European Journal of Wildlife Research

, Volume 60, Issue 2, pp 271–278

Measuring effects of linear obstacles on wildlife movements: accounting for the relationship between step length and crossing probability

  • Sindre Eftestøl
  • Diress Tsegaye
  • Ivar Herfindal
  • Kjetil Flydal
  • Jonathan E. Colman
Original Paper

DOI: 10.1007/s10344-013-0779-7

Cite this article as:
Eftestøl, S., Tsegaye, D., Herfindal, I. et al. Eur J Wildl Res (2014) 60: 271. doi:10.1007/s10344-013-0779-7


Animal movements in the landscape are influenced by linear features such as rivers, roads and power lines. Prior studies have investigated how linear features, particularly roads, affect movement rates by comparing animal's movement rate measured as step lengths (i.e., the distance between consecutive observations such as GPS locations) before, during and after crossing of a linear feature. The null hypothesis has been that the length of crossing steps should not differ from other steps, and a deviation from this, mainly that steps are longer during crossing, has been taken as support for a disturbance effect of the linear feature. However, based on the simple relationship between the length of a step and its probability to cross a linear feature, we claim that this assumption is inappropriate to test for behavioural responses to linear features. The probability is related to the proportion of the total length of the trajectory (i.e., the path of movement) a step constitutes. Consequently, care should be taken when formulating hypotheses about how animal moves in relation to linear features in the landscape. Statistical tests should be set up with respect to the expected length based on the distribution of step lengths in the trajectory. We propose two methods that accounts for the bias in crossing frequency that is caused by step lengths, and illustrates their applications by using simulated animal trajectories as well as empirical data on reindeer in an area with a power line.


Animal movement Bias Crossing probability Linear features Random trajectories Step length 


Animals' habitat includes natural physical obstacles, such as lakes, rivers or hillsides, and areas of avoidance due to increased predation risk such as river banks or other open areas without refuge, all which affect animals' movement patterns (With et al. 1997; Brown and Kotler 2004). In addition, habitats are often intersected by human made linear structures, such as roads, railways, power lines and pipe lines, or linear features associated with high human activity, like tourist paths or ski trails (Fahrig and Rytwinski 2009; Klein 2000). Such structures often involve increased caution by animals, particularly if they are associated with human activity (e.g., Stankowich 2008; Anttonen et al. 2011; Eldegard et al. 2012; Polfus et al. 2011). However, areas along linear structures may also favour some animals. Many species, both predator and prey, have higher movement abilities along linear structures (James and Stuart-Smith 2000), and food availability is often higher due to clear cutting and edge effects (James and Stuart-Smith 2000; Gill et al. 2001; Heithaus and Dill 2002; Small and Hunter 1988). Moreover, scavengers are found to follow linear structure such as fences and power lines due to the increased abundance of carcasses from collisions (e.g., Smith et al. 2008; Lambertucci et al. 2009; Knight and Kawashima 1993). For prey species, there is a trade-off between predator avoidance and foraging (Kuijper et al. 2009; Frair et al. 2005; Hebblewhite and Merrill 2009), but if human or predator activity is high, such as is the case for many linear structures, it is expected that prey species would decrease their use of nearby areas (Forman and Alexander 1998; Colman et al. 2012; Beyer et al. 2013).

From the factors listed above, it is expected that under equal resource levels, movement rate is higher in areas with high predation risk, including human activity resulting in disturbance because disturbance stimuli can be viewed as analogous to predation risk from an evolutionary perspective (e.g., Murphy and Curatolo 1987; Frid and Dill 2002). A physical obstacle is expected to result in a decrease in movement rate when crossing it. A barrier of fear, or a linear disturbance that is avoided by the animals, may result in an increase in movement rate when crossing, while the movement rate decreases when the animal gets outside the avoided and perceived as risky area (Dussault et al. 2007; Forman and Deblinger 2000; Leblond et al. 2013; Panzacchi et al. 2013).

Technological developments over the last 15 years have enabled a large number of studies on habitat use and movement patterns from a wide range of species (Frair et al. 2010). Specifically, GPS data have frequently been used to investigate how wildlife behaviour is affected by anthropogenic activities (e.g., Leblond et al. 2013). GPS technology provides data for both area use and movement rate from long series of consecutive animal positions taken at constant time intervals. The movement rate can simply be calculated as the distance divided by time between two consecutive animal positions (i.e., the step length, Dussault et al. 2007).

