graph and circuit theory connectivity models of conservation
TRANSCRIPT
Ecological Applications, 23(7), 2013, pp. 1554–1573� 2013 by the Ecological Society of America
Graph and circuit theory connectivity models of conservationbiological control agents
INSU KOH,1,3 HELEN I. ROWE,2 AND JEFFREY D. HOLLAND1,4
1Department of Entomology, Purdue University, 901 West State Street, West Lafayette, Indiana 47907- 2089 USA2School of Life Sciences, Arizona State University, 427 East Tyler Mall, Tempe, Arizona 85287 USA
Abstract. The control of agricultural pests is an important ecosystem service provided bypredacious insects. In Midwestern USA, areas of remnant tallgrass prairie and prairierestorations may serve as relatively undisturbed sources of natural predators, and smallerareas of non-crop habitats such as seminatural areas and conservation plantings (CP) mayserve as stepping stones across landscapes dominated by intensive agriculture. However, littleis known about the flow of beneficial insects across large habitat networks. We measuredabundance of soybean aphids and predators in 15 CP and adjacent soybean fields. We testedtwo hypotheses: (1) landscape connectivity enhances the flow of beneficial insects; and (2)prairies act as a source of sustaining populations of beneficial insects in well-connectedhabitats, by using adaptations of graph and circuit theory, respectively. For graphconnectivity, incoming fluxes to the 15 CP from connected habitats were measured using anarea- and distance-weighted flux metric with a range of negative exponential dispersal kernels.Distance was weighted by the percentage of seminatural area within ellipse-shaped landscapes,the shape of which was determined with correlated random walks. For circuit connectivity,effective conductance from the prairie to the individual 15 CP was measured by regarding theflux as conductance in a circuit. We used these two connectivity measures to predict theabundance of natural enemies in the selected sites. The most abundant predators wereAnthocoridae, followed by exotic Coccinellidae, and native Coccinellidae. Predatorabundances were explained well by aphid abundance. However, only native Coccinellidaewere influenced by the flux and conductance. Interestingly, exotic Coccinellidae werenegatively related to the flux, and native Coccinellidae were highly influenced by theinteraction between exotic Coccinellidae and aphids. Our area- and distance-weighted flux andthe conductance variables showed better fit to field data than area-weighted flux or Euclideandistance from the prairie. These results indicate that the network of seminatural areas hasgreater influence on the flow of native predators than that of exotic predators, and that theprairie acts as a source for native Coccinellidae. Managers can enhance conservationbiocontrol and sustain the diversity of natural enemies by optimizing habitat networks.
Key words: Anthocoridae; Aphis glycines; circuit theory; Coccinellidae; conductance; conservationbiocontrol; conservation planting; correlated random walk; dispersal; insect predator; soybean aphid.
INTRODUCTION
Control of pests is an important ecosystem service
provided by natural enemies. Natural enemy popula-
tions are supported by complex landscapes with a high
proportion of natural or seminatural vegetation (Bian-
chi et al. 2006, Gardiner et al. 2009b). The influence of
landscape composition on the abundance and diversity
of natural enemies has been studied by several research-
ers; however, the effect of the spatial arrangements of
large networks of habitat has received less attention
(Bianchi et al. 2006). For example, habitats that are well
connected to each other may enhance the population of
predators, but little is known about how predators move
across the networks of these habitats. In this paper, we
adopted graph and circuit theoretical frameworks to
measure connectivity of the networks and then tested the
effects of connectivity on the abundance of natural
enemies of soybean aphids.
The soybean aphid, Aphis glycines Matsumura, is an
invasive exotic agricultural pest and a major threat to
soybean production in the USA. This pest was
discovered in Wisconsin in 2000 and has spread
throughout the Midwest, much of the Great Plains
states, and southern Canada (Venette and Ragsdale
2004). In eastern Asia, the soybean aphid is controlled
by a variety of natural enemies including predators,
parasitoids, and fungal pathogens (Chang et al. 1994,
Wu et al. 2004). In North America, native and exotic
generalist predators can be effective at controlling A.
glycines (Rutledge et al. 2004). The exotic ladybird
beetle, Harmonia axyridis (Pallas) was introduced as a
Manuscript received 18 September 2012; revised 13 March2013; accepted 8 April 2013. Corresponding Editor: C. Gratton.
3 Present address: Gund Institute for Ecological Econo-mies, University of Vermont, Johnson House 207, 617 MainStreet, Burlington, Vermont 05405 USA.
4 Corresponding author. E-mail: [email protected]
1554
biological control agent to control exotic pests in pecans
and red pines and is the most efficient biocontrol agent
for A. glycines (Koch 2003), but it has been argued that
H. axyridis has a negative impact on native ladybird
populations (Harmon et al. 2007). The regulation of
aphid populations by natural enemies provides biolog-
ical control (or conservation biological control in the
case of native predator species), an important ecosystem
service. However, in landscapes dominated by intensive
agriculture, natural and seminatural areas that may
serve as relatively undisturbed refuges for these natural
enemies can be sparse.
Landscape composition has been studied as a main
factor affecting the diversity and abundance of the
natural enemies in agricultural landscapes. In 74% of
studies reviewed, natural enemy populations were
enhanced in complex landscapes compared to simple
landscapes that are mostly dominated by agricultural
croplands (Bianchi et al. 2006). Within wheat fields, for
example, populations of beneficial predators are posi-
tively correlated with vegetation diversity and the
proportion of non-crop area in the surrounding
landscape (Elliott et al. 1999). Recently, Gardiner et
al. (2009a) showed that pest control service provided by
Coccinellidae in soybean fields is positively influenced by
increasing forest and grassland habitats in the surround-
ing landscapes.
Compared to the study of the effects of landscape
composition, the effects of landscape configuration on
natural enemy populations are little studied (Bianchi et
al. 2006). A few studies address the effects of spatial
configuration of non-crop habitats on pest control. In a
landscape with insufficient non-crop habitats, higher
parasitism rates have been observed in the edges than in
the centers of crop fields (Tscharntke et al. 2002). A
simulation study showed that the control of aphids by
Coccinella septempunctata L. is maximized when the
non-crop habitats are evenly distributed (Bianchi and
van der Werf 2003). These studies indicate the impor-
tance of spatial arrangement of the non-crop habitat
elements for pest control. From this perspective, the
relationship between the connectivity of these elements
and the movement of natural enemies in agriculturally
intensive agroecosystems should lead to better under-
standing of this conservation biocontrol service.
To examine the flow of natural enemies across a
complex habitat network, we applied an adaptation of
graph theory that includes circuit theory to measure the
connectivity of non-crop habitat elements. Graph
theory, which originates from mathematics and sociol-
ogy, is associated with measuring flow efficiency in
networks. It was first introduced in landscape ecology to
measure either structural or functional connectivity
because of its representation of patchy landscape by
nodes (e.g., habitats or patches) and edges (e.g.,
connections or paths between habitats) and its efficient
computation (Urban and Keitt 2001). With its use of
graph edges to represent individual flows and dispersal,
which imply ecological fluxes, the graph-connectivity
model is particularly linked to metapopulation models
(Hanski 1994, Urban et al. 2009). In metapopulation
models, source and sink can be represented by connec-
tions (i.e., edges) with different inward and outward
fluxes (i.e., emigration and immigration). Thus, graph
edges can be implemented to model differential move-
ment rates of individuals into and out of a habitat/node
(Minor and Urban 2007, Urban et al. 2009). Since its
first introduction, graph-connectivity has been applied
rapidly for many organisms including plants, insects,
amphibians, vertebrates, and coral reefs (O’Brien et al.
2006, Jordan et al. 2007, Treml et al. 2008, Minor et al.
2009, Awade et al. 2012, Decout et al. 2012, Kang et al.
2012). More recently, the graph model has incorporated
electrical circuit theory by replacing graph edges with
resistors conducting current flow. Circuit theory pro-
vides electrical connectivity measurements such as
effective resistance or conductance, which measure a
network’s ability to carry current between paired source
and sink nodes separated by a network of resistors
(McRae et al. 2008). These measurements account for
parallel and serial connections of resistors, and alterna-
tive and least resistance pathways, between a source and
a sink across a landscape network (McRae et al. 2008).
Circuit-connectivity shows a better prediction of species
dispersal than Euclidean distance or the least-cost
analysis (McRae and Beier 2007). Therefore, circuit
theory offers an advantage because different flows along
all possible paths can be modeled in measuring
connectivity between a paired source and sink through
a network with multiple intermediate nodes, which
cannot be measured with graph theory alone (McRae et
al. 2008, Urban et al. 2009).
