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Edited by: Mary I. O'Connor, University of British Columbia, Canada

Reviewed by: Matthew Barbour, University of Zurich, Switzerland; Nelson Valdivia, Universidad Austral de Chile, Chile

This article was submitted to Biogeography and Macroecology, a section of the journal Frontiers in Ecology and Evolution

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Understanding constraints on consumer-resource body size-ratios is fundamentally important from both ecological and evolutionary perspectives. By analyzing data on 4,685 consumer-resource interactions from nine ecological communities, we show that in spatially complex environments—where consumers can forage in both two (2

For at least a century, biologists have wondered why “

Several studies have developed mathematical models to understand how body size determines the feasibility of consumer-resource size-ratios in specific taxa and trophic interaction types (e.g., McArdle and Lawton,

Arguably, the key to a more nuanced understanding of variation in community size-ratios is to incorporate community- and environment-specific biomechanical constraints into models of consumer-resource interactions (Vucic-Pestic et al.,

An illustration of components of consumption rate and environmental constraints on them. The parameters shown belong to our model for size-mediated consumer-resource dynamics. Feasible body size-ratios depend on consumer and resource body velocities (v_{R} and v_{C}), reaction distance (

We develop a mathematical model to predict the feasibility of community-wide resource to consumer size-ratios. To this end, we first incorporate body size constraints on components of consumer-resource interactions—relative velocity, detection distance, attack success, and handling time (

We begin with a general equation for consumption rate ^{−1}) (Pawar et al.,

Here, _{R} is resource number density (individuals × m^{−2} or m^{−3}), _{R} is resource body mass, ^{2} or m^{3} × s^{−1}),

where

We now define size-dependence of the components of

Here, _{0} is a constant that includes effects of temperature and dimensionality, _{v} is the scaling exponent for consumer body velocity, _{d} is the scaling exponent for reaction distance between consumer and resource, _{R}/_{C} is body size-ratio. We emphasize that this simple definition of interaction dimensionality arises because resource detection typically occurs in Euclidean space, regardless of which sensory modality is used. Later, we discuss how our model can be extended to more complex definitions of dimensionality by considering non-sensory components (such as relative velocity) of consumer-resource interactions. As such, Equation (3) is a scaling model for grazing (i.e., consumer searching for sessile resources) but also well-approximates the scaling of search rate in active-capture interactions (i.e., both consumer and resource moving actively across the landscape) when _{C} > _{R} (_{C} > _{R} (

Next, for attack success probability

where γ is a constant that governs the decrease in attack success as resources get very large relative to consumer size (_{R} ≫ _{C}). The exponent γ in Equation (4) captures biomechanical constraints that appear at upper size-ratios (McArdle and Lawton,

Substituting Equations (3, 4) into (1) and rearranging to gives the scaling of per-capita (biomass) consumption rate:

Note that here the resource mass term _{R} from Equation (1) has been absorbed into the size-ratio term. This equation captures four essential features of consumption rate:

For a given resource size and therefore size-ratio _{C} because larger consumers have greater body velocity,

Consumption rate _{R} < _{C} (i.e., _{R} due to increasing reaction distance (and for active-capture, also increasing relative velocity;

When resource mass far exceeds consumer mass (^{γ})^{−1} term. That is, the product of per-capita search rate (a monotonically increasing function with respect to size and size-ratio; Equation 3) and attack success probability

Consumption rate _{C} and size-ratio ^{−3}) allows higher detection probability than 2^{−2}) (Pawar et al.,

Finally, for handling time we use another empirically well-supported model (Pawar et al.,

where _{0} is a constant and β_{h} is the scaling of the metabolic rate of a consumer during pursuit, subjugation, and ingestion of resources.

We first derive feasible ranges of size-ratios that meet consumer energy requirements for somatic maintenance, by setting a lower bound on energy gain from resource consumption (Carbone et al.,

Here, _{C} is the rate of the consumer's energy use converted to mass units (kg/s) while resting (resting metabolic rate, RMR), _{c} to mass units (like the quantity ^{6} J (the combustion energy content per unit of wet biomass) (Peters, _{C} is RMR, which is an underestimate of maintenance energy needs because it typically does not include the energy required for somatic growth, producing offspring, storage, and bursts of activity (such as during foraging). These may cause significant additions to the energy needs of adult animals in certain periods of their lifetime (Rizzuto et al.,

