Dispersal success—the ability of organisms tolocate suitable habitat on a landscape—is a fun-damental property of most spatially structuredpopulation models. For metapopulation models,
A.W. King, K.A. With / Ecological Modelling 147 (2002) 23–3924
in which the extinction–colonization dynamics ofpopulations occupying habitat patches are mod-eled, dispersal success is equivalent to, or deter-mines, the patch colonization rate. Dispersal isthus modeled phenomenologically as the proba-bility that a given patch will be colonized. Inspatially implicit metapopulation models, thisprobability may be a function of the size orrelative isolation of the patch (i.e. landscape struc-ture; see Hanski, 1999 for a comprehensive treat-ment of metapopulation models). In contrast,spatially explicit population models (SEPMs), inwhich a population simulation model is coupledwith a landscape map (Dunning et al., 1995),often simulate dispersal explicitly using an indi-vidual-based approach with movement rules de-scribing how organisms interact with the spatialcomplexity of the landscape. The landscape isusually represented as a lattice of habitat cells,such as a raster-based map of a real landscape ina geographical information system (GIS) or as ahypothetical habitat distribution (neutral land-scape models; Gardner et al., 1987; With, 1997;With and King, 1997). At a minimum, the disper-sal modules of individual-based SEPMs includeinformation on (1) the type of movement (e.g.random, correlated random walk, self-avoidingwalk) and (2) either the dispersal distance ornumber of sites that can be searched by individu-als (e.g. Doak et al., 1992; Lamberson et al., 1992,1994; Lindenmayer and Possingham, 1996; Withand King, 1999a) or a dispersal-distance functionfor the distribution of propagules in the case ofplants (e.g. Lavorel et al., 1995). More elaboratedispersal modules may additionally include habi-tat-specific rates of movement or mortality, habi-tat-specific movement behaviors (such as morelinear movement in habitat matrix) and density-dependent dispersal (e.g. Pulliam et al., 1992;Lamberson et al., 1994; Liu et al., 1995; Linden-mayer and Possingham, 1996). Dispersal successis calculated as the fraction of individuals thatsuccessfully locate and occupy new habitat (Doaket al., 1992; With and King, 1999b).
Because habitat destruction and fragmentationare cited as the major causes of species endanger-ment (e.g. Meffe and Carroll, 1997), SEPMs havebecome increasingly popular in conservation ap-
plications. One of the more immediate conse-quences of habitat loss and fragmentation shouldbe a disruption of dispersal or movement betweenhabitat patches, which may lower colonizationsuccess and lead to local extinction. Conse-quently, ecology and conservation biology hasbecome preoccupied with space and modeling dis-persal success (Dunning et al., 1995; Wennergrenet al., 1995; Ruckelshaus et al., 1997; South,1999a). Spatial structure may not always be im-portant, however, for predicting dispersal success.Whether or not dispersal success is affected bylandscape pattern depends upon the scale ofmovement relative to the scale of fragmentation(Doak et al., 1992; With and King, 1999b). Infact, definitions of landscape connectivity—whether a given landscape is perceived as frag-mented by a particular species—are based onhow organisms interact with spatial pattern (Tay-lor et al., 1993; Pearson et al., 1996; With, 1997;With et al., 1997). Detailed information on disper-sal of real organisms is often lacking, however,particularly for parameters such as dispersal mor-tality and thus, it can be very difficult to parame-terize the individual-based dispersal algorithmsused in SEPMs (Wennergren et al., 1995). Incor-rect estimates of dispersal distances and dispersalmortality resulted in the greatest prediction errorsfor dispersal success (Ruckelshaus et al., 1997). Ifthese errors propagate in SEPMs, then estimatesof population viability (patch occupancy andprobability of persistence) may be so seriouslybiased as to be worthless (but see South, 1999a).
Because of these concerns and the central im-portance dispersal has in spatial population mod-els, it is important to understand the conditionsunder which landscape structure affects dispersalsuccess, as a means of identifying when a spatiallyexplicit approach is necessary. Other investigatorshave used simple landscapes to investigate theeffects of landscape structure on dispersal success(e.g. Doak et al., 1992; Ruckelshaus et al., 1997),but have not explored the full range of spatialcomplexity across a gradient of habitat abun-dance and fragmentation, as we do here usingneutral landscape models. It is also important tounderstand how sensitive dispersal success is todifferent movement algorithms, particularly those
encountered in the lattice-based SEPMs that havebecome so popular. Many of these lattice-basedapproaches use simple movement rules, whichmay lack realism but which therefore do notrequire detailed empirical behavioral data to im-plement. If dispersal success is less sensitive to theway in which dispersal is modeled than to theeffects of landscape structure, then concerns overthe realism of movement rules become less criticaland would be welcome news given the paucity ofempirical data on dispersal. We therefore under-took a simulation study to address the relativeeffects of landscape structure and dispersal behav-ior on dispersal success.
2. Methods
2.1. Neutral landscape models
We used the program RULE (Gardner, 1999)to generate spatially structured landscapes (i.e.neutral landscape models). Binary landscapeswere generated as either a random or fractaldistribution of habitat and non-habitat on a two-dimensional square grid measuring 128×128 cells(Fig. 1). Random landscapes were created byrandomly assigning some fraction (h) of the cellsto be habitat. Fractal landscapes were generatedusing the mid-point displacement algorithm(Saupe, 1988) in which both the fraction (h) andthe contagion of habitat (H) were varied to createcomplex landscape patterns across a gradient ofhabitat fragmentation (see With, 1997; With et al.,1997 for further details). Habitat abundance (h)was simulated at 12 levels (h=0.001, 0.01, 0.05,
0.10, 0.20, . . . , 0.90) for random and fractal land-scapes. For fractal landscapes, we also varied thecontagion of habitat (H) across three levels (H=0.0, 0.5, 1.0) for each level of h. This resulted inlandscapes that varied in the degree of habitatfragmentation (Fig. 1) and also allowed us totease apart the relative effects of habitat abun-dance (h) and fragmentation (H) on dispersalsuccess. Ten replicate maps were generated foreach landscape type (random: 12 h-levels×10=120 maps; fractal: 12 h-levels×3 H-levels×10=360 maps; total=480 landscape maps). We thensimulated individual dispersal and calculated dis-persal success for different dispersal algorithms onthese landscape maps.