In much of the midwestern USA, there are few
patches of natural or seminatural habitat remaining to
support native natural enemy populations. Larger areas
such as prairie remnants may be the most stable source
of native natural enemies. Smaller such habitats and on-
farm conservation practices and seminatural areas may
function as stepping-stone habitats between the larger
remnant sources and farms, but may be more prone to
extirpation. The flow of natural enemies from a large
stable source habitat across agricultural landscapes will
be complex and follow multiple paths. Understanding
this flow may allow managers and farmers interested in
conservation biological control to place conservation
practices strategically across the landscape to enhance
the benefits of this ecosystem service. In this study we
hypothesized that high connectivity between these
habitats and agricultural fields enhances the flow of
natural enemies and that prairie remnants act as a
source of predators that sustain populations in well-
connected habitats. We tested the following predictions
for different predator insect families: (1) the abundance
of natural enemies will increase with increasing flux, a
graph-connectivity measure, in a network; and (2) the
abundance of natural enemies will increase with
October 2013 1555CIRCUIT FLOW FOR PREDATORS
increasing conductance, a circuit-connectivity measure,
between a prairie remnant and crop fields (see Plate 1 foran abstract illustration). We tested these predictionsseparately for native and exotic Coccinelidae as these
may use and move across habitat networks differently.The other predator families studied consisted of native
species.
METHODS
Study region, study sites, and habitat maps
Our study area was Newton County in Indiana, USA,which has an agroecosystem landscape representative of
much of Indiana and Midwestern USA (Fig. 1A).Newton County is dominated by corn and soybeanagriculture, and also includes an extensive tallgrass
prairie restoration complex, forests, urban areas, andmany small conservation plantings. We targeted two
common United States Department of Agriculture(USDA) Natural Resource Conservation Service(NRCS) conservation planting types for sampling in
our study: CP21 plantings are low-diversity filter stripsof cool-season grasses, and CP33 plantings are moder-
ate-diversity wildlife buffers of grass and floweringplants. In this region, we selected 15 conservationplantings (hereafter CP): 10 CP21s and 5 CP33s adjacent
to soybean fields for predator sampling. AdditionalCP33s next to soybean fields were not available in 2011
due to the increase in corn planted that year at theexpense of soybean acreage. The study sites were
selected to span as much of the county as possible.As a first step toward graph and circuit analyses, we
created a map of seminatural habitat. Seminatural
habitat is less subject to disturbance such as tilling andpesticide application, and we therefore assumed that
these areas are less hostile over an annual cycle thancrop fields and act as refuges for beneficial predatorinsects. Forest, shrub, grassland, pasture, and woody
wetland of 16 land cover classes in the 2001 30-mresolution national land cover data (NLCD) as well as
polygons of CP21, CP33, remnant tallgrass prairie, andprairie restoration were considered seminatural habitat
and mapped. We aggregated the prairie remnant andadjacent prairie restorations into one prairie feature(hereafter prairie complex). We also aggregated semi-
natural habitat grid pixels when they were within 30 mof each other. The resulting map of seminatural habitat
was binary, with seminatural habitat pixels having avalue of 1 and all other pixels 0. The extent of this mapwas Newton County plus a 10-km buffer (Fig. 1A).
From this map of seminatural habitat, we created ahabitat polygon map and extracted 490 patches that met
at least one of the following criteria: larger than 5 ha,and CP21s or CP33s within 7 km of the prairie complexand study sites (Fig. 1A). Only a subset (n ¼ 15) of the
CP21 and CP33 patches were sampled.
Graph-connectivity model
A graph is defined by two types of elements: nodes
and edges. These nodes and edges can represent discrete
habitat patches and their connectivity in a landscape.
The edges can be defined in binary and probabilistic
ways. A binary connection model defines an edge from
patch i to patch j as 1 (connected) or 0 (not connected).
A probabilistic connection model can define an edge as a
dispersal probability between two habitat patches
(Urban and Keitt 2001). This probability can be
calculated by a dispersal kernel, a negative exponential
function of inter-patch distance, for example:
Pij ¼ e�kdij
where k is a distance–decay coefficient (k . 0), and dij is
the distance between two patches, i and j. For both
binary and probabilistic approaches, when edges are
based on distance alone, an undirected graph is
produced in which the value of an edge from patch i
to patch j is the same as the value of an edge from patch j
to patch i. However, when these edges are redefined
based on dispersal fluxes, i.e., the rate or relative number
of individual dispersing, these fluxes may be influenced
by habitat size or quality, or other factors leading to
asymmetrical dispersal. Therefore, a flux value from a
large patch i to a small patch j will be greater than that
from patch j to patch i. This directed edge graph can be
measured by area-weighted flux (Urban and Keitt 2001,
Minor and Urban 2007) as follows:
Fluxaij ¼ Aie�kdij
where Ai is area of patch i (ha). Area-weighted flux
shows the incoming flux from patch i to patch j (Fluxaij)
is different from outgoing flux from patch j to patch i
(Fluxaji ), when the areas of patch i and j are different. As
a result, the matrix of this flux between every pair of
patches is asymmetric. This flux measure employs an
approach similar to Hanski’s meta-population model.
The flux measure we used, based on Euclidean
distances between patches, may have less biological
meaning because it does not consider the interruption of
movement, or resistance to movement, by nonnatural
areas. To take into account these effects, a few studies
have measured least-cost path distance and counted
successful dispersal events (Bunn et al. 2000, Lookingbill
et al. 2010). A least-cost path is calculated by an
algorithm to find the optimal, least costly, single path
for an animal movement on a map representing
resistance to movement (e.g., Foley and Holland
2010). However, it is often indicated that a least-cost
path is not necessarily the path used and does not allow
for more realistic movement along many alternative
paths simultaneously (Fahrig 2007, Pinto and Keitt
2009). Counting successful dispersal events between all
pairs of patches has been conducted using a correlated
random walk simulation (e.g., J-walk software) on a
heterogeneous landscape with different probabilities of
movement and mortalities according to land cover
(Gardner and Gustafson 2004, Lookingbill et al. 2010,
Morzillo et al. 2011). Circuit theory is another approach
INSU KOH ET AL.1556 Ecological ApplicationsVol. 23, No. 7
to modeling dispersal, and it allows for estimation of
movement by individuals along an unlimited number of
different pathways within a network simultaneously
(McRae et al. 2008). All of these approaches have
strengths and we used a combination of them in this
study.
Correlated random walk simulation
We assigned resistance to edges between nodes based
upon the nature of the area over which dispersing insects
are likely to have encountered. To estimate the area over
which dispersing predators moving from a patch i to
another patch j are likely to have encountered, a
correlated random walk (CRW) simulation was per-
formed on a homogeneous landscape. The CRW
simulation required three parameters: step lengths,
turning angles, and a maximum total steps per dispersal
event (Kareiva and Shigesada 1983, Turchin et al. 1991).
To determine these parameters, we used published data
for Coccinellidae species because this family has been
well-documented in the literature on dispersal compared
to the other families of natural enemies in our study. We
defined the step length as a movement distance for an
hour to facilitate the use of published movement data.
We chose a gamma distribution with mean and standard
deviation (6SD) of 50 6 35 m, for the step lengths in
order to match the movement distance reported from
Coccinellidae species (Appendix A). For turning angles,
we selected a normal distribution with mean 6 SD (0 6
27 degrees), closely corresponding to the observed
probability distribution of the species’ turning angles
reported for Coccinella septempunctata (Banks and
Yasenak 2003), which was the only species for which
we could find data (Appendix A). Last, for the
maximum number of steps, we referred to the life cycle
of Harmonia axyridis, which typically lives for 30–90
days as an adult (Koch 2003). We determined the total
number of steps to be 1000 steps, which is approximately
the same as 900 movement hours, when we assumed that
FIG. 1. Study sites and insect sampling design. (A) Land cover map of Newton County, Indiana, USA. The Inset map showsthe location of Newton County within Indiana. Study sites 1–10 and 11–15 soybean fields are located beside CP21 and CP33 fields,respectively. Two USDA Natural Resource Conservation Service (NRCS) conservation planting (CP) types include low-diversityfilter strips of cool-season grasses (CP21) and moderate-diversity wildlife buffers of grass and flowering plants (CP33). (B)Transects for insect sampling in study sites.
October 2013 1557CIRCUIT FLOW FOR PREDATORS
the species lives 60 days and can move 15 hours per day
during a summer season.
Using these three parameters, CRW simulations were
performed between two virtual round-shaped patches,
corresponding to source and target patches. The patches
were separated by different distances of 250–4000 m.
