## Introduction

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.