2.2. Dispersal algorithms
We modeled dispersal as a random process, anearest-neighbor process (NND) or as a percola-tion process (PD).
2.2.1. Random dispersalIn random dispersal, the disperser moved with
random direction and distance to any point (cell)on the landscape. If dispersal is modeled in thisfashion, spatial structure does not matter andonly the amount of habitat and the number ofcells searched affect dispersal success:
P(success)=1−um (1)
where u=1−h and m is the number of cells thatcan be searched (Lande, 1987; With and King,1999a). This is equivalent to the mean-field ap-proximation for the probability of dispersal suc-cess and serves as a baseline for assessing how
Fig. 1. Neutral landscape models. All four maps contain the same amount of habitat (h=0.5, black cells). Fractal landscapes varyin the spatial contagion (H) of habitat, which produces a gradient of fragmentation.
A.W. King, K.A. With / Ecological Modelling 147 (2002) 23–3926
spatial structure and different dispersal behaviorsaffect dispersal success (Bascompte and Sole,1998). Note that in this definition of randomdispersal, m does not define a local neighborhood.It simply defines the number of cells across theentire landscape that can be visited in seekinghabitat. The distance between visited cells and thedistance from the cell of origin are random vari-ables. Random movement of the type describedby our random dispersal algorithm has beenshown to be a reasonable estimate of dispersal inmany insects.
(NND), individual dispersers (n=1000) were in-dependently initialized at randomly chosen habi-tat cells on the landscape. At each time step, thedispersers were constrained to move with equalprobability into one of the four neighboring cells(the ‘nearest neighbors’). The direction of move-ment was random, but the dispersal step lengthwas limited to one cell. This is a non-self-avoidingrandom walk of fixed step-length and the ‘area-limited dispersal’ (ALD) of With and King(1999b). On average, this movement rule producesmore localized dispersal than a random dispersalalgorithm, in which both direction and step lengthare random. Dispersers were permitted to take upto m steps in their search for a suitable habitat cell(m=1, 2, 5, 10, 20 or 50 steps), and could movethrough non-habitat cells. Upon encountering acell of suitable habitat, the disperser stopped andwas scored as a success. The number of steps adisperser can take (m) may be interpreted aseither the innate dispersal ability of the species oras an indirect measure of matrix quality thataffects dispersal for a given species in differentlandscape contexts (e.g. dispersal distances m maybe reduced in landscapes with hostile matrix habi-tat, perhaps because dispersers are unwilling tocross such inhospitable habitat or suffer highermortality if they do). While a theoretical abstrac-tion, NND might approximate dispersal by largermammals or birds willing to cross unsuitablehabitat to find appropriate habitat in which toestablish dens or nests. The edge of the grid was areflective barrier, but edge effects were insignifi-
cant because of the large size of the grid (16,384cells), the relatively limited number of grid cellsdispersers could search (m�50) and the largenumber of independent dispersal events (n=1000)(With and King, 1999b).
2.2.3. Dispersal by percolationDispersal as a percolation process (PD) is simi-
lar to NND except individuals were constrainedto move only through habitat cells. This is basi-cally the With (1997) ‘Rule 1’ movement, whichdescribes organisms that are unable or unwillingto cross unsuitable habitat (i.e. a species that lacksgap-crossing abilities; Dale et al., 1994). PD mightapproximate the dispersal of voles or other smallanimals unwilling to leave the cover of suitablehabitat and risk predation or environmentalstress. Dispersal by percolation (PD) is the mostrestrictive movement algorithm of the three weexplored, and it has the potential to result in themost localized dispersal if habitat is not wellconnected and landscapes lie below the percola-tion threshold (i.e. the threshold amount of habi-tat, hcrit, at which a continuous cluster of habitatno longer occurs on the landscape; see With,1997). As with NND, the map boundaries (edgesof the grid) were reflective barriers.
2.3. Simulation design
Simulations were conducted in a factorial de-sign, in which landscape pattern (random, H=0.0, H=0.5 and H=1.0), habitat abundance (h)and dispersal ability (m) were varied, resulting in2880 simulation runs (480 landscape maps×6m-levels). The probability of NND and PD dis-persal success on a given landscape for a givenlevel of m was calculated as the proportion ofindividuals (n=1000) that located a suitable habi-tat cell within m steps. We compared dispersalsuccess for random dispersal and for NND andPD on both random and spatially structuredlandscapes. We also examined variability in dis-persal success among replicate maps for a givenlandscape (With and King, 1999a calculated dis-persal success on fractal landscapes for only asingle map for each landscape combination of hand H).
Fig. 2. Mean dispersal success (N=10 replicate landscapes) asa function of dispersal strength (m) for different dispersers onrandom and fractal landscapes with 0.1, 1 or 5% habitat.NND, nearest-neighbor dispersal; PD, percolation.
NND on random landscapes and for randomdispersal when dispersal ability is limited (m=1,2) (Fig. 6). Dispersal success is non-linear andasymptotic for NND and PD on fractal land-scapes (Figs. 6 and 7) and for greater dispersalstrengths by random dispersers and NND and PDon random landscapes (Fig. 7). The function isgenerally monotonic, except for PD on random orhighly fragmented fractal landscapes (H=0.0)(Figs. 6 and 7), but these exceptions are perhaps aconsequence of stochasticity. Dispersal as a perco-lation process (PD) produces greater variability indispersal success among replicate landscapes, par-ticularly if the habitat is randomly distributed(Fig. 8). This is attributable to the greater vari-ability in the clustering of habitat cells into con-tiguous patches among replicate random maps.Contagious clustering greatly influences the suc-cess of PD since those dispersers cannot cross
Fig. 3. Mean dispersal success (N=10 replicate landscapes) asa function of dispersal strength (m) for different dispersers onrandom and fractal landscapes with 10, 20 or 30% habitat.NND, nearest-neighbor dispersal; PD, percolation.
3. Results
Dispersal success is an increasing function ofdispersal strength (m) for all landscape patternsand dispersal behaviors (Figs. 2–5). The relation-ship is typically non-linear, with a rapid increasein dispersal success at small m followed by anasymptotic approach to a maximum probabilityof dispersal success. The asymptote is achievedmost rapidly in landscapes with abundant habitat(Fig. 5). When habitat is very rare, the rate ofapproach to that asymptote is very slow for bothrandom dispersal and NND on random maps; PDon random maps asymptotes quickly to low levelsof success (Fig. 2).