We randomly placed 100 000 random walkers along the
edge of the source patch. For each step, a step length
and a turning angle were sampled from the Gamma and
the Normal distributions, respectively, except for the
first step’s turning angle, which was determined from a
random selection from�180 to 180 degrees. The turning
angles were relative to the direction of the previous step.
When a random walker hit the target patch, this
successful path from the source to the target was
recorded. For each successful path, we calculated two
maximum perpendicular distances from the line linking
the centers of the source and the target and from the
center line separating them corresponding to a major
and minor radius for drawing an ellipse with a center
point on the intersection between the two lines (Fig. 2;
Supplement). The ellipse was regarded as a potential
region in which a successful random walker is likely to
pass between the source and the target for each patch
separation distance. A dispersing individual that leaves
this area is less likely to arrive at the target patch j,
although it may end up at another patch node in the
study region. This process was repeated for distances
250, 500, 750, 1000, 1500, 2000, 3000, and 4000 m.
The major and minor radii of all successful random
walkers were averaged within each distance, and the
averaged ellipse polygon was defined as a final potential
region. As distances between source and target patches
increase, the relative lengths of the averaged major and
minor radii of the final ellipse decline toward 60% and
40% of the distance separating the patches, respectively
(Fig. 3). This trend was not sensitive to changes in the
size of source and target patches, nor to changes in the
distribution of turning angles or step length. However, it
largely depended on the total step count, which, in this
study, was limited to the largest possible step count
based on the insect’s longevity.
Using this relationship between distance between a
pair of patches and the size and shape of the area likely
encountered, we calculated the proportion of seminat-
ural area within the resulting ellipse encompassing every
pair of connected nodes. We made the assumption that
altering the movement parameters for different insect
predators would not have a large influence on the shape
of the resultant elliptical landscape, i.e., the ratio
between the major axis (fixed by inter-node distance)
and minor axis will not be greatly affected. Finally,
fluxes were measured by weighting distance by the
naturalness (% seminatural) as follows:
Fluxadij ¼ Aie�kð1�NijÞdij
where Nij is the proportion of seminatural area within
the ellipse delineating likely successful paths of preda-
tors between two patches. We refer to this estimate of
dispersal as area- and distance-weighted flux (Fluxad).
This measure decreases with i�j distance and increases
as natural and seminatural areas increase within the
elliptical area likely crossed. In this way, all seminatural
FIG. 2. A successful path of a random walker based oncorrelated random walk simulation for source and targetpatches with a distance of 3000 m and an ellipse-shaped regioncovering the successful path. The points in the boundaries ofthe source and target patches show the starting point andhitting point.
FIG. 3. The relationship between the radii of the averagedellipse polygon and distance between source and target patches,dij (m). The averaged ellipse polygon was obtained for allsuccessful random walkers of 100 000 walkers. The majorradius can be calculated by the equation (e(�0.001dij þ 5.294) þ60)/100 3 dij. The minor radius can be calculated by theequation (e�0.001dij þ 5.557 þ 40)/100 3 dij.
INSU KOH ET AL.1558 Ecological ApplicationsVol. 23, No. 7
areas, regardless of patch size, were incorporated into
the final model either as nodes or as determining
resistance.
Using this flux, we calculated incoming flux for a
certain habitat node. Because incoming flux can be
ecologically interpreted as the number of immigrants
(Minor and Urban 2007), it was used for measuring
graph-connectivity as follows:
inFluxadj ¼Xn
i¼1
Fluxij
where inFluxadj is incoming flux for patch j, and n is the
total number of habitat patches (in this study, n¼ 490).
Circuit-connectivity model
For applying circuit theory to a graph, edges of a
graph are defined as resistors, the reciprocal of
conductors (McRae et al. 2008). Conductance (G, in
Siemens) is a measure of a resistor’s ability to carry
electrical current, and resistance (R, in Ohms) is the
opposite. According to Ohm’s law, the current I (in
amperes) flows directly proportional to its applied
voltage V and its conductance (the reciprocal of
resistance): I ¼ VG ¼ V/R. In a circuit, when two
conductors are connected in series, equivalent or effective
conductance Ge is calculated by Ge¼G1G2/(G1þG2). In
the case of parallel connection, the effective conductance
is calculated by Ge ¼ G1 þ G2. As a result, effective
conductance between current source and sink nodes
considers additional available pathways. Therefore, this
circuit connectivity measure can be ecologically inter-
preted as the flow capacity for a species between pairs of
habitat nodes along many pathways across the land-
scape.
To apply this electric circuit concept to our graph, we
regarded the flux value as conductance between two
habitat nodes. Because the flux is a measure of
movement rate between two habitat nodes, this substi-
tution is ecologically feasible. However, the flux graph is
a directed graph, where conductance in each direction is
not necessarily the same, so that effective conductance
will not be calculated by a single algebraic operation
(Thomas 2009). Instead, we followed the iterative
procedure developed by Thomas (2009) to calculate an
effective conductance in a directed circuit graph by
searching equilibrium voltages or currents with no
conflicting current flows.
The graph- and circuit-connectivity concepts are
summarized and represented in Fig. 4. The flux between
nodes depends on the size of the patch of origin, so that
the flux can be represented as directed edges. Even
though node b in Fig. 4C has one more link than in Fig.
4A, the incoming flux for node b in Fig. 4A and C are
the same because fluxes are weighted by area. When one
node is connected to the current source and a second
node is tied to the current sink or ground, current flows
without conflict from the source to the ground node
(Fig. 4B, D). Using this current flow path, effective
conductance can be calculated. Effective conductance is
larger in a circuit with an additional path (Fig. 4D).
Graph- and circuit-connectivity computing
based on threshold distances
We did not know the shape of the dispersal kernel for
our species, so we ran parameter sweeps on a range of
threshold distances and dispersal probability cutoffs.
The threshold distance was the distance beyond which
nodes were not connected in the graph. At the threshold
distance, if the dispersal probability is given as 5%, i.e.,
5% dispersal probability cutoff, the distance–decay
coefficient k of the negative exponential dispersal kernel
can be calculated such as�ln(0.05)/(threshold distance).
For graph- and circuit-connectivity calculations, we
used different shapes of the negative exponential
dispersal kernel based on a range of threshold distances
from 500 m to 7 km, every 100-m distance (i.e., 66
threshold distances) with 5% dispersal probability cutoff
(Appendix B). In addition, we also varied the dispersal
probability cutoff across a large range of values
(0.001%–100%) at threshold distances of 500 m to 5
km, every 500 m, to determine if this influenced the
graph and circuit theory model fit to our empirical data
because rare long-distance dispersal events can lead to
colonization and influence occurrence (e.g., Fitzpatrick
et al. 2012) (Appendix B). The dispersal probability and
threshold distance are related such that a 5% dispersal
probability occurs at a given threshold distance for a
given dispersal kernel, suggesting that 5% of dispersing
individuals travel at least this far and that this is just
sufficient for successful colonization and future dispersal
stemming from that colonization event. The parameter
sweeps indicated that the threshold distance had a much
larger influence on the model fit than did the dispersal
probability cutoff (Appendix C). For a given dispersal
kernel, the dispersal probability and threshold distance
are intricately related in that they can be thought of as
different axes or dimensions through which the kernel is
described.
Area- and distance-weighted flux matrices, Fluxadij,
for all pairs of habitat nodes were calculated using the
distance–decay coefficients corresponding to all 66
threshold distances. For calculating this flux, edge-to-
edge distances for all pairs of habitat patches were
measured on the habitat polygon map. Then, the sums
of incoming flux were measured for the selected 15 CP
sites. This flux value was used in statistical analysis to
test the first hypothesis. Procedures for the graph-
connectivity model including CRW simulation and
naturalness calculation were conducted with R software
(R Development Core Team 2011) with R packages
‘‘raster’’ for raster calculation (Hijmans and van Etten
2011) and ‘‘igraph’’ for graph visualization (Csardi
2010).
Using the area- and distance-weighted flux, effective
conductance was calculated according to the different
October 2013 1559CIRCUIT FLOW FOR PREDATORS
threshold distances. Regarding the flux as conductance,
15 effective conductance values were calculated for the
pairs of the source and sink habitat nodes: the prairie
complex habitat was connected to current source, and
each of the 15 CP nodes was connected to ground, or
the current sink. The iterative procedure and effective
conductance calculations were conducted using R
package ‘‘ElectroGraph’’ (Thomas 2012). Because the
calculation of effective conductance for each threshold
distance takes a long time (i.e., in the case of 1-km
threshold, it takes ;3 h on a 2.33 GHz Intel E5410
central processing units (CPU), and we had 66
threshold distances and a range of dispersal probabil-
ities), we used a supercomputer, Radon, which is a
computer cluster operated by the Rosen Center for
Advanced Computing, Purdue University (West La-
fayette, Indiana, USA), for the parameter sweep
analysis. Lastly, the effective conductance for each site
was multiplied by the sampling days corresponding to
each site. This value was used in statistical analysis to
test the second hypothesis.
