Emergence of stable motifs in consumer-resource communities

Understanding how and why complex communities can be stable has preoccupied ecologists for over a century. Data show that real communities tend to exhibit characteristic motifs and topologies. Despite a large body of theory investigating both ecological (niche partitioning) and evolutionary (speciation and extinction) mechanisms, a general explanation for why particular motifs are more common than others remains elusive. Here we develop a mechanistic framework that investigates the set of possible motifs that can emerge under minimal conditions of a nutrient-limited system with no external inputs, and no spatial heterogeneity. Focusing on consumer-resource communities structured by competition and predation, we find that the emergent motifs under these minimal conditions are vertical trophic chains that maximize energy transfer and biomass production. Not only are such motifs stable to perturbations of species’ abundances, but they are also robust to species additions and removals. Our findings provide a mechanistic explanation for why tri-trophic chains are overrepresented in real food webs. They suggest that, because they maximize energy transfer, and can emerge and persist under minimal conditions, vertical trophic chains may constitute the fundamental architecture of consumer-resource communities.

As might be expected, given the diversity of approaches, these studies report conflicting 74 findings. Rule-based structural models suggest that omnivory is unlikely to be a stable motif (e.g., Johnson et al. (2014)), while dynamical models suggest it to be strongly stabilizing (e.g., 76 McCann (1997); McCann et al. (1998). Some structural models find apparent competition to be overrepresented (e.g., Bascompte and Melian (2005)), while others find it to be underrepresented 78 (e.g., Camacho et al. (2007)). Random matrix approaches that assume stable point equilibria generated by self-limitation in all species find consumer-resource communities to be more stable 80 than mutualistic communities (e.g., Allesina and Tang (2015)). In contrast, dynamical models that explicitly consider the oscillatory nature of consumer-resource interactions find consumer-82 resource interactions to be highly unstable (extinction-prone) and that weak horizontal links (e.g., competitive interactions) are required to reduce the oscillatory tendency and increase persistence  2019)). Some studies find consumer-resource communities to be compartmentalized (e.g., Stouffer and Bascompte (2011)), but others find them to be nested 90 (e.g., Kondoh et al. (2010)), or exhibiting a combination of the two topologies (e.g., Kondoh et al. (2010); Thébault and Fontaine (2010)). The diversity of approaches and outcomes has 92 been greatly beneficial in enhancing our understanding of how complex communities can be stable. Further progress, however, requires reconciling these differences to find common ground. anistic framework that combines the topological features of networks with biologically realistic dynamics of consumer-resource interactions. On the topological side we consider trophic interactions as feed-forward loops in which energy is transferred from primary producers to secondary 98 consumers and top predators. On the dynamical side we consider these interactions as feedback loops in which producers have positive effects on consumers, while consumers have negative ef-100 fects on producers. We use this framework to predict the types of network motifs that emerge under minimal conditions of a constant nutrient input, a single niche axis, no external inputs 102 of nutrients or species, no self-limitation other than that induced by nutrient limitation, and no spatial heterogeneity. Starting from the very basal level -nutrient uptake by primary produc-104 ers -we investigate which, if any, motifs emerge from the interplay between competition and predation, and whether they are robust to perturbations. 106