Dispersal success is also an increasing functionof habitat abundance (h) for all landscapes anddispersal behaviors (Figs. 6 and 7). The functionis linear or approximately linear for PD and
A.W. King, K.A. With / Ecological Modelling 147 (2002) 23–3928
Fig. 4. Mean dispersal success (N=10 replicate landscapes) asa function of dispersal strength (m) for different dispersers onrandom and fractal landscapes with 40, 50 or 60% habitat.NND, nearest-neighbor dispersal; PD, percolation.
uncommon (10–30%), success is not assured ex-cept when habitat is highly clumped (H=1.0)until m=20 (Fig. 3). Even then, at 10% habitat,dispersal success is �90% for random dispersaland for NND and PD dispersal on random andhighly fragmented fractal landscapes (H=0.0;Fig. 3). When habitat is very rare (�10%), levelsof success as high as 90% may not be achieveduntil m�50, except for NND and PD onclumped fractal landscapes (H=0.5, 1.0; Fig. 2).At 0.1% habitat, mean dispersal success rarelyexceeds 90%, although it may exceed that forNND and PD on some clumped fractal land-scapes (Fig. 2).
Thus, dispersal success tends to converge withincreasing dispersal strength (m) and habitatabundance (h), but differences emerge amonglandscape patterns and dispersal behaviors whendispersal is limited (small m) and habitat is scarce
Fig. 5. Mean dispersal success (N=10 replicate landscapes) asa function of dispersal strength (m) for different dispersers onrandom and fractal landscapes with 70, 80 or 90% habitat.NND, nearest-neighbor dispersal; PD, percolation.
non-habitat to find habitat. Hence, variability inclustering is reflected in variability in dispersalsuccess.
When habitat is abundant (e.g. �70%), theprobability of success is high (�70%) even forthe most limited dispersers (m=1; Fig. 5) andsuccess is virtually guaranteed (�95%) within fivedispersal steps regardless of dispersal behavior orlandscape structure (Fig. 5). Dispersal success isnot, however, independent of landscape structureand dispersal behavior at intermediate levels ofhabitat abundance. At intermediate levels of habi-tat abundance (40–60%), success is not generallyguaranteed (�95%) until dispersal strength ap-proaches m=10 (Fig. 4). However, on clumpedfractal landscapes (H=0.5, 1.0) of intermediatehabitat abundance, dispersal success for NNDand PD is still high even for weak dispersers(m�5, Figs. 4 and 6). When habitat is rare to
Fig. 6. Mean dispersal success (N=10 replicate landscapes) asa function of habitat abundance (h) for different disperserswith limited dispersal strength (m=1, 2 or 5) on random andfractal landscapes. NND, nearest-neighbor dispersal; PD, per-colation.
especially if dispersal strength is limited (Fig. 3) orhabitat is scarce (h�0.1 for all fractal landscapes;Tables 1 and 2, Fig. 6). Because individuals begintheir search from a habitat cell, even individualsconstrained to move only to an adjacent cell(m=1) have a higher probability of encounteringhabitat in fractal landscapes with high spatialcontagion than in landscapes where habitat israndomly distributed. This effect of contagiouspattern is greatest when habitat is rare (low h).Not unexpectedly, landscape pattern is unimpor-tant if dispersal strength is high or if habitat isabundant (Fig. 7). Differences between randomand fractal landscapes exist even for strong dis-persers (m=50), however, when habitat is scarce.For NND, this occurred in landscapes with �5%habitat (Table 1, Figs. 2 and 7). For PD, strongdispersers had lower success in random than frag-
Fig. 7. Mean dispersal success (N=10 replicate landscapes) asa function of habitat abundance (h) for different disperserswith moderate to high dispersal strength (m=10, 20 or 50) onrandom and fractal landscapes. NND, nearest-neighbor dis-persal; PD, percolation.
and fragmented. The probability of successful dis-persal is thus often a consequence of interactionsbetween dispersal behavior, habitat abundanceand landscape structure. It is useful to try andseparate these influences into landscape effectsand behavioral effects.
3.1. Landscape effects: random �ersus fractallandscapes
Recall that dispersal success for random dis-persers is independent of landscape pattern anddepends only upon the amount of suitable habitat(h) and the number of dispersal steps (m) (Eq.(1)). In contrast, landscape pattern does affectdispersal success for NND and PD. Dispersalsuccess is generally higher for both NND and PDon fractal landscapes than on random landscapes,
A.W. King, K.A. With / Ecological Modelling 147 (2002) 23–3930
Fig. 8. Dispersal success (mean�S.D., N=10 replicate land-scapes) as a function of habitat abundance (h) for disperserswith different dispersal strength (m=1, 10 or 20) on randomand fractal landscapes of H=0.5. NND, nearest-neighbordispersal; PD, percolation. Note the greater variability amongreplicate maps for PD on random landscapes.
tal landscapes with minimum contagion (H=0.0)is not present at 50% habitat, but is present at 40and 60% habitat (Fig. 6). This may be a stochasticresult owing to high variability in dispersal suc-cess among maps for PD on random and highlyfragmented fractal landscapes (H=0.0), or it mayindicate a complex relationship between PD andspatial pattern. In general, however, dispersal suc-cess of both NND and PD tended to increase withincreased spatial contagion of habitat.
The RULE-generated landscape metrics for therandom and fractal landscapes used in this studyare given in With and King (1999b). Theoreti-cally, dispersal success for PD should be highlycorrelated with the RULE calculated correlationlength (a measure of patch contagion). Correla-tion length is itself correlated with H (as H in-creases, correlation length increases) and thatcorrelation is reflected in the greater dispersalsuccess for PD with increasing H (Figs. 2–7).However, With and King (1999b) found that cor-relation length does not predict dispersal successfor NND. Indeed, dispersal success was not re-lated to any of the patch-based measures of land-scape structure (including correlation length), butwas correlated with lacunarity thresholds, a mea-sure of gap structure or inter-patch distances (cf.Figs. 4 and 5 of With and King, 1999b).