We already have the size scaling of _{C} and biomass abundance _{R} _{R}. For _{C}, we use the scaling of basal or resting metabolic rate (Peters,

where _{0} is a constant that includes the effect of temperature and converts metabolic rate units (J/s) to mass use rate units, and β is the scaling exponent of metabolic rate. For biomass abundance we use (Peters,

Where _{0} is a normalization constant that includes the effect of temperature, and β_{x} is the scaling exponent of numerical abundance. Substituting the scaling (Equations 5, 6, 8, 9) into (7) and solving for _{R} gives the bounds on resource mass _{R} and therefore size-ratios that guarantee a balanced energy budget. To obtain an exact solution for this we set _{R}, which gives:

Where

where _{0} = (_{0}/_{0}_{0})^{2.22} in 2_{0}/_{0}_{0})^{1.54} in 3

The smaller _{C} and

Within either 2_{0}) through the term _{0}. In particular, following empirical data (Peters, _{0}) is about two orders of magnitude higher in 3

The upper bound on size ratios (where _{R} > _{C} so

Predicted effects of interaction dimensionality on consumer energetic feasibility and consumer-resource coexistence _{0} (because baseline abundances tend to be higher by orders of magnitude in 3_{10} number density (darker means more abundant). These results are for _{0} = 10^{4} s (Equation 6) and γ = 2 (Equation 4; cf. _{R} ≫ _{C}, possibly due to decrease in gape-limitation, coexistence becomes possible at those extreme ratios, illustrated by the dotted γ = 1 line (at _{0} = 1, with other parameter values remaining the same) in the 2

The above theory based upon the consumer's energetic considerations does not account for consumer-resource population dynamics. Therefore, we consider whether accounting for population dynamics changes our predictions about the effect of dimensionality on feasible size-ratios. Using a general consumer-resource model, in

where _{x}. That is, the above predictions (i)–(iv) from the energetic model also hold for the population dynamics model. We also show that local asymptotic stability to small perturbations around equilibrium abundances of consumer and resource (Equations S8–S9) differs between 2

Next, we calculated community-specific predictions about the magnitude of difference in central tendency of 2_{10}-transformed 2_{10}-transformed) size ratio of each local community as a measure of central tendency because most communities exhibit skewed and multimodal size-ratio distributions (

Effect of interaction dimensionality on species' log_{10} size-ratio and size (mass, kg) distributions across communities. All pairs of 2_{10} size in 2_{R≪}m_{C}; ^{−10}) in the Eastern Weddell Sea, and Deer Flies on Roe Deer (m_{R≫}m_{C}; ^{6.5}) in Grand Cariçaie Marsh.

To study size-ratio distributions and test our theoretical predictions we compiled published data on interacting consumer-resource pairs for nine communities (four terrestrial, five aquatic;

We tested whether, as predicted by our theory, 2_{10}) size-ratio distributions are often right-skewed with long tails and/or multi-modal (^{5} lists of random 2_{10}-transformed size-ratios (3^{5} random lists. The distribution of these 10^{5} differences is an approximation of the sampling distribution of differences assuming random partitioning of the community into 2

As an even more stringent test in the face of non-independent size-ratios, we also re-analyzed the data for differences between 2

Finally, to determine whether size ranges [m_{C}_{, min}, _{C}_{, max}] and [_{R}_{, min}, _{R}_{, max}] are influenced by factors independent of dimensionality, such as oxygen limitation, physical medium for locomotion, and phylogenetic history (Allen et al.,

We find strong and statistically significant empirical evidence that median 3

Differences between 2

_{10} |
_{10} |
||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

All communities | −1.07 | −3.59 | −2.52^{*} (−2.60) |
−4.79 | −4.05 | 964 | 704 | 463 | 4685 | 3055 | 1630 | 0.09 | 0.20 |

Eastern Weddell Sea | −1.37 | −3.65 | −2.28^{*} (−2.58) |
−2.78 | −2.51 | 314 | 137 | 270 | 979 | 258 | 721 | 0.11 | 0.30 |

Estero de Punta Banda | −2.81 | −4.60 | −1.79^{*} (−1.13) |
−2.48 | −2.73 | 105 | 102 | 47 | 1388 | 1086 | 302 | 0.34 | 0.19 |

Grand Cariçaie Marsh | −0.34 | −1.14 | −0.80^{*} (−1.36) |
−5.55 | −5.44 | 88 | 86 | 45 | 623 | 460 | 163 | 0.00 | 0.46 |

