Journal of Animal
Ecology 2007
76, 30– 35
Scale dependence of immigration rates:
models, metrics and data
Blackwell Publishing Ltd
GÖRAN ENGLUND and PETER A. HAMBÄCK*
Department of Ecology and Environmental Science, Umeå Marine Science Center, Umeå University, SE-901 87
Umeå, Sweden; *Department of Botany, Stockholm University, SE-106 91 Stockholm, Sweden
Summary
1. We examine the relationship between immigration rate and patch area for different
types of movement behaviours and detection modes. Theoretical models suggest that
the scale dependence of the immigration rate per unit area (I/A) can be described by
a power model I/A = i*Areaζ, where ζ describes the strength of the scale dependence.
2. Three types of scaling were identified. Area scaling (ζ = 0) is expected for passively
dispersed organisms that have the same probability of landing anywhere in the patch.
Perimeter scaling (− 0·30 > ζ > − 0·45) is expected when patches are detected from a very
short distance and immigrants arrive over the patch boundary, whereas diameter
scaling (ζ = − 0·5) is expected if patches are detected from a long distance or if search is
approximately linear.
3. A meta-analysis of published empirical studies of the scale dependence of immigration
rates in terrestrial insects suggests that butterflies show diameter scaling, aphids show
area scaling, and the scaling of beetle immigration is highly variable. We conclude that
the scaling of immigration rates in many cases can be predicted from search behaviour
and the mode of patch detection.
Key-words: dispersal, insects, meta-analysis, meta-populations, patch size
Journal of Animal Ecology (2007), 76, 30–35
doi: 10.1111/j.1365-2656.2006.01174.x
Introduction
Many organisms live in spatially structured habitats
that can be described as a mosaic of patches rich in
resources, such as food and mating opportunities, and
a matrix that offers few resources. Movements between
patches have profound effects on the dynamics of populations living in spatially structured habitats, controlling
local short-term dynamics as well as persistence at larger
spatial scales (Englund 1993; Hanski & Gilpin 1997;
Thomas & Kunin 1999). The rate of successful migration between patches is a function of behavioural traits
such as search behaviour, mode of patch detection and
overall mobility (Bowman, Cappuccino & Fahrig 2002;
Bukovinszky et al. 2005), but also of physical properties
of patch networks, such as the size of patches, distances
between patches, the presence of dispersal barriers and
corridors, and network dimensionality (Levine 2003;
Haddad & Tewksbury 2005). Thus it is expected that
© 2006 The Authors.
Journal compilation
© 2006 British
Ecological Society
Correspondence: Göran Englund, Department of Ecology and
Environmental Science, Umeå Marine Science Center, Umeå
University, SE-901 87 Umeå, Sweden. Tel.: +46907869728.
Fax: +46907866705. E-mail: [email protected]
the relationships between migration rates and network
properties have important effects on the dynamics of
patchily distributed populations. For example, the relationship between migration rates and patch area, which
is the focus of this paper, can influence the persistence
of metapopulations (Kindvall & Petersson 2000), and
it is a key mechanism underlying density–area relationships for many organisms (Hambäck & Englund 2005).
One approach to the study of movements between
patches assumes that organisms have a mental map of
the landscape and select patches to maximize expected
fitness (Milinski & Parker 1991). However, in many
situations organisms are not able to make well-informed
decisions because they lack the required cognitive
capacity, or because information gathering is too costly
(Morris 1992). For such situations it has proven useful
to assume that movements have a random component
and therefore that immigration and emigration rates
are determined by factors such as patch geometry and
mode of patch edge detection (Bowman et al. 2002;
Englund & Hambäck 2004a,b). Specifically, theoretical
analyses and empirical data both suggest that simple
power models based on patch geometry describe the
scaling of emigration rates for patches larger than the
31
Scale dependence
of immigration
rates
scale of animal movements (Englund & Hambäck 2004a,b).