3.2. Beha�ioral effects: comparison of dispersalalgorithms
There is essentially no difference between PDand NND on fractal landscapes with H�0.5(Figs. 2–7, Table 3). On highly fragmented fractallandscapes (H=0.0) with 5–20% habitat, there issome tendency for NND to become more success-ful than PD as dispersal strength increases. Thiscrossover occurs, for example, at m�17 for 20%fragmented fractal landscapes (Fig. 3) or at m�25 for 5% fragmented fractal landscapes (Fig. 2).
The comparison between PD and NND onrandom landscapes is more complicated. Whenhabitat is extremely rare (0.1%) (Fig. 2) or disper-sal is limited to a single neighboring cell (m=1, 2)(Fig. 6), there is no difference between PD andNND, although dispersal success is more variablefor PD on random landscapes (Table 3). Simi-
mented fractal landscapes (H=0.0) with �30%habitat (Table 2, Figs. 2, 3 and 7). Alternatively,for weak dispersers (m=1, 2), landscape patterngenerally affected dispersal success even whenhabitat was abundant (Tables 1 and 2, Figs. 5 and6). When habitat was abundant (�80%) buthighly fragmented (H=0.0), weak dispersers con-strained to move through habitat (PD), generallyhad lower success than in random landscapes(Figs. 5 and 6), reversing the general trend ofgreater success on fractal landscapes. A similarexception apparently occurs for weak PD onhighly fragmented landscapes (H=0.0) with 50%habitat. Dispersal success did not increasemonotonically as a function of habitat abundancefor PD with extremely localized dispersal (m�2);a difference in success between random and frac-
larly, allowing for the greater variability in PD,there is no difference between PD and NND onrandom landscapes with �40% habitat (Table 3,Figs. 4, 5 and 8), although there is some tendencyfor PD to be more successful than NND whendispersal is limited (Figs. 4 and 8). Within therange of 1 and 30% habitat, however, differencesin the success of these two types of dispersaldepend upon dispersal strength. For example, dis-persal success is higher for NND on randomlandscapes with 5% habitat for m�6 and contin-ues to increase with increasing dispersal strength,whereas dispersal success plateaus for PD at �18% (Table 3, Fig. 2). For strong dispersers (m=50), this results in a difference in dispersal successof 57% on these 5% random landscapes (Fig. 7).Depending upon the dispersal strength, differ-ences may occur between PD and NND at vari-ous points within or across the entire range of1–40% habitat on random landscapes (Table 3,Figs. 6 and 7).
Random dispersal does not differ from NNDor PD on random landscapes for weak dispersers
(m=1) or when habitat is exceedingly rare (0.1%;Table 4). Again, the higher mean success of PDover random and NND dispersal at h=0.4–0.5and m=1 (Fig. 6) may be a stochastic effect.Differences between random dispersal and NNDon random landscapes emerge across particularranges of habitat depending upon the dispersalstrength (Table 4). For example, dispersal successis higher for random dispersers than NND onrandom landscapes with 5–60% habitat for indi-viduals constrained to move up to five steps (Figs.2–4, Table 4). There is no difference betweenrandom dispersers and NND above or below thishabitat range on random landscapes. Similarcomplex interactions emerge at particular levels ofhabitat abundance on random landscapes, inwhich differences in dispersal success betweenrandom dispersal and NND occur within a partic-ular range of dispersal strengths. For example,dispersal success is �10% greater for randomdispersers that can move two to 14 steps on 30%random landscapes (Fig. 3). Below this habitatrange, however, dispersal success is greater for
Table 1Summary of simulation results comparing the difference in dispersal success (probability of encountering a suitable habitat cell)between random and fractal (H=0.0, 0.5 and 1.0) landscapes for nearest neighbor dispersers (NND). Is there a difference betweenrandom and each of the fractal landscapes for NND?
H=1.0H=0.5H=0.0
Dispersal strengthm=1 Yes, if h�0.9 Yes Yes
Yes, if h�0.8 Yes, if h�0.9 Yes, if h�0.9m=2Yes, if h�0.5 Yes, if h�0.6m=5 Yes, if h�0.7
Yes, if h�0.5Yes, if h�0.4Yes, if h�0.3m=10Yes, if h�0.2m=20 Yes, if h�0.3Yes, if h�0.2
Yes, if h�0.2 Yes, if h�0.2 Yes, if h�0.2m=50
Habitat abundanceh=0.001 YesYesYes
Yesh=0.01 Yes YesYes Yesh=0.05 Yes
YesYesh=0.10 YesYes, for m�17 Yes, for m�20h=0.20 Yes, at least for m�20Yes, for m�9h=0.30 Yes, for m�11 Yes, for m�15Yes, for m�7 Yes, for m�9h=0.40 Yes, at least for m�10
h=0.50 Yes, for m�4 Yes, for m�7 Yes, for m�8No, except for m�3h=0.60 Yes, for m�8Yes, for m�6
Yes, for m�5Yes, for m�4h=0.70 No, except for m�2No, except for m�2 No, except for m�2 No, except for m�3h=0.80No, except for m�2 No, except for m=1 No, except for m=1h=0.90
A.W. King, K.A. With / Ecological Modelling 147 (2002) 23–3932
Table 2Summary of simulation results comparing the difference in dispersal success (probability of encountering a suitable habitat cell)between random and fractal (H=0.0, 0.5 and 1.0) landscapes for dispersal as a percolation process (PD). Is there a differencebetween random and each of the fractal landscapes for PD?
H=0.5H=0.0 H=1.0
Dispersal strengthm=1 YesYes, for h�0.4 and h=0.7 Yesm=2 Yes, for h�0.9Yes, for h�0.5 and h=0.7 Yes, for h�0.9
Yes, for h�0.7Yes, for h�0.3 Yes, for h�0.7m=5Yes, for h�0.5 Yes, for h�0.5m=10 Yes, for h�0.3Yes, for h�0.5Yes, for h�0.3 Yes, for h�0.5m=20
YesYes Yesh=0.05Yesh=0.10 YesYesYesYes Yesh=0.20Yesh=0.30 YesNo, except for m�3YesNo Yesh=0.40
Noh=0.50 No, except for m=1 No, except for m�3Noh=0.60 No, except for m�10 No, except for m�10
No, except for m�5No, except for m�2 No, except for m�5h=0.70Noh=0.80 No, except for m�4 No, except for m�4
No, except for m�2No No, for m�2h=0.90
random dispersal on random landscapes with 1–20% habitat, once sufficient dispersal strength hasbeen attained (m�3 for 20% random landscapes,m�17 for 1% landscapes; Table 4).
