Intransitivity as a dynamic assembly engine of competitive communities

Significance

Some ecological communities are thought to gain structure through competition among species, frequently assuming transitive competition (species A beats B, B beats C, C beats D, and so on). Recently, the field has become excited about the prospect of intransitive competition (A beats B, B beats C, but C beats A). In this article, we explore the consequences of connecting a group of three species in such an intransitive loop, to a group of transitive competitors, devising a theory showing many species can coexist through this mechanism. This transitive/intransitive modality is reflected in a community of ants in Puerto Rico, where at least 13 species seem to coexist through their connection to a triad of species forming an intransitive loop.

Abstract

Historically, those ecological communities thought to be dominated by competitive interactions among their component species have been assumed to exhibit transitive competition, that is, a hierarchy of competitive strength from most dominant to most submissive. A surge of recent literature takes issue with this assumption and notes that some species in some communities are intransitive, where a rock/scissors/paper arrangement characterizes some components of some communities. We here propose a merging of these two ideas, wherein an intransitive subgroup of species connects with a distinct subcomponent that is organized hierarchically, such that the expected eventual takeover by the dominant competitor in the hierarchy is thwarted, and the entire community can be sustained. This means that the combination of transitive and intransitive structures can maintain many species even when competition is strong. Here, we develop this theoretical framework using a simple variant on the Lotka–Volterra competition equations to illustrate the process. We also present data for the ant community in a coffee agroecosystem in Puerto Rico, that appears to be organized in this way. A detailed study on one typical coffee farm illustrates an intransitive loop of three species that seems to maintain a distinct competitive community of at least 13 additional species.

The underlying structure of transitive competition initially seems natural – species, much like sports teams, can be arranged from good to bad competitors (1–4). This framing suggests that excluding all but the best competitor can be avoided only if competition is weak, as suggested long ago by Gause (5). This generalization has been substantially weakened through theoretical analyses of networks that incorporate intransitivities as part of an overall coexistence framework (5–10) and empirically through the use of sensitive statistical approaches to field data (11, 12). Nevertheless, explicit examples of intransitive triplets exist (13–15) although they are rare, partially due to inherent methodological problems in recognizing them (16). Assumptions of transitivity remain common (1, 17, 18), although frequently tacit. Here, we propose a unique theoretical framework that combines both intransitive and transitive competitive interactions and provide empirical evidence that a specific ant community is so structured—a specific intransitive triplet of ant species connects with a presumably transitive set of other ant species which, when taken in a spatial context, can support the coexistence of a relatively large number of species.

Intransitive competitive structures (e.g., the rock/scissors/paper metaphor) have been well documented in the literature. They are thought to be sometimes (19, 20), or frequently (12), at the foundation of community assembly. Theoretically, their basic structure provides opportunities for maintaining a large number of associated competitors in a never-ending cycle of competition without extinction (8, 21), even if that competition is strong. Few empirical examples involving insects have been reported and, to our knowledge, none involving ants (22). We here propose a framework that connects the intransitivity of three ant species to a larger ant community presumed to be organized in a competitive hierarchy (2, 23–26), suggesting that species diversity beyond the three species intransitive loop is enhanced through the periodic cycling of the species in that intransitive loop.

Theoretically, we follow a long tradition of assuming that much of what structures this ant community is interspecific competition (27, 28) (see ref. 25 for a different view, regarding ants in particular). At the most elementary level, when two species compete only weakly, they may coexist in the environment, but if they compete too strongly, one or the other must win and exclude the other, as in Gause’s principle. The idea easily generalizes to the multiple species scale, giving rise to, for example, May’s paradox (29). Yet embedded in the extension to multiple species is the possibility that the simple, and frequently unstated, transitive assumption of interspecies competition is violated. Even in a completely random assembly of (n−1)² pairs of competition coefficients among n species, in repeatedly sampling three random species, 25% of these triplets will have an intransitive structure, on average (21). On the contrary, if transitive, any triplet must be either a “chain” (a_i,i+1 > 0 and a_i,i+j+1 = 0 for j arbitrarily large), or a “hierarchy” (a_i,i+1 > 0 and 0 < a_i,i+j+1 ≤ a_i,i+1) (Fig. 1 A and B). Qualitatively, we propose an “organizing metaphor,” (30) for competitive communities as collections of intransitive loops (Fig. 1C) connected to transitive chains or hierarchies. We thus loosely follow the conceptual framework of community structure as effectively a simplified combination of intransitive loops (as in Fig. 1C) and transitive chains and hierarchies, the latter two being extremes on a continuum (as in Fig. 1 A and B). We then present data for a well-known ant community in a coffee farm in Puerto Rico that appears to fit this structure.

