–1999, 1990–2000, 2001–2010 and 1992–2001 for the Morava-south, Morava-central, Berounka and Elbe populations respectively. Although there were significant differences among the regression slopes (see Table 2), the range diameter growth rates of the Morava-south, Morava-central and Berounka populations did not differ significantly, being all about 15–20 km yr−1 (15·4, 18·2 and 20·8, respectively; see Tables 1 and 3). The rates of all these populations differed significantly, however, from the diameter growth rate of the Elbe population, which was near to stagnation (0·8 km yr−1).
For the reason described above (see 'Method'), we removed the first data point from the Berounka and Elbe time series, as well as the first two data points from the Morava-south and Morava-central series (see Fig. 6). There was no linear trend in the population density of the Morava-central population and significant, but very slow increase (0·0003) of the Morava-south population density (see Table 1). The density values of both these populations were very similar (the average values were 0·113 and 0·147 territories per km for the Morava-south and Morava-central populations respectively). The population density of the Berounka population was decreasing at the rate of −0·006, ranging roughly from 0·3 to 0·1 territories per km, and those values were fluctuating much more around the trend or mean than were the values of any other populations (see Fig. 6). The population density of the Elbe population was greatest (ranging roughly from 0·2 to 0·35 territories per km) and rapidly increasing (with the rate of 0·015). Almost all differences between slopes were significant, exceptions being the differences between Morava-south and Morava-central and between Morava-central and Berounka.
Comparison with classical methods
For linear regression of two-dimensional range diameter (Fig. 7), we used periods of linear growth from 1992 to 1999 for the Morava-south population; from 1998 to the end of the series for the Berounka population; and the entire series for the Morava-central population. The Elbe series could not be used for linear regression due to its nonlinear growth. The diameter growth rate values were 2·3, 3·4 and 1·5 km yr−1 for the Morava-south, Morava-central and Berounka populations, respectively, and they all differed significantly from one another.
The presented method is based on two important assumptions about the data and the organism, implying limitations on its use. First, the method is only designed for species dispersing entirely along a river network, and moreover, both upstream and downstream. Secondly, the method only deals with point observations.
Obviously, the method can be easily modified for downstream or upstream dispersal by using an oriented graph for river network representation, in which the edge orientation would reflect the direction of the flow. Another possibility would be to include a preference for down or upstream movement by weighting the distances between individuals during estimation of dispersal trajectories (i.e. distances downstream would have different weight than distances upstream). Although there is evidence that beavers disperse further downstream than upstream (Heidecke 1984), we have no information as to how strong is such a preference and whether it varies among populations, and thus we could not include it into our dispersal model.
The restriction of the dispersal to a river network, i.e. to a particular watershed, is crucial, as it represents a basis for graph representation of the dispersal trajectories. In the case of beavers, one can argue that such assumption neglects an occasional overland dispersal. As overland dispersal typically occurs either during initial spread or from a fully saturated watershed (Hartman 1994, 1995), and as beavers are subject to much higher mortality when moving over land than by streams (Halley & Rosell 2002), we argue that the overland dispersal can be ignored.
The use only of point observations is not very restrictive, as these represent a highly frequent type of data (e.g. Sharov et al. 1995; Veit & Lewis 1996; Wehtje 2003; Tobin, Liebhold & Roberts 2007; Aikio, Duncan & Hulme 2010). Although the method does not require information about the actual dispersal trajectories, it does depend heavily on their proper estimation. While we used the simplest approach, approximating the dispersal trajectories by shortest paths to the nearest already settled neighbours, this could obviously be replaced by other dispersal scenarios in accordance with available knowledge about the true dispersal process (for use of different dispersal scenarios, see Suarez et al. 2001 and Aikio et al. 2010). Inasmuch as beavers tend generally to colonize in close vicinity to their parental territories (Saveljev et al. 2002), we reason our approach to be appropriate for them. This is, in fact, a simplifying assumption of continuous dispersal without significant long-distance jumps.
