Introduction

Movement of organisms is a key process maintaining connectivity between subpopulations in fragmented landscapes and is thus of critical importance in the design and maintenance of reserve networks to conserve threatened species (e.g. Crooks & Sanjayan 2006; Hilty, Lidicker, & Merenlender 2006). Moreover, as environmental and climatic conditions change, species may need to undergo range-shifts to track the conditions to which they are adapted (e.g. Hanski 1999; Kokko & Lopez-Sepulcre 2006; Reusch & Wood 2007; Ronce 2007), and they will only be able to do so if at least some individuals are capable of dispersing to suitable habitat beyond their current range limit. To assess whether threatened and endangered species will be able to do this, conservation managers need reliable and accurate methods for determining the degree of connectivity within fragmented habitat networks. The simplest measure of connectivity between any two patches, the Euclidean distance, will frequently be inadequate for this purpose, as dispersing individuals do not necessarily move in straight lines through a heterogeneous and often hostile landscape (e.g. Wiens 2001).

The least cost path (LCP) approach aims to improve upon Euclidean distance as a measure of connectivity by taking account of the character of landscape elements lying between isolated habitat patches. This method traditionally determines, for a pair of patches within a heterogeneous landscape, the optimum route between the patches incurring the least cumulative cost according to a cost grid (Adriaensen et al. 2003; Chardon, Adriaensen, & Matthysen 2003; Driezen et al. 2007). The approach may be extended by summing cost distance maps for the different potential sources of dispersers to evaluate cumulative and synoptic connectivity (e.g. Compton et al. 2007) or by evaluating LCPs among networks of points for species which view the landscape as a gradient of habitat quality rather than discrete patches (e.g. Cushman, McKelvey, & Schwartz 2009). The cost has been taken to represent the estimated resistance to movement (e.g. Sutcliffe et al. 2003; Vignieri 2005; Broquet et al. 2006; Finn et al. 2006; Lada et al. 2008; Koscinski et al. 2009; Hokit et al. 2010), the time taken to move through a habitat (e.g. Stevens et al. 2006) or an index representing some combination of more than one cost (e.g. Stevens et al. 2006; Nichol et al. 2010). Further possible costs could be the energetic demands of moving through a habitat or the risk of mortality, although we know of no examples in the literature of their use. The relative connectivity between different pairs of patches is then estimated by comparing the lengths and/or the costs of the LCPs among all possible pairs of patches.

Whatever the cost metric employed, however, the underlying assumption of the LCP approach is that of perfect knowledge of the entire landscape (omniscience) on the part of the moving individual. Although animals may indeed have excellent knowledge of a territory or home range, this assumption is presumed to be highly unrealistic for most animals during dispersal, and especially so for long-distance dispersal (LDD) and/or post-natal dispersal. Thus, animals presumably have to make frequent movement decisions during dispersal based on the limited information they have about the surrounding habitat types within their perceptual range (be that visual, olfactory or a combination of sensory mechanisms) (Zollner & Lima 1999; Conradt et al. 2000; Merckx & Van Dyck 2007), and which may be context-dependent (Olden et al. 2004). Even if an animal has a fundamental tendency to follow a highly correlated (i.e. fairly straight) dispersal path, it may still change its direction of movement based on choices that relate to the composition of habitat in the proximity of its route. Thus, animals may follow routes through inhospitable habitat which differ considerably from the optimum (in terms of least distance or cost), and as a result the effective (functional) connectivity between habitat patches may differ substantially from the LCP connectivity. Moreover, we predict that the smaller a species’ perceptual range in relation to the grain of the landscape, the more the functional connectivity will deviate from the LCP connectivity, i.e. the correlation between LCP and functional connectivity will itself be positively correlated with perceptual range. For these reasons, the legitimacy of LCP-based decisions in land management has been questioned (Fahrig 2007; Baguette & Van Dyck 2007).

Stochastic modelling of organism movement has a long history much of which builds on early work using random walk models (e.g. Skellem 1951). In these first models, Brownian motion was assumed where the direction of movement in one step is completely independent of that in the previous step. For most organisms, it is more realistic to incorporate correlation between steps (Codling & Hill 2005) and such correlated random walks have become widely used in modelling animal movement (e.g. Bovet & Benhamou 1988; Byers 2001). Importantly in the context of our work, landscape ecology theory highlights that the degree of correlation in a movement trajectory is key in determining the likelihood that an emigrating individual successfully arrives at a non-natal habitat patch (Zollner & Lima 1999). More recently, some studies have begun to incorporate biased movement either towards the centre of a home range (Morales et al. 2005) or towards a habitat patch (Bartońet al. 2009). However, these models tend to assume movement in a largely homogenous environment with, at any one point in time, a positive bias towards a single point (but see Vuilleumier & Metzger 2006 for an alternative object-oriented approach to representing landscape features). In reality, an individual will frequently be moving through a complex landscape surrounded by a variety of matrix elements imposing different biases. By integrating concepts from LCP modelling with recent progress in modelling movement, we believe it should be possible to construct a relatively simple model that captures much of the complexity in terms of both the rules that an individual follows and the structure of the landscape.