Dispersal success is generally higher for NNDthan random dispersal on fractal landscapes, par-ticularly if habitat is rare (h�0.05) or dispersal islimited to the neighboring cell or two (m=1, 2)and habitat is sufficiently abundant (h�0.6;Table 4). Increasing dispersal strength mitigatesany differences between NND and random dis-persal on highly clumped fractal landscapes (H=0.5, 1.0), except when habitat is scarce. Forexample, dispersal success is 10–70% higher fornearest-neighbor dispersers that can move up tofive steps on highly clumped (H=1.0) fractallandscapes with �50% habitat (Fig. 6). Forstrong dispersers (m=50), however, the highersuccess of NND over random dispersal occursonly with 10% habitat on clumped fractal land-scapes (Fig. 7). The relationship is more complexon highly fragmented (H=0.0) fractal landscapes.Differences between NND and random dispersal
generally occur at particular levels or ranges ofhabitat abundance once a limited amount of dis-persal has been attained (e.g. m�5), although thegeneral effect of convergence with increasing dis-persal strength still holds (Table 4). Alternatively,for a given level of habitat abundance between 20and 70% on fragmented fractal landscapes, con-vergence between NND and random dispersalmay occur within or across a particular range ofdispersal strengths (Table 4). The relationship ismore straightforward on clumped fractal land-scapes (H=0.5, 1.0), although a significant differ-ence between NND and random dispersal persistseven when the landscape is mostly habitat (90%)for individuals constrained to move only a singlestep (m=1; Figs. 5 and 6).
4. Discussion
Because dispersal is featured in most spatialpopulation models, it is important to understand(1) the conditions under which landscape struc-
ture affects dispersal success and (2) the depen-dency of dispersal success on the choice of disper-sal algorithm (movement rule). We thereforeundertook this simulation study to address therelative effects of landscape structure and disper-sal behavior on dispersal success. The mainresults are summarized in Table 5. A comparisonof local dispersal on spatially structured land-scapes with the mean-field approximation (ran-dom dispersal) provides the most general evidencefor the importance of landscape structure on dis-persal success. Differences between random andfractal landscapes for local dispersal processesrepresent a finer distinction regarding the impor-tance of spatial structure in predicting dispersalsuccess.
When do spatial pattern (landscape structure)and dispersal behavior really matter? The mean-field approximation (random dispersal) was apoor predictor of dispersal success for local dis-persal processes (NND or PD) on spatially struc-tured landscapes when habitat abundance was
�40% and when dispersal was limited. Land-scape structure (as measured by H) was alwaysimportant for predicting the success of weak dis-persers constrained to move within a local neigh-borhood (m=1, 2 for NND and PD) unlesshabitat was very abundant (�80%). In general,dispersers attained highest success on landscapesin which habitat had a high degree of spatialcontagion (H=0.5, 1.0). Habitat clumping maythus mitigate the negative effects of habitat losson dispersal success, a result also reported fromother studies (e.g. Doak et al., 1992; Lambersonet al., 1994; Wennergren et al., 1995; Ruckelshauset al., 1997). In our study, for example, even weakdispersers (m=1) could achieve high levels ofsuccess (�70%) in landscapes with as little as 1%habitat when it was clumped (H=1.0). The effectof habitat clumping is particularly notable fordispersers that are constrained to move onlywithin a particular habitat type (PD). Such dis-persers were very sensitive to habitat fragmenta-tion, and even strong dispersers (m=20, 50)
Table 3Summary of simulation results comparing the difference in dispersal success (probability of encountering a suitable habitat cell)between nearest-neighbor dispersal (NND) and dispersal as a percolation process (PD) on random and fractal (H=0.0, 0.5 and 1.0)landscapes. Is there a difference between NND and PD in each of these landscapes?
H=0.5H=0.0 H=1.0Random
Dispersal strengthNo, although PD more variable Nom=1 No No
m=2 NoNoNoNoNoNoNo, except for h=0.3 Nom=5
m=10 No, except for h=0.05 and h=0.2–0.3 No No NoNo Nom=20 NoYes, for 0.01�h�0.4
Yes, for 0.01�h�0.4 NoYes, for 0.05�h�0.2m=50 No
Habitat abundanceNo Noh=0.001 NoNo
No Noh=0.01 Yes, if m�16 NoYes, if m�6 Yes, if m�25h=0.05 No No
NoNoYes, if m�19h=0.10 Yes, if m�10Yes, if m�9 Yes, if m�17h=0.20 No No
No Noh=0.30 Yes, if m�3 No, PD success becomes more variableNo Noh=0.40 No No
h=0.50 No No No NoNo Noh=0.60 NoNo
NoNoNoh=0.70 NoNo No No Noh=0.80No No No Noh=0.90
A.W. King, K.A. With / Ecological Modelling 147 (2002) 23–3934
Table 4Summary of simulation results comparing the difference in dispersal success (probability of encountering a suitable habitat cell)between random dispersal and nearest-neighbor dispersal (NND) on random and fractal (H=0.0, 0.5 and 1.0) landscapes. Is therea difference between random dispersal and NND?
H=0.0 H=0.5Random H=1.0
Dispersal strengthYes, if h�0.9 YesNo Yesm=1Yes, if h�0.6 Yes, if h�0.8m=2 Yes, if h�0.9Yes, if 0.3�h�0.8Yes, except for h=0.3–0.4 and h�0.6 Yes, if h�0.4Yes, if 0.05�h�0.6 Yes, if h�0.5m=5
m=10 Yes, except for h=0.2 and h�0.4Yes, if 0.05�h�0.4 Yes, if h�0.3 Yes, if h�0.3Yes, except for h=0.1 and h�0.3 Yes, if h�0.2Yes, if 0.05�h�0.3 Yes, if h�0.2m=20
m=50 Yes, except for h=0.05 and h�0.2Yes, if 0.05�h�0.2 No, except for h�0.05 No, except for h�0.1
Habitat abundanceYes Yes Yesh=0.001 NoYes YesYes, if m�17 Yesh=0.01Yes Yesh=0.05 YesYes, if m�5Yes, if m�15 YesYes, if m�4 Yesh=0.10Yes, except if 8�m�14 Yes, if m�11 Yes, if m�15h=0.20 Yes, if m�3Yes, except if m=5–9 and m�13 Yes, if m�7Yes, if 2�m�14 Yes, if m�9h=0.30Yes, except if m=4–5 and m�10 Yes, if m�6h=0.40 Yes, if m�7Yes, if 2�m�10No, except for m=1–2 and m=5–6 Yes, if m�4Yes, if 2�m�7 Yes, if m�5h=0.50
No, except for No, except for m=1 and 4 No, except if m�3h=0.60 No, except if m�4m=2–4No, except for No, except for m=1 and m=3–4h=0.70 No, except if m�3 No, except if m�4m=2–4No, except forH=0.80 No, except if m�3 No, except for m�2 No, except if m�2m=2–3
No, except for m=1h=0.90 No, except for m=1No, except if m=2 No, except for m=1
experienced low success on random and frag-mented fractal landscapes (H=0.0) that lay be-low the percolation threshold (hcrit) where habitatconnectivity had been disrupted (hcrit=0.59 forrandom landscapes, hcrit=0.54 for fractal H=0.0landscapes; With and King, 1999b).