Moreover, analyses of animal density by Hambäck &
Englund (2005) suggest that similar scaling relationships exist in the density response to patch size as a
consequence of a balance between immigration and
emigration. Scale-dependent densities may also arise
because within-patch processes such as predation and
resource acquisition is scale-dependent. In Hambäck
& Englund (2005) we theoretically identified three
broad classes of scaling patterns for immigration rates;
area dependence, perimeter dependence and diameter
dependence, and suggested that differences among
species in these patterns could account for differences
in density–area relationships. In this paper, we will
review the literature on immigration rate and evaluate
the predictive capacity of immigration scaling patterns.
In addition, we compare the field methods and metrics
used to estimate immigration rates in different studies
and show that some methods and metrics give biased
estimates. When correcting for this bias, and comparing
among terrestrial insect taxa, we find good agreement
between theoretically predicted and empirically observed
scaling patterns of immigration rates.
  
 
The predictions for immigration rates are based on
Hambäck & Englund (2005), using an immigration
model analogous to the emigration model in Englund
& Hambäck (2004a,b). The model used in the former
paper examines the effect of immigration on local
density, and a general formulation for this effect is
the following power model
I/A = iAζ,
© 2006 The Authors.
Journal compilation
© 2006 British
Ecological Society,
Journal of Animal
Ecology, 76,
30–35
eqn 1
where I is the number of immigrants per time unit, A
is patch area, i is the immigration rate into a patch
with unit area and ζ describes the strength of the
scale dependence. A positive value of ζ means that the
number of immigrants per area increases with patch
size. Immigrants are defined as individuals present in
a patch that were not present in the patch at an earlier
point in time. Based on this model it is possible to
identify three general patterns of immigration scaling.
First, area scaling (ζ = 0) arises when individuals have
an equal probability of immigrating into any part of the
patch and has the consequence that the immigration
rate per unit area is independent of patch size. This
occurs when colonizing individuals arrive to the patch
falling from the sky, or when patches are small enough
that single movements are longer than the linear dimension of patches (Englund & Hambäck 2004b). For
example, it has been proposed that area scaling applies
for aphids, mites, spiderlings, small seeds, spores, and
stream insects dispersed by wind or water (Bowman
et al. 2002; Englund & Hambäck 2004b; Strengbom,
Englund & Ericson 2006). Second, perimeter scaling
arises when individuals arrive across the patch boundary
and the patch can only be detected at a very short
distance (Englund & Hambäck 2004a; Hambäck &
Englund 2005). For such movements, the number of
immigrants is proportional to the length of the perimeter
and therefore to the perimeter-to-area ratio. In general,
it can be shown, analogous to perimeter-dependent
emigration, that the immigration rate for patches with
a fractal perimeter of dimension d is
I/A = iA(d/2−1)
eqn 2
(Englund & Hambäck 2004a). In natural systems, edge
dimensions (d ) typically vary between 1·1 and 1·4,
suggesting that scale coefficients should vary between
−0·45 and −0·3 (Krummel et al. 1987; Rex & Malanson
1990). Third, diameter scaling arises when patches are
detected from a distance that is large compared with
the patch diameter or when the search path is approximately linear (Dusenbery 1989). As the patch diameter
is proportional to Area0·5, the predicted scaling coefficient for immigration per unit area is ζ = −0·5.
Thus, as a null model based on patch geometry
and random walk assumptions we expect the scaling
of immigration rate in natural systems to vary between
−0·5 and 0. Values close to zero are expected when
organisms have an equal probability to immigrate into
all parts of the patch, whereas large negative values
are expected when organisms search along linear paths
or when patches are located from a distance. Empirical
observations of values outside this range indicate that
additional assumptions are required. These predictions
will be compared with data on immigration rates in
terrestrial insects.