Fig. 1.

The fundamental elements in the transitive/intransitive conceptualization of community structure, where species 1, 2, and 3 are the species of the intransitive loop and species 3, 4, 5, and 6 are the transitive species (clearly species 3 is a member of both sets). The strength of competition is conceptualized as the overlap between two fitness curves divided by the appropriate breadth of the purveyor of competition—see SI Appendix, Supplementary Material S1 for more detail. (A) Transitive chain. The graphs on the left represent the fitness functions on a niche axis, where the overlap of the fitness functions of two species, divided by the breadth of the function (the niche breadth), is taken to be the competition coefficient. For example, overlap [peak(sp3) − peak(sp4)/(breadth of niche of sp3) = a(3, 4), the competitive effect of species 4 on species 3. Right-hand side of A] is the interaction graph where a negative sign (small lines above or beside the species symbol) signifies competitive dominance (e.g., species 3 competitively dominates species 4). (B) Transitive hierarchy, where species 3 dominates species 4, 5, and 6; species 4 dominates species 5, and 6, etc. Right-hand side of B) is the interaction graph with the addition of the red lines indicating the hierarchies. (C) Intransitive loop. Note it is not possible to have an intransitive loop on a single niche axis. Here, on the first niche axis, species 1 competitively dominates species 2 and species 2 competitively dominates species 3. However, on the second niche axis, species 3 competitively dominates species 1 and species 1 competitively dominates species 2. Thus, overall, species 1 dominates species 2 (both niche axes), species 2 dominates species 3 (first niche axis), and species 3 dominates species 1 (second niche axis), forming an intransitive loop.

Theoretically, each species of an intransitive triad could be linked to an independent transitive chain or hierarchy of competitors, wherein the cycle of intransitivity delays or cancels the exclusions expected from the basic transitive structure. Although the wiring of a complex competitive community can require substantial theoretical stamina to comprehend in any but the simplest of cases, the elementary idea of independent transitive chains being periodically reset by species involved in an intransitive loop provides an elegant framework for studying the maintenance of biodiversity in a competitive community.

Previous studies of the ant community throughout the coffee-growing region of Puerto Rico suggest a heterogeneous structure, where each coffee farm houses a subset of species drawn from a pool of about 25 species (31). One typical pattern, albeit not wholly universal across the island, is the seeming codominance of three species, the red imported fire ant, Solenopsis invicta, the electric ant, also known as the little fire ant, Wasmannia auropunctata, and the flower ant, Monomorium floricola. S. invicta (native to South America) and W. auropunctata (native to Central and South America) are both regarded as agricultural pests due to their painful sting, but they also contribute to the control of the coffee berry borer and the coffee leaf miner (32–34). M. floricola (native to South Asia) is best known as a household pest, but not recognized as significant for agriculture. These three species appear to form an intransitive loop in which S. invicta dominates W. auropunctata (35), which dominates M. floricola (36), which dominates S. invicta.