Scotch Broom | −0.28 | −0.91 | −0.63^{*} (−1.01) |
−5.44 | −5.28 | 150 | 147 | 11 | 362 | 347 | 15 | 0.00 | 0.22 |

Skipwith Pond | −0.71 | −3.01 | −2.3^{*} (−0.47) |
−4.69 | −4.55 | 33 | 31 | 17 | 321 | 284 | 37 | 0.78 | 0.23 |

Broadstone Stream | −1.09 | – | – | −6.71 | – | 28 | 28 | 0 | 138 | 138 | 0 | – | – |

Gearagh Woodland | −0.46 | – | – | −5.56 | – | 113 | 113 | 0 | 370 | 370 | 0 | – | – |

UK Grasslands | 0.22 | – | – | −5.40 | – | 61 | 61 | 0 | 112 | 112 | 0 | – | – |

_{10}(Size-ratio) column shows observed medians of log_{10} transformed size-ratios, and their observed and predicted (in parentheses) difference in medians (3D−2D). All observed and predicted differences are significantly different from 0 (p < 0.05; flagged with an asterisk) based upon a randomization test (see main text). Note that although median 2D and 3D size-ratios are significantly different in each community, median 2D, and 3D consumer and resource sizes are not (p > 0.05; Wilcoxon–Mann–Whitney test with shared taxa removed). The 2D/3D overlap column shows proportion of consumers in each community feeding on both 2D and 3D resources (Jaccard index) (Con), and proportion of resources exploited by both 2D and 3D consumers (Res). If such an overlap exists, the total number of taxa (Taxa-All) within a community will be smaller than the sum of 2D and 3D taxa

We also found multimodalities in 2_{R} ≪ _{C}) and another at extremely large ratios (m_{C} ≪ _{R}). The lower 2_{R} ≪ _{C}) found in several communities corresponds to grazing. Scotch Broom, UK Grasslands, and Estero de Punta Banda also each have a peak at very high 2_{R} ≫ _{C}), corresponding to macroparasites, parasitoids, herbivores, and micropredators. Indeed, these types of interactions are why only 87.8% of 2_{0} case in _{R} ≫ _{C}) (

Eastern Weddell Sea, Grand Cariçaie Marsh, and Scotch Broom also show a secondary 2_{R} ≪ _{C}) (_{R} ≪ _{C} (

In communities that have both interaction dimensionalities, median body size distributions of species in 2

By combining theory with extensive empirical data, we have shown that interaction dimensionality strongly constrains resource-to-consumer size ratios in ecological communities. Specifically, 3

Our results provide an explanation for three important empirical patterns in the body size structure of communities. First, our theory predicts that wider ranges of size-ratios become feasible as consumer size increases in both 2

Our theory also predicts that irrespective of dimensionality, size-ratios will be smaller in magnitude (closer to ^{2} and 1 kg/m^{3}), irrespective of whether abundance was high or low.

Hairston and Hairston (

Our theory can partly explain multimodalities found in 2_{R} ≪ _{C} corresponds to grazing. Our theory predicts that grazing allows a wider range of size-ratios (_{R} ≫ _{C}), we are able to explain the 2_{R} ≫ _{C} (corresponding to macroparasitism, parasitoidism, herbivory and micropredation) seen in several communities. This is also consistent with the fact that the empirical data on consumption rates we used to obtain estimates of γ are only from predator-prey interactions, not macroparasitism, parasitoidism, herbivory or micropredation. At the same time, we did not find multimodalities in 3

Empirical biases also need to be considered while interpreting our results. For example, the fact that no observed species pairs lie in the predicted feasible regions at smallest and largest consumer sizes in 2

Our theoretical analysis assumes that the criteria for energy balance and stable coexistence of two-species systems approximately hold even when these pairwise interactions are embedded in food webs. We are encouraged by the fact that we are able to correctly predict the differences between median 2

Our classification of interactions according to dimensionality of the search and interaction space is appealingly simple, and necessarily so because detection typically occurs in Euclidean space (McGill and Mittelbach,

In conclusion, our study helps explain a number of empirical observations in which community size structure varies with habitat, type of consumer-resource interaction, and foraging strategy (Elton,

SP and VS developed the theory. SP and TL conducted the theoretical and numerical analyses. SP and AD collected empirical data for the main analysis and meta-analysis. SP, AD, and DW analyzed the empirical data. SP, AD, TL, DW, and VS wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank P. Amarasekare, C. Johnson, and the reviewers for helpful discussions and comments.

The Supplementary Material for this article can be found online at: