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e c o l o g i c a l m o d e l l i n g 2 1 0 ( 2 0 0 8 ) 104–114
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Cost distance defined by a topological function of landscape
J.C. Foltetea,∗, K. Berthierb, J.F. Cossonb
a TheMA-UMR 6049 CNRS, University of Franche-Comte, 32 rue Megevand, F-25030 Besancon, Franceb Centre de Biologie et de Gestion des Populations, INRA, Campus International de Baillarguet, F-34980 Montferrier-sur-Lez, France
a r t i c l e i n f o
Article history:
Received 22 May 2006
Received in revised form
29 June 2007
Accepted 9 July 2007
Published on line 20 August 2007
Keywords:
Cost distance
Resistance
a b s t r a c t
Distance is a basic concept in the domain of animal species motion. Cost distances, rather
than Euclidian distances, are more and more used in order to have a more realistic mea-
sure, on the basis of resistance values assigned to each landscape class. We propose here
a method to compute resistance values by using topological functions of landscape, i.e. by
taking account of the proximity of habitat/non-habitat edges, with continuous functions.
An example is given when comparing cost distances and the propagation of water vole in
the massif of Jura (France). The comparison with usual cost distances gives information
about the ecological assumptions. The results show also that the statistical behaviour of
the distances depending of the parameters of the functions allows to precise the influence
of edges in terms of spatial range.
Edge
Habitat
Spread
© 2007 Elsevier B.V. All rights reserved.
ever its use assumes a neutral spatial framework, which is
Water vole
1. Introduction
Distance is a basic concept inherent to any geographical space.This notion is a key factor in population ecology and especiallyin animal movement analysis. In the well-known theory of“isolation by distance” (Wright, 1943), it is considered a centralfactor playing of role on species invasions or species extinc-tions. According to this theory, the genetic difference betweenpopulations increases with geographical distance when con-sidering a spatial scale higher than the average dispersaldistance of the species. The relationship between genetic dis-tance and geographical distance (Rousset, 1997, 2000) is thena proxy used to infer the individual movement ability of agiven species (Arter, 1990; Michels et al., 2001; Arnaud, 2003;
Coulon et al., 2004; Berthier et al., 2005; Broquet et al., 2006). Inlandscape ecology, spatial distance is of a primary importancein the concepts of connectivity and fragmentation (Forman∗ Corresponding author.E-mail addresses: [email protected] (J.C. Folte
(J.F. Cosson).0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2007.07.014
and Godron, 1986; Forman, 1995). Distance is also explicitlyincluded in the formulation of some landscape metrics such asthe proximity index proposed by Gustafson and Parker (1994)and in the use of graph theory applied to the spatial relationsbetween landscape patches (Urban and Keitt, 2000; Bunn etal., 2000). The application of diffusion-reaction models in thecontext of population movements (Turchin, 1998; Okubo andLevin, 2001) requires the use of the distance between placesand a central point (as a “release point”). All these examplesof analysis based on the notion of spatial distance show thegreat importance of its measurement.
Euclidian distance, “as the crow flies” when applied to theflat space of maps, is the simplest measure of a distance. How-
te), [email protected] (K. Berthier), [email protected]
not very realistic when representing effective spatial acces-sibility. In the context of animal population movements, theassumption of a homogeneous space where all places are
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qually accessible is rarely justified (Matthiopoulos, 2003): theabitat of the studied species, the configuration of habitatatches, resources, corridors or displacement constraints leado a very heterogeneous space and finally challenges the rel-vance of Euclidian distance. Following these observations,everal authors have shown the interest of least-cost distancesor cost distances) that take into account the spatial hetero-eneity. These distances are computed from raster data onlyfor example from a given cell to a target cell) and allow to allotmovement resistance value to each landscape class called a
friction”. Cost distances are sometimes called “effective dis-ances” (Ferreras, 2001), in contrast to the Euclidian distance.
The use of cost distance computations is more and moreopular in ecology (Knaapen et al., 1992; Yu, 1996; Pain et al.,000; Bunn et al., 2000; Halpin and Bunn, 2000; Ray et al., 2002;hardon et al., 2003; Adriaensen et al., 2003; Ray and Burgman,006), since they are implemented in several GIS softwaresESRI, 1996, 2001; Eastman, 1998; Ray, 2005). The choice of theesistance values assigned to each landscape class is usuallyased on a specific knowledge of the mobility behaviour of thetudied population. A resistance of 1 is quite often assigned tohe habitat class and to the displacement corridors; a higherevel is given to the other classes in relation to their degree ofhostility” (presence of potential predators) or their ability toimit the movements (physical barriers, lack of resources).
Verbeylen et al. (2003) have shown a simple method toetermine an optimal combination of resistance values whensing the cost distance in prediction model of species abun-ance. However, the assignment of uniform resistance valueso all landscape classes implicitly means that a given land-over class involves a uniform displacement behaviour. Thiss in contradiction with the well-known “edge effect” in ecol-gy (see for example Paton, 1994; Reese and Ratti, 1988; Fagan,999). Following this effect, the proximity to different land-over classes can lead to a modification of this behaviour foreveral reasons: change of accessibility to resources, increasef predator risk, and so on. For example, for a species subject toredators living outside its habitat, the land-cover class defin-
ng this habitat does not necessarily correspond to a uniformontext of mobility, because of the varying ability of preda-ors to reach these areas. In an ecological context where the
ain moving factor of a given species is the accessibility toesources, the inhospitable land cover classes may be par-ially crossed by the individuals, provided they remain nearhe boundary, because in this case their resources remainccessible.
In consequence, we argue that proximity to the limitetween hospitable and inhospitable areas can play a greatole in the permeability of landscape to animal movements.owever, the classical approach of cost distances does not take
his phenomenon of progressive permeability into account: itimply replicates the discrete structure of land cover classesn the spatial distribution of resistance values. The aim of theresent paper is to show that a specific method of allocationf resistance can be implemented in order to improve the rel-vance of cost distance when the limits between certain land
over classes play an effective ecological role.Starting from a simplified landscape divided into twolasses – habitat (H) non-habitat or inhospitable (NH) – aethod is proposed to assign variable resistance values to
0 ( 2 0 0 8 ) 104–114 105
each class. Such values are defined by a function of the posi-tion in relation to the boundary between H and NH and byapplying what we call “a topological function of landscape”.An additional class designated “neutral class” or N is added toinclude areas which are not under the influence of the previ-ous boundary. After the presentation of the data set and theused method (2) we will show the results obtained in the con-text of the spread of a rodent population in a part of the Juramassif (France) (3).
2. Materials and methods
2.1. Research context and data set
In the Jura massif, the grasslands are regularly swarmingwith a rodent species, the common vole (Arvicola terrestris).The spread of the vole populations makes a travelling wavewith a period of about 5 years (Giraudoux et al., 1997). Thistravelling wave ruins the grasslands (Quere et al., 1999) andpromotes the transmission of diseases (Delattre et al., 1998).Furthermore, the use of pesticide leads to many environ-mental consequences especially by hitting vole predators(Delattre et al., 2000). In this context, biologists and geogra-phers seek to understand the role of landscape structures onthe vole spread. As several epicentre zones have been identi-fied (Duhamel et al., 2000), we focus here on the “plateau ofNozeroy”, a zone influenced by a single epicentre in order toanalyze the spread phenomenon without interferences.
This zone is a plateau of approximately 200 km2 com-posed of a matrix of grassland surrounded by large forestsof conifers, which may be considered as almost impassablebarriers for vole populations (Fig. 1). Different spatial con-figurations of grasslands are present in the plateau: certainparts contain very large and continuous grassland patches(openfield) whereas other parts can be considered as bocagestructures, i.e. fragmented by hedges and some linear forestelements. The relief is rather homogeneous, at an altituderanging between 700 and 900 m and without significant topo-graphic accidents.
Starting on April 2002, estimations of A. terrestris densitywere done every 6 months in a set of cells of observation ina regular grid of 1 km2. One cell out of two was analyzed, giv-ing a total of 92 sites of observation. For each site, the densitywas estimated on the basis of vole tumuli count along two250 m diagonal segments (Fig. 2a). Using this count, a surfaceindex, expressed as a percentage, was calculated as describedin Giraudoux et al. (1995); a resulting value of 0 means theabsence of vole and a value of 100 means a maximal voledensity.
This paper analyzes the initial phase of the invasion of theplateau by the A. terrestris. At the start of the observation (April2002), the high densities are located in a very small area in thenorth of the zone. The unique site of observation with themaximal value of 100 is considered here as the local epicen-tre of the diffusion (Fig. 2b). The challenge is to explain the
tion (September 2002) through a function of spatial distancefrom the epicentre. As different types of spatial distance maybe used (Euclidian distance, cost-distances), the resulting spa-
106 e c o l o g i c a l m o d e l l i n g 2 1 0 ( 2 0 0 8 ) 104–114
e pla
Fig. 1 – Experimental zone: thtial spread represents an interesting test of these types of costdistances. Because of the presence of vole predators nestingin hedges and linear forest elements, we assume that the per-meability of grassland areas (the habitat of A. terrestris) variesdepending on the distance to these landscape elements.
Landscape classes were defined from remotely senseddata. An Indian Remote Sensing LISS-III image acquired inSeptember 1997 was merged with the panchromatic band(IRS-1c PAN) to have a multispectral image of 7 m-spatial res-olution. Then a maximum likelihood classifier was applied toobtain a categorical image with four land cover classes: grass-lands, wooded areas, bare grounds and water. Because of therelative stability of land cover structures in this zone, the tem-poral delay between demographic data and landscape data isconsidered acceptable for cross-analysis.
2.2. General framework
The assumptions presented above lead to implement a contin-uous function of resistance instead of the dichotomy betweenthe uniform absence of constraint of H and the uniform con-straint of NH. This function depends on the boundary betweenH and NH; it is applied to each cell of a grid to define resis-tance surfaces that are used to compute the cost distances.
The global principle of the method is thus based on a particulardefinition of the resistance values.The third spatial class N is added to account for areas whichare not concerned by the opposition between H and NH. It is
teau of Nozeroy (Jura, France).
called “neutral class” because it is not linked to the ecologi-cal assumptions about the progressive permeability of certainlandscape classes, but it is not necessarily neutral in termsof resistance. For example, class N may be represented by aland cover type which is impassable for the species in ques-tion and which is not in keeping with the notions of resourcesor vulnerability to predators. It may also be a class of poten-tial movement (resistance value of 1) without any relationshipto an exogenous influence. The resistance value of the classN depends on each ecological context and its specificity liesin its independence from the spatial influence of the otherclasses H and NH. The method presented here is thereforespecific only with relation to the allocation of resistance ofthe other landscape classes, and that is why the presentationof the computations will not mention class N.
The computation of cost distances is accomplishedthrough four successive steps. In the first step, the minimaldistance between each cell and the H–NH boundary is com-puted by gradually increasing a buffer zone from the cellslocated at this boundary. An image of boundary distances isthen generated (Fig. 3b). The transformation of these distancevalues by a topological function (that will be detailed later)allows us to obtain an image of resistance values (Fig. 3c).For a given source zone which can be a single cell or a set
of cells, this image is then used for computing the minimalcost by the use of the classical growth algorithm implementedfor example in Arcview (ESRI, 1996) and Idrisi (Eastman, 1989)(Fig. 3d).
e c o l o g i c a l m o d e l l i n g 2 1 0 ( 2 0 0 8 ) 104–114 107
Fig. 2 – Demographic data for analysing the spread of vole swarming.
108 e c o l o g i c a l m o d e l l i n g 2 1 0 ( 2 0 0 8 ) 104–114
com
Fig. 3 – Successive steps ofAt the end of this process, the minimal cost can be com-puted for any cell of the image. The creation of the image ofboundary distances and the cost distances computation areinvariant steps but the definition of the topological functionrequire several parameters. The choice of these parameters isexplained below.
2.3. Different kind of topological distance
In order to yield relative resistance values in relation tothe N–NH boundary, several continuous functions can beconsidered according to biological assumptions. Here thesefunctions are based on two parameters when allocating a spe-cific status (for example impassable) to the neutral class N:
• rm: the maximal resistance value that corresponds to thegreatest constraint level of the inhospitable class;
• dm: the maximal distance of influence of the boundarywhich corresponds to a given spatial range.