Despite underestimating distance moved (Rowcliffe et al. 2012), methods based on data of straight lines between consecutive GPS positions provide relative estimates of movement rate that are convenient for relating animal movement to factors such as individual characteristics, predator level, human activities, vegetation or other landscape features (Dyer et al. 2002). For instance, studies on how linear features affects animal behaviour compare the length of the crossing step with other steps in general, or the steps nearest in time prior to and after crossing (e.g., Dussault et al. 2007; Leblond et al. 2013; Panzacchi et al. 2013). Steps crossing a linear feature, such as a road, are found to be significantly longer than other steps, which is taken as evidence for a behavioural change supporting the hypothesis that animals associate the linear feature with fear (Dussault et al. 2007; Leblond et al. 2013). We argue that such results may contain a systematic bias, based on the relationship between the length of a line (here, the step length) and its probability to intersect another line (here: the linear feature). In a random setting, we expect and test whether the probability of intersecting a linear feature is positively related to step length. If this is the case, then the assumption must be adjusted for the relationship between step length and crossing probabilities, and the statistical test modified accordingly.

We present mathematically how the length of a step is related to whether it crosses a linear feature or not, given that the trajectory the step belongs to has crossed the feature, and propose methods to assess whether the length of crossing steps deviates from what is expected based on theory. By simulating movement trajectories that are random with respect to linear features, we show how previous findings may contain a methodological bias regarding the relationship between step length and crossing probability. Finally, we use empirical data from wild reindeer to compare movement rate before, during and after crossing both for artificial linear features placed randomly in the landscape and for real power line located within the home range of animals.


Theoretical basis

We consider the relationship between step length and crossing frequency to be a probability distribution. Consider a trajectory (i.e., the curve or path with consecutive GPS positions of an animal movement) consisting of j = 1 to n steps in an area that has one linear feature. If all steps are of equal length, the probability that each step has crossed the linear feature, given that the linear feature has been crossed one time, is 1/n. However, animals move at various speed, which generates variation in step lengths if they are sampled at equal time interval. If we know that the linear feature has been crossed once by the trajectory, the probability that step i is the one that has crossed the linear feature is:
$$ P\left({X}_i/1\kern0.5em \mathrm{crossing}\right)=\frac{L_i}{{\displaystyle \sum_{j=1}^n{L}_j}}, $$
where Li is the length of step i, and the sum of Lj in the denominator is the total length of the trajectory. Accordingly, the probability that a step has crossed the linear feature is proportional to the part the step constitutes of the trajectory, and longer steps have a higher probability to cross the linear feature (Fig. 1a). If we know that the linear feature has been crossed several times by the trajectory, and that each step can cross a linear feature several times, we can write this as:
Fig. 1

The probability that a step crosses a linear feature in relation to its length. a The probability is shown for trajectories with different number of steps (N), but all trajectories have crossed the linear feature only once. b All trajectories consist of N = X steps, but the number of times the trajectory has crossed the linear feature (y) varies. The gray lines are probabilities from a distribution where a step can cross the linear feature more than once (Eq. 2), whereas the black lines are from the Wallenius non-centralised multi-hypergeometric distribution where each step can cross the linear feature only one time

$$ P\left({X}_i/y\kern0.5em \mathrm{crossing}\right)=1-{\left(1-\frac{L_i}{{\displaystyle \sum_{j=1}^n{L}_j}}\right)}^y, $$
where y is the number of times a linear feature is crossed by the trajectory. The relationship between Li and P(Xi|y crossing) will then be an asymptotic increase towards 1, where the increase is more rapid for high values of y (Fig. 1b). Thus, longer steps that constitute larger parts of the total length of the trajectory will have a higher probability of crossing the linear feature than shorter steps that only make up low proportions of the trajectory. Equation 2 assumes that a step can cross the linear feature several times. However, animal steps are normally represented as straight lines between two observations (e.g., GPS locations), and often the shape of the linear feature also is close to a straight line. In such instances, a step can only cross the linear feature only once. This gives a crossing probability with respect to proportion of the total length of a step of the type Wallenius non-centralised multi-hypergeometric distribution (WNCMH; Fog 2008). The difference between WNCMH and the distribution following Eq. 2 is that longer steps get a higher probability with the WNCMH, particularly when the trajectory has crossed the linear feature many times (Fig. 1b). When number of crossings (y) = 1, the two distributions are identical and the relationship between step length (measured as the proportion a step constitutes of the total trajectory) and crossing probability is linear (Fig. 1b). As y increases, the relationship becomes more non-linear. However, at low probabilities (P < 0.3), both distributions show close to linear relationships between step length and crossing probabilities (Fig. 1b). In general, the theory predicts either a linear (for a trajectory with only one crossing) or nonlinear (for a trajectory with several crossing) relationship between the step length and probability of crossing.