 -   
    
 
We searched biological abstracts and reference lists in
published papers and found 23 studies, for a total of 48
observations, that reported data on immigration rates
from natural or man-made patches of at least three
different sizes. Extracted data are listed in Appendix S1
(Table S1). The studied organisms were: butterflies
(n = 31), moths (n = 3), beetles (n = 6), planthoppers
(n = 1), dipterans (n = 1) and aphids (n = 6). The number
of immigrants to a patch were measured by recording
densities early during colonization (n = 17), or by mark–
recapture methods (n = 31), i.e. recaptured individuals
that had been marked in other patches were recorded as
immigrants. For studies that did not provide estimates
of scaling coefficients we used a quasipoisson regression
of I/A on log(area) and a log link function to estimate
scaling coefficients. The quasipoisson procedure
provides identical parameter estimates as a Poisson
regression and handles the effect of overdispersion on
estimates of standard errors (R Development Core Team
2005). In four cases zero values were reported for a
32
G. Englund &
P. A. Hambäck
number of patches that all were smaller than those
receiving immigrants. We excluded these patches, as we
expect that small unused patches were present, but not
reported, in most of the analysed systems. Weighted
means of the scaling coefficients were calculated using
the inverse of the sample variances as weights (Hedges
& Olkin 1985). Statistical tests of differences of mean
values were calculated using a random model and the
bootstrap procedure provided in MetaWin (Rosenberg,
Adams & Gurevitch 1997). This test produces a test
statistic, Q, for the amount of heterogeneity within and
between groups.
   
© 2006 The Authors.
Journal compilation
© 2006 British
Ecological Society,
Journal of Animal
Ecology, 76,
30–35
When examining the materials and methods of included
papers, it was obvious that different studies had used
different metrics for immigration rates. Because the
scaling of some metrics may deviate from those predicted
by our theoretical analysis of the scaling of immigration
per unit area (I/A), we first identify these differences,
but also one problem associated with estimating I/A
in the field. Metrics used in the various studies were, in
addition to I/A, immigration per capita in the patch
(I/N), the immigration fraction (Ifract) and the probability
that a potential immigrant arrives in the patch (Iprob).
First, estimates of I/A are of course in concordance
with our model but may be problematic when the
supply of potential immigrants is correlated with
patch area. In natural systems such correlations are not
expected to be a problem, but they can bias estimates
that are based on mark–recapture methods. The reason
is that a migration event is only recorded when a marked
individual moves between patches and not if it returns
to the same patch, and the probability for betweenpatch movements is higher from a large to a small
patch for the simple reason that the number of marked
individuals is often higher in large patches. The resulting
bias is strongest in networks with few patches and those
with large size variation. To obtain a rough estimate of
the bias caused by this method we assumed that the
supply of immigrants for a patch was given by emigration from all other patches in the network and that
the number of emigrants entering the pool of potential
immigrants from a particular patch was proportional
to its Area0·8. This value is based on measurements of
emigration rates from the mark–recapture studies that
provided data on emigration rates (Table S1, Appendix
S1). A power model was then fitted to the data for each
network. A detailed presentation of this method is made
in Appendix S2.
Second, I/N has the same scaling as I/A if densities
are scale independent. However, as the densities of many
organisms are indeed related to patch size (Hambäck
& Englund 2005), it is not recommended to estimate
the scaling of immigration from this metric. Third, the
metric used in many mark–recapture studies, the
immigration fraction, is given by Ifract = Ir/(Ir + Nr), were
Ir is the number of individuals captured in a patch that
were marked in other patches and Nr is the number of
same-patch recaptures (e.g. Kuussaari, Nieminen &
Hanski 1996). This metric has the same scaling as I/N
for values close to zero but approaches ζ = 0 when Ifract→1.
Thus it is expected to show weaker scale dependence
than per capita immigration rate. Finally, studies using
the virtual migration model (Hanski, Alho & Moilanen
2000) estimate a scaling coefficient (ζim) for the probability that a potential immigrant arrives in the patch
(Iprob), where Iprob/A is expected to scale as I/A. The
scaling coefficient used by us (ζ) and the coefficient
estimated by the virtual migration model (ζim) are therefore related as ζ = ζim − 1. To summarize, I/N and Ifract
are clearly unsuitable for studying the scale dependence
of immigration rates. In this paper, we therefore only
use studies estimating immigration rate per unit area
(I/A) and immigration probability (Iprob) and attempt to
evaluate bias affecting I/A in mark–recapture studies.
Results
The mean scaling coefficient for all studies was −0·39
(CI = ±0·12), which is within the range predicted by
our models (−0·5 ≤ ζ ≤ 0). There was substantial heterogeneity within the data (Fig. 1, Table S1 in Appendix
S1). We expect that some of this variation can be
attributed to the different search behaviours used
by the studied taxa. Because information about movement patterns and patch detection modes are lacking
for most of the studied species, we opted instead for an
analysis based on broad taxonomic groups. We found
significant variation between taxonomic groups (Q =
23·6, d.f. = 3, P < 0·005).