We use a sigmoid-shaped function which smoothes thetransitions between the extreme resistance values. Startingfrom this generic shape, several functions can be preciselydefined by changing the position of the function in relationto the boundary and by modifying the values of rm and dm
(Fig. 4).The usual definition of resistances values for the computa-
tion of cost distances is called “function 0”. It will constitute areference function in order to evaluate the relevance of otherfunctions. For a cell of landscape class p, the resistance is given
by r = f0(p) where f0 is a discrete positive function: f0(1) = �;f0(2) = �;. . .; f0(k) = � and f0(i) ≥ 1∀i. With only two classes H andNH and a resistance of 1 for the class H, it only necessaryto assign a resistance value rm to the class NH, giving a veryputation of cost distances.
simple definition of f0 as follows:
f0(p = H) = 1
f0(p = NH) = rm (2)
The following functions depend both on the classes H andNH and on the distance d to the boundary between them.
For the function f1, the class H is assigned a uniform resis-tance of 1 whereas the class NH takes an increasing resistanceas one goes away from the H–NH boundary, up to the maximaldistance where the resistance becomes uniform:
f1(d, p = H) = 1,
f1(d, p = NH) =
⎧⎪⎨⎪⎩
rm − (rm − 1)
(1 − d2
dm
)2
if d ≤ dm
rm if d > dm
(3)
Using this function assumes that the areas of NH closeto H (i.e. not more distant than dm) can be used for move-ments, for example thanks to a possible and rapid accessto the resources. If the individual goes beyond the distancedm, this accessibility becomes more and more difficult andthe environment becomes rapidly more restrictive, yielding anavoidance behaviour. On the other hand, the resistance valueof 1 of the H areas means that very few movement constraintsare added to the physical distance.
In the function f2 the class NH is assigned a uniform valueof constraint rm. In the areas close to the H–NH boundary theclass H presents a strong resistance which decreases as onegoes away from this boundary until the value becomes 1 for
e c o l o g i c a l m o d e l l i n g 2 1
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t
f
f
cbtcmta
topfnti
correlation obtained from the Euclidian distance. For bothkinds of distance (least cost and least distance) the changeof the resistance value allocated to the neutral class does notshow additional results.
Table 1 – Relationship between usual cost distances andvole density
rm value Least cost Least distance
5 −0.7646 −0.764110 −0.7635 −0.7657
ig. 4 – Functions used to compute the resistance values.
he distance dm.
2(d, p = H) =
⎧⎪⎨⎪⎩
1 + (rm − 1)
(1 − d2
d2m
)2
if d ≤ dm,
1 if d > dm,
2(d, p = NH) = rm (4)
In comparison with the preceding function, this functionorresponds to a relative vulnerability of the habitat near theoundary. It can represent for example the pressure of preda-ors living outside the habitat of the species in question. In thisase the distance dm means the range of potential displace-ent of these predators. The habitat areas located far from
he inhospitable areas will be in consequence favorite transitreas.
In the example above both functions can be compared toest the assumptions of permeability of inhospitable areas orf vulnerability of habitats. The cost distances are thus com-uted from the epicentre point by defining values of rm and dm
or all others sites of observation. Starting from a given combi-ation of parameters, several distances can be recorded for arip between two points: (1) the accumulation of costs, whichs the most common computed measure and which is called
0 ( 2 0 0 8 ) 104–114 109
here “least cost”; (2) the distance of the least-cost trip, usedfor example in Broquet et al. (2006) and called “least distance”.The first measure explicitly includes the notion of cost givenin a somewhat abstract unit whereas the second, expressed ina metric unit, is perhaps easier to interpret. Both will be usedin this study.
The comparison of combinations of parameters is done byusing a Pearson correlation coefficient (r) between resultingleast cost or least distances and the distribution of vole den-sity in September 2002. As densities are expected to decreasewith the distance from the epicentre, the weakest correlationswill be obtained from the more relevant type of distance, i.e.the distance whose spatial distribution is the most realisticrepresentation of the spread phenomenon.
3. Results
The knowledge about vole swarming leads to assume thatgrasslands will always be preferred to other landscape classesand wooded areas involve a repulsion effect. Hence grasslandsconstitute the habitat of voles (H), wooded areas are the habi-tat of most vole predators that gives inhospitable areas (NH),and the other classes are defined as neutral areas (N).