Conceptual framework
We use theory and data on transcription and signal transduction networks to generate hy-108 potheses about feasible network motifs in ecological communities. As noted above, transcription networks in unicellular organisms (e.g., yeast and E. coli; (Alon, 2007)) exhibit feed-forward 110 loops (FFLs) that resemble tri-trophic food chains or a closed loop with omnivory ( Fig. 1(a) and (b)). For instance, when the biomass of a primary producer exceeds the level at which a 112 secondary consumer (e.g., herbivore) can persist on it, it opens up the possibility of a primary producer-secondary consumer interaction. Similarly, when primary productivity is sufficiently 114 high to generate a secondary consumer biomass exceeding the level at which a tertiary consumer (e.g., predator) can persist, a primary producer-secondary consumer-tertiary consumer interac-116 tion can form, with the primary producer "controlling" the secondary consumer directly, and the tertiary consumer indirectly, through the provision of energy. The persistence of such loops are 118 enhanced when there is negative autoregulation (self-limitation) at one or more trophic levels.
When the tertiary consumer can derive energy directly from the primary producer, the latter 120 directly "controls" both consumer trophic levels (Fig. 1). Unicellular transcription networks also exhibit single input modules (SIM), which are akin to exploitative and apparent competition 122 motifs in ecological communities ( Fig. 1(c)). When species engage in both exploitative and apparent competition, we get the bi-parallel (diamond) motif found in signal transduction networks 124 ( Fig. 1(d)). When two primary producers each support two secondary consumers, we get the bi-fan motif found in neural networks.

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Of note, ecological networks are distinct from other biological networks in two ways. First, it is energy, rather than information, that is transferred through the network. Second, species 128 interactions constitute both feed-forward loops and feedback loops. Energy is transferred unidirectionally from primary producers to consumers, with species at lower trophic levels having a 130 positive effect on those at higher trophic levels. At the same time, by extracting energy through direct feeding or other means, species at higher trophic levels have a negative effect on those 132 at lower trophic levels. Similarly, species at the same trophic level compete to acquire energy from the lower trophic level, thus leading to mutually negative effects on one another. It is these 134 feedback loops that define the dynamical nature of consumer-resource communities, with species' abundances changing as a result of interactions within and between trophic levels.

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The next step is to make the connection between motifs and modularity. Networks that are locally cohesive, i.e., the fraction of the feasible edges (links) that occur around a given 138 node (molecules, species), exhibit a high degree of clustering (Watts and Strogatz, 1998). High clustering is a signature of modularity, a group of linked nodes whose collective action achieves a 140 particular function (Milo et al., 2002). In transcription networks, a module is a set of co-regulated genes that share a common function; in signaling pathways, a module is a chain of interacting 142 proteins propagating a signal within a cell (Alon, 2003(Alon, , 2007. In ecological communities, a module is a group of interacting species whose collective action (energy acquisition) leads to 144 production of biomass.
One could hypothesize that motifs such as FFLs, SIMs and bifans are common in unicellular 146 organisms because they represent the set of minimal modular configurations that can both emerge in a closed system and are robust to perturbations. If this is the case, the preponderance of tri-148 trophic chains, and to some extent omnivory, in ecological communities could be because these motifs constitute the feasible configurations that can both emerge in closed communities and are 150 robust to species invasions. Below we develop this hypothesis in more detail.
Consider a community with a constant nutrient input, in which the total nutrient availability 152 sets the upper limit to the total biomass, and hence the number of species the community can contain. The ways in which species apportion the available biomass determines the number of 154 species and the types of interactions that the community contains. The basal unit is a nutrientprimary producer interaction. In what follows we refer to the primary producer as a plant, but 156 the ideas we develop are general and can apply equally well to other primary producers such as phytoplankton. The plant species' growth and reproduction depends on an essential nutrient 158 (e.g., Nitrogen, Phosphorous), which it converts into biomass, and for which the individuals in the population compete. The plant will compete with other plant species for a common nutrient 160 pool, and be subject to attack by herbivores. These herbivores in turn are consumed by predators that can also be omnivorous (Fig. 1).