Aphids and beetles had mean scaling coefficients
within the expected range, whereas coefficients for butterflies and moths were more negative than predicted
(Fig. 2). There was a significant bias in the 16 mark–
recapture studies (mean ± SE = −0·10 ± 0·02, max =
−0·51, min = −0·04, Appendix S1, Table S1). Subtracting this bias from observed mean coefficients resulted
in weaker scale dependence for moths and butterflies
(Fig. 2). Notably, the adjusted mean value for butterflies
Fig. 1. Histogram of scaling coefficients for the relationship
between patch area and immigration rate. The grey area indicate
the range predicted by our models.
33
Scale dependence
of immigration
rates
Fig. 2. (a) Scaling coefficients (mean ± CI 95%) for four broad taxonomic groups. The range predicted by our models is indicated
by grey shading. Filled diamonds indicate observed means, whereas open diamonds denote mean values adjusted for the effect
of an area dependent supply of marked immigrants in mark–recapture studies. Mark–recapture was not used in the studies of
beetles and aphids. (b) Scaling coefficients (mean ± CI 95%) estimated for the same data but using different metrics. I/A is the
number immigrating per unit patch area, I/N is the number of immigrants per capita in the patch, and Ifract is the fraction of all
marked individuals captured in a patch that was marked in a different patch.
is −0·51, which is close to the value expected for diameter
scaling. Scaling coefficients for aphids and beetles
were not affected as they were not studied with mark–
recapture methods.
Studies using mark–recapture methodology reported
larger negative scaling coefficients than those using
natural colonization (mean ± CI = −0·53 ± 0·13 and
−0·15 ± 0·18, respectively, Q = 15·7, d.f. = 1, P < 0·005).
It should be noted that this comparison is confounded
with the one between taxonomic groups as most butterfly
studies used mark–recapture methods (29 of 31) and
most studies of other organisms report data for natural
colonization (15 of 17).

To evaluate how the choice of metric affects estimated
scaling coefficient we first used a subset of studies for
which three metrics could be calculated, i.e. the number
of immigrants per unit area I/A, the number of immigrants per capita in the patch (I/N), and the immigration fraction (Ir/(Ir + Nr)). In general, I/A produced
more negative coefficients than I/N and the immigration fraction (Randomized block , F2,19 = 7·62,
P < 0·005, Fig. 2b). Studies using the virtual migration
model did not provide the data needed to calculate
other metrics. Thus to evaluate the performance of
Iprob/A, we compared the results of studies employing
the virtual migration model and studies for which we
could estimate the scaling coefficient for I/A. To minimize
confounding the analysis was restricted to studies of
butterfly migration that used mark–recapture methodology (n = 29). The mean values of the scaling
coefficients measured using I/A and Iprob/A did not
differ significantly (mean ± CI = −0·58 ± 0·21, n = 9 and
−0·62 ± 0·18, n = 20, respectively, Q = 0·03, NS).
© 2006 The Authors.
Journal compilation
© 2006 British
Ecological Society,
Journal of Animal
Ecology, 76,
30–35
Discussion
Although immigration and emigration, in many respects,
are analogous processes, there are also important
differences. Metrics such as per capita emigration rate
and the emigration fraction are well suited for studying
the area scaling of emigration, but the corresponding
rates for immigration cannot be recommended. This
difference occurs because immigration rates, but not
emigration rates, are influenced by the supply of potential
immigrants. This supply is primarily affected by the
distances to source patches, and the sizes and numbers
of such patches, i.e. patch connectivity (Moilanen &
Nieminen 2002). Differences between patches in their
connectivity are likely to cause variation in immigration rates (Winfree et al. 2005), but this variation
should be averaged out when estimating scaling
coefficients for large patch networks. Thus we do not
expect that connectivity is a source of bias in the data
analysed here.
