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ARTICLE doi:10.1038/nature11131
Dimensionality of consumer search space
drives trophic interaction strengths
Samraat Pawar
1
, Anthony I. Dell
1,2
& Van M. Savage
1,3,4
Trophic interactions govern biomass fluxes in ecosystems, and stability in food webs. Knowledge of how trophic
interaction strengths are affected by differences among habitats is crucial for understanding variation in ecological
systems. Here we show how substantial variation in consumptionrate data, and hence trophic interaction strengths,
arises because consumers tend to encounter resources more frequently in three dimensions (3D) (for example, arboreal
and pelagic zones) than two dimensions (2D) (for example, terrestrial and benthic zones). By combining new theory with
extensive data (376 species, with body masses ranging from 5.24 310
214
kg to 800 kg), we find that consumption rates
scale sublinearly with consumer body mass (exponent of approximately 0.85) for 2D interactions, but superlinearly
(exponent of approximately 1.06) for 3D interactions. These results contradict the currently widespread assumption of a
single exponent (of approximately 0.75) in consumer–resource and foodweb research. Further analysis of 2,929
consumer–resource interactions shows that dimensionality of consumer search space is probably a major driver of
species coexistence, and the stability and abundance of populations.
Understanding how physical differences between habitats, such as dif
ferences in precipitation, temperature and spatial dimensionality, affect
trophic interactions is key to predicting stability and diversity in eco
logical systems
1–6
. By assuming a simple relationship between con
sumption rate (energy acquisition) and metabolic rate (energy use),
most studies assume that percapita consumptionrates scale with con
sumer body size (m) to an exponent of approximately 0.75, irrespective
of taxon, environment or dimensionality
7–13
. Consequently, mass
specific production rates
8,14
scale as m
20.25
, including biomass flow rate
and perlink trophic interaction strengths in food webs
10,11,13,15,16
.
Deviations from quarterpower scaling can arise for at least two reasons.
First, foraging is constrained by traits, such as length of locomotory
appendages or visual acuity, that do not scale directly with metabolic
rate
8,17–20
. Second,species interactions in the fielddo not occur under the
idealized conditions at which metabolic and ingestion rates are usually
measured, in which individuals are not foraging, growing or repro
ducing
8,18,19
. Therefore, consumptionrate scaling may be more closely
tied to field or maximal metabolic rate (exponent greater than 0.85),
rather than resting metabolic rate (exponent of approximately 0.75)
8,21
.
From a biomechanical perspective, both nonmetabolic and
metabolic constraints on consumption rate should depend on the
habitat’s spatial dimensionality because it strongly influences the
energetic costs of locomotion (for example, to overcome gravity)
18,19
and the probability either of a consumer detecting a resource or vice
versa
17,20
. Indeed, over two decades ago, habitat dimensionality was
proposed as a major factor driving foodweb structure and ecosystem
dynamics
1,4,22
. Subsequent studies have further elucidated the effects
of habitat dimensionality
3,6,23–25
. Notably, previous models suggest
that grazers (one type of consumer; Fig. 1 and Supplementary Fig. 1)
are constrained by how resources are distributed in space
3,24,25
. These
studies are foundational, but do not apply to the full diversity of
foraging strategies and interactions in natural communities.
Here we show that shifting focus from dimensionality of the
habitat
3,4,6,23–25
to the dimensionality of each trophic interaction yields
a new, mechanistic theory for trophic interaction strengths (Figs 1
and 2). Our approach allows both 2D and 3D interactions within the
same habitat to be considered, and can be applied to the wide range of
foraging strategies found in nature (Fig. 1 and Supplementary Fig. 1).
To test our predictions, we compiled a data set that contains a per
capita consumption rate of 255 consumer–resource interactions
covering 230 species, 12 orders of magnitude in body size, and aquatic
(189 interactions) as well as terrestrial (66 interactions) habitats
(Methods).
1
Department of Biomathematics, David Geffen School of Medicine, University of California, Los Angeles, California 900951766, USA.
2
School of Marine and Tropical Biology, James Cook University,
Townsville QLD 4811, Australia.
3
Department of Ecology & Evolutionary Biology, University of California, Los Angeles, California 90095, USA.