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A motif that emerges out of such an interaction has to satisfy two criteria. The first is its feasibility. The possible set of species interactions have to follow the order of nutrient and energy 164 flow, and trophic status. To give an obvious example, we cannot have a consumer without a basal resource, just as we cannot have a top predator without an intermediate consumer. The 166 second criterion is robustness. A stable motif is one that is both (i) stable to perturbations of its constituents' abundances, and (ii) cannot be replaced by another configuration, i.e., it is 168 robust to species additions or removals. Of note, perturbations may lead to substitutions of species occupying a particular position of the motif, but they will not alter its configuration. For 170 instance, if a tri-trophic food chain is a stable motif, a second herbivore can either invade and exclude the resident herbivore or be excluded by the latter; it cannot invade and coexist with the 172 resident herbivore species. We quantify stability in terms of permanence (long-term persistence of all interacting species), which also encompasses the notion of mathematical stability (return 174 to a non-trivial steady state following a perturbation of species' abundances).
We make two predictions. First, in a closed community with a constant energy input in 176 which the primary producer's growth depends on a single limiting nutrient, the emergent motifs are vertical chains (nutrient-plant-herbivore, nutrient-plant-herbivore-predator) with the single Second, because vertical chains exhibit high cohesiveness (i.e., the ratio of realized to allowable 182 links approaches 1), they also constitute modules that achieve the common function of biomass production. Modules also tend to localize perturbation impacts, thus increasing the community's 184 robustness to perturbations. Below we explain the rationale underlying these expectations.
We know from competition theory that, in the absence of local niche partitioning via multiple 186 limiting factors (Tilman, 1982;Chesson, 2000), environmental heterogeneity that allows for spatial or temporal niche partitioning (Chesson, 2000;Amarasekare, 2003), allochthonous nutrient 188 inputs, or dispersal that allows for species recolonizations or source-sink dynamics (Chase and Leibold, 2003;Leibold et al., 2004;Amarasekare, 2003), the species that reduces its resource to 190 the lowest level will exclude all other species (R ; Tilman (1982)). This is a mechanistic interpretation of the competitive exclusion principle that applies to producers and consumers alike.

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In the absence of ameliorating factors, there can be only as many species at any given trophic level as there are resources at the level below.

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The R rule means that, in a closed community with a single limiting factor, a second plant species cannot invade and coexist with the resident. Even in the case that the plant species 196 are active at different times and partition the nutrient supply in time, a herbivore entering the community, save in the unlikely event of identical preferences for both plant species, will cause 198 the exclusion of the plant species more susceptible to herbivory (P ; (Holt, 1977)). Hence, diversity can increase only through the addition of a vertical link to the initial nutrient-plant 200 interaction. This link can be either mutualistic (e.g., pollinator, seed disperser) or antagonistic (e.g., herbivore). Here we focus our attention to antagonistic interactions.

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The nutrient-plant community can be invaded by a herbivore if standing plant biomass exceeds that required for the herbivore to maintain itself. The R rule ensures that only a single species 204 can occupy the secondary consumer trophic level: the herbivore that reduces the plant biomass to a lower level will exclude all other invaders. Temporal partitioning may allow two herbivores to 206 coexist on the plant, but the arrival of a top predator will exclude the herbivore more susceptible to the predator (P rule). A second vertical link is therefore the most likely configuration in a 208 closed community. Omnivory (i.e., a species feeding on both plant and herbivore trophic levels) can convert the tri-trophic chain into a closed loop (Fig. 1). Below we formalize these predictions 210 mathematically.
The dynamics of a consumer-resource community are given by: where S is the total nutrient content in the system, b is the nutrient turnover rate, N is the 214 nutrient availability at any given time, and P i , H j and C k are, respectively, the biomasses of the i th plant species, j th herbivore and k th predator/omnivore. Since the system is closed, the total 216 nutrient content S remains constant over time, imposing a mass balance constraint (Loreau, 1994, 1995) on the system such that The function a X (X) X = P i , H j , C k is the per capita uptake rate, which can be linear where h X is the handling time). Importantly, 220 a X (X) represents matter and energy flow through the system (Loreau, 1994(Loreau, , 1995. Although the model does not explicitly consider energy, the flow of matter to producers and consumers is dependent on the flow of energy from photosynthesis (Loreau, 1995); the amount of energy available to producers and consumers is, therefore, encapsulated in a X (X). The parameters 224 d X (X = P, H, C) and e X depict respectively, the per capita mortality rate and the unit (e.g., gram) of biomass generated per unit of nutrient. The fraction X e X is, then, the total nutrient 226 amount contained in species X. Note that q is the proportion of the predator's biomass from feeding on the herbivore, and 1 − q, the proportion from feeding on the plant. The magnitude 228 of q determines the strength of omnivory.