One minor source of bias could, however, be identified
for data from mark–recapture studies. The supply of
marked individuals that are potential immigrants to
a patch is slightly larger for small than for large patches
in small patch networks (Appendix S2). When we
approximated the magnitude of this bias and adjusted
observed immigration scaling, we arrived at a mean
coefficient for butterflies that is close to the value
predicted for diameter scaling (ζ = −0·5).
For animals that rely on visual cues, such as most
butterflies, diameter scaling is expected if patch
detection distance is so large that the patch diameter
does not cover the width of the visual field, which raises
the question at what distance butterflies can detect
and orientate towards patches. In many of the studies
included in our meta-analysis, patches were open areas
in otherwise forested landscapes, or areas with diverse
vegetation in agricultural landscapes dominated by
monocultures. The latter patch type may be identified
from a distance by their contrasting colour, or because
they present a silhouette visible against the sky, whereas
long distance detection of open patches in forested
areas supposedly requires that the butterflies search
above the canopy. Some studies report that butterflies
have limited ability to detect individual host plants or
small patches of host plants from distances exceeding
a metre (Fahrig & Paloheimo 1987; Capman, Batzli
34
G. Englund &
P. A. Hambäck
© 2006 The Authors.
Journal compilation
© 2006 British
Ecological Society,
Journal of Animal
Ecology, 76,
30–35
& Simms 1990). However, these observations may
be examples of search for host plants within patches.
Other studies suggest that orientation towards large
habitat patches can occur from distances exceeding
100 m (Conradt et al. 2000; Conradt, Roper & Thomas
2001; Cant et al. 2005). Diameter scaling may also result
if search paths are linear at the scale of typical patches
(Cant et al. 2005) or if search is guided by linear elements
such as roads, stream corridors and forest edges
(Sutcliffe & Thomas 1996; Haddad & Tewksbury 2005).
As such search strategies appear to be common in
butterflies we conclude that diameter scaling of immigration rates in butterflies are in agreement with known
behavioural mechanisms, as well as observed scaling
coefficients after adjustment of bias. However, as
the scaling coefficients predicted for diameter scaling
(ζ = −0·5) is not very different from perimeter scaling
(−0·3 > ζ > −0·45), this alternative can not be rejected
at this stage.
The studied aphid species had area scaling (ζ ≈ 0)
as suggested by several authors (Bowman et al. 2002;
Bukovinszky et al. 2005). This type of scaling is predicted
when organisms have an equal probability to land in
any part of the patch (Englund & Hambäck 2004b;
Hambäck & Englund 2005), and this seems reasonable
for aphid species that arrive to the patch mainly through
passive dispersal in the air column. The scaling of
immigration by beetles was highly variable. Phyllotreta
spp. (flea beetles), are of particular interest because they
show a strong positive density scaling in experiments
where there is no reproduction in the patches (0·65–
1·38) (Cromartie 1975; Kareiva 1985). In Hambäck &
Englund (2005) we suggested that this pattern is caused
by a positive immigration scaling, possibly caused by
intraspecific attraction based on chemical cues. This
interpretation is supported by the fact that the shortterm study (24 h) by Kareiva (1985) showed strong
negative scaling coefficients, whereas the longer study
(3–6 days) by Cromartie (1975) showed a positive
scaling. Intraspecific attraction requires that some
individuals arrive without this attraction, and as
chrysomelid beetles often have very weak attraction
to undamaged plants (Hambäck, Petterson & Ericson
2003) this could imply a perimeter- or diameter-scaling
in short-term experiments.
A consequence of scale-dependent immigration
rates can be scale dependent densities. Recently, we
demonstrated that if migration processes control local
densities, then the scaling of densities at equilibrium is
given by a simple power model of the form n ∝ Areaβ–ζ,
where β is the scaling coefficient for emigration
(Hambäck & Englund 2005). This means that the
scaling of immigration and emigration cause densities
to change with patch area when the two rates have
different scaling, which is expected when emigrants
and immigrants have different movement behaviours
and when they detect patch boundaries from different
distances. In a previous paper, we hypothesized that
such differences could explain the strong negative
scaling of butterfly densities observed in field studies
(Hambäck & Englund 2005). If the long-distance edge
detection observed for some butterflies (Conradt et al.