4
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico
87501, USA.
2D
3D
3D
2D
3D
2D
Figure 1

Consumer–resource interactions can be classified by
dimensionality. If the consumer searches for resources (by flying, swimming,
or sitting and waiting) on habitat surfaces (for example, on the water surface,
benthos or in grassland), the interaction is 2D, and if it searches habitat volume,
the interaction is 3D. A consumer or resource may be involved in both 2D and
3D interactions, corresponding to different consumer–resource combinations
and foraging strategies.
28 JUNE 2012  VOL 486  NATURE  485
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©2012
Empirical patterns
Using our comprehensive data set, we first demonstrate strong
empirical differences between 2D and 3D interactions in the scaling
of search and consumption rate with consumer body size (Fig. 3).
When resources are scarce, more closely resembling field conditions,
the observed scaling exponent for consumption rate in 3D interac
tions (1.06 60.06 (95% confidence intervals)) is significantly higher
than in 2D (0.85 60.05) (likelihood ratio test, P,0.001) (Fig. 3a, c).
These scaling exponents are significantly higher than the currently
used exponent of 0.75 (onesample Ftest P,0.01). Furthermore,
apart from organisms that are much smaller than a honeybee (weigh
ing less than 3 310
24
kg, where 2D and 3D scaling lines would
intersect), 3D consumption rates are higher than in 2D (Fig. 3a, b).
For a 1kg organism, 3D consumption rate is ten times higher than in
2D (6.30 63.01 versus 0.63 60.24 mg s
21
) (Fig. 3a, b).
When resources are abundant, typical of laboratory conditions,
consumption rates still scale more steeply (1.00 60.06 versus
0.85 60.05) and show higher baseline values in 3D than 2D
(19.95 611.00 versus 3.16 61.30 mg s
21
for a 1kg organism)
(Fig. 3c, d). Thus, even at high resource densities at which searching
for resources is expected to be less constraining, dimensionality
remains important. The canonical 0.75 scaling exponent for con
sumption rate is excluded from the 95% confidence intervals of the
observed scaling exponents under all conditions (Fig. 3).
We also analysed the scaling of search rates. The rate at which a
consumer searches for a resource limits consumption rates when
resources are scarce (Figs 1, 2 and 3e, f). For activecapture and
grazing foragers, search rate (area/time or volume/time) is the speed
at which a consumer moves through the landscape to find food,
whereas for sitandwait foragers, it is the speed at which resources
move through the consumer’s attack space (Figs 1 and 2). We find
that search rates have a scaling exponent of 1.05 60.08 in 3D
and 0.68 60.12 in 2D (Fig. 3e, f), indicating that differences in
consumptionrate scaling are primarily driven by differences in
search rate. This result is a key validation of our model below.
A mechanistic model for search rate
Our empirical analysis reveals that search and consumptionrate
scaling vary systematically with the dimensionality of search space
(that is, interaction dimensionality). We now present a model that
predicts these empirical patterns by focusing on three key compo
nents of search rate: relative velocity, reaction distance and handling
time
13,17,26
(Fig. 2). Relative velocity (v
r
) is the rate at which consumer–
resource pairs converge across the landscape, and it is the rootmean
square of their body velocities. A potential encounter occurs when
either the resource or consumer comes within the distance (d)at
which one can detect and react to the other. Because each individual
moving through the landscape maintains a search space enclosed by a
surface with radius d, we can derive (Supplementary Information)
that the search rate (a) increases with dimensionality (D):
a~sDvrdD{1ð1Þ
where s
D
52 in 2D and pin 3D. Based on biomechanical principles,
we obtained predictions for the scaling exponents p
v
and p
d
(of v
r
and
d, respectively; Fig. 2), and validated them empirically using another,
independent data set that we compiled (Table 1 and Supplementary