Here, we demonstrate how we may model the dispersal of animals through a heterogeneous landscape, in which different components are associated with different costs of movement, in such a way that the path taken depends on a sequential series of movement decisions each based on the information about the landscape within the animal’s perceptual range. We also aim to show how the perceptual range, the way in which an animal evaluates habitat quality and its tendency to move in a particular direction correlated with its previous direction interact to govern the efficiency with which it is able to locate suitable habitat patches, and hence how the functional connectivity of landscape patches is governed by these movement parameters.

Materials and methods

The landscape

We derived our example virtual landscape from that used by Chardon, Adriaensen, & Matthysen (2003), in which the source patch is located at the centre of the square landscape, and four target patches are located in the four cardinal directions, each separated from the source patch by a landscape feature which is either beneficial (a corridor or a series of stepping stones) or inhibitory (a barrier or an area of poor habitat) to movement. We rescaled the landscape to a grid of 300 × 300 cells and replaced the central patch by a single source cell, but otherwise retained the original landscape characteristics, including the relative cost values of the various landscape features (Fig. 1). We recalculated the LCP distances between the source and the four target patches using standard methods in ArcGIS v9·2 (ESRI, Redlands, CA, USA).

Animal movement

PR was varied incrementally from 1 to 10 cells, and then in steps of five cells up to a maximum of 30 cells (i.e. 10% of the landscape dimension). Each animal moved through the landscape by choosing to move at each step from its current cell to one of the eight neighbouring cells. The probability that a cell was chosen depended on its cost value, the cost of neighbouring cells lying beyond it within the PR and the direction which the animal had taken to reach its current location.

If the PR was one cell, then the cost value of a neighbouring cell was simply its cost as determined by the landscape grid. When the PR was greater than one cell, the animal still had to make a choice between the eight neighbouring cells, but their effective cost had also to take into account the cost of potential future moves from the chosen cell on the basis of information available from all cells within the PR. We achieved this by taking as the effective cost of a neighbouring cell the mean of its own cost and of the costs of its own neighbours surrounding it in an array which extended to the edge of the perceptual range (Fig. 2). Three alternative methods were used for evaluating the effective cost: the (unweighted) arithmetic mean, the harmonic mean¹ and a weighted arithmetic mean (in which the contribution of each cell within the PR towards the effective cost was weighted by the inverse distance of the cell to the current cell).

There is good evidence from theoretical studies (Zollner & Lima 1999; Heinz & Strand 2006; Bartońet al. 2009) and from empirical observations (Schultz & Crone 2001; Fahrig 2007 and citations therein; Schtickzelle et al. 2007; del Mar Delgado & Penteriani 2008) that dispersing animals tend to follow a highly correlated path, i.e. they make few large turns. We implemented this constraint within our model by increasing the probability of selection for cells in the direction of the animal’s movement and decreasing it for those in the direction from which it had come. Specifically, we applied a weighting to the effective costs of neighbouring cells (prior to calculating cell probabilities), which was lowest in the direction taken in the previous step and highest in the opposite direction (i.e. a 180° turn). The weighting was controlled by a single parameter (hereafter ‘directional bias’), which was raised to the power zero in the direction of previous travel, and to powers increasing from 1 to 4 in the direction of turns from 45° through to 180°; larger values of the parameter reduce the tendency for the path to deviate from a straight line.²

Finally, the reciprocals of the weighted effective costs were taken and scaled to sum to unity to yield selection probabilities for the eight neighbouring cells (Fig. 2); the next cell in the path was chosen stochastically in proportion to these selection probabilities. Thus, our model bears strong similarities to the self-avoiding random walk of Gustafson & Gardner (1996), but with a non-zero probability of returning to the previous cell and, critically, with increased PR and directional bias. Example paths generated using our movement rules are shown in Fig. 3. The accumulated cost of the path taken was then incremented by the cost of the chosen move (i.e. the mean cost of the original cell and the chosen destination cell, weighted by √2 for diagonal moves). This is exactly the same method as the LCP algorithm uses to calculate the cost of movement between adjacent cells. No special recognition of the targets was implemented, but because they each comprised a block of low-cost cells, animals tended to move towards a target once it was within the individual’s PR.