How sensitive is dispersal success to differentdispersal algorithms, especially in terms ofwhether movement is modeled as a percolationprocess (restricted to move only within habitat)versus localized dispersal in which dispersers alsomove through non-habitat (NND)? PD and NNDwere very similar, especially in fractal landscapesbecause of the higher relative spatial contagion ofhabitat (measured by H) and the high probabilitythat adjacent cells would be habitat (Fig. 1).Divergence between these two localized dispersalprocesses occurred primarily in random land-scapes, with 1–30% habitat, for which sufficient
dispersal strength had been attained (Table 5).Thus, the main difference is between local disper-sal and random dispersal (the mean-field approxi-mation). The difference between these algorithmsis in step length relative to the spatial grain andextent of the landscape pattern. To what extent,then, can dispersal success be modeled as a ran-dom process that ignores localized interactionswith spatial structure? Random dispersal is gener-ally a good approximation in landscapes with�40% habitat unless dispersal strength is limited(m�5).
Many dispersal models are not based on themovement of individuals (representing the spreadof populations instead), or are not spatially ex-plicit, or if they are, do not consider heteroge-neous or non-random spatial patterns (e.g. Wilderet al., 1995; Blackwell, 1997; Johst and Brandl,1997; Torres-Sorando and Rodriguez, 1997; Mc-
A.W
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Table 5Summary of the effects of landscape structure and choice of dispersal algorithm on dispersal success1
Is landscape Yes Yes Yes Yes Yes, unless No, unlessYes, unless No, unlessYes, unless Yes, unless No, unless No, unlessm�10m�20 for m�7m�15 for m�5structure m�4 m�2m�3
NNDimportant NNDinpredictingdispersalsuccess?
No, except ifYes, unless No, except ifIs dispersal Yes, unless No, exceptNo, except if No, exceptUsually Usually Usually No, except if No, except ifm�5m�8 for PD m�3random m�3randomsuccess m�2 for m=1for m=1
landscape landscape andaffected by with m�5choice of m�10dispersalalgorithm?
50m1 2 5 10 20Yes, unless Yes, unlessIs landscape Yes, unless No, except No, except No, except ifh�0.9 h�0.7 h�0.4structure when h�0.3 h�0.3when h�0.3
importantinpredictingdispersalsuccess?
Yes, unless No, unlessYes, unlessIs dispersal No, unless No, unless No, unlesssuccess h�0.4h�0.8 h�0.4 h�0.4h�0.9 h�0.4affected bychoice ofdispersalalgorithm?
1 To address the effect of landscape structure, comparisons are made between landscape types (random versus fractal) for a given type of dispersal algorithm and particularly against the mean-field approximation (randomdispersal) which is unaffected by landscape structure. To assess the importance of the choice of dispersal algorithm on dispersal success, comparisons are made among dispersal algorithms for a particular landscape typeand with the mean-field approximation of random dispersal.
A.W. King, K.A. With / Ecological Modelling 147 (2002) 23–3936
Carthy, 1999; South, 1999b). Consequently, thesemodels are not appropriate dispersal submodelsfor the individual-based SEPMs we are concernedwith here. Other models describe the dispersal ofplant seeds as algebraic or exponential decayfunctions of distance (e.g. Malanson and Arm-strong, 1996; Latore et al., 1994), while our dis-persal algorithms address animal movement andbehavior. Moreover, few analyses of dispersal suc-cess systematically vary both spatial pattern anddispersal behavior as we have done here. Directcomparisons of our results with earlier results arethus difficult. Nevertheless, a variety of modelingstudies have shown dispersal success to be influ-enced by landscape pattern or animal behavior.For example, Byers (1996) found that relativelyfew simulated bark beetles with very low flightspeeds found suitable host trees when the treeswere widely distributed, a situation comparable toour relatively weak dispersers (m�5) on randomlandscapes of rare habitat (h�0.30). Gustafsonand Gardner (1996) found that most of the vari-ability in the dispersal success of a self-avoidingrandom walker (similar to our NND, althoughour NND is not self-avoiding) was accounted forby differences in the size and isolation of patches.They also found that dispersal success was verysimilar on random and homogeneous landscapes,but significantly reduced on ‘curdled’ maps ex-hibiting patch contagion or aggregation(Gustafson and Gardner, 1996). In a simulationof forest beetles, Tischendorf et al. (1998) foundthat the proportion of individuals arriving at asink patch varied with the length and width ofmodeled hedgerows. Other modeling studies,while not explicitly quantifying dispersal success,have shown that migration rates, species diversity,resource consumption, patterns of species distri-bution consumption and the survival of individu-als, populations and metapopulations are allinfluenced by spatial heterogeneity (Dyer, 1995;Anderson, 1996; Baker, 1996; Malanson andArmstrong, 1996; Swart and Lawes, 1996; Withand Crist, 1996; Blaine and DeAngelis, 1997;Fahrig, 1998; Carter and Finn, 1999; With andKing, 1999b). That influence arises, at least inpart, from the impacts of spatial pattern andmovement behavior on dispersal success. These
results, combined with arguments for the criticalimportance of movement rules in individual-basedSEPMs (Railsback et al., 1999), argue the needfor a fundamental understanding of when andhow spatial structure interacts with dispersal be-havior to affect dispersal success.