3.1. Relationships between population density anddistances to the epicentre
Correlation coefficients are calculated between the densityvalues of September 2002 and the different compared set ofdistances. The different resistance values rm allocated to theunfavourable landscape class are 5, 10, 20, 40, 60, 80 and 100.The neutral class is successively affected a resistance value of1 (favourable), rm (also depending on each parameter set) and+∞ (symbolic value representing an impassable barrier).
The first result concerns the Euclidian distance, whichshows a strong relationship, with an r value of −0.76(p < 0.0001). The cost distances are then applied following theusual manner (f0 function). With a neutral class considered asan impassable zone, correlations are globally in the same areaas the values obtained from the Euclidian distance (Table 1).For the least cost, such a relationship decreases when the rm
value increases. For the least distance, the level of the corre-lation is very stable and not significantly different from the
20 −0.7613 −0.764440 −0.7580 −0.764160 −0.7542 −0.764280 −0.75023 −0.7644
100 −0.7462 −0.76455
110 e c o l o g i c a l m o d e l l i n g 2 1 0 ( 2 0 0 8 ) 104–114
f2-b
Fig. 5 – Relationships between f1 andNext functions f1 and f2 are applied using the same set ofrm values and by successively increasing distances dm of 5 pix-els (35 m), from 5 (35 m) to 50 pixels (350 m). Results obtainedwith function f1 are similar to the earlier results for all val-ues of distance dm and resistance rm, whatever the kind ofdistance used (Fig. 5a). On the other hand, the application offunction f2 leads to higher absolute values of Pearson corre-lations with some combinations of parameters, especially forleast cost (Fig. 5b). Globally the curves are very similar and allshow the same shape with the minimal correlation for dm = 20(140 m). When inspected in more detail, it becomes appar-ent that the level of relationship increases from resistancerm = 5 to 40 and becomes quite stable for the upper values.For the smaller distance, the difference between the correla-tions obtained with function f2 and using Euclidian distanceor the usual cost distances are not as high as seen previously.
Several maps are used to compare the different distancesfrom a spatial point a view. First, a spatial interpolation ofthe vole density is applied by means of an ordinary kriging(Cressie, 1993: 119). We note that the spatial precision of theground observations and the spatial resolution of the inter-polation grid (7 m) are not in agreement. Consequently, eventhough the accuracy of the density computed locally cannotbe validated, the resulting map can be considered as the ref-erence spatial distribution in order to evaluate the ability ofcost distances to represent the spreading process (Fig. 6a).
Second, the logarithmic transformation of the Euclidiandistances from the epicentre shows regular aureoles (Fig. 6b)which do not match the previous distribution. However, theapplication of usual cost distances (Fig. 6c) leads globally to
ased cost distances and vole density.
a similar distribution, where the shape of aureoles is weaklymodified near wooded patches. Only the cost distances basedon function f2 provide a global distribution which is more rad-ically different from an isotropic shape (Fig. 6d). In this case,the preferential orientation North-East/South-West is repro-duced even if several details in the maps Fig. 5a and d stronglydiffer.
4. Discussion and conclusion
The resistance values allocated to the landscape classes forthe cost distances computations have been defined here byusing topological functions of landscape. The main assump-tion explaining this proposal is the role of edges as interfacein the operation of ecosystems. Several kinds of functionsare possible, corresponding to assumptions on the movementforms of the studied species.
The case of the spread of vole has provided an examplewhere the method is applied by comparing distances froman epicentre to density measured after the start of a diffu-sion process. Compared to the Euclidian distance, the costsdefined by uniform resistances do not contribute to increasethe explanatory power of the resulting distances. In the sameway the costs defined by function f1 do not bring any change.This means that the wooded class cannot be considered as
partially permeable areas. On the other hand, the use of func-tion f2 leads to a higher level of statistical relationship, whenthe statistical criterion used is the least cost. This means thatthe grasslands near the wooded areas (in a spatial range of
e c o l o g i c a l m o d e l l i n g 2 1 0 ( 2 0 0 8 ) 104–114 111
Fig. 6 – Cost distances generated by function f2 in the studied zone.
i n g
112 e c o l o g i c a l m o d e l l140 m) are avoided, and this can be interpreted as some kindof spatial vulnerability. Predator constraints and accessibil-ity to the resources can be modelled as resistance functionincreasing when approaching an edge with a wooded element.This result is consistent with precedent conclusions obtainedfrom other analysis based on different methods (Foltete et al.,2005).