Natural enemies and aphids sampling
During late June to mid-September in 2011, the
abundance of natural enemies was examined in the
selected 15 CP and the abundance of natural enemies
and soybean aphids were examined in the 15 soybean
fields adjacent to these CP. We set up two transects with
sampling points at 30 m, 60 m, and 150 m along each
transect in each soybean field and each CP. In soybean
fields, the two transects ran perpendicular to the edge
between the soybean fields and the CP, whereas in the
CP, transects ran parallel to the edge because the CP are
almost always long, narrow plantings (Fig. 1B). Two
transects sampled at the same three distances were also
set up in 10 prairie restorations and five pairs in the
tallgrass prairie remnant. In the crop fields, four tall
wooden stakes were used to surround each transect 25 m
from the transect to indicate where farmers should not
spray. However, in the only four fields that were sprayed
(fields 1, 2, 12, and 15), the aphid abundance data
indicate that our instructions were likely disregarded.
Natural enemy surveys were conducted every two
weeks using sweep-net sampling. Sweep-net sampling is
FIG. 4. Simple circuit with a series and a parallel connection using an area-weighted flux graph. If we assume that the size ofnodes is 1 (for b, c1, c2, and d) or 2 (for a, c, and d) and distances among nodes a, b, c (c1 and c2), d, and e are the same, the fluxesdepend on the size of nodes: Here, we assign 1 or 2 for the fluxes. When we regard these fluxes as conductances, the edges can berepresented by resistances, the reciprocal of the fluxes. Even though the distance from a to d in the circuits depicted in panels (B)and (D) is the same, the effective conductance between a and d of the circuit in panel (D) is larger than that of the circuit in panel(B) because of the alternative paths of the circuit in panel (D). In panels (B) and (D), A is the current in amperes.
INSU KOH ET AL.1560 Ecological ApplicationsVol. 23, No. 7
the best sampling method for capturing aphid predatory
insects in the families Anthocoridae, Nabidae, Chry-
sopidae, Hemerobiidae, and Coccinellidae (Roth 2003,
Kriz et al. 2006, Harwood 2008). We used a 20 pace–
sweep sample perpendicular to the transect at the three
sampling distances in soybean fields, CP, prairie, and
restorations. Samples were bagged and immediately
placed on ice in an ice box. At the end of each sampling
day, the samples were placed in a freezer. Natural
enemies in the samples were later identified to species
level for Anthocoridae and Coccinellidae and to family
level for others. All natural enemies were divided into
adult and immature categories. Aphids were counted on
two randomly selected soybean plants at each transect
point each week using the procedures developed by
Ragsdale et al. (2007) beginning when the plants reached
the V3 growth stage (three nodes of trifoliate leaves fully
developed).
For each study site, the predator abundance was
determined as the sum from all samplings on a given
date from the 12 points in each soybean field and
adjacent CP pair. Although there may be movement
between the CP and the crop field through the season,
we combined the predator data from field and CP at
each site because in this study we were interested in the
total predator community regardless of microhabitat. A
future study will examine farm-scale heterogeneity in
predator abundance. Soybean aphid abundance was
determined as the averaged number of aphids per plant
in each soybean field across the 12 plants on each
sampling date. Finally, we calculated the integrated area
under the abundance curves over time for the predators
and aphids for each site. These integrated predator-days
and aphid-days measures of the abundances were used
in statistical analysis. Hereafter, we refer to the
integrated abundance of predators and aphids as
predator-days and aphid-days, respectively. There was
a range of first sampling dates due to the difficulty of
finding appropriate fields. While this may introduce
some bias into the Anthocoridae abundances because
they were present early in the season, aphids and
Coccinellidae did not appear in our samples until after
surveys in all fields had begun.
PLATE 1. Abstract composition of seven-spotted ladybird beetles moving across a landscape upon which resistance values(resistors) have been superimposed. Image was created using POV-Ray and a ladybird beetle model by Giles Tran distributed underthe Creative Commons Attribution License. Image credit: J. D. Holland.
October 2013 1561CIRCUIT FLOW FOR PREDATORS
Statistical analysis
To examine the performance of graph and circuitconnectivity models, we compared the difference be-
tween the area-weighted incoming flux (Fluxa) andarea- and distance-weighted incoming flux (Fluxad)
determined by the CRW in this study based on differentthreshold distances. Because the circuit connectivity
model used Fluxad as conductance between patches, weexamined the relationship between Fluxad and the
effective conductance from the prairie complex toindividual patches (Conduct) based on different thresh-
old distances. We also compared Conduct with Euclid-ean distance from the prairie complex to individual
patches (Dist) to examine the degree to which these arecorrelated.
We used a model selection approach to test our twohypotheses (the effects of landscape connectivity on the
flow of natural enemies and the role of prairie as asource in sustaining the natural enemies) at the same
time to examine whether our flux and conductancevariables (Fluxad and Conduct) in this study show better
fits than Fluxa and Dist. Because Fluxad (or Fluxa) andConduct were dependent on threshold cutoff distances,we first examined the threshold distances with 5%probability cutoff at which each predator-days (Antho-coridae-days [Anth], exotic Coccinelldiae-days [Ecocc],
and native Coccinellidae-days [Ncocc]) was best predict-ed by aphid-days (Aphid) and flux or conductance
variables (i.e., Aphid þ Fluxad and Aphid þ Conductregression models, respectively). At the selected thresh-
old distances, we applied model selection techniques tofind the best model. Additionally, we examined how the
model fits of the selected best models varied withdispersal probability cutoffs.
For the first hypothesis, we tested whether fluxvariables influenced the predator-days by comparing
all possible linear model combinations including anintercept-only model to predict predator-days. In the
linear regressions, variables were transformed usingsquare-root or ln (x þ 1) to meet the assumptions of
normality and homogeneity of variance and to increaselinearity. We considered the flux variables as alternativevariables (either of Fluxad or Fluxa) to see which served
as better predictors. Particularly in the regressionmodels for predicting Ecocc and Ncocc, we included
as a predictor the other Coccinellidae group because ofthe reported displacement of native Coccinellidae by
exotic Coccinellidae (Evans 2004, Gardiner et al. 2009b).Coccinellidae–Aphid interaction terms were included
when both main predictors were included in the model.We then examined the correlation coefficients between
predictors in multiple linear regressions to avoidpossible problems of collinearity, such as when Pearson
correlation jrj . 0.7 (Dormann et al. 2012). In addition,to avoid the potential problem of multicollinearity
between main effects and interaction, we rescaled themain predictor variables by centering. We used these
centered variables and their interactions for predicting
Ecocc and Ncocc, which has been suggested as a
solution of reducing multicollinearity without altering
regression slopes or hypothesis tests (Jaccard et al. 1990,
Quinn and Keough 2002). To compare models using
these transformed and centered predictors, we calculated
corrected Akaike information criterion (AICc) values for
small sample size (Burnham and Anderson 2002). AICc
values were also used to derive AICc differences (DAICc)
for each model against the best model with minimum
AICc value. Finally, we presented all models with DAICc
, 4 to compare the models DAICc , 2, which are
generally considered to have substantial support (Burn-
ham and Anderson 2002). We also calculated model
weights, w, which were used to compare the strength of
evidence of the models from those with DAICc , 4
(Burnham and Anderson 2002). For the selected models
with more than one predictor, we calculated partial
correlations for all variables to assess the importance of
individual independent variables after controlling for
additional variables in the model. We applied this
procedure for testing our second hypothesis and we
compared Conduct and Dist with competing models.
All statistical analyses were conducted in R version
2.14.0 (R Development Core Team 2011). We examined
the normality and homoscedasticity of the errors of
regression models. We examined spatial autocorrelation
in the residuals of the regression models using a global
Moran’s I test (Dormann et al. 2007) and a spline
correlogram for each predator group (Bjornstad and
Falck 2001). Because of the low number of study sites
(driven by a lack of available plantings), we had a low
overall power to detect strong significance. Therefore,
we also presented marginally significant effects with P ,
0.1 when they were observed.