Although the relationship between landscapestructure and dispersal success may not be asimple matter to resolve (Doak et al., 1992), oursimulation results indicate that both dispersal be-havior and the explicit arrangement of habitat(landscape structure) are predicted to affect dis-persal success in landscapes with �30–40% habi-tat. Given that the amount of suitable habitatavailable to species of conservation concern maygenerally fall within this range (e.g. Ruckelshauset al., 1997), details of how organisms interactwith spatial pattern may be essential to the selec-tion or development of dispersal modules inSEPMs implemented on landscapes with �40%habitat (Fig. 9). In other words, representationsof dispersal based on mean-field approximationswill not suffice under these circumstances andmay provide biased estimates of dispersal success.Furthermore, errors in estimating dispersalparameters may be most severe in landscapes witha limited amount of habitat (Wennergren et al.,1995; Ruckelshaus et al., 1997). For example,Wennergren et al. (1995) found that ‘modest’errors in estimating attributes of dispersal behav-ior propagated into huge errors in predicting dis-persal success for a variety of habitatconfigurations in landscapes with 2–25% habitat.If the per-step mortality rate during dispersal isoverestimated by 16–24%, the prediction errormay be as high as 90% (Ruckelshaus et al., 1997).Prediction errors resulting from errors in estimat-ing dispersal distances tended to be less than thoseassociated with dispersal mortality, however, andwere generally �10% in landscapes with �16%habitat (Ruckelshaus et al., 1997). Thus, it islandscapes in which habitat has been drasticallyreduced (e.g. to 10%) in which all these sources oferror have the greatest potential to be com-pounded and where details on dispersal behaviormay be most critical for predicting dispersalsuccess.
As has been pointed out recently by South(1999a), however, such concerns over accuratepredictions of dispersal success may be misplaced.These concerns emanate from two assumptions:(1) errors in predicting dispersal success propagateinto similar errors in estimates of population vi-ability within SEPMs, and (2) dispersal success isimportant for predicting population viability inthe first place. Although dispersal figures promi-nently in SEPMs, dispersal may not be the key topopulation persistence, contrary to conventionalwisdom (e.g. Opdam, 1990). Studies are emergingthat demonstrate demographic factors may, insome circumstances, be more important than dis-persal for population persistence in fragmentedlandscapes (Pulliam et al., 1992; Liu et al., 1995;With and King, 1999a; South, 1999a). South
(1999a) found, for example, that if the intrinsicpopulation growth rate (�) was sufficiently high(e.g. ��1.12), populations persisted in the major-ity of cases irrespective of dispersal success. Ifdispersal is generally less important than demo-graphic factors for population persistence, thenSEPMs may not require detailed dispersal mod-ules. Fairly simple movement rules, such as thosefeatured in our study, or even patch incidencefunctions (With and King, 2001) may suffice tocapture the essential properties of how organismsredistribute in space. The choice of movementrule should not be made, however, without con-sideration of the influence of spatial pattern ondispersal success. As we have shown here, model-ing dispersal as a random process (the mean fieldapproximation) is not likely to be sufficient whenhabitat is rare and fragmented (i.e. for species andlandscapes of conservation concern). Localizeddispersal algorithms, such as NND and PD withtheir coarse distinctions in movement behavior,may adequately capture the interaction betweenspatial structure and dispersal success such thatmore detailed movement rules, with their con-comitant data requirements, may not be needed inmany applications.
Acknowledgements
This research was supported by a grantawarded by the National Science Foundation toKAW (DEB-9532079). AWK received supportfrom the Strategic Environmental Research andDevelopment Program through military intera-gency purchase requisition no. W74RDV53549127and the Office of Health and Environmental Re-search of the US Department of Energy undercontract DE-AC05-96OR22464 with LockheedMartin Energy Research Corporation.
References
Anderson, D.C., 1996. A spatially-explicit model of searchpath and soil disturbance by a fossorial herbivore. Ecolog-ical Modelling 89, 99–108.
Fig. 9. An example of a screening process to determine whenchoice of a dispersal algorithm other than the mean-fieldapproximation of random dispersal is appropriate. YES meansselection of a localized dispersal algorithm is indicated (NNDor PD in our examples) because dispersal success is affected bychoice of dispersal and interactions with landscape pattern(Table 5); NO means the mean field approximation (randomdispersal) is sufficient. On random landscapes with �40%habitat, the difference in dispersal success of NND or PD formoderate to strong dispersers (m�5) suggests that attentionshould be given to selecting between these alternative behav-iors (right of figure). When habitat is abundant, these differ-ences are unimportant in determining dispersal success (left offigure).
A.W. King, K.A. With / Ecological Modelling 147 (2002) 23–3938
Baker, B.D., 1996. Landscape pattern, spatial behavior, and adynamic state variable model. Ecological Modelling 89,147–160.
Bascompte, J., Sole, R.V., 1998. Effects of habitat destructionin a prey–predator metapopulation model. Journal of The-oretical Biology 195, 383–393.
Blackwell, P.G., 1997. Random diffusion models for animalmovement. Ecological Modelling 100, 87–102.
Blaine, T.W., DeAngelis, D.L., 1997. The interaction of spatialscale and predator–prey functional response. EcologicalModelling 95, 319–328.
Byers, J.A., 1996. An encounter rate model of bark beetlepopulations searching at random for susceptible host trees.Ecological Modelling 91, 57–66.
Carter, J., Finn, J.T., 1999. MOAB: a spatially explicit, indi-vidual-based expert system for creating animal foragingmodels. Ecological Modelling 119, 29–41.
Dale, V.H., Pearson, S.M., Offerman, H.L., O’Neill, R.V.,1994. Relating patterns of land-use change to faunal biodi-versity in the central Amazon. Conservation Biology 8,1027–1036.
Doak, D.F., Marino, P.C., Kareiva, P.M., 1992. Spatial scalemediates the influence of habitat fragmentation on disper-sal success: implications for conservation. Theoretical Pop-ulation Biology 41, 315–336.
Dunning, J.B., Stewart, D.J., Danielson, B.J., Noon, B.R.,Root, T.L., Lamberson, R.H., Stevens, E.E., 1995. Spa-tially explicit population models: current forms and futureuses. Ecological Applications 5, 3–11.