We simulated animal movement by generating 100 correlated random walk (CRW; Kareiva and Shigesada 1983) trajectories using the function simm.crw in the package adehabitatLT (Calenge 2006, 2011) in R (R Development Core Team 2012). The step lengths were drawn from an exponential distribution as this was more similar to our reindeer data (see below). The correlation parameter for the turning angle (r) was set to 0.2. The length of each CRW was 999 steps. On each CRW trajectory, we added one random straight line that entered the centre of the random walk in a random direction (Fig. 2a). These represented linear features, which the CRW was not influenced by.
Fig. 2

Results from the simulated correlated random walks. a Example of a simulated CRW and a linear feature (gray line) added after the CRW is created. b Frequency distribution of step lengths from 100 CRW, each with 999 steps. Gray bars are the total distribution of step lengths. Hashed bars are the length distribution for the steps that crossed the randomly placed line. The solid and dashed gray vertical lines indicates the mean step length of all steps and crossing steps, respectively, whereas the vertical black solid line shows the predicted length based on theory. c The relationship between proportion of total trajectory length a step constitutes, and the probability that it crossed the random line, fitted with a generalised linear mixed model with binomial family and cloglog link, and trajectory identity as a random factor. d The length of crossing steps (C) and the three preceding (C − 1, C − 2, C − 3), and following steps (C + 1, C + 2, C + 3); with standard errors

Reindeer data

We used data from 19 GPS-collared female reindeer from the Setesdal Aust wild reindeer area (7°42′E and 59°10′N), Southern Norway, during the years 2007–2012. The animals were captured (six animals in March 2007, four in March 2009, three in March 2011, and six in March 2012) and marked with GPS collars following procedures described in Reimers et al. (2012). The GPS collars were programmed to register the animals' position every 3 h, and the average lifetime for the GPS collars was 3 years. To avoid effects from more directional and higher movement rate during migration, we limited this study to only include data from the summer range, i.e., 1 June until 30 August. A total of 30 trajectories (mean number of steps per trajectory, N = 712.5, range 375–734) from 14 reindeer were used. A 132-kV power line crosses the summer range centrally and there are several trails in the area, but no roads intersect it. We tested if reindeer steps that crossed the power line were significantly longer or shorter than expected. Moreover, we added ten power lines (same length as the real one) randomly with respect to direction within the study area, but aimed to cover the entire study area, and assessed whether the steps that crossed the random power lines deviated from what was expected.

Data analysis

We propose two methods for assessing if the length of crossing steps is different from the expected based on the theory outlined above. The first is a simple comparison of the length of crossing steps against the predicted length given the distribution of step lengths and number of crossings by the trajectory. The predicted length is based on the probability that a step in a trajectory should cross a barrier, given the distribution of step lengths and total number of crossing. We used the WNCMH distribution as this best corresponds to our simulated and the reindeer data. Crossing probabilities (pi) were obtained using the function dMWHDSL in package BiasedUrn (Fog 2011), modified to allow for up to 10,000 different types of steps (i.e., step lengths). The proportion a step constitutes of the total trajectory length was entered as the steps bias in the function. The predicted length of crossing steps \( \left({\widehat{L}}_{{}_{\mathrm{c}}}\right) \) for a trajectory with N steps and y crossings is obtained by weighing the length of each step by its probability for crossing, divided on the total number of crossings:
$$ {\widehat{L}}_{{}_{\mathrm{c}}}={\displaystyle \sum_{i=1}^N}\frac{p_i{L}_i}{y}, $$
where pi and Li is the crossing probability and length of step i. We compared predicted and observed length of crossing steps using t-tests, where we ln-transformed the observed and predicted length to reduce heteroscedasticity.

The second approach utilises the linear relationship between step lengths (proportions of total length) and crossing probabilities when the probabilities are low. In a landscape with one or a few barriers, the number of crossing steps is likely to be low compared to the total number of steps in the trajectory. We modelled the probability of crossing in relation to the proportion a step to the total length using a logistic regression with each steps "fate" (crossed or not) as the dependent variable. For low probabilities, the relationship between log(proportion) and predicted probabilities is approximately linear, particularly when using a complementary log–log link. Deviation from such linearity indicates that the relationship between the log(proportion), and the crossing probability, deviates from the predicted linear relationship. This is indicated by a regression coefficient for log(proportion) different from 1. The intercept from such a regression indicates the slope of the relationship, which is affected by number of crossing (y) by the trajectory (see Fig. 1b). In case of several trajectories (e.g., many individuals), a mixed model (Zuur et al. 2009) was appropriate with trajectory identity as a random factor on the intercept and on the slope (log(proportion)) to assess population specific relationships between step lengths and crossing probabilities.

In addition to the two approaches described above, we applied a method that compares the step length of the crossing step with the length of the three previous and following steps (e.g., Dussault et al. 2007). The method takes significant differences in step length between the crossing step and the six closest steps, as evidence for a behavioural response to the linear feature or associated landscape characteristics. Distances were ln transformed prior to statistical tests. All analyses and simulations were done in R version 2.15.2 (R Development Core Team 2012).


Simulated CRW

Steps from the simulated CRW that crossed the randomly added line was significantly longer than steps that did not cross (crossing steps: mean = 2.02, sd = 1.43; non-crossing steps: mean = 0.98, sd = 0.98; test of the mean difference of each of the 100 CRW: t = 26.15, df = 99, P < 0.001). The difference in lengths between crossing and non-crossing steps appears because the length distribution of crossing steps differs from the total length distribution (Fig. 2b). In contrast, the length of crossing steps did not deviate from the predicted length based on the WNCMH distribution (t = 0.58, df = 99, P = 0.560). The relationship between log(proportion of trajectory length) and probability of crossing at the cloglog scale did not deviate significantly from 1 (β = 1.05, se = 0.03, z = 1.67, P = 0.095). There was thus a linear relationship between proportion of total length and the probability to cross, suggesting that longer steps did not have a higher or lower probability to cross compared to the predicted (Fig. 2c). However, the predicted probabilities were for some CRW quite high, which may violate the approximation of linearity. The crossing steps were longer than the three preceding and succeeding steps (Fig. 2d, test of difference between C and other steps: all P < 0.001), further emphasizing that in a trajectory that is random with respect to crossing linear features, crossing steps will be longer than non-crossing step.

Reindeer data

Reindeer steps that crossed randomly located power lines were longer than the non-crossing step lengths (crossing steps: mean = 1.38 km, sd = 1.24, non-crossing steps: mean = 0.60 km, sd = 0.67, test of the mean difference for each reindeer: t = 29.71, df = 29, P < 0.001, Fig. 3a), but not different from what was predicted from the WNCMH distribution given the trajectories and total number of crossings (t = 0.49, df = 29, P = 0.628, Fig. 3a). In contrast, the steps that crossed the real power line were longer than both the non-crossing step lengths (crossing steps: mean = 1.94 km, sd = 1.56, non-crossing steps: mean = 0.60 km, sd = 0.67, test of the mean difference of each reindeer: t = 19.25, df = 29, P < 0.001, Fig. 3b), as well as longer than the predicted length (t = 6.89, df = 29, P < 0.001, Fig. 3b), indicating that reindeer behaviour is not random with respect to the power line or some confounding factor associated with the location of the power line. For the random power lines, the relationship between crossing probability (at the cloglog scale) and log(proportion) did not deviate significantly from one (β = 1.03, se = 0.03, z = 1.044, df = 30, P = 0.305; Fig. 3c). For the real power line the relationship was significantly higher than 1 (β = 1.54, se = 0.08, z = 20.31, df = 30, P < 0.001; Fig. 3c), indicating that the longer steps (i.e., made up a high proportion of the total trajectory length) had higher probabilities for crossing the power line than predicted from theory. However, both with respect to random power lines (all P < 0.001) and the real power line (all P < 0.001), the crossing step was significantly longer than the three preceding or succeeding steps (Fig. 3d).
Fig. 3

Results based on GPS data from 14 female reindeer. a Distribution of all reindeer step lengths (gray bars) and steps that cross the random power lines added in the study area (hashed bars). The solid and dashed gray vertical lines indicate the mean step length for all steps and crossing steps, respectively. The vertical black solid line shows the predicted step length based on theory. b Same as a, but for the real power line that crosses the study area. c The relationship between step length (measured as proportion of the total trajectory length) and probability of crossing the real power lines (dashed lines) and the random barriers (solid lines). The thick black lines are the population estimates, whereas the thin gray lines represent individual estimates based on the population estimate and best linear unbiased predictions (BLUPs) from the random factors (individual and barrier identity) in the mixed model. d The length of crossing steps (C) and the three preceding (C − 1, C − 2, C − 3), and following steps (C + 1, C + 2, C + 3); with standard errors. Gray bars are for the real power line, hashed bars are the randomly located power lines

For the empirical reindeer data, independent of crossing or not, there was a positive correlation for step length between consecutive steps, and this autocorrelation in step length decreased with increasing time interval between the steps: t0 vs. t1 (r = 0.20), t0 vs. t2 (r = 0.04), and t0 vs. t3 (r = 0.03).