RESULTS
Graph- and circuit-connectivity model performance
Graph- and circuit-connectivity measurements, in-
coming flux and effective conductance, depended
considerably on the threshold distance that determined
which nodes were connected. At a 1.5-km threshold
distance, most nodes were connected to the network and
the all study sites had at least one incoming flux edge
(Fig. 5A). At threshold distances shorter than 1.5 km,
some sites had zero incoming flux (e.g., site 2). Between
2- and 3-km threshold distances, the study sites were
completely connected to the prairie complex network
(Fig. 5B).
When the threshold distance increased, the flux value
of the study sites also increased, and two different
patterns of increase appeared showing rapid and slow
changes. The rapid flux increase was caused by
proximity to large habitat patches, that is, when a
relatively large habitat patch was connected to the study
sites, as for sites 1, 5, 8, 10, and 15 (Fig. 6A). The slow
flux increase was related to increasing dispersal proba-
bility. In most study sites, the slow increase in flux with
threshold distance was shown before and after the rapid
INSU KOH ET AL.1562 Ecological ApplicationsVol. 23, No. 7
FIG. 5. Area- and distance-weighted (A, B) flux and (C, D) circuit graphs; the width of gray edges is proportional to flux in panels(A) and (B), and current in panels (C) and (D). (A) Flux graph based on a 1.5-km threshold distance; node circle size is relative topatch size; the width of edge is relative to averaged flux (directed edges are shown in themagnified box). (B) Flux graph based on a 3.0-km threshold distance. (C) Example of current flow from the prairie complex to site 8 based on the flux in panel (A); width of edge isproportional to current flow rate from source to sink; effective conductance calculated from the source to sink is 1.8. (D) Current flowrate from the prairie complex to site 11 based on the flux in panel (A); effective conductance is 0.7. ‘‘Nature’’ in the key refers to nodesof natural and seminatural areas, and prairie refers to the prairie restoration complex. Site numbers are shown in brown (see Fig. 1).
October 2013 1563CIRCUIT FLOW FOR PREDATORS
increases in all the study sites for all threshold distances.
However, when a habitat (e.g., site 9) was located near a
large habitat patch the incoming flux was maintained at
a high level at all threshold distances rather than
showing a rapid increase because the large patch
remained connected and contributed a large proportion
of the flux for the habitat (Fig.6B). The rapid increase in
flux with increasing graph connectedness, which was
displayed for most sites in the graph-connectivity model,
became even more apparent in the circuit-connectivity
analysis. Below a threshold distance of ;2 km, some
sites, such as 2 and 3, were not connected to the prairie
complex network and therefore had a predicted effective
conductance of zero (Figs. 5C, D, and 6B). When they
were connected to the prairie network, at ;2 km, the
zero conductances sharply increased to levels similar
with other sites (Fig. 6B). For the study sites, the
area- and distance-weighted flux (Fluxad) was highly
correlated with area-weighted flux (Fluxa). As the
threshold distance increased, this correlation increased
from 0.77 to 0.97 (Fig. 6C). Up to a threshold distance
of ;2 km, the correlation coefficient rapidly increased to
high levels (.0.9) before leveling off with increasing
threshold distance. These nonlinear increases, along
with the shape of the connected network, at least at the
1.5-km threshold (Fig. 5), limits correlation between
Euclidean distance and our two network-based predic-
tors. As the threshold increases and the network
becomes more connected, the correlation between these
three increases.
At a threshold distance of 1.5 km, effective conduc-
tance from the prairie complex to the other habitat
nodes (489 patches) including the selected CP fields was
negatively correlated to Euclidean distances to the
prairie complex (Fig. 6D). Although the trend became
quite linear once ln (x þ 1)-transformed (r ¼ �0.66),there was a high variance within the negative correla-
tion.
Aphid and natural enemies
The first Aphis glycines, a single individual on a
soybean plant, was observed on 19 July of 2011. The
total population of aphids increased to a peak between
late August and early September (Fig. 7). The two
highest field averages of aphids per plant, ;2520 and
870, were observed in site 12 on 31 August and in site 13
on 13 September 2011, respectively. At some fields (sites
1, 2, 12, and 15), pesticides were sprayed by famers
around the peak dates, so the aphid population
FIG. 6. (A, B) The comparison of area- and distance-weighted flux (Fluxad) and effective conductance (Conduct) connectivityvalues for selected 15 study sites based on threshold distances with 5% probability cutoff. (C) The relationship between area-weighted flux (Fluxa) and Fluxad and the relationship between Fluxad and conductance for the study sites. (D) The relationshipbetween conductance that was calculated at 1.5-km threshold distances with 5% dispersal probability cutoff and Euclideandistances from the prairie remnant to all other patches including the selected study sites. All variables are ln(xþ 1)-transformed.Numbers beside the lines in panels (A) and (B) and in gray circles in panel (D) indicate the selected 15 study sites.
INSU KOH ET AL.1564 Ecological ApplicationsVol. 23, No. 7
dropped. During the sampling period in 2011, the mean
aphid-days per plant per field were 9371 (SD ¼ 5003).
A total of 2573 individual natural enemies were
collected in soybean fields and adjacent CP in the
sampling period. This number was ;10 times the
number of predators collected in the prairie remnant
and restorations (Appendix C). Based on the number of
adult insects collected, within family-level abundances
were dominated by (most to least) Anthocoridae,
Coccinellidae, Chrysopidae, Nabidae, and Hemerobii-
dae in soybean fields and adjacent CP. The two most
abundant families, Anthocoridae and Coccinellidae,
comprised ;90% of the total natural enemies collected
and were selected to test our hypothesis. Four species of
coccinellids were collected and divided into exotic
(Harmonia axyridis and Coccinella septempunctata) and
native species (Coleomegilla maculate and Cycloneda
munda). Interestingly, we did not observe Coccinella
septempunctata in the prairies and restorations, but we
did observe two additional native species there, Hippo-
domia convergens and Hippodomia parenthesis. Harmo-
nia axyridis adults comprised ;70% of the coccinellid
adults in soybean fields and CP. We focused our analysis
on the adult predators because the nymphs and larvae
are unlikely to move between patches. Anthocoridae
adults were found in most study sites before aphids
appeared, and their population increased immediately
with the increasing of the aphid population (Fig. 7).
Exotic and native Coccinellidae adults were less
common in soybean fields before the aphids arrived,
but populations increased when the aphid population
rapidly increased. In the two sites where the highest
numbers of aphids were observed, a large number of
exotic Coccinellidae were observed, but no native
Coccinellidae were observed. In only three sites (4, 7,
and 14) were native Coccinellidae found with no exotic
species collected. Anthocoridae-days, exotic, and native
Coccinellidae-days per field were (mean 6 SD) 1627 6
827, 74 6 110, and 30 6 31 within sites, respectively.
Predicting the abundance of predators using graph-
and circuit-connectivity
For each predator, linear regression models using
area- and distance-weighted incoming flux (Fluxad) and
effective conductance from the prairie complex to
individual study sites (Conduct) detected the best
threshold distances with a 5% dispersal probability
cutoff (Fig. 8). The alternative predictors, which were
area-weighted incoming flux (Fluxa) and Euclidean
distance from the prairie complex to individual study
FIG. 7. The number of soybean aphids and natural enemies (Anthocoridae, exotic and native Coccinellidae) in 15 study fieldsfor the sampling period. For aphids, at least 6–10 sampling surveys, and for natural enemies, at least 4–8 sampling surveys, wereconducted in each field. Aphid number is the averaged number of aphids per plant, and natural enemy number is the total numberof each predator collected on each sampling date in the 12 sampling points of each field (i.e., sum of predators per 20 sweeps at 12sampling points).
October 2013 1565CIRCUIT FLOW FOR PREDATORS
sites (Dist), were compared with Fluxad and Conduct,
respectively, in model selection.
For Anthocoridae-days (Anth), the regression model
with aphid-days (Aphid) þ Fluxad, showed relatively
high r2 values in a range of threshold distances between
1.4 km and 2.7 km (Fig. 8A). In this range, the AphidþFluxad model at 2.3 km showed the highest adjusted r2
value of the regressions. Although the coefficient of
Fluxad showed positive relationship with Anth (Fig. 8E),
it was not significant and the model weight was much
less than the simpler Aphid only model (Table 1).