Equation
(1) provides a mechanistic representation of exploitative competition based on the 230 R rule (Tilman, 1982). Species at each level (plant, herbivore, predator) compete for a resource whose dynamics are explicitly modelled. For instance, plants compete for nutrients, herbivores 232 compete for plants, predators compete for herbivores, etc. Competition is experienced through the effects that other species have on the abundance of the common "resource". For brevity, we 234 will refer to the primary producer as plant, and to herbivores and predators as antagonists.
Our approach is three-fold. First, we investigate the emergence of stable motifs via community 236 assembly from the ground up, starting with a plant species that colonizes an empty habitat whose establishment facilitates subsequent invasions by competitors and antagonists. We define 238 a stable motif as one whose configuration, once attained, is robust to species additions and removals. We use mathematical invasion analysis (i.e., the ability of an incoming species to 240 increase from initially small numbers when the resident community is at equilibrium) to quantify robustness. Invasion analyses have the drawback that they focus on the stability of a resident 242 community to a single invader. The mathematical methods involved do not easily lend themselves to investigating the outcomes when more than one species can simultaneously enter a community.

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In our second approach we use numerical analyses to determine which species combinations persist in the long term when two or more species simultaneously invade a community. If a given 246 motif were truly robust to perturbations, we would expect it to be stable to invasions by single and multiple species. Our third approach to quantifying robustness is species sorting and community 248 disassembly. In the case of sorting, we initiate each community with the full complement of species, allow interactions to proceed, and determine which configurations are persistent. In 250 the case of disassembly, we start with the full complement of species and sequentially remove competitors, antagonists, and primary producers. We determine which motifs remain stable to 252 species removals.

Model analysis 254
We use a combination of analytical methods and numerical simulations to investigate community assembly, species sorting, and community disassembly. In the simpler cases of community 256 assembly (e.g., two and three-species interactions), we use mathematical invasion analyses to derive the conditions under which an incoming species can maintain a positive per capita growth 258 rate when the resident species are at equilibrium. Details of these analyses are given in the online Appendix A. We investigate the more complex cases of community assembly and all cases of sort-260 ing and disassembly using extensive numerical simulations of the biologically feasible parameter space. In the case of species invasions, we initiate the community at the boundary equilibrium 262 in the absence of the invader(s), and introduce the invader(s) once the resident community has reached its steady state. In all cases the initial invader abundance was set to one individual.

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In the case of sorting, we initiate each community with the full complement of species (e.g., two plant species, two herbivore species, one predator/omnivore), allow interactions to proceed  (Fig. 2(a), Appendix A, Fig. S1).

Nutrient-plant-herbivore (NPH) community: invasion by a top predator
A predator can invade a plant-herbivore community provided the herbivore biomass at the 290 nutrient-plant-herbivore steady state exceeds the level to which the predator would depress it ( Fig. 2(b),Appendix A, Fig. S1).
biomass at the nutrient-plant-herbivore steady state exceeds the level that the omnivore requires to maintain itself (Fig. 2(c), Appendix A, Fig. S1). However, invasion leads to 296 the exclusion of the herbivore. Coexistence via omnivory is much less frequent (Fig. 2(c)).
This is because the omnivore has the advantage of feeding on two trophic levels, while the 298 herbivore has the constraint of feeding only on the level below while being fed on by the level above. Exclusion of the herbivore occurs even when relative non-linearity (Armstrong 300 and McGehee, 1980), mediated via Type II functional responses in the plant, herbivore and omnivore, allows an additional coexistence opportunity (Fig. S2).