2000, 2001; Cant et al. 2005) is a general phenomenon,
it seems reasonable to assume that emigrants often can
detect patch boundaries from long distances, which
should lead to area-scaling of emigration (Englund &
Hambäck 2004a). Combined with the diameterscaling of immigration rates, observed in this study, this
leads to a density–area scaling of the observed magnitude. To test this hypothesis we used bias adjusted data
from studies that provide estimates of both emigration and immigration for butterflies (n = 27, data in
Appendix S1, Table S1). The mean scaling coefficients
for immigration and emigration are ζ = −0·54 and
β = −0·17. The coefficient for emigration is larger
than expected, suggesting a mixture of area and
perimeter scaling, but the observed difference (β −
ζ = −0·37) is reasonably close to the observed scaling
coefficient for density (−0·44). This result supports
the idea that the area dependence of butterfly densities
reflects the scaling of emigration and immigration
rates.
In conclusion, we found that scaling coefficients were
largely within the range predicted by simple models
and differences between taxonomic groups could in
many cases be related to their movement behaviours
and modes of patch detection. Thus, we suggest that
the results of this study and previous studies of emigration (Englund & Hambäck 2004a,b) can be used when
deciding how to represent scale-dependent migration
in models of spatially structured populations. Earlier
models have often used a phenomenological representation of the interplay between patch geometry and
movement behaviours (e.g. Hanski et al. 2000; Heinz
et al. 2005). Other analyses show that details about
movement and location behaviours are indeed critical
for meta-population persistence (Kindvall & Petersson
2000; Heinz, Wissel & Frank 2006). Thus we emphasize
that future models need to investigate the effects of
these details.
Acknowledgements
We thank Michel Baguette, Ulf Norberg and Niklas
Wahlberg for sharing their knowledge of butterfly search
behaviour and field methods.
References
Bowman, J., Cappuccino, N. & Fahrig, L. (2002) Patch size
and population density: the effect of immigration behavior.
Conservation Ecology, 6.
Bukovinszky, T., Potting, R.P.J., Clough, Y., van Lenteren, J.C.
& Vet, L.E.M. (2005) The role of pre- and post-alighting
detection mechanisms in the responses to patch size by
specialist herbivores. Oikos, 109, 435 – 446.
Cant, E.T. Smith, A.D. Reynolds, D.R. & Osborne, J.L.
(2005) Tracking butterfly flight paths across the landscape
with harmonic radar. Proceedings of the Royal Society,
Series B, 272, 785 –790.
35
Scale dependence
of immigration
rates
© 2006 The Authors.
Journal compilation
© 2006 British
Ecological Society,
Journal of Animal
Ecology, 76,
30–35
Capman, W.C., Batzli, G.O. & Simms, L.E. (1990) Responses
of the common sooty wing skipper to patches of host
plants. Ecology, 71, 1430 –1440.
Conradt, L., Bodsworth, E.J., Roper, T.J. & Thomas, C.D.
(2000) Non-random dispersal in the butterfly Maniola
jurtina: implications for metapopulation models. Proceedings
of the Royal Society, Series B, 267, 1505 –1510.
Conradt, L., Roper, T.J. & Thomas, C.D. (2001) Dispersal
behaviour of individuals in metapopulations of two British
butterflies. Oikos, 95, 416 – 424.
Cromartie, W.J.J. (1975) The effect of stand size and vegetational background on the colonization of cruciferous plants
by herbivores. Journal of Applied Ecology, 12, 517– 533.
Dusenbery, D.B. (1989) Ranging strategies. Journal of
Theoretical Biology, 136, 309 –316.
Englund, G. (1993) Interactions in a lake outlet stream community: Direct and indirect effects of net-spinning caddis
larvae. Oikos, 66, 431– 438.
Englund, G. & Hambäck, P.A. (2004a) Scale dependence of
emigration rates. Ecology, 85, 320 –327.
Englund, G. & Hambäck, P.A. (2004b) Scale dependence of
movement rates in stream invertebrates. Oikos, 105, 31– 40.
Fahrig, L. & Paloheimo, J.E. (1987) Interpatch dispersal of
the cabbage butterfly. Canadian Journal of Zoology, 65,
616 – 622.
Haddad, N.M. & Tewksbury, J.J. (2005) Low-quality habitat
corridors as movement conduits for two butterfly species.
Ecological Applications, 15, 250 –257.
Hambäck, P.A. & Englund, G. (2005) Patch area, population
density and the scaling of migration rates: the resource concentration hypothesis revisited. Ecology Letters, 8, 1057–
1065.
Hambäck, P.A., Petterson, J. & Ericson, L. (2003) Mechanism
underlying reduced herbivory on purple loosestrife in shrubby
thickets: Is associational resistance species-specific. Functional Ecology, 17, 87– 93.
Hanski, I.A. & Gilpin, M.E. (1997) Metapopulation Biology:
Ecology, Genetics, and Evolution. Academic Press, London.
Hanski, I., Alho, J. & Moilanen, A. (2000) Estimating the
parameters of survival and migration of individuals in
metapopulations. Ecology, 81, 239 –251.
Hedges, L.V. & Olkin, L. (1985) Statistical Methods for
Meta-Analysis. Academic Press, New York.
Heinz, S.K., Conradt, L., Wissel, C. & Frank, K. (2005)
Dispersal behaviour in fragmented landscapes: Deriving a
practical formula for patch accessibility. Landscape Ecology,
20, 83 – 99.
Heinz, S.K., Wissel, C. & Frank, K. (2006) The viability of
metapopulations: Individual dispersal behaviour matters.
Landscape Ecology, 21, 77– 89.
Kareiva, P. (1985) Finding and losing host plants by Phyllotreta:
patch size and surrounding habitat. Ecology, 66, 1809–1816.
Kindvall, O. & Petersson, A. (2000) Consequences of modelling interpatch migration as a function of patch geometry
when predicting metapopulation extinction risk. Ecological
Modelling, 129, 101–109.
Krummel, J.R., Gardner, R.H., Sugihara, G., O’neill, R.V. &
Coleman, P.R. (1987) Landscape patterns in a disturbed
environment. Oikos, 48, 321–324.
Kuussaari, M., Nieminen, M. & Hanski, I. (1996) An
experimental study of migration in the Glanville fritillary
butterfly Melitaea cinxia. Journal of Animal Ecology, 65,
791– 801.
Levine, J.M. (2003) A patch modeling approach to the
community-level consequences of directional dispersal.
Ecology, 84, 1215 –1224.
Milinski, M. & Parker, G.A. (1991) Competition for resources.
In: Behavioural Ecology: an Evolutionary Approach (eds
J.R. Krebs & N.B. Davies), pp. 137–168. Blackwell, Oxford.
Moilanen, A. & Nieminen, M. (2002) Simple connectivity
measures in spatial ecology. Ecology, 83, 1131–1145.
Morris, D.W. (1992) Scales and costs of habitat selection in
heterogeneous landscapes. Evolutionary Ecology, 6, 412–432.
R Development core team (2005) R: A language and environment for statistical computing. R foundation for statistical
computing. Vienna, Austria.
Rex, K.D. & Malanson, G.P. (1990) The fractional shape of
riparian forest patches. Landscape Ecology, 4, 249–258.
Rosenberg, M.S., Adams, D.C. & Gurevitch, J. (1997) Metawin:
Statistical Software for Meta-Analysis, Version 2.0. Sinauer
Associates, Sunderland, MA.
Strengbom, J., Englund, G. & Ericson, L. (2006) Experimental
scale and precipitation modify effects of nitrogen addition
on a plant pathogen. Journal of Ecology, 94, 227–233.
Sutcliffe, O.L. & Thomas, C.D. (1996) Open corridors appear
to facilitate dispersal by ringlet butterflies (Aphantopus
hyperantus) between woodland clearings. Conservation
Biology, 10, 1359 –1365.
Thomas, C.D. & Kunin, W.E. (1999) The spatial structure of
populations. Journal of Animal Ecology, 68, 647–657.
Winfree, R., Dushoff, J., Crone, E.E., Schultz, C.B., Budny,
R.V., Williams, N.M. & Kremen, C. (2005) Testing simple
indices of habitat proximity. American Naturalist, 165,
707–717.
Received 27 February 2006; accepted 31 August 2006