Information). Using these, we predict:
a~a0mpvz2pd(D{1)
Cf(kRC)ð2Þ
where m
C
is consumer body mass. For active foraging, the constant
a0is 2v0d0in 2D and pv0d2
0in 3D. The function f(k
RC
) isolates
dependence of aon consumer–resource size ratio k
RC
(that is,
m
R
/m
C
(where m
R
is resource body mass)) from its direct dependence
on consumer mass. Both a
0
and f(k
RC
) vary weakly with foraging
strategy (Supplementary Information). To relate equation (2) directly
to previous studies by expressing it solely in terms of consumer mass,
we determine how k
RC
scales with consumer mass using our con
sumptionrate data set. Substituting this scaling together with values
for p
v
and p
d
(Table 1) into equation (2) gives:
–15
–10
–5
0
–10
–5
0
log10(Consumer mass)
–15
–10
–5
log10(Consumption rate)
0.75 power
scaling
log10(Search rate)
b Scarce resources
c Abundant resources d Abundant resources
y = 0.85x – 6.2
R2 = 0.84
y = 1.06x – 5.2
R2 = 0.85
Aquatic
Terrestrial
Aquatic
Terrestrial
e Scarce resources f Scarce resources
a Scarce resources
2D 3D
y = 0.85x – 5.5
R2 = 0.86
y = 1.00x – 4.7
R2 = 0.84
y = 0.68x – 3.08
R2 = 0.60
y = 1.05x – 1.77
R2 = 0.77
–8 –6 –4 –2 0 2 –8 –6 –4 –2 0 2
0
Figure 3

Effect of interaction dimensionality on scaling of search and
consumption rate. a–d, Scaling of percapita consumption rate (kg s
21
)with
consumer body mass (kg) at different resource densities. e,f, Scaling of search
rate (m
2
s
21
in 2D, and m
3
s
21
in 3D). See Table 1 for sample sizes. Solid black
lines were fitted using OLS regression (see Methods). Exponents in all panels
except eare significantly different from the canonical 0.75 value (dotted line).
Consumptionrate scaling shows less variance than search rate, possibly
because consumers choose resources that maximize biomass consumption rate
(product of search rate, resource density and resource mass; equation (4)),thus
minimizing scatter.
Detection
Searching
dth
Capture and
subjugation
Ingestion
De
d
mR, mC = Resource (R) and consumer (C) body masses
kRC = mR/mC = Body mass ratio
β = Exponent for handling time (th)
pv = Exponent for consumer or resource velocity (vR,vC)
pd = Exponent for consumer–resource reaction distance (d)
Scaling parameters
D = Interaction dimensionality (2 or 3)
d (mRmC)
th mC
vR,vC mC
D = 2
vC > 0
vR > 0
pv
pd
–β
Figure 2

Model for scaling of search and consumption rate with body size.
This model (2D active capture isshown here) can also be used to predict search
and consumption rates for grazing and sitandwait foraging strategies
(Supplementary Information).
RESEARCH ARTICLE
486  NATURE  VOL 486  28 JUNE 2012
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a<a2Dm0:63
Cin 2D
a<a3Dm1:03
Cin 3D ð3Þ
where a
2D
and a
3D
are dimensionspecific constants. These exponents
match our empirical results extremely well (Fig. 3e, f and Table 1).
Even if the weak contribution of f(k
RC
) (Supplementary Information)
to the scaling is ignored, the predicted search rate exponents (p
v
1
2p
d
(D–1)) would be 0.68 in 2D and 1.06 in 3D. These exponents are
extremely close to the empirical estimates of 0.68 60.12 in 2D and
1.05 60.08 in 3D (Table 1; Fig. 3).
Predictions for consumption rate
The product of search rate, a, and resource density, x
R
(individuals per
area or volume), yields encounter rate. Consumption rate is con
strained by this encounter rate and by handling time; that is, the
duration of time to pursue, subdue and ingest each resource
(Fig. 2). Together, these components give a saturating percapita bio
mass consumption rate (c) (Holling’s type II functional response
27
)in
terms of spatial dimension (D):
c~amRxR
1zthamRxR
~s’DvrdD{1mRxR
1zths’DvrdD{1mRxR
ð4Þ
Here, m
R
is the average mass of the resource, x
R
m
R
is resource biomass
density, and t
h
is conventional handling time divided by resource
mass (Supplementary Information 1.4). The constant s9
D
includes a
roughly constant attack success probability. Our results are robust to
changes in this probability for resource items common in the con
sumer’s diet (Supplementary Information).
