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Dimensionality of consumer seach space drives tropic interaction strengths

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Dimensionality of consumer seach space drives tropic interaction strengths

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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 consumption-rate 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 × 10(-14) 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 food-web 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.
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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 consumption-rate 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 food-web 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 per-capita 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 per-link trophic interaction strengths in food webs
10,11,13,15,16
.
Deviations from quarter-power 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, consumption-rate 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 non-metabolic 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 food-web 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 90095-1766, 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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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 (one-sample F-test 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 1-kg 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 1-kg 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 active-capture 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 sit-and-wait 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
consumption-rate 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 consumption-rate
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 root-mean-
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-
sumption-rate 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. ad, Scaling of per-capita 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).
Consumption-rate 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 sit-and-wait 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 dimension-specific 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 per-capita bio-
mass consumption rate (c) (Holling’s type II functional response
27
)in
terms of spatial dimension (D):
c~amRxR
1zthamRxR
~sDvrdD{1mRxR
1zthsDvrdD{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 dimension-specific 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 mass-specific
handling rate (1/t
h
) (Fig. 2):
c~t{1
h,0mb
Cð7Þ
where bis the scaling exponent of the consumer’s whole-body
metabolic rate and t
h,0
is a body-temperature and metabolic-state-
dependent constant. We find that mass-specific handling time, t
h
,
scales as 1.1 60.07 in 3D and 1.02 60.08 in 2D (Supplementary
Information). However, the observed consumption-rate 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 non-directed 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 food-web 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 body-size
Table 1
|
Empirical and predicted scaling exponents of consumption rate and its components with interaction dimensionality (D)
DSearch and consumption rate (n 5255) Consumption-rate 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
28 JUNE 2012 | VOL 486 | NATURE | 487
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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 top-down 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 food-web 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 pair-wise trophic inter-
actions is key to understanding higher-order 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 food-webs.
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
488 | NATURE | VOL 486 | 28 JUNE 2012
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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 food-web 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 mass-specific 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 Rosenzweig-MacArthur 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 start-up 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).
ARTICLE RESEARCH
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Supplementary resource (1)