Addition of horizontal links
Consistent with expectations, when plant growth is limited by a single essential nutrient, 304 the operation of R and P rules prevent the formation of horizontal links even when plant and herbivore species exhibit trade-offs in resource acquisition ability or susceptibility to 306 herbivory. This is because such trade-offs can only increase fitness differences between species (i.e., differences in per capita growth rates in the absence of density-dependent 308 feedbacks; Chesson (2000)); they cannot in themselves generate the stabilizing negative feedbacks that allow species to limit themselves more than they do others (Chesson, 2000).

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Generating such feedbacks requires more than one niche dimension. In a closed system with a single limiting nutrient and no spatial heterogeneity, the only possible dimension is 312 time. As noted above, temporal partitioning of the basal nutrient by two plant species may generate a horizontal link, but this link is susceptible to invasion by a herbivore, which 314 would set the P rule in motion.
To see this consider the two possible mechanisms of temporal partitioning. First, if the 316 two plant species respond differentially to temporal variation such that one species has a high nutrient uptake rate during periods of the year when the other species exhibits 318 little or no activity (e.g., spring and summer annuals), they may coexist via temporal niche partitioning (Fig. 3, Appendix B, Fig. S2). Second, if the plant species differ in the 320 non-linearity of their resource uptake functions such that the species with the more nonlinear response generates oscillations in nutrient-plant abundance, a second plant species 322 with a less non-linear response can invade and coexist through the mechanism of relative non-linearity (Armstrong and McGehee, 1980). Coexistence via relative non-linearity can 324 occur when the species with the less non-linear response is better at utilizing the nutrient when it is rare, and the species with the more non-linear response is better at utilizing the 326 nutrient when it is abundant (Fig. 3, Appendix B, Fig. S2). However, neither coexistence mechanism is stable to invasion by a herbivore. Such invasion leads to the exclusion of the 328 plant species more susceptible to herbivore attack (Fig. 3, Fig. S2).
As expected, R and P rules prevent the formation of horizontal links when the nutrient- (i) When the N P H interaction is invaded by a second plant or herbivore species, two 334 outcomes are possible: the original N P H interaction is stable to invasion, or the invading plant or herbivore species replaces the resident species (Fig. 4). There are (ii) The same two outcomes occur when the tri-trophic food chain (N P HC) is invaded by 340 a second plant or herbivore species. However, invasion failure is more frequent than replacement of the resident plant and herbivore species by the invaders (Fig. 5).

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(iii) In contrast to the linear chains (N P, N P H, N P HC), omnivory is not stable to invasions by additional plant or herbivore species (Fig. 5). This is because omnivory 344 itself is rare in closed systems with a limited nutrient supply (see above). Invasion by a second plant species does not alter the omnivore's advantage of being able to 346 feed on two trophic levels. The herbivore is excluded, and the plant species that is less susceptible to omnivore attack will exclude the other. The overall outcome is 348 a nutrient-plant-omnivore interaction with the omnivore relying solely on herbivory.
Similarly, invasion by a second herbivore species results in the exclusion of the inferior 350 competitor for the common plant resource (Fig. 5).
6. Community assembly via invasion by single species: summary of results