With scarce resources (x
R
Rx
R,min
) the second term in the denom
inator of equation (4) becomes much smaller than 1, and thus c<
ax
R
m
R
. Substituting the scaling for a(equation (2)) gives:
c~a0mpvz2pdD{1ðÞ
Cfk
RC
ðÞxRmRð5Þ
To convert this into a scaling relationship solely with consumer mass,
we use our functional response data set (Supplementary Information)
to quantify the scaling of x
R
and m
R
with consumer mass (Table 1).
Substituting these along with the previously determined scaling of size
ratio (k
RC
) in equation (5) gives:
c<c2Dm0:78
Cin 2D
c<c3Dm1:16
Cin 3D ð6Þ
where c
2D
and c
3D
are dimensionspecific constants. Equation (6)
predicts the steeper and superlinear scaling that is empirically
observed in 3D for consumption rate (Fig. 3a, b and Table 1). Note
that the scaling of consumption rate, c, closely matches the scaling of
search rate, a(compare equations (3) and (6)). The existing small
difference arises because of the weak scaling of the product (x
R
m
R
)
of resource density and mass with consumer mass (Table 1 and
Supplementary Information).
When resources are unlimited (x
R
R‘), the term s0DvrdD{1xRmR
dominates both the numerator and denominator of equation (4),
resulting in a value of 1. Consequently, search and detection become
instantaneous, and consumption rate depends only on massspecific
handling rate (1/t
h
) (Fig. 2):
c~t{1
h,0mb
Cð7Þ
where bis the scaling exponent of the consumer’s wholebody
metabolic rate and t
h,0
is a bodytemperature and metabolicstate
dependent constant. We find that massspecific handling time, t
h
,
scales as 1.1 60.07 in 3D and 1.02 60.08 in 2D (Supplementary
Information). However, the observed consumptionrate scaling in 2D
is 0.85 60.05, and is 1.00 60.06 in 3D, both closer to predictions for
scarce rather than unlimited resources (Table 1). Therefore, even
when functional responses seem to saturate and resources are
considered abundant, consumption rate does not scale like handling
time, and must therefore continue to be constrained by search
dimensionality. This also explains why most previous studies have
reported 0.75 power scaling of consumption rate
7,8,19
. The data in
these previous studies are actually maximal ingestion rates collected
from sedentary individuals that are provided with unlimited
resources
7,8,19
. Our data, for both scarce and abundant resources, are
more representative of field conditions because they are extracted
from functional response data.
Although our theory predicts that a
3D
and c
3D
are larger than a
2D
and c
2D
, respectively (Supplementary Information), the magnitude of
the observed difference is much larger than predicted (Fig. 3). One
explanation is that most 3D interactions are aquatic, and most 2D
interactions are terrestrial. The energetic cost for swimming is about
ten times lower than for running
18,19
, probably increasing encounter
rates for nondirected movement. This difference could elevate the
intercept (but not exponent), contributing to the observed ten times
larger baseline consumption rates in 3D. Nevertheless, 2D aquatic and
2D terrestrial interactions scale similarly (Fig. 3a–c), indicating other
differences between pelagic (3D) and benthic (2D) aquatic zones, and
highlighting the need for further study.
Dimensionality and trophic interaction strengths
By deriving the scaling of search rate (a), a fundamental parameter in
consumer–resource and foodweb models, we have provided a mech
anistic basis for linking interaction dimensionality with trophic inter
action strengths, which are proportional to ax
R
m
R
/m
C
(refs 11, 13, 15,
16, 28, 29). In contrast to current theories, our results show that
scaling of trophic interaction strength can deviate substantially from
m{0:25
C. Specifically, if resource size (m
R
) and resource density (x
R
)
are decoupled from consumer size, consumption rate scales like
search rate (equation (3)), and thus interaction strength scales as
axRmR=mC!m{0:32
Cin 2D, and m0:05
Cin 3D. Even when m
R
and x
R
scale with consumer mass (Table 1 and Supplementary Fig. 2), trophic
interaction strengths scale as m{0:15
C(2D) or m0:06
C(3D) when
resources are scarce, and as m{0:15
C(2D) or m0
C(3D) when resources
are abundant. This variation in the scaling of trophic interaction
strengths implies that consumer–resource dynamics are likely to be
constrained by interaction dimensionality.