For all the types of cost distance, the resistance valueallocated to the neutral class has never played a role in theglobal relationship between distances and densities. This factis probably the consequence of a very weak surface taken upby this class; in other applications its role might depend on itssurface in the studied zone.
Concerning the results obtained with the function f2, it isinteresting to note that both “least cost” and “least distance”criteria provide different levels of statistical relationship. How-ever, the present analysis does not allow interpreting thisdifference from a general point of view, because the greaterrelevance of the least cost criterion may be linked to the spe-cific context of the present example. The diffusion process ofvole swarming involves a very complex combination of repro-duction, colonization, relation to predators and resources, and
cannot be compared to a simple movement of a set of individ-uals seeking to minimize their distance of moving. The costunit depends on the series of resistance values, making itsecological meaning difficult to interpret.Fig. 7 – Cost methods and rela
2 1 0 ( 2 0 0 8 ) 104–114
The fundamental contribution of the topological functionsof landscape used in the computation of cost distances is liesin the possibility they offer to successfully model a kind ofbarrier effect which is not directly visible in the land covermap. To be more precise, the classical approach of cost dis-tances involves no difference of habitat permeability betweena large zone and a “corridor” surrounded by inhospitablepatches, while the method presented here allows to distin-guish between these two cases. In other words, even if volepopulations live in the landscape matrix (i.e. in the studiedzone where nearly all grassland areas are inter-connected),the specific allocation of resistances according to the proposedmethod tends to gradually decrease the level of permeabilitynear inhospitable patches until absolute barriers are gen-erated when the density of these patches exceeds a giventhreshold. A theoretical example of this case is illustrated inFig. 7: it shows how the influence of some landscape elementson potential trips of individuals may be different according tothe method used for allocation of resistances. This means thatthe use of topological functions of landscape which does notrely only on strict localization but takes into account the spa-tial context results in a modification of the scale of influence
of landscape on the movement.Globally the method used here allows to substantiate someassumptions about the landscape-species relationships andto produce new knowledge elements. It can complete the
tive barrier of landscape.
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sual analysis of edge effects based on ground data by mak-ng use of GIS tools, especially when generalization of edgeffect patterns are difficult (Paton, 1994; Batary and Baldi,004). The method also provides useful data with the prospectf use of cost distances in other approaches: for example,he cost distances can be incorporated into certain diffu-ion model instead of Euclidian distances; the resistanceaps can be used in individual-based movement models.
he present example is based on a single point source butan easily become widespread with any set of spatial tar-ets. In the same way other types of function can also besed.
It remains to bring up an important limit inherent to theethod. The main interest of cost distance computations is
he possibility of taking into account the spatial heterogene-ty of the landscape matrix, especially in the case of speciesiving in habitat patches. The present method required a sim-lification of landscape with only two classes where resistancean be defined by a topological function, and another classeserved to a neutral behaviour towards the limit between thewo previous classes. It is theoretically possible to take intoccount more “neutral” classes by allocating several differentesistances in relation to land cover, but the number of resis-ance values to be defined is in reality difficult to manage, andorking on a very simplified list of land cover classes is muchore practical.One can certainly imagine an extension of the topological
unctions to more classes in order to combine several ecolog-cal effects but it would lead to a very complex method. Inddition, the presence of multiple edge effects proves to beifficult to interpret (Fletcher, 2005) and the validation of theesistance values which is time-consuming in the usual com-utations (Verbeylen et al., 2003) would be even longer in thisase. Consequently, the present method can be applied to rel-tively simple landscape contexts or to landscapes that cane simplified without significant loss of information.
cknowledgements
unding was provided by the Regional Council of the Franche-omte and the French Ministry of Environment. The authorsre grateful to F.P. Tourneux for helping with image prepro-essing and to R. Barzel for reviewing the language of theanuscript.
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