Habitat fragmentation is a global threat to the distribution of animal populations (Fahrig 2003). Factors that influence animal movement rates and directions affect both their ability to utilize resources within the available habitat, and migrate into neighbouring areas (Johnson et al. 1992). When studying animal behaviour in relation to linear landscape features like human infrastructure, GPS data enables us to test whether the movement rate or general habitat use is affected (Dyer et al. 2002; Leblond et al. 2013; Montgomery et al. 2013; Beyer et al. 2013). However, when assessing change in movement rate when crossing a linear feature, no studies have to our knowledge included a model for how the probability of crossing depends on the step length. We have shown mathematically how this probability increases with increasing step length, and how the difference between observed and predicted values can be tested statistically.

We found that that the step length during crossing was as predicted from theory for the simulated data and for the random lines from the empirical data. On the contrary, the steps that crossed the real power line were longer than predicted. However, linear features are seldom randomly located in the landscape, and the observed relationship between animal movement and the linear feature may be confounded by other factors such as habitat type, topography, migration pattern, or time of day when linear feature is crossed. Indeed, the importance of accounting for other environmental factors when drawing inferences about animal behaviour towards specific characteristics or features has recently been emphasised (Beyer et al. 2013; Van Moorter et al. 2013). Only then can we disentangle the effect of the linear feature from environmental or other effects on animal movement rates. Our suggestion to use generalized linear mixed models will allow future studies to account for such factors by including environmental conditions, time period, individual states or other values for each step as explanatory variables in addition to the step length. Doing so would most likely also account for large parts of the autocorrelation in the data, as autocorrelation in animal movement data to a large extent is shaped by spatial autocorrelation in the landscape characteristics or temporal autocorrelation in animal behavior (Fieberg et al. 2010). Not surprisingly, we also found a positive correlation between the length of a step and its adjacent step for the empirical data, while this was not the case for simulated data. This explains why also C − 1 and C + 1 in Fig. 3d is longer than C − 2 and C + 2, and is as expected from what is known about temporal and spatial autocorrelation in animal movements (Cushman 2010; Otis and White 1999). The length of the time interval between consecutive GPS locations is of crucial importance for the ability to detect dependency in the individual movement rate, both in time and space (Boyce et al. 2010; Cushman 2010; Horne et al. 2007) while this length also must be chosen based on expected spatial and temporal animal response distances to the disturbance studied (Stankowich 2008). Hence, autocorrelation needs to be adjusted for to see the real effect of the disturbance on steps adjacent to the actual crossing step.

With increasing human activities in wildlife habitats, the importance of studying interactions between wildlife and human development also underlines the need for accurate methods for assessing interactions between wildlife and human. The rapidly increasing number of GPS studies provides an immense and invaluable amount of data (Frair et al. 2010; Olsson and Widen 2008; Tomkiewicz et al. 2010). However, there are several challenges associated with the large amount of data (e.g., Cagnacci et al. 2010). In particular, when analyzing ecological patterns it is important to develop appropriate predictions to test hypotheses. If one fails to do so, erroneous conclusions about mechanisms driving animal movement patterns can be drawn (e.g., Van Moorter et al. 2013). Our results show that in order to test whether animal movement behaviour is affected when crossing a linear feature, one has to account for the bias in a steps crossing probability that is connected to its length. The prediction from earlier studies that crossing steps have the same length as the non-crossing steps simply does not hold. Instead, the predication should be that crossing probability follows the distribution shown in Fig. 1. Both our simulation and empirical data on reindeer movement in relation to linear features in the landscape clearly confirms the mathematical evidence and, as a consequence, crossing steps will on average be longer than other steps, even if animals do not change behaviour towards a feature.


The reindeer data used here was generated by the GPS project in Setesdal, and we thank the leader of this project, Olav Strand, as well as the organizing board responsible for the project. The manuscript was improved by comments from two anonymous reviewers.

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sindre Eftestøl
    • 1
  • Diress Tsegaye
    • 1
    • 2
  • Ivar Herfindal
    • 3
  • Kjetil Flydal
    • 1
  • Jonathan E. Colman
    • 1
    • 2
  1. 1.Department of BiosciencesUniversity of OsloOsloNorway
  2. 2.Department of Ecology and Natural Resource ManagementNorwegian University of Life SciencesÅsNorway
  3. 3.Department of Biology, Centre for Biodiversity DynamicsNorwegian University of Science and TechnologyTrondheimNorway

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