Implications for population dynamics
By incorporating our scaling equations for a(equation (3)) into a
population dynamics model (Methods), we now show that dimen
sionality can affect populations in three fundamental ways. First, 3D
interactions allow a larger range of viable consumer–resource bodysize
Table 1

Empirical and predicted scaling exponents of consumption rate and its components with interaction dimensionality (D)
DSearch and consumption rate (n 5255) Consumptionrate components
Search rate
(scarce resources)
Consumption rate Relative velocity
(n521)
Reaction distance
(n539)
Handling time
(n578)
Resource mass
(n5255)
Resource density
(n5255)
Scarce resources Abundant resources
2D 0.68 60.12*(0.63) 0.85 60.05 (0.78) 0.85 60.05 (0.78) 0.26 60.04*(0.27) 0.21 60.08 (0.3 3) 21.0260.08 (20.75) 0.73 60.10 20.79 60.08
3D 1.05 60.08*(1.03) 1.06 60.06 (1.16) 1.00 60.06 (1.16) 0.26 60.04*(0.27) 0.20 60.06 (0.3 3) 21.1 60.07 (20.75) 0.92 60.08 20.86 60.07
For search and consumption rate, if the 3D exponent is significantly larger than 2D as predicted (likelihood ratio test), both are shown in bold. There are no predicted exponents for resource mass and resource
density scaling because they depend upon experimental design (Supplementary Information). Steeper than predicted exponents of handling time may arise because pursuit and subjugation scale with maximal
rather than resting metabolic rate
8,21
.
*Empirical exponent is statistically indistinguishable (P50.05 for all significance tests) from the predicted value (in parentheses).
ARTICLE RESEARCH
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©2012
combinations than in 2D, primarily because 3D consumption rates
scale more steeply and have higher baseline values. Depending upon
baseline carrying capacity (K
0
, defined as maximal biomass density for
a 1 kg organism; Supplementary Information), the majority of 2,929
species pairs from seven communities fall within our predicted
coexistence domains (Fig. 4a), with upper and lower limits of observed
size ratios closely matching predicted extinction boundaries. In 2D,
when K
0
ranges from 0.01 to 1 (kg
0.75
m
2
) the predicted coexistence
domains contain 88.8% to 99.8% of the empirical data. In 3D, when K
0
ranges from 3 to 300 (kg
0.75
m
3
), 74.3% to 99.8% of the data are within
the predicted domain (we explain below why carrying capacity is
typically higher in 3D than 2D). Thus, interaction dimensionality
may explain why consumer–resource interactions with larger size
ratios (for example, filter feeding
30
) and larger consumers are more
common in pelagic environments compared to benthic or terrestrial
environments
1,8,31
(Fig. 4a).
Second, because strong trophic interactions can destabilize com
munities
15,16, 28,29
, communities dominated by 3D interactions (for
example, pelagic or aerial habitats) may be inherently unstable. Indeed,
we find that persistent consumer–resource boom–bust dynamics are
more likely in 3D than in 2D (Fig. 4b and Supplementary Fig. 3). In
nature, these instabilities may be partly offset by larger regions of
coexistence that are possible in 3D (Fig. 4a) or by negative consumer
density dependence
3,24
. Nevertheless, our results are consistent with
empirical observations that pelagic communities appear less stable
than terrestrial communities
5
. They also suggest that 3D aquatic eco
systems may experience more frequent topdown regulation than 2D
terrestrial ecosystems
32,33
.
Third, we predict that population densities across consumer–
resource pairs scale with body size more steeply in 3D (exponent of
–1.12) than 2D (exponent of –0.76) (Fig. 4c). Only 2D scaling matches
Damuth’s 20.75 rule, which was derived from data on terrestrial
mammals (that is, 2D consumers)
14,34
. Thus, for a given carrying
capacity (maximal abundance of resources), steeper size–abundance
scaling of consumers in 3D habitats relative to 2D habitats should be
expected, and this helps to explain deviations from Damuth’s rule in
local communities
6,14,34–36
.