Another simplification is that we ignore the inflow of dispersers from surrounding areas, and particularly from the source population. Once we have determined the ‘source’ records, we expect that these are predecessors of all other, ‘non-source’ records. Thus, the extent of inflows can be underestimated. As we have no information about actual dispersal trajectories, however, we have no real possibility to quantify the actual inflow of dispersers, and therefore we must accept some simplifying assumption. Moreover, most of the populations studied here have been founded either by pioneers coming long distances from their parental populations or they have been reintroduced, translocated individuals. Hence, at least in the early stage of colonization, we would not expect substantial inflow that would violate our assumptions.
There is also a fundamental question as to the necessity of measuring the range size on a river network instead of using classical two-dimensional measures. If the rates of spread measured on the network would not differ significantly from those measured on the plane, there would be no reason for using a more sophisticated method requiring additional information about the network position. Comparing the results of range diameter analysis with analogous findings from the classical, quadrat-based (Table 1), one can see that the values of the latter are only one-third to one-half of those from the network-based analysis. Hence, it is clear that the methods yield substantially different values.
In fact, these two approaches yield rather different kinds of information. The result of the classical two-dimensional range-radius analysis indicates how quickly the dispersal front (assuming continuous radial spread) moved over a certain Euclidian distance on the plane. Since in reality, however, the process runs on a network, such a value would vary on different networks, even when the process is the same. Therefore, we need another, network-independent measure of range size which should describe the process as it really works, i.e. along a network. The proposed network-based measure satisfies such a requirement, and we suggest it is appropriate for measuring the rate of spread in river networks.
Measuring spatial progress on the kind of data we used is obviously dependent on the quality of the data recording process. Although we used relatively rigorous restrictive assumptions for incorporating the published data into our data sets (see 'Method'), there are certain limitations. Mainly, the majority of records are not based on regular and systematic monitoring. On the one hand, this might cause some occupations to be missed both in space and time, and hence our population and range values are possibly underestimated. On the other hand, as will be seen from the following discussion, all our populations (except that for the Elbe with highly specific circumstances) exhibited very similar population developments that are in general accordance with the literature. Furthermore, there is no reason for the recording rates to be substantially different in the various areas. Hence, the possible underestimation is likely to be regularly distributed over time and without significant impact on the comparisons among populations. Moreover, the great advantage of recording beaver settlement is that their activity is well visible in the terrain. This helps to ensure proper and early documentation, and it minimizes the potential for misinterpretation of the studied processes.
The linear trends in most of the series in Fig. 3 (see also Table 1) indicate that beavers (at least in a certain phase) follow an exponential population growth pattern, which is in accordance with previous studies (Heidecke 1984, 1991; Balodis 1990). Some studies (Heidecke 1984, 1991) also have reported a characteristic slowdown coming after the exponential phase of beaver population growth. This indicates that the growth is in fact logistic rather than exponential, which is rather to be expected for any population not having interspecific interactions and is well documented for a number of species (see e.g. Krebs 1985; Mduma, Sinclair & Hilborn 1999; and many others). In the Morava-central and Morava-south populations, we distinguished such slowing in 2001 and 2002 respectively (see Fig. 4). The similar timing of these slowdowns, although possibly indicating sampling bias, corresponds with the time when the two populations met one another, as well as the time when the Morava-central population reached the boundary of the Morava watershed. Therefore, the slowdowns can be regarded as expected results of the finite area's carrying capacity being approached. In the Elbe population, the initial growth slowed after a short time of 4 years and then remained almost unchanging. Here, the explanation by way of carrying capacity is obvious, as both the population growth and spatial spread were constrained by physical and orographic barriers.
As can be seen in Fig. 4, range expansion also seems to follow exponential or logistic growth. Also here, we identified final slowdowns after exponential growths in both Morava series, as well as analogous slowing in the Elbe population. In fact, these periods of slowing were even more apparent and lasted longer than those during population growth, which indicates that the effect of a finite area first terminates the spatial spreading and then reduces the population increase, a pattern already documented by Hartman (1995).