Dyer, J.M., 1995. Assessment of climate warming using amodel of forest species migration. Ecological Modelling 79,199–219.
Fahrig, L., 1998. When does fragmentation of breeding habi-tat affect population survival? Ecological Modelling 105,273–292.
Gardner, R.H., 1999. RULE: map generation and a spatialanalysis program. In: Klopatek, J.M., Gardner, R.H.(Eds.), Landscape Ecological Analysis: Issues and Applica-tions. Springer, New York, pp. 280–303.
Gardner, R.H., Milne, B.T., Turner, M.G., O’Neill, R.V.,1987. Neutral models for the analysis of broad-scale land-scape pattern. Landscape Ecology 1, 19–28.
Gustafson, E., Gardner, R.H., 1996. The effect of landscapeheterogeneity on the probability of patch colonization.Ecology 77, 94–107.
Hanski, I., 1999. Metapopulation Ecology. Oxford UniversityPress, Oxford.
Johst, K., Brandl, R., 1997. The effect of dispersal on localpopulation dynamics. Ecological Modelling 104, 87–101.
Lamberson, R.H., McKelvey, K.S., Noon, B.R., Voss, C.,1992. The effects of varying dispersal capabilities on thepopulation dynamics of the northern spotted owl. Conser-vation Biology 6, 505–512.
Lamberson, R.H., Noon, B.R., Voss, C., McKelvey, K.S.,1994. Reserve design for territorial species: the effects ofpatch size and spacing on the viability of the northernspotted owl. Conservation Biology 8, 185–195.
Lande, R., 1987. Extinction thresholds in demographic modelsof territorial populations. American Naturalist 130, 624–635.
Latore, J., Gould, P., Mortimer, A.M., Lavorel, S., O’Neill,R.V., Gardner, R.H., 1994. Spatio-temporal dispersalstrategies and annual plant species coexistence in a struc-tured landscape. Oikos 71, 75–88.
Lindenmayer, D.B., Possingham, H.P., 1996. Modelling theinter-relationships between habitat patchiness, dispersal ca-pability and metapopulation persistence of the endangeredspecies, Leadbeater’s possum, in south-eastern Australia.Landscape Ecology 11, 79–105.
Liu, J., Dunning, J.B. Jr, Pulliam, H.R., 1995. Potential effectsof a forest management plan on Bachman’s sparrows(Aimophila aesti�alis): linking a spatially explicit modelwith GIS. Conservation Biology 9, 62–75.
Malanson, G.P., Armstrong, M.P., 1996. Dispersal probabilityand forest diversity in a fragmented landscape. EcologicalModelling 87, 91–102.
McCarthy, M.A., 1999. Effects of competition on natal disper-sal distance. Ecological Modelling 114, 305–310.
Meffe, G.K., Carroll, C.R., 1997. Principles of ConservationBiology, second ed. Sinauer, Sunderland, MA.
Opdam, P., 1990. Dispersal in fragmented populations: the keyto survival. In: Bunce, R.G.H., Howard, D.C. (Eds.),Species Dispersal in Agricultural Habitats. Belhaven Press,New York, pp. 3–17.
Pearson, S.M., Turner, M.G., Gardner, R.H., O’Neill, R.V.,1996. An organism-based perspective of habitat fragmenta-tion. In: Szaro, R.C., Johnston, D.W. (Eds.), Biodiversityin Managed Landscapes: Theory and Practice. OxfordUniversity Press, Oxford, pp. 77–95.
Pulliam, H.R., Dunning, J.B. Jr, Liu, J., 1992. Populationdynamics in complex landscapes: a case study. EcologicalApplications 2, 165–177.
Railsback, S.F., Lamberson, R.H., Harvey, B.C., Duffy, W.E.,1999. Movement rules for individual-based models ofstream fish. Ecological Modelling 123, 73–89.
Ruckelshaus, M., Hartway, C., Kareiva, P., 1997. Assessingthe data requirements of spatially explicit dispersal models.Conservation Biology 11, 1298–1306.
Saupe, D., 1988. Algorithms for random fractals. In: Petigen,H.-O., Saupe, D. (Eds.), The Science of Fractal Images.Springer, New York, pp. 71–113.
South, A., 1999a. Dispersal in spatially explicit populationmodels. Conservation Biology 13, 1039–1046.
South, A., 1999b. Extrapolating from individual movementbehavior to population spacing patterns in a ranging mam-mal. Ecological Modelling 117, 343–360.
Swart, J., Lawes, M.J., 1996. The effect of habitat patchconnectivity on samango monkey (Cercopithecus mitis)metapopulation persistence. Ecological Modelling 93, 57–74.
Taylor, P.D., Fahrig, L., Henein, K., Merriam, G., 1993.Connectivity is a vital element of landscape structure. Oikos68, 571–573.
Tischendorf, L., Irmler, U., Hingst, R., 1998. A simulationexperiment on the potential of hedgerows as movementcorridors for forest carabids. Ecological Modelling 106,107–118.
Torres-Sorando, L., Rodriguez, D.J., 1997. Models of spatio-temporal dynamics in malaria. Ecological Modelling 104,231–240.
Wennergren, U., Ruckelshaus, M., Kareiva, P., 1995. Thepromise and limitations of spatial models in conservationbiology. Oikos 74, 349–356.
Wilder, J.W., Christie, I., Colbert, J.J., 1995. Modelling oftwo-dimensional spatial effects on the spread of forest pestsand their management. Ecological Modelling 82, 287–298.
With, K.A., 1997. The application of neutral landscape modelsin conservation biology. Conservation Biology 11, 1069–1080.
With, K.A., Crist, T.O., 1996. Translating across scales: simu-lating species distributions as the aggregate response ofindividuals to heterogeneity. Ecological Modelling 93, 125–137.
With, K.A., King, A.W., 1997. The use and misuse of neutrallandscape models in ecology. Oikos 79, 219–229.
With, K.A., King, A.W., 1999b. Dispersal success on fractallandscapes: a consequence of lacunarity thresholds. Land-scape Ecology 14, 73–82.
With, K.A., Gardner, R.H., Turner, M.G., 1997. Landscapeconnectivity and population distributions in heterogeneousenvironments. Oikos 78, 151–169.
With, K.A., King A.W., 2001. Analysis of landscape sources andsinks: the effect of spatial pattern on avian demography.Biological Conservation 100, 75–78.