In our population model, we assume resource carrying capacities
scale with a 0.75 exponent (Supplementary Information), as expected
when food supply to resources is unlimited (equation (7))
26
. For
example, maximal abundance of primary producers in 2D (for
example, terrestrial plants) and 3D (for example, pelagic phytoplankton)
should scale as metabolic rate (that is, Damuth’s rule) irrespective of
dimensionality, which is well supported empirically
6,8,37,38
. Future studies
should incorporate potential differences in scaling of carrying capacity
across trophic levels. We also assume higher baseline carrying capacities
(K
0
)in3Dthan2D(Fig.4a)becausepelagic(3D)phytoplanktonhave
2–3 orders of magnitude higher turnover rates than terrestrial plants and
form a less variable and more nutritious autotroph base than plants in
2D terrestrial ecosystems such as grasslands
6,32,39
. This is an important
difference between habitats because it helps to explain the potential
advantage of 3D interactions. If resources had the same numbers
(but not densities) in 2D and 3D habitats (for example, 1 kg m
22
and
1kgm
23
), resources would probably be too sparse for a 3D search space
to be advantageous.
The consequences of interaction dimensionality for population
dynamics may also be mediated by other abiotic differences between
aquatic and terrestrial habitats. For example, 2D habitats such as
benthic zones may have a greater potential for prey refuges than 3D
habitats such as pelagic zones. Structural complexity reduces con
sumer search rates, potentially resulting in type III functional res
ponses instead of type I or II (refs 30, 40). We find no significant
propensity for type III functional responses in 2D relative to 3D in our
data set (Supplementary Information), probably because laboratory
experiments typically use habitats that are simpler than real habitats.
Even if type III responses are more common in 2D, results for the effects
of dimensionality on consumer–resource population dynamics remain
qualitatively unchanged (Supplementary Information). Nevertheless,
an important future direction will be to understand how habitat com
plexity affects search and consumption rates. Synthesizing our model
with previous work on fractal dimensionality of resource disper
sion
3,22,25
should be an important step in this direction. Perception of
structural complexity also scales with body size
3
.Grasslandsmaybe
structurally simple for a bison, but complex for a nematode.
Conclusion
Our study provides new and more accurate scaling relationships for
consumer–resource interactions
11,16,31
, gives novel insights into con
sumer–resource dynamics, and offers a mechanistic model that incor
porates dimensionality and foraging strategy into foodweb dynamics.
Our results help to explain why aquatic environments generally show
higher energy fluxes and lower stabilitythan terrestrial environments
5
,
why they often show inverted biomass pyramids
5,32
, and why larger
consumers have a relative advantage in pelagic (3D) versus terrestrial
(2D) environments
1,6
. Predicting strengths of pairwise trophic inter
actions is key to understanding higherorder effects, including indirect
interactions and polyphagy
5,28,29
. Our model for pairwise interactions
should provide a starting point for studying how the effects of
dimensionality propagate through entire community foodwebs.
Studying communities with mixtures of 2D and 3D interactions will
3D
b
3D exponent = –1.12
–10 –5 05
0
5
10
15
3
4
c
log
10
(Consumer mass)
log
10
(Resource mass)
log
10
(Body mass)
1
Fixed
point
Stable
limit cycle
2D 3D
2
34
Resource abundance
−9 –4 1
Extinction
Extinction
–14
–9
–4
1
6
Extinction
2D
1
2
Consumer
Resource
Consumer abundance
log
10
(Abundance)
–14
a
Extinction
6
High densityLow density
–9 –4 1–14 6
2D exponent = –0.76
Damuth's rule = –0.75
/
K
0
= 3
K0 = 30
K0 = 300
K
0
=
0
.
0
1
K
0
=
0.1
K
0
=
1
/
Figure 4

Effects of interaction dimensionality on consumer–
resource dynamics. a, Intensity map of logarithm of total consumer–resource
equilibrium densities, ranging from coexistence at high (dark) to low (yellow)
densities, or extinction (white). Black dots are real 2D (n51,627) and 3D
(n51,302) consumer–resource pairs (Supplementary Table 8). Consumer and
resource sizes are equal along the diagonal line. Lower extinction boundaries
(dashed lines) correspond to different baseline carrying capacities (K
0
); the
outermost boundary corresponds to empirical estimates. Predicted 2D
coexistence regions that lack observed species pairs probably represent under
sampling of interactions for the smallest consumers (for example, micro
predators) and largest consumers (for example, large mammalian
herbivores)
31
.b, Comparison of population dynamics of two 2D (1 and 2 in
a) and two 3D (3 and 4 in a) species pairs. c, Scaling of equilibrium abundance
across all 3D (blue) and 2D (red) consumer–resource pairs plotted in a.The
variation and discreteappearance of the data arises mainly because a consumer
may feed on multiple resource species of different sizes and vice versa.
RESEARCH ARTICLE
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©2012
be particularly revealing in this context. We conclude that interaction
dimensionality is a critical factor driving consumer–resource
dynamics. A better understanding of the effects of dimensionality will
lead to better predictions of foodweb and ecosystem dynamics, and
how these complex systems might respond to environmental change.
METHODS SUMMARY
Functional response data were compiled from the literature (Supplementary
Table 5). Interaction dimensionality was assigned according to consumer search
space (Fig. 1). The minimum resource density in each study was classified as
scarce, and the density corresponding to the maximum consumption rate was
classified as abundant. The search rate (a) in each functional response was cal
culated at each scarce density by dividing the associated consumption rate (c)by
the associated density. The scaling of ais our fundamental theoretical result
(equation (2)) and is based on derived scalings for v
r
,dand t
h
. We verified
predicted scalings of these components by compiling an additional data set of
136 interactions between 157 taxa. To move from predicted scaling exponents of
a(equation (3)) to predictions for scaling exponents of c(equation (4)), we
calculated the scaling of resource number density (x
R
) and mass (m
R
) across
studies in the functional response database. All exponents were estimated using
ordinary least squares regression (OLS) of log trait value versus log body mass.
Major axis regression yields steeper exponents than OLS but does not qualita
tively alter our results. We also tested for robustness of predictions to realistic
variation in body velocity scaling. All data were standardized to 15 uC using the
Boltzmann–Arrhenius model
9,14
. For population dynamics we used the
Rosenzweig–MacArthur model for the rate of change in time, t, for the resource
(R5x
R
m
R
) and consumer (C5x
C
m
C
) biomass densities
13,26
:
dR
dt~rR 1{R
K
{a=mC
ðÞRC
1zthaR
dC
dt~ea=mC
ðÞ
RC
1zthaR{zC
Here, ris the resource’s intrinsic biomass production rate, Kis resource’s biomass
carrying capacity, zis the consumer’s biomass loss rate, eis the consumer’s
biomass conversion efficiency, and t
h
is the resource massspecific handling time.
Size scaling for aand t
h
were based on our results, and that for r,zand Kwere
based on previous work
8,9,14
. We tested robustness of our results by varying model
structure between the RosenzweigMacArthur model and the Lotka–Volterra
predator–prey model, and also by using a type III instead of a type II functional
response.
Received 13 December 2011; accepted 3 April 2012.
Published online 30 May 2012.
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Supplementary Information is linked to the online version of the paper at
www.nature.com/nature.
Acknowledgements We thank the authors who contributed data (Supplementary
Tables 5–8), and P. Amarasekare, J. H. Brown, E. Economo, A. Mikheyev, C. Estrada,
C. Johnson, M. Johnson and K. Lafferty for helpful discussions and comments. S.P.,
A.I.D. and V.M.S. were supported by University of California, Los Angeles
Biomathematics startup funds andby the US National Science Foundation Division of
Environmental Biology award 1021010.The data reported in thispaper are available in
the Supplementary Information online.
Author ContributionsS.P., A.I.D. and V.M.S. contributed equally to this work.All authors
discussed the results and commented on the manuscript.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. The authors declare no competing financial interests.
Readers are welcome to comment on the online version of this article at
www.nature.com/nature. Correspondence and requests for materials should be
addressed to S.P. (samraat@ucla.edu).
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