Progress along the main directions
An apparent and expected feature of the range diameter curves (Fig. 5) is terraced growth, especially in the two Morava series. This can easily be explained by the nature of the process of progressing along the main stream (i.e. the process to be described by range diameter growth), which includes long-distance dispersal events. In searching for optimal habitat or for possible partners, some animals can travel relatively long distances along the main stream and thus increase the range diameter, while other animals colonize the remaining habitats lying along the path travelled by that pioneer and without increasing the range diameter. In other words, the terraced nature of the range diameter growth reflects a jump-dispersal at smaller spatio-temporal scale, whereas the overall linear trend reflects a continuous dispersal at larger scale.
The lag phases distinguished in the beginning of the Morava-south and Berounka series (1977–1985 and 1980–2000, respectively; see Fig. 5) indicate that beavers needed a certain time before they reached the main stream and started to progress along it. Our results suggest, however, that without considerable barriers (such as are those facing the Elbe population), and once the main stream is reached, the progress along that stream has a roughly linear trend with a rate of about 15–20 km yr−1. It would be desirable to compare this finding with those of other populations to verify whether or not it represents a general pattern.
Although our density values represent only rough and probably undervalued estimates, they are in general agreement with the range of values reported from previous studies (reviewed in Novak 1987). A notable feature of the density plot (Fig. 6) is similarity between the two Morava populations, as both fluctuated in a similar range (roughly from 0·1 to 0·15 territories per km), with either no or just a slight trend, and with similar variance. This suggests that during the spatial spread in a homogeneous environment without spatial constraints, the population density tends to be relatively low and stable across populations as well as over time. This can be regarded as another consequence of preferential habitat choice or the mating process described above, as by searching for optimal habitats (or partners) beavers first progressively fill the watershed, thus forming a relatively sparse population, and then while the remaining habitats are filled, the density gradually increases (Hartman 1995).
In the Berounka basin, by contrast, there was a substantially decreasing trend in population density and with large fluctuations around that trend. Moreover, the initial density was relatively high, which seems to contradict previous beaver studies reporting lower densities in poorer habitat conditions (e.g. Novak 1987; Hartman 1994; Müller-Schwarze & Schulte 1999). This course is an effect of fragmented habitat, which first caused an accumulation (and thus high density) of beavers in the relatively suitable habitat at the upper part of the basin, and, after overcoming unsuitable parts downstream (roughly around 2000), it led to accelerated spread over long distances between habitat patches.
The method has been proven usable for species occupying river networks and dispersing along them. Moreover, it yields substantially higher spread rate values than do classical, two-dimensional methods. The explanation is that in our method, the population progress is described along the actual network rather than over a plane, i.e. it reflects the particular network geometry. As a result, the spread rate values are much less dependent on the actual network geometry than are the values obtained by classical methods, and therefore they are better applicable for predictions in another network.
Beavers' population growth as well as range expansion seems to follow a general logistic pattern, with the carrying capacity determined by the finite area of the particular watershed. In the presence of one or several main preferential directions of spread, the progress along these seems to be generally linear, with a speed of about 15–20 km yr−1. Although the spatial spread can be described by means of general trends, at a smaller scale which depends on the level of habitat fragmentation it consists of fluctuations probably caused by long-dispersal events. In a homogeneous environment, the overall population density seems to be maintained at some roughly constant and relatively low level during the process of spread – at least until the population reaches the watershed boundary (or another barrier). This can be also explained by long-dispersal, leading to progressive space filling that precedes the increase in local population densities. In a heterogeneous environment, however, the density can be either high and increasing, as the population accumulates in an area of suitable habitat, or low and decreasing, as preferential habitat choice leads to rapid range expansion of a small population.
This work was supported by the grant of the Ministry of the Environment of the Czech Republic, no. SP/2D4/52/07, Analysis of parameters for prediction of dispersion and dispersion model of European beaver in the ecosystems of Central Europe. We thank Vítek Moudrý for his help with transcribing the data into GIS form. We also thank Lenka Válková, Lenka Hamšíková and Jana Korbelová-Křováková for searching out the oldest beaver distribution data. We also thank two anonymous reviewers for their valuable comments which greatly improved the manuscript.