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When community assembly occurs in the absence of niche partitioning mechanisms that allow the addition of horizontal links, increase in diversity can occur only through the 354 addition of a vertical link. If the plant species' per capita growth rate does not depend on a mutualist (e.g., because it is obligately selfing or has wind-dispersed seeds), the first 356 vertical link is most likely be a herbivore, which opens up the possibility of invasion by a top predator or an omnivore. Vertical chains (N P, N P H, N P HC) are more robust to 358 species invasions than omnivory. This is because omnivory involves resource partitioning, the opportunity for which is constrained in a closed system with a limiting nutrient supply 360 that sets the upper limit to community biomass.
As noted previously, mathematical invasion analysis focuses on the conditions under which a single species can increase from initially small numbers when the rest of the community is at a 364 steady state. In reality, more than one species can enter a community at any given time. Numerical simulations spanning a large parameter space show that vertical chains (N P H, N P HC) 366 are robust to the simultaneous invasion of competitors and antagonists, but omnivory (N P HO) is not (Fig. 5). Below we explain these results in detail. When plant and herbivore species simultaneously invade the N P H community, the outcome is invasion failure or the replacement of resident species by invaders (Fig. 5). This is a direct The outcome is the same as that for the NPH community. The plant species that extracts 382 more of the nutrient in the face of herbivory will exclude the other, and the herbivore that can consume as much plant biomass as possibly while evading predation will exclude the 384 other (Fig. 5). Invasion by a second top predator in combination with a plant or herbivore species leads to the same outcome. In contrast to the vertical chains, omnivory proves to be unstable to simultaneous invasion by multiple species. Regardless of which combination of species invades (plant and 390 herbivore, plant and omnivore, herbivore and omnivore), the outcome is the exclusion of the herbivore and the emergence of a vertical chain with the omnivore acting as a top 392 predator (Fig. 5). This is because, as shown above in the single invasion case, the omnivore has the advantage of feeding on multiple trophic levels while the herbivore has the dual 394 disadvantage of competing with the omnivore for the plant resource while also being fed on by omnivore. In a closed system with a limiting nutrient supply and no external inputs, 396 the herbivore being a superior competitor for the plant species does not generate sufficient niche partitioning opportunities to allow for herbivore-omnivore coexistence.

Species sorting
In this step we start with the full assemblage of species for each of the three communities 400 (N P H, N P HC and N P HO) and allow the dynamics to proceed naturally. We find that species sorting occurs via the operation of R and P rules, with the result that the stable motifs to 402 emerge are, again, the vertical chains (N P H, N P HC and N P O).

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Starting from the full community, species sorting leads to the emergence of the N P H chain (Fig. 5). Which plant species persists depends on the cumulative effect of resource acquisition ability and susceptibility to herbivory. Which herbivore species persists depends on the level to which each herbivore can depress plant biomass. 408 2. Tri-trophic chain (N P 1 P 2 H 1 H 2 C) Starting from the full community, species sorting leads to the emergence of the N P HC 410 chain as the dominant motif (Fig. 5). Other motifs that occur in low frequency (e.g., As with assembly, species sorting leads to the exclusion of the herbivore by the omnivore, with the vertical chain (N P O) being the emergent outcome (Fig. 5).

Community disassembly
Here we start with the full complement of species for each community, and sequentially remove 418 species starting with the highest trophic level. We find that, across all community types, the motifs that are robust to disassembly are the vertical chains (N P H, N P HC and N P O). When 420 fitness differences between species are strong, transient coexistence of plant or herbivore species can occur, but this outcome is restricted to the nutrient-plant-herbivore community (N P H); it 422 is not observed in N P HC or N P HO interactions.

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When one herbivore species is removed, the community simplifies to two nutrient-plantherbivore chains (N P i H, i = 1, 2) ( Fig. 6(a) and (b)). Removal of one plant species also 426 leads to the formation of nutrient-plant-herbivore chains (N P H i i = 1, 2; Fig. 6(c) and (d)).
Simultaneous removal of one plant and one herbivore species leads to the same outcome 434 (Fig. 6(e) and (f)).

Omnivory (N P
When one herbivore species is removed, the community simplifies to two nutrient-plantomnivore chains (N P i O i = 1, 2) with the omnivore excluding the remaining herbivore 438 ( Fig. 7(a) and (b)). When one plant species is removed, we get a single nutrient-plantomnivore chain (N P O; Fig. 7(c) and (d)). Simultaneous removal of one plant and one 440 herbivore species leads to the same outcome (Fig. 7(e) and (f)). When the omnivore itself is removed, the community simplifies to one of four nutrient-plant-herbivore chains 442 (N P i H j i, j = 1, 2).

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Taken together, the outcomes of community assembly, species sorting, and community disassembly show that in a closed system with a constant supply of a limiting nutrient, no spatial 446 heterogeneity, and no external nutrient inputs or immigration, the only motifs that can emerge are vertical chains of nutrient-plant-herbivore or nutrient-plant-herbivore-predator. This is true 448 even in the presence of temporal niche partitioning at the plant level and relative non-linearity at plant and herbivore trophic levels.

Discussion
There is strong empirical evidence that ecological communities exhibit recurrent elements We find that, in a closed community in which the total nutrient availability sets the upper 498 limit to the total biomass, the only motifs that can emerge are vertical chains of nutrientplant, nutrient-plant-herbivore, and nutrient-plant-herbivore-predator interactions. Although 500 temporal variation, either through relative non-linearity in functional responses or temporal nutrient partitioning can allow the coexistence of plant species, such coexistence is not robust 502 to invasion by a herbivore. The plant species whose susceptibility to herbivory, when averaged over the year (in temporal niche portioning) or nutrient cycle (in relative non-linearity) is lower 504 will exclude the other plant species. The same process happens at the herbivore level. In the absence of a top predator, the herbivore species that depresses plant biomass to the lowest level 506 will exclude all other species; in the presence of a top predator, the herbivore, that can extract more energy from the plant while withstanding predation will exclude all other species.

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On the face of it, this result may seem trivial. After all, what we are seeing is the operation of the R and P rules. However, looking beneath the surface reveals several important insights.

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First, just as feed-forward loops maximize information transfer in transcription networks, vertical chains maximize energy transfer in ecological communities. Since the total amount of energy is 512 constant in a closed community, coexistence at any trophic level means the apportionment of the same amount of energy amongst species (since the amount of energy at that level cannot 514 exceed the amount procured by the species best at resource acquisition) with extra losses during conversion of energy to biomass (determined by the conversion efficiency parameter) and mor-516 tality. In a vertical chain, more energy is transferred from one level to another because having a single species at each level minimizes energy loss due to biomass conversion and mortality. Thus, 518 a vertical chain also leads to greater overall biomass and hence productivity of the community.
The key point is that the ecological constraint imposed by the R and P rules not only serves 520 to make the linear chains more robust to perturbations, but they also maximize energy transfer.
The second insight is that the operation of the R and P rules increases the trophic coherence 522 of the community. Coherence is determined by the number of trophic levels that a given consumer occupies (Johnson et al., 2014); a top predator feeds only on the trophic level below it (herbivore) 524 but an omnivore feeds on two trophic levels below it (herbivore and plant). The fewer trophic levels a given consumer extracts energy from, the more coherent a network is, and the less self-526 regulation required to stabilize it (Johnson et al., 2014). This is why we observe the vertical chains to be not only persistent but also stable to perturbations of species' abundances. Vertical 528 trophic chains not only maximize energy transfer and biomass production, but they are also stable both in the ecological sense (long-term persistence) and the mathematical sense (recovery from 530 perturbations to species' abundances). What is notable is that stability is achieved through the minimum possible level of self-regulation: a single negative feedback loop at the primary 532 producer level.
Importantly, the vertical trophic chains that emerge as stable motifs also confer modularity.

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Vertical chains exhibit high clustering coefficients (i.e., the ratio of realized to allowable links approaches 1; Watts and Strogatz (1998)). Clustering is a signature of modularity, a group of a 536 linked nodes with strong interactions (Alon, 2003). Vertical trophic chains satisfy these criteria.
They are highly cohesive (i.e., all allowable links are realized), and they achieve the common

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It is also noteworthy that our findings contradict the empirical findings that omnivory, exploitative and apparent competition, and the diamond web can be common in natural commu-566 nities. These discrepancies serve to illustrate important biological realties. Consider first, the omnivory motif. There is disagreement in the empirical literature as to whether omnivory con- negative feedbacks is also why we do not observe exploitative and apparent competition. The minimalist scenario we explore provides no opportunities for the niche partitioning mechanisms 578 required to sustain such interactions. This is particularly interesting because we do not observe coexistence above the primary producer (plant) level even in the presence of temporal coexis-580 tence mechanisms such as relative non-linearity and temporal niche partitioning. When multiple species occupy multiple trophic levels, we need more than two niche dimensions; resource/natural 582 enemy and time are no longer sufficient. We need to invoke space, external resource inputs, and immigration.