... Previous studies that have considered the surrounding medium have usually focused on specific aspects of predation or on specific taxa (Domenici et al., 2011), or have investigated one specific aspect of the medium such as dimensionality (Pawar et al., 2012(Pawar et al., , 2015 or habitat complexity , more rarely two factors simultaneously (Wasserman et al., 2016). But the overall role played by the surrounding medium acting on the predator-prey relationship, which drives the functional response, remains to be explored. ...
... Several studies have begun to investigate this promising avenue. For example, the dimensionality of the physical medium was shown to constrain predator-prey interactions since predators are expected to capture pelagic and flying prey more efficiently than benthic and terrestrial prey (Pawar et al., 2012). Extending this framework to predict pairwise trophic interactions in natural situations, Pawar et al. (2019) fall short of deriving the parameters of their functional response model from physical factors other than dimensionality. ...
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First derivations of the functional response were mechanistic, but subsequent uses of these functions tended to be phenomenological. Further understanding of the mechanisms underpinning predator-prey relationships might lead to novel insights into functional response in natural systems. Because recent consideration of the physical properties of the environment has improved our understanding of predator-prey interactions, we advocate the use of physics-based approaches for the derivation of the functional response from first principles. These physical factors affect the functional response by constraining the ability of both predators and prey to move according to their size. A physics-based derivation of the functional response should thus consider the movement of organisms in relation to their physical environment. One recent article presents a model along these criteria. As an initial validation of our claim, we use a slightly modified version of this model to derive the classical parameters of the functional response (i.e., attack rate and handling time) of aquatic organisms, as affected by body size, buoyancy, water density and viscosity. We compared the predictions to relevant data. Our model provided good fit for most parameters, but failed to predict handling time. Remarkably, this is the only parameter whose derivation did not rely on physical principles. Parameters in the model were not estimated from observational data. Hence, systematic discrepancies between predictions and real data point immediately to errors in the model. An added benefit to functional response derivation from physical principles is thus to provide easy ways to validate or falsify hypotheses about predator-prey relationships.
... Interactions between predator and prey in natural communities are difficult to describe quantitatively. Various mathematical models have been used to quantify prey acquisition by the predator and to investigate the nature and the strength of species interactions within food webs (Abrams et al., 1998;Baudrot et al., 2016;Chan et al., 2017;Pawar et al., 2012). Several studies have compared how a predator acquisition rate varies with prey density using statistical approaches (reviewed by Novak and Stouffer 2020), but few of them explicitly tackled the underlying mechanisms. ...
... A potential encounter occurs between the predator and a prey item i when the predator is at a distance (d i ), being defined as the maximum distance at which the predator can detect a prey item i (in 2D, detection region = 2d i ; Pawar et al., 2012). ...
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How can geckoes walk on the ceiling and basilisk lizards run over water? What are the aerodynamic effects that enable small insects to fly? What are the relative merits of squids' jet-propelled swimming and fishes' tail-powered swimming? Why do horses change gait as they increase speed? What determines our own vertical leap? Recent technical advances have greatly increased researchers' ability to answer these questions with certainty and in detail. This text provides an up-to-date overview of how animals run, walk, jump, crawl, swim, soar, hover, and fly. Excluding only the tiny creatures that use cilia, it covers all animals that power their movements with muscle--from roundworms to whales, clams to elephants, and gnats to albatrosses. The introduction sets out the general rules governing all modes of animal locomotion and considers the performance criteria--such as speed, endurance, and economy--that have shaped their selection. It introduces energetics and optimality as basic principles. The text then tackles each of the major modes by which animals move on land, in water, and through air. It explains the mechanisms involved and the physical and biological forces shaping those mechanisms, paying particular attention to energy costs. Focusing on general principles but extensively discussing a wide variety of individual cases, this is a superb synthesis of current knowledge about animal locomotion. It will be enormously useful to advanced undergraduates, graduate students, and a range of professional biologists, physicists, and engineers.
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Question: Can we develop simple allometric relationships based on predator and prey body size to more easily parameterize optimal foraging models and thereby make them more useful to community ecologists interested in studying species interactions? Model: The rate at which a predator encounters its prey is often the most difficult parameter to estimate in any foraging model. We develop a simple geometric model to predict prey encounter rates as a function of predator mass, prey mass, and prey density using allometric relationships between predator search velocity and vision as a function of body size. Empirical test: We suggest that the model has both strategic and tactical uses. Tests geared towards both uses are performed and these tests validate the model within the limits of existing data. Conclusions: It appears possible to parameterize optimal foraging models through easily measured variables such as body size. This provides hope that Lotka-Voltera style community matrix models could be replaced with more mechanistic models based on optimal foraging that are easy to parameterize for an entire community. If so, this research agenda holds promise for developing the link between foraging models and species interactions that the original inventors of optimal foraging theory envisioned.
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The size structure of aquatic communities is generally measured using size spectra, an approach which is tedious or inapplicable in benthic and terrestrial communities. This has inhibited comparison of size structure of aquatic and terrestrial communities. This study uses an approach more common among terrestrial ecologists to develop a general density-body size relationship for lacustrine communities, based on mean annual population densities for dominant species of phytoplankton, zooplankton, zoobenthos and fish measured in 18 lakes worldwide. Overall, mean annual population density (D, individuals m-2) decreases log-linearly with increasing species body size (M, μg fresh mass) as D = 4 x 105 · M-0.89 (n = 280, r2 = 0.92), although the exponent appeared smaller (-0.55 ± 0.04) within broad taxonomic groups (algae, invertebrates). We found that density-body size relationships for dominant species are quantitatively similar to size spectra, a pattern which suggests that density-body size relationships may provide an interesting alternative to size spectra for the prediction of ecosystem processes. These relationships also suggest that aquatic species reach, on average, 6-60 times higher densities than terrestrial species, depending on their body size and on their thermoregulatory system (ectotherms vs endotherms). The implications of these differences in size structure for size-related patterns of energy use and other processes depend on which physiological groups (unicells, ectotherms, endotherms) are being compared.
Article
SUMMARY Carbon stocks and flows give a picture of marine and continental biotas different from that based on food webs. Measured per unit of volun1e or per unit: of surface area, biomass is thousands to hundreds of thousands of times more dilute in the oceans than on the continents. The number of described species is lower for the oceans than for the continents. One might expect that each species of organism would therefore feed on or be consumed by fewer other species in the oceans than on the continents. Yet in reported food webs, the average oceanic species interacts trophically with more other species than the average terrestrial or aquatic species. Carbon turnover times imply that the mean adult body length of oceanic organisms is 240 to 730 times shorter than that of continental organisms. By contrast, in reported food webs, marine anirnal predators are larger than continental animal predators, and marine animal prey are larger than continental animal prey, by as much as one to two orders of magnitude. Estimates of net primary productivity (NPP) per unit of surface area or per unit of occupied volume indicate that the oceans are several to hundreds of times less productive than the continents, on average. If NPP limited mean chain length in food webs, oceanic food chains should be shorter than continental chains. Yet average chain lengths reported in published food webs are longer in oceans than on land or in fresh water. In reconciling these unexpected contrasts, the challenge is to determine which (if any) of the many plausible explanations is or are correct.
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The Late Cretaceous Oldman Formation comprises sediments that were deposited along the margin of a great inland sea that covered much of the western interior of North America. The environment of deposition appears to have been tracts of fluvial marshes that separated "islands" of higher, drier ground. The climate was probably warm-temperate, and it is suggested that upland plant communities were parkland-like in aspect. The large dinosaurs of this community comprised animals that were between a hippopotamus and a large African elephant in adult weight. Some workers have suggested that dinosaurs had metabolic rates comparable to those of living birds or mammals. By extrapolating from the food consumption rates of these living endotherms it is possible to obtain crude estimates of the ingestion rates of endothermic dinosaurs. Similar extrapolations from the ingestion rates of living reptiles and amphibians provide estimates of the ingestion rates of ectothermic dinosaurs. By deriving an empirical equation relating the ratio of annual secondary productivity/average annual biomass to adult weight in living mammals, and employing estimates of adult weight and biomass for the herbivorous dinosaur populations, it is possible to estimate the annual secondary production of endothermic Oldman herbivorous dinosaurs. If the body weight vs. production/biomass relation derived for mammals can be applied to ectothermic tetrapods, it is possible to estimate annual secondary production of ectothermic dinosaur populations. These calculations suggest that the annual secondary production of endothermic herbivorous dinosaurs would have been insufficient to meet the food requirements of an endothermic carnivorous dinosaur population as large as is preserved in the Oldman Formation. However, ectothermic carnivorous dinosaurs would have been easily able to make energetic ends meet. Unfortunately, the situation is complicated by the possibility that carnivores are overrepresented in collection from the Oldman. Because of this, I cannot presently decide between ectothermy and endothermy in dinosaurs on the basis of methods presented in this paper. Alternative methods that may be more successful in this regard are discussed. It is hoped that as paleontologists collect fossils from an ecological point of view the methods presented in this paper can be employed to make realistic statements about the trophic dynamics of ancient vertebrate communities.