Compared to area-weighted incoming flux (Fluxa),
Fluxad increased the model fit slightly (Table 1). Similar
to the AphidþFluxad model, the adjusted r2 of AphidþConduct model was much less than that of the simpler
Aphid model, which was the best model, in most
FIG. 8. Linear regression model results and partial residual plots. Panels (A–C) show the adjusted r2 of regression models foreach predator-days at different threshold distances with 5% dispersal probability cutoff. Different shades of symbols indicate thesignificance levels of coefficients of Fluxad or Conduct in models (black shows P , 0.05; gray shows P , 0.10; white shows P �0.10). Abbreviations are: Aphid, soybean aphid (Aphis glycines Matsumura); Anth, Anthocoridae-days; Ecocc, exoticCoccinelldiae-days; and Ncocc, native Coccinellidae-days. Panels (D–K) show the relationship between predator-days and AphidþFluxad (or AphidþConduct) at the threshold distance showing the highest adjusted r2. All variable were ln (xþ 1)-transformed,except for Anthocoridae-days and native Coccinellidae-days, which were square-root(x)-transformed. Centered values areindicated. The pair of letters in parenthesis in panels (A–C) with arrows indicate the points of the highest adjusted r2 of regressionmodels.
INSU KOH ET AL.1566 Ecological ApplicationsVol. 23, No. 7
threshold distances and the coefficient of Conduct and
Dist did not show any significance (Fig. 8A, Table 1).
The exotic Coccinellidae-days were, interestingly,
negatively correlated with flux variables. The highest
r2 value appeared at the 2.3-km threshold distance (Fig.
8B), but the coefficient of Fluxad in the model was not
significant (Fig. 8G, Table 1). Similar to Anthocoridae-
days, the simpler Aphid model was the best model for
exotic Coccinellidae-days. However, the AphidþFluxad(or Fluxa) model was also strongly supported by data
(DAIC , 2), although the flux variables did not show
any significance. Similar to Anthocoridae-days, Fluxadincreased the model fit slightly compared to Fluxa(Table 1). Conduct and Dist variables did not show any
significance in the Aphid þ Conduct (or Dist) models,
which were not considerably supported by data (DAIC
. 2).
Native Coccinellidae-days were significantly positively
influenced by the flux and conductance (Fig. 8C). The
coefficients of Fluxad in the FluxadþAphid model were
significant at all threshold distances. Relatively high
peaks of adjusted r2 values of the FluxadþAphid model
appeared at the three different threshold distances of
0.9–1.3 km, 2.8–2.9 km, and 4.5–4.6 km. The highest r2
value appeared at the 1.1-km threshold distance (Fig.
8C). Interestingly, native Coccinellidae-days was nega-
tively correlated to Aphid (Fig. 8H, Table 2). The Aphid
þFluxad model was selected as the best model, although
the Aphid þ Ecocc þ Aphid 3 Ecocc model showed a
higher r2 value. In this interaction model, native
Coccinellidae-days was not significantly correlated with
Aphid and Ecocc, but was considerably negatively
correlated with their interaction (Table 2). Similar to
the other two predator groups, Fluxad increased the
model fit slightly in predicting native Coccinellidae-days.
The coefficient of Conduct in the Aphid þ Conduct
model was significant at a range of threshold distances
beyond 4.5 km (Fig. 8C). The highest r2 appeared at the
5.2-km threshold distance. At this threshold distance,
the Aphid þ Conduct model showed the lowest AICc,
which was selected as a better model than the interaction
model. The Conduct variable in Aphid (or Ecocc) þConduct models showed better model fit than the Dist
variable in the Aphid (or Ecocc) þ Dist models. The
Aphid (or Ecocc) þ Conduct models were considerably
supported by data (DAICc , 2) more than the model
with Dist (DAICc . 2; Table 2).
The model fit for all predator groups based on a 5%
dispersal probability cutoff were similar to the results
from the application of different dispersal probability
TABLE 1. Summary of the model selection statistics for Anthocoridea-days (Anth) and exoticCoccinellidae-days (Ecocc).
Response and model DAICc wi Adjusted r2 Partial correlation
Anth
Fluxad vs. FluxaAphid* 0.00 0.62 0.31Aphid* þ Fluxad2.3km 2.95 0.14 0.30 Aphid ¼ 0.55*, Fluxad ¼ 0.24Aphid* þ Fluxa2.3km 3.06 0.13 0.29 Aphid ¼ 0.56*, Fluxa ¼ 0.22Intercept 3.58 0.10 0.00
Conduct vs. Dist
Aphid* 0.00 0.65 0.31Aphid* þ (Conduct0.5km) 3.02 0.14 0.30 Aphid ¼ 0.62**, Conduct ¼ �0.23Intercept 3.58 0.11 0.00Aphid* þ Dist 3.72 0.10 0.26 Aphid ¼ 0.58*, Dist ¼ 0.08
Ecocc
Fluxad vs. FluxaAphid* 0.00 0.43 0.34Aphid** þ (Fluxad2.3km) 0.95 0.27 0.41 Aphid ¼ 0.70***, Fluxad ¼ �0.42Aphid** þ (Fluxa2.3km) 1.25 0.23 0.40 Aphid ¼ 0.69***, Fluxa ¼ �0.40Aphid* þ (Ncocc) 3.36 0.08 0.31 Aphid ¼ 0.59*, Ncocc ¼ �0.17
Conduct vs. Dist
Aphid* 0.00 0.64 0.34Aphid* þ (Dist) 3.20 0.13 0.32 Aphid ¼ 0.54*, Dist ¼ �0.20Aphid* þ (Ncocc) 3.36 0.12 0.31 Aphid ¼ 0.59*, Ncocc ¼ �0.17Aphid* þ Conduct0.5km 3.46 0.11 0.31 Aphid ¼ 0.62**, Conduct ¼ 0.15
Notes: Variables compared: incoming area- and distance-weighted flux (Fluxad) vs. incomingarea-weighted flux (Fluxa); effective conductance (Conduct) vs. Euclidian distance from prairiecomplex (Dist). The first model listed at each predator is the minimum AICc model, and boldfacetype indicates models with DAICc , 2. The intercept-only model in the regression models isindicated as Intercept. Negative relationships between variables and predator-days are indicated inparentheses. The selected best threshold distances of Flux and Conduct are indicated as subscripts.Aphid-days (Aphid) and native Coccinellidae-days (Ncocc) were centered when the interactionterm was included in the model. All variables were ln(x þ 1)-transformed except for Anth andNcocc, which were square root(x)-transformed. Aphid and Ncocc were centered when theinteraction term was included in the model.
* P , 0.05; ** P , 0.01; *** P , 0.001.
October 2013 1567CIRCUIT FLOW FOR PREDATORS
cutoffs because this had less influence than the threshold
distance (Appendix D). For Anthocoridae-days and
exotic Coccinellidae-days, the r2 values appeared to
continue increasing beyond the boundaries of the ranges
tested, although these further values are not considered
biologically realistic. For native Coccinellidae-days, the
highest r2 values of the Aphid þ Fluxad (or Conduct)
occurred with a 1-km threshold with 30% dispersal
probability cutoff, and at 4.5 km with a 15% dispersal
probability cutoff (Fig. 9). However, the overall trends
of the model fits were very similar to the threshold
distances with a 5% dispersal probability cutoff and
changed little as the values moved away from this,
compared to altering the threshold distance (Fig. 9).
We did not find collinearity among variables and
between the centered variables and their interactions in
multiple linear regression models (all jrj , 0.7). We also
did not find significant spatial autocorrelation in the best
regression models for any predator group (all jMoran’s
I j , 0.75, P . 0.19; 95% point-wise bootstrap
confidence intervals of spline correlograms included 0
at all scales). There is the possibility that this is the result
of a Type II error in our test, but as we sampled all of
the CP33s next to soybean in the entire county (and a
large proportion of the CP21s next to soybean), our
small sample size was unavoidable.
DISCUSSION
The graph- and circuit-connectivity models success-
fully tested our two hypotheses regarding the effects of
landscape connectivity and the prairie complex as a
source for the flow of natural enemies. Graph- and
circuit-connectivity predicted native natural enemy
pressures within a certain range of thresholds in edge
distance. The similarity of the threshold distances for
Anthocoridae and exotic Coccinellidae (2.3 km and 2.9
km) suggests that this may partially be an artifact of the
specific region of the study or of the parameters
considered.
Limitations of graph- and circuit-connectivity models
The graph- and circuit-connectivity models did not
incorporate information on long-distance dispersal
TABLE 2. Summary of model selection statistics for native Coccinellidae-days.