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The need to move beyond the minimal conditions of limited energy takes us to considerations of the conditions required for the assembly and persistence of complex ecological communities. 586 We propose that, just in the way that macromolecules (e.g., DNA, protein) are formed by the intertwining of molecular chains that subsequently fold into complicated structures held together 588 by relative fragile bonds, complex communities are formed by the coming together of vertical chains that are then held together by relatively fragile horizontal links that can break when 590 the energy inputs that make them possible are removed, or when perturbations to the existing structure occur in terms of species additions or removals. Let us consider a simple example, 592 the coming together of two trophic chains, each supported at the base by a different nutrient or a different supply of the same nutrient separated in space (e.g., soil space occupied by the 594 root systems of individual plants). Since each chain has an independent base (energy input), the R and P rules no longer hold, resulting in coexistence at primary producer and secondary 596 consumer levels. A top predator that attacks secondary consumers of both chains, or an omnivore that feeds on a plant species from one chain and a herbivore from the other, will be able to do 598 so without species losses at lower trophic levels, leading to a compartment that now contains multiple motifs: tri-trophic chain, omnivory, exploitative and apparent competition, and the 600 diamond web. Extending our framework to incorporate multiple vertical chains into models with energy limitation and mass balance constraints is an important next step.  Single Input Modules (SIMs) found in transcription networks, and the bi-parallel (diamond) motif found in signal transduction networks. In the FFLs, X and Y are transcriptional activators and Z is a promotor (Alon, 2007); In SIMs, X is a regulator and Z i i = 1, · · · , 3 are a group of target genes. In the bi-parallel motif, X, Y i ( i = 1, 2), Z represent signaling proteins and the arrows represent processes such as phosphorylation (Alon, 2007). The bottom row depicts the equivalent food web motifs: try-trophic food chain, omnivory, exploitative competition and the diamond motif arising from the combination of exploitative and apparent competition. Note that the ecological motifs constitute feed-forward loops in terms of energy transfer but feedback loops in terms of interactions among consumers and resources.
Panels (a)-(c) depict the frequency distribution of emergent motifs when a nutrient-plant interaction is invaded by a herbivore (panel (a)), a nutrient-plant-herbivore interaction is invaded by a top predator (panel (b)) and nutrient-plant-herbivore interaction is invaded by an omnivore (panels (c)). Parameter definitions and values are given in Tables 1 and 2

Simultaneous invasions
Figure 5: Simultaneous invasion by multiple species and species sorting in vertical trophic chains. Panels in the left column ((a)-(c)) depict simultaneous invasion by a second plant and herbivore species in nutrient-plantherbivore, tri-trophic, and omnivory interactions. Panels in the right column depict the outcome of species sorting.
In the case of sorting, each community is started with the full complement of species (N P 1 P 2 H 1 H 2 , N P 1 P 2 H 1 H 2 C, N P 1 P 2 H 1 H 2 O), and allowed to interact for 50, 000 time units. All panels depict the frequency distributions of species in the three communities at the long-term steady state. Parameter definitions and values are given in Tables 1 and 2 Figure 6: Disassembly of nutrient-plant-herbivore (panels (a)-(c)), tri-trophic (panels (d)-(f)), and omnivory ((g)-(j)) communities. Each community is started with the full complement of species (N P 1 P 2 H 1 H 2 , N P 1 P 2 H 1 H 2 C, N P 1 P 2 H 1 H 2 O), allowed to reach steady state, and subjected to sequential species removals (omnivore, predator, herbivore, plant). Parameter definitions and values are given in Tables 1 and 2.