For each combination of PR, evaluation method and directional bias, we released sequentially a sample of 1000 animals into the landscape from the point source. An animal’s path was terminated as soon as it entered any one of the cells of a target patch, and the animal was then classed as a successful disperser. We treated the landscape as a torus for the purpose of evaluating the effective cost if the animal was within its PR of the boundary, but once an animal made a move which took it beyond the boundary, its path was terminated and it was classed as an emigrant. If neither successful dispersal nor emigration occurred within 2000 steps, the animal was regarded as having run out of time and classed as dead. For each animal, its fate, the path length (number of steps), incurred total cost, target patch reached (if any) and number of steps spent in each of the four landscape feature types were recorded for subsequent analysis.

Results

Dispersal success

Directional bias (χ² = 28713) and PR (χ² = 9501) had the greatest effect on the probability of successful dispersal, and there was also a substantial interaction between PR and evaluation method (χ² = 1202). At low PR and low directional bias, very few individuals located a target patch within the prescribed number of steps, whereas at high values of these parameters, almost all did so (Fig. 4). At all but the lowest PR, individuals were more likely to locate a target patch if employing the harmonic mean method of evaluating landscape cost rather than either of the arithmetic mean methods. Moreover, as directional bias (F = 7384) and/or PR (F = 613) increased, the length of the path followed by successful dispersers decreased (Fig. 5a). The length of the path was also lower if using the harmonic mean method rather than either of the arithmetic mean methods at all but very low PR. Similarly, as directional bias (F = 1018) increased, the total path cost incurred by successful dispersers decreased, but the decrease was substantial only if the harmonic evaluation method were used (PR × method interaction F = 39·5) and especially if there were high directional bias (PR × bias interaction F = 273) (Fig. 5b).

The increase in successful dispersal with increasing PR and directional bias was largely accounted for by increasing numbers of dispersers reaching the western and northern target patches lying beyond the corridor and stepping stones, respectively (supplementary material Fig. S1). These numbers reached asymptotes at high levels of directional bias (above 3·5), but continued to increase as PR was increased up to its maximum value. In contrast, as PR increased, the numbers reaching the eastern and southern patches decreased, which is attributed to individuals turning round on encountering inhibitory landscape features and thereby increasing their chance of subsequently locating the beneficial features and the patches to which they led. However, despite the increased efficiency afforded by high PR and directional bias, some successful dispersers incurred relatively high costs, which could be more than an order of magnitude greater than the LCP cost to the same patch (e.g. Fig. 6).

Relative connectivity

If the relative connectivity of the source patch to the four target patches were determined on the basis of the reciprocal of Euclidean distance, then all target patches would be equal as they were equidistant from the source (Fig. 7a). Replacing Euclidean distance by the LCP length made little difference (roughly 10% increases to the northern and southern patches and similar decreases to the other two patches; data not shown), as the LCPs did not deviate very far from straight lines. Replacing Euclidean distance by the LCP cost increased the relative connectivity to the northern and western patches as the stepping stones and corridor (especially) provide relatively low-cost paths (Fig. 7b). In contrast to this deterministic method, connectivity estimated by the simulation of individual movement depended on the parameters controlling the movement. Here, we use the relative numbers of successful dispersers to each target patch under a particular scenario as the measure of connectivity and illustrate this for four examples of combinations of parameters. An example of a simulation with low PR and low directional bias shows the western patch linked by the corridor being favoured, but the stepping stones to the north provided poor connectivity as individuals tended to become trapped within them (Fig. 7c). Increasing the directional bias could counteract the trapping problem, leading to improved relative connectivity to the northern patch (Fig. 7d). In an example where PR was high but directional bias was low, no individuals located the southern patch in the poor matrix habitat, and very few the eastern patch behind the barrier (Fig. 7e), whereas high directional bias combined with high PR enabled more dispersers to either pass around or break through the barrier to the east and to locate and move towards the southern patch even though higher cost matrix cells have to be crossed (Fig. 7f). Further examples of how the relative connectivity changed in relation to changing directional bias and PR are presented in the supplementary material (Fig. S2).

In order to check that our results were unlikely to be artefacts of assumptions we had made in constructing the model, we ran a series of additional separate variants as follows: (i) the corridor and stepping stones had a cost of two units rather than 1 (i.e. to be more costly than the target patches), (ii) the maximum number of steps was increased from 2000 to 5000 and (iii) once a target patch was within the PR, the individual moved directly towards it. In all cases, there were trivial and expected effects, such as an overall change in the proportion of successful paths and their mean costs, but the patterns of interaction between PR, directional bias and evaluation method were, broadly speaking, unaltered. One point of note, though, is that variant (ii) was initially attempted with no step limit, in which case one individual finally completed a path of over 125 million steps, of which 99·98% were spent stuck in the stepping stones. This implies that the inclusion of some form of step-dependent mortality risk or resource loss will be crucial in more realistic versions of the IBM.

Discussion

There has been substantial recent progress in modelling animal movement (e.g. Zollner & Lima 1999; Morales 2002; Morales & Ellner 2002; Morales et al. 2004, 2005; Heinz & Strand 2006; Vuilleumier & Metzger 2006; Bartońet al. 2009), much of which uses stochastic models incorporating correlation and bias in the movement rules. We believe that there is considerable potential in integrating these concepts and methods in landscape ecology, in particular for improving estimates of connectivity. Here, we have presented a deliberately simple method, which utilises the same landscape information as LCP modelling, i.e. resistance parameters for each of the matrix elements in the landscape, and models movement in discrete space (in this respect our model is distinct from many of the recent models of movement). Our method requires just two additional parameters to the standard LCP algorithm, one specifying the PR and the other the strength of correlation in the movement. A third can be optionally used to specify the method employed to determine the mean effective cost within the PR. Later, we explain the key results generated from our model and highlight key future applications and further developments of the approach.

Our novel stochastic rule-based model simulates the inter-patch movement of individuals through a landscape matrix. It clearly demonstrates how the probability of successful dispersal depends on both the movement correlation of the path and on the perceptual range of the individual. Zollner & Lima (1999) have previously demonstrated the effect of movement correlation on success and shown how it interacts with the landscape configuration, the risk of mortality and the energy reserves of the disperser. Even if all these factors are the same for two different species, and even if they have the same habitat preferences and incur the same movement costs, their probability of successful dispersal will differ if their perceptual ranges differ. Thus, the functional connectivity of the habitat network will also differ for the two species.

For successful dispersers, the total path length and total cost incurred are also affected by the way in which the animal evaluates the landscape within its perceptual range. Undoubtedly, animals actually use some form of rule-based method to decide amongst the alternatives available at any point in their path, rather than the mathematical methods adopted here. The benefits gained by increasing PR in our model were greater using the harmonic mean method than under the unweighted or weighted arithmetic mean methods, as these diluted the influence of low-cost cells on the probability of movement. Weighting the arithmetic mean by inverse distance actually had only a very limited effect in our simulations (compared to the unweighted arithmetic mean), but in part this may have been because the matrix we used was largely homogeneous (except for the specific features) in terms of cost value. Further evaluation of the relative benefits of various potential weighting methods would undoubtedly be worthwhile, but, nevertheless, our results suggest that a rule-based method which gives a high weighting to beneficial landscape features, such a corridors or stepping stones, is likely to be the most efficient. For simplicity, we did not include energy reserves or mortality risk explicitly in our model, although we acknowledge that it would typically be more realistic to do so. However, it is intuitive that increasing the perceptual range will tend to lead to a more efficient path through the landscape and hence a reduced likelihood of mortality through exhaustion of energy reserves or predation.

Recent empirical studies have demonstrated a closer correlation with LCP connectivity than with Euclidean connectivity of genetic distance among populations or among individuals (e.g. Coulon et al. 2004; Cushman et al. 2006; Stevens et al. 2006; Hokit et al. 2010; but see Greenwald, Purrenhage, & Savage 2009). This suggests that differential resistance values of matrix elements do indeed influence gene flow across landscapes. However, the improvement in the correlation is often very limited. The fact is that, for most animal species, LCPs are a poor representation of movement behaviours, especially when individuals are engaging in dispersal behaviour, as opposed to regular foraging activity. It is only if an individual knows its intended destination and has information on matrix quality for all potential routes to it that the individual is likely to take the LCP. Otherwise, the paths taken are substantially longer and less efficient than the LCP (Russell, Swihart, & Feng 2003; Driezen et al. 2007). Our results clearly illustrate this. Imagine that a real species moves according to the set of rules used in producing Fig. 7c. The relative connectivity of the four destination patches is poorly approximated by Euclidean distance (Fig. 7a) and is not much better approximated by LCP (Fig. 7b), which provides an improved estimate for the West patch, but a poor estimate for the East patch. In other cases, the relative connectivity obtained by Euclidean distance and LCP are both very different to those obtained using simulations of movement rules (compare Fig 7e and f with 7a and b). An important next step will be to see how much better a fit to genetic distances can be obtained using a model of movement that incorporates not only different matrix resistance values but also correlation and bias. Another reason for the poor improvement provided by LCPs is that LCP distance can depend critically on (i) exactly how the landscape is represented in the form of discrete cells, especially with regard to beneficial or inhibitory linear features (Adriaensen et al. 2003) and (ii) on the attribution of cost values to the different landscape elements, which is more often based on expert opinions than on experiments or on field data (but see the approach of Verbeylen et al. 2003); note, though, that these two factors may be expected to influence the individual-based approach too. Recent theoretical attempts to improve on estimates of connectivity have included extending the LCP method stochastically to generate a set of possible paths forming a corridor of variable width between patches (Pinto & Keitt 2009) and combining the standard incidence function model (Hanski 1994) with the LCP distance between patches (Watts & Handley 2010). Another approach has been to draw a parallel between the flow of individuals and electrical current and hence borrow from electrical circuit theory to estimate landscape connectivity: sources of individuals are represented as nodes linked by electrical resistors of conductance proportional to the number of migrants between the sources. The main advantage of this method is that it allows incorporating multiple pathways (as several resistors can link the same two sources) and evaluating the relative importance of each of them for landscape connectivity (McRae 2006; McRae & Beier 2007; McRae et al. 2008). However, none of these methods allows for limited perceptual range. The method of Vuilleumier & Metzger (2006) incorporates perceptual range, and functional connectivity estimates derived from it correlated better than Euclidean distance with observed genetic distance (Vuilleumier & Fontanillas 2007). However, their model does not include any tendency to move in the same direction as previously, and therefore patch-location success was poor if perceptual range was limited, leading to a tendency to become trapped in the matrix (Vuilleumier & Perrin 2006), as we have also found in the present study.

Conservation managers desire simple connectivity indices that can inform their selection of sites for reserves. Clearly, although it remains relatively simple, the method proposed here is more complex than existing structural (e.g. Euclidean) or functional (LCP) estimates. Gaining estimates for the two additional parameters comes at a cost, and it is important to understand how much of an improvement our more complex method provides. Future work should establish for which types of species and for which types of landscapes we should invest the extra effort required to parameterise the more complex model. And, conversely, we should be able to recommend when the existing methods are likely to do a sufficiently good job that the extra investment of time and money is not worthwhile.

Future testing and development of the model can progress in several directions. In terms of model testing, an important next step will be to evaluate the connectivities for a broad range of landscapes with different proportions and spatial arrangement of habitat and various compositions of matrix. The estimates of connectivity generated by our stochastic movement simulator for a range of different animal behaviours can then be compared against those produced by alternative structural and functional connectivity indices, with the objective being to establish whether there are particular types of landscapes, or particular types of species, for which the much-used existing methods are likely to be most erroneous. In terms of future model development, it is increasingly recognised that all individuals in a population, or belonging to a species, do not behave identically. Instead, there is often substantial inter-individual or inter-population variability in dispersal behaviour (e.g. Stevens, Turlure, & Baguette 2010). The distribution of behaviours is likely to be important in determining relative patch connectivity, spatial population dynamics and gene flow. A simple extension to the existing model would be to consider how variation in movement rules influences the connectivity between patches and to establish the degree to which relatively few individuals with ‘rare’ movement behaviours make the LDD events that are so important in terms of range expansion and genetic dynamics. Recent work has begun to consider how inter-patch movement rules should be expected to evolve (Heinz & Strand 2006; Bartońet al. 2009). Both of these two models incorporate correlation, but only Bartońet al. (2009) also incorporates bias towards habitat patches, and neither model includes a heterogeneous matrix. There is clearly scope to develop new theory related to the evolution of movement rules in complex landscapes by integrating the method presented here with dynamic evolutionary modelling.

Acknowledgements

We are grateful to Frank Adriaensen for providing us with data for virtual landscapes and to Thomas Hovestadt and three anonymous reviewers for comments on previous versions of the paper. This publication issued from the project TenLamas funded by the French Ministère de l’Energie, de l’Ecologie, du Développement Durable et de la Mer through the EU FP6 BiodivERsA Eranet.