Model DAICc wi Adjusted r2 Partial correlation
Fluxad vs. Fluxa(Aphid)� þ Fluxad1.1km* 0.00 0.15 0.33 Aphid ¼ �0.51*, Fluxad ¼ 0.61**(Aphid)� þ Fluxa1.1km* 0.34 0.13 0.31 Aphid ¼ �0.50*, Fluxa ¼ 0.60**Fluxad 1.1km� 0.77 0.10 0.16Aphid þ (Ecocc) þ (Aphid 3 Ecocc)** 0.79 0.10 0.44 Aphid ¼ 0.26, Ecocc ¼ �0.26, Aphid 3 Ecocc ¼
�0.71***Fluxa 1.1km� 0.95 0.10 0.15(Aphid) þ (Ecocc) þ (Aphid 3 Ecocc)* þ
Fluxad1.1km�0.98 0.09 0.57 Aphid ¼ 0.00, Ecocc ¼ �0.20, Aphid 3 Ecocc ¼
�0.68**, Fluxad ¼ 0.56*Intercept 1.33 0.08 0.00Aphid þ (Ecocc) þ (Aphid 3 Ecocc)* þ
Fluxa1.1km�1.37 0.08 0.56 Aphid ¼ 0.01, Ecocc ¼ �0.21, Aphid 3 Ecocc ¼
�0.68**, Fluxa ¼ 0.55*(Ecocc) þ Fluxad1.1km� 2.15 0.05 0.23 Ecocc ¼ �0.38, Fluxad ¼ 0.52*(Ecocc) þ Fluxa1.1km� 2.33 0.05 0.22 Ecocc ¼ �0.39, Fluxad ¼ 0.51*(Ecocc) 2.98 0.03 0.03(Aphid) 3.17 0.03 0.01
Conduct vs. Dist
(Aphid)� þ Conduct5.2km* 0.00 0.16 0.30 Aphid ¼ �0.51*, Conduct ¼ 0.58*Aphid þ (Ecocc) þ (Aphid 3 Ecocc)** 0.03 0.16 0.44 Aphid ¼ 0.26, Ecocc ¼ �0.26, Aphid 3 Ecocc ¼
�0.71***Intercept 0.57 0.12 0.00Conduct5.2km 0.69 0.12 0.12(Ecocc) þ Conduct5.2km� 1.37 0.08 0.23 Ecocc ¼ �0.43�, Conduct ¼ 0.52*(Ecocc)� þ (Dist)� 2.09 0.06 0.19 Ecocc ¼ �0.49*, Dist ¼ �0.48�(Ecocc) 2.22 0.05 0.03(Aphid)� þ (Dist)� 2.22 0.05 0.18 Aphid ¼ �0.49�, Dist ¼ �0.49�(Aphid) 2.41 0.05 0.01(Dist) 2.47 0.05 0.01Aphid þ (Ecocc) þ (Aphid 3 Ecocc)* þ
Condcut5.2km
2.49 0.05 0.50 Aphid ¼ 0.04, Ecocc ¼ �0.27, Aphid 3 Ecocc¼ �0.63*, Conduct ¼ 0.45
Aphid þ (Ecocc) þ (Aphid 3 Ecocc)* þ(Dist)
2.66 0.04 0.50 Aphid ¼ 0.08, Ecocc ¼ �0.37, Aphid 3 Ecocc¼ �0.66**, Dist ¼ �0.44
Notes: Variables compared: incoming area- and distance-weighted flux (Fluxad) vs. incoming area-weighted flux (Fluxa); effectiveconductance (Conduct) vs. Euclidian distance from prairie complex (Dist). The first model listed is the minimum AICc model, andboldface type indicates the models with DAICc , 2. The intercept-only model in the regression models is indicated as Intercept.Negative relationships between variables and predator-days are indicated in parentheses. Native Coccinellidae-days (Ncocc) wassquare-root(x)-transformed, and other variables were ln(x þ 1)-transformed. Aphid-days (Aphid) and exotic Coccinellidae-days(Ecocc) were centered when the interaction term was included in the model. The selected best threshold distances of Flux andConduct are indicated as subscripts.
* P , 0.05; ** P , 0.01; *** P , 0.001; � P , 0.1.
INSU KOH ET AL.1568 Ecological ApplicationsVol. 23, No. 7
beyond a threshold distance, which was applied to
identify graph edges, allowing for an efficient calculation
of flux and conductance. This threshold partially
determined graph connectedness and the shape of the
dispersal curves (dispersal kernel). However, this was a
limitation because it assumes that there is no dispersal
leading to successful colonization beyond the threshold
distance with the 5% dispersal probability cutoff. This
assumption ignores less common long-distance dispersal
beyond the threshold distance, which does not ideally
model the behavior of dispersing individuals (Moilanen
2011). This loss of long-distance dispersal information
was evident in the rapid increase of the incoming flux
and effective conductance values with increasing thresh-
old distance (Fig. 6A, B). This pattern of rapid increase
within a short range of threshold distances indicates that
certain important habitats with a high potential to
contribute dispersers were not considered to contribute
to the dispersal of natural enemies until these habitats
were connected at a threshold distance. The rate of
increase in flux and effective conductance was reduced
as habitats were connected by edges, around the 1.5-km
threshold distance for incoming flux and 2.0 km for
effective conductance. While an ideal model may use a
very large threshold (i.e., beyond the 7 km that we
focused on in this study), such as the predicted distance
for 0.001% of dispersers for successful colonization, this
would lead to a very large number of connected nodes
(i.e., edges) and become much more computationally
intensive. Because the 5% dispersal probability cutoff at
each threshold distance can represent the trend of the
lower dispersal probability and large threshold distance,
intensive computation seems to be not efficient (Fig. 9).
Additionally, such a large graph network may also be
unrealistic for some species as it would imply that
dispersers pass by many habitats to colonize a distant
patch. Another solution may be to develop area-
weighted connectedness thresholds.
Because the circuit-connectivity model built upon the
graph-connectivity measure, the two were quite corre-
lated (Fig. 6C). The sum of conductances at the
electrical ground node in a parallel connection is same
as the incoming flux (Ge ¼ G1þG2). Therefore, the
correlation of these two measures increased as the
threshold distance increased and the network became
more connected. The rapid increase in this correlation
with threshold distances greater than 2 km seems to have
been caused by large patches joining the network and
FIG. 9. Linear regression model results of native Coccinellidae-days at different threshold distances with different dispersalprobability cutoffs. The boxes outlined in black indicate the highest adjusted r2 within the two dimensions of threshold distancesand dispersal probability cutoffs. Asterisks indicate that the coefficients of the relationship between threshold distance and Fluxador conductance in the model are significant at P , 0.05.
October 2013 1569CIRCUIT FLOW FOR PREDATORS
greatly increasing both measures and by the ‘‘short-
circuiting’’ network across the gap in the southwestern
area (Fig. 6D).
Natural enemy populations
As in previous work (e.g., Rutledge et al. 2004), the
field surveys of natural enemies and aphid populations
showed that Anthocoridae was the most abundant
natural enemy in Indiana soybean fields, followed by
Coccinellidae (Fig. 7). In most of the study sites
Anthocoridae were present before aphids were detected
(Fig. 7). Orius insidious (Anthocoridae) can survive in
soybean crops before aphid arrival by feeding upon
soybean plants and soybean thrips (Isenhour and
Marston 1981, Yoo and O’Neil 2009).
Predator populations were largely driven by aphid
populations (Fig. 8D, F, Table 1), but native coccinellids
were also considerably influenced by the interaction
between aphids and exotic Coccinellidae (Table 2).The
peak arrival of H. axyridus (Ecocc) in fields occurs
around of the time of peak aphid population because
adult females are able to search for the best field to
maximize their offspring’s fitness (Koch 2003). This can
result in aggregation of exotic coccinellids in fields with
high numbers of aphids. Another reason for the
aggregation may be the trapping effect. The trapping
effect occurs when these exotic coccinellids change their
search mode from long-range movement to within
habitat foraging (Kawai 1976, Koch 2003). As a result,
they may show a strong ability to follow aphid
populations as indicated in Osawa (2000) and With et
al. (2002). These behaviors of H. axyridis might exclude
native Coccinellidae from these fields. In our study
native Coccinellidae were negatively influenced by the
interaction term between aphids and exotic Coccinelli-
dae (Table 2). This suggests that as exotic species
increase in response to aphid populations the native
species suffer. This could be due to direct predation
(Evans 2004, Yasuda et al. 2004) or competitive
exclusion (Michaud 2002). Therefore, the native species
were correlated with aphid populations in the regression
model with aphids and flux (or conductance; Fig. 8H, J,
Table 2). However, we did not explicitly test the
biological interactions of these two groups. The
ecological consequences of these interactions on native
species populations have not been clearly revealed
(Harmon et al. 2007).
Effects of landscape connectivity and prairie on natural
enemies’ dispersal
The graph- and circuit-connectivity models success-
fully examined the different responses of three natural
enemy groups to landscape connectivity and the prairie
complex. For Anthocoridae, the results indicated that
landscape connectivity may enhance the flow of this
natural enemy, but that the prairies do not act as a
source for them (Fig. 8E, Table 1). From the best
threshold distance shown in the graph-connectivity
model (2.3 km), we obtained the distance–decay
parameter determining the shape of dispersal curves, k
¼ 0.0013. The adults of O. insidious, the most dominant
species of Anthocoridae in this study, are known to
disperse in order to forage. In early spring, they forage
on flowering herbaceous plants, and in summer many
may move onto crop plants and come back to the
perennial forbs for overwintering (Saulich and Musolin
2009). In our study area, conservation plantings may
have provided overwintering habitat for them.
We failed to reject the null hypothesis that the
abundance of exotic Coccinellidae, mostly H. axyridis
in this study, is not related to the connectivity of
seminatural areas (Fig. 8G, Table 1). This lack of a
relationship may suggest that H. axyridis are not
sensitive to landscape fragmentation at large spatial
scales because of their high dispersal ability. They are
known to not be affected by fragmentation at small
spatial scales (With et al. 2002). Exotic Coccinellidae did
not show a response to circuit connectivity (Table 2).
This may mean that the prairie complex was not a
source for them or they are less dependent on the prairie,
as indicated by Gardiner et al. (2009b). It is also possible
that populations of exotic coccinellids are bolstered by
the prairie complex, but they move across the landscape
using habitats in a different way than we have modeled
to construct our network. However, this would not
entirely explain the negative relationship between exotic
coccinellids and graph connectivity (Fig. 8G, Table 1).
Native Coccinellidae were positively influenced by
seminatural areas and connectivity to the prairie
complex (Fig. 8C, Table 2). This result highlights that
the movement of native Coccinellidae depended on the
configuration of seminatural habitats and the role of the
prairie complex as source habitat. Well-connected
seminatural habitats enhanced the flow of these natural
enemies. This result is supported by other studies, which
have showed that native Coccinellidae are more sensitive
to habitat fragmentation than exotic Coccinellidae
(With et al. 2002) and that grasslands enhance their
abundance (Gardiner et al. 2009b). The best threshold
distances for flux (1.1 km) and conductance (5.2 km)
were different. However, this large threshold distance
for conductance was similar to the threshold distances
(4.5–4.6 km) showing one of peaks of flux. The small-
scale and large-scale ranges of good model fit for flux
may suggest that habitat fragmentation is important to
these predators at multiple spatial scales. Alternatively,
it could be indicating the result of different ecological
processes such as long-range dispersal and small-scale
foraging movements, or within-generation dispersal and
longer term patterns of movement over many genera-
tions. The second hypothesis was supported by the
insect sampling data from the prairie complex. The
sampling data showed two more native Coccinellidae
species, Hippodomia convergens and Hippodomia paren-
thesis, in the prairie than in the selected soybean field
and CP sites. The data also showed that native
INSU KOH ET AL.1570 Ecological ApplicationsVol. 23, No. 7
Coccinellidae was dominant in this habitat compared to
exotic Coccinellidae (Appendix C).
The numbers of Coccinellidae were low. About half of
the zero values occurred before soybean aphids arrived.
During this period, soybean fields are almost devoid ofinsects. We have confidence that our data did capture
real trends in abundance. The abundance of these
predators followed that of the prey aphids as one would
expect. These may actually be quite realistic numbers ofCoccinellidae predators. In a recent study, Costamagna
et al. (2007) found aphid abundances per plant similar to
ours, and a maximum average of about nine Coccinelli-
dae (all species combined) per 1 m2 detected with anexhaustive search with caged plants. We swept a much
larger area, but only sampled the tops of plants (the
accepted method). Adult Harmonia can eat 90–270
aphids per day, and larvae (not included in our analysis)
can eat 600–1200 aphids every day (Knodel andHoebeke 1996). They will therefore, be at much lower
densities than the aphids. For these reasons, we believe
that our abundance data are reflective of the true
abundances within the fields.
The developed flux measure, area- and distance-weighted flux, showed slightly better prediction accuracy
for all the selected three predator groups than the model
using a flux without distance weighting (Tables 1 and 2).
Weighting distance using the adaption of CRW simu-
lation considered a potential region of insect’s move-ment while traveling between a pair of seminatural
habitats (Figs. 2 and 3). However, because the study
region is highly dominated by agriculture, its weighting
effect on calculating flux was limited. This could limitthe potential improvement in model fit and explain the
only slight improvement in the prediction that we
observed. The circuit-connectivity model explained the
relationship between source and sink, i.e., prairie
complex and CP in this study, better than Euclideandistance. Compared to the Euclidean distances between
source and sink, using circuit theory certainly resulted in
better predictions (Tables 1 and 2). Although aphid
pressure was the factor that most influenced thepressures of natural enemies, incoming flux of the
graph-connectivity model and effective conductance of
the circuit-connectivity model were also selected as
important factors for predicting the pressures of natural
enemies, especially for native species.
Implications for conservation biocontrol
Our study found that well-connected habitat networksin landscapes enhance the flow of native natural enemies
such as Anthocoridae and native Coccinellidae, and that
prairies act as a source for native Coccinellidae. Exotic
Coccinellidae were not positively related to the large-scale connectivity of seminatural habitats. Therefore, if
landscape fragmentation of seminatural habitat occurs
heavily on agricultural landscape, the control of soybean
aphids will depend more on exotic natural enemies,
especially H. axyridis, than on native species. Relying on
one dominant species for an ecosystem service rather
having redundancy including multiple native species
carries a risk. If H. axyridis ever fell victim to
unexpected disease, predator, or parasitoid, we may
need to look to our native species for control of pests
such as A. glycines (Koch 2003). Diversity is insurance
against future changes in ecosystem services (Kremen et
al. 2002). From this perspective, how to arrange habitat
networks to support native predator species becomes
important. Further study should focus on how to
optimize landscape connectivity to maximize biocontrol
service. In this optimization, the estimated individual
natural enemy’s dispersal parameter will be useful.
Although Anthocoridae is the most abundant pred-
ator, they are individually less efficient in aphid control
than many other predators, and they often leave
soybean fields early. However, we did find that they
were present at the crucial period of colonization by A.
glycines (Yoo and O’Neil 2009). As suggested by Yoo
and O’Neil, it is possible that having Anthocoridae as
part of the predator community in soybean fields is
important because their presence at the time of soybean
aphid arrival may help control populations before they
grow rapidly. Even a slight delay in the approach to the
economic threshold of pest population (250 aphids/
plant) can be of great benefit, especially if this occurs
after the soybeans have reached the R6 growth stage
because yield is then not negatively affected and
pesticide application is no longer necessary (Krupke et
al. 2013).
ACKNOWLEDGMENTS
This research was supported by a USDA-NIFA EcosystemServices Grant. The authors thank Derek Calhoun, MonicaMain, Stephanie Hathaway, Sage McCafferty, Tommy Mager,John Shukle, and Caoilin Hoctor for field work that formed thebasis of this study. Carol Ritchie at the Newton County IndianaFarm Service Agency office was a great help with locatingfarmer cooperators and with spatial data for the study sites.Stephanie Frischie was a great help with fieldwork logistics.Many farmers graciously allowed us access to their fields. TheNature Conservancy allowed us to sample at their prairie andrestoration sites. Two reviewers provided suggestions thatgreatly improved the analysis and text. Carol Sung, JohnMichael, and Brian Raub of the Purdue SupercomputingCenter were instrumental in running parameter sweeps.Christian Krupke shared his knowledge of soybean agriculture.The authors thank Andrew Thomas for his technical help withthe ElectroGraph package.
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SUPPLEMENTAL MATERIAL
Appendix A
Two parameters for correlated random walk simulation: reported data and distributions of step length and turning angles(Ecological Archives A023-081-A1).
Appendix B
Dispersal kernels used for the parameter sweep (Ecological Archives A023-081-A2).
Appendix C
Insect sampling count data in 15 CP and prairies (Ecological Archives A023-081-A3).
Appendix D
Linear regression model results of Anthocoridae-days (Anth) and exotic Coccinellidae-days (Ecocc) with area- and distance-weighted flux (Fluxad) and effective conductance (Conduct) at different threshold distances with different dispersal probabilitycutoffs (Ecological Archives A023-081-A4).
Supplement
Detailed description of correlated random walk simulation and how to obtain an ellipse-shaped region covering successfulpathway of a random walker, including R code (Ecological Archives A023-081-S1).
October 2013 1573CIRCUIT FLOW FOR PREDATORS