Brief natural history notes on the three “intransitive” species are warranted. All the three species produce multiple female reproductives in each nest, and territorial expansion is mainly through spatial movement of these individual female reproductives along with brood. For the ground-nesting S. invicta, underground foraging tunnels extend from nest mounds approximately 10 m in every direction, with exit holes to the surface of the ground at discrete intervals (37–39). Foraging workers emerge from those holes and forage within a small territory surrounding the exit holes and ascend coffee bushes regularly within that foraging territory. From previous studies, we have evidence that S. invicta dominates in competition against W. auropunctata (35), displacing it spatially. W. auropunctata nests in small arboreal and terrestrial cavities and forages both arboreally and terrestrially. This species has been reported to win in laboratory competition experiments against another species of Monomorium, by following their trails, entering their nest, and killing the brood (36). M. floricola nests in small arboreal cavities and forages actively only arboreally (in the citrus and coffee trees, at our field site). We have routinely observed M. floricola encounters with S. invicta. When this happens, M. floricola rotates its abdomen forward and sprays a substance directly at the head capsule of S. invicta, causing the latter to immediately withdraw from the encounter, which explains repeated observations of M. floricola eventually dominating any sampling site that had originally been occupied by S. invicta. All the three species make extensive use of sugary exudates of hemipteran insects and extrafloral nectaries, when available, and are generalist omnivores. Other species in the system (the presumed transitive component of our system) are listed in Table 1.

Table 1.

List of all species found in the study, the three intransitive dominants above the double underscore, and the 13 additional species, along with natural history notes (T = terrestrial, A = arboreal)

Species	Origin	Nests	Forages	N2019	N2020	N202	N2022
S. invicta	South America	T	T and A	55	113	344	1,130
W. auropunctata	America mainland	T and A	T and A	1,073	2,072	993	585
M. floricola	South Asia	A	A	148	46	91	73
Nylanderia fulva	Northern South America	T and A	T and A	65	79	87	59
Brachymyrmex heeri	Native	T	T and A	46	27	57	41
Pheidole moerens	Native	T	T and A	39	18	304	101
Pheidole megacephala	Africa	T	T and A	35	1	52	56
Tapinoma melanocephala	Old World Tropics	T and A	T and A	23	2	42	58
Brachymyrmex obscurior	Native	T	T and A	15	0	23	28
Solenopsis zeteki	Native	T and A	T and A	8	0	1	2
Pseudomyrmex simplex	Native	A	A	6	0	0	2
Pheidole exigua	Native	T	T	1	0	2	1
Camponotus sexguttatus	Native	T and A	T and A	0	0	0	11
Odontomachus bauri	Native	T	T	0	0	2	1
Paratrechina longicornis	Native	T	T	0	0	11	22
Technomyrmex difficilis	Native	A	A	0	0	112	292

N 20ii refers to the total number of baits on which each of the species was observed during the course of the study in that year. Transitive species are organized according to their abundance in the year 2019.

Theory

The combination of an intransitive competitive loop connected to a transitive hierarchy is illustrated in Fig. 2, using the names of the three species that we propose form an intransitive loop in our system (a coffee farm in Puerto Rico). Here, we elaborate this theoretical formulation specifically as adapted to the community of ants on one specific farm in Puerto Rico, this being only one community example, albeit a common one, of several observed on coffee farms throughout the coffee-producing region (31).

Fig. 2.

Proposed theoretical structure applied to the ant community of a Puerto Rican coffee farm. M. floricola (species 1), S. invicta (species 2), and W. auropunctata (species 3) (X₁, X₂, and X₃ from Fig. 1C) form an intransitive loop. Smaller lines beside or above variables refer to negative (usually competition) effects. Species 4, 5, and 6 form a transitive hierarchy when h is large. However, if h is small and much less than “a” (see equation set 1 below), the transitive hierarchy transforms into a transitive chain as h decreases. Connecting an intransitive loop with a transitive hierarchy is the proposed structure of the competitive community studied.

Given the rather complicated dynamics of intransitive competition (40–43), we guide our thoughts with a simplified version of the theoretical system, as follows:

$$\frac{d{x}_1}{\mathit{dt}}=x_1(1-x_1-a{x}_3),$$ [1a]

$$\frac{d{x}_2}{\mathit{dt}}=x_2({1-x}_2-a{x}_1),$$ [1b]

$$\frac{d{x}_3}{\mathit{dt}}=x_3(1-x_3-a{x}_2),$$ [1c]

$$\frac{d{x}_4}{\mathit{dt}}=x_4(1-x_4-a{x}_3),$$ [1d]

$$\begin{array}{c}\frac{d{x}_i}{\mathit{dt}}=x_i\left(1-x_i-a{x}_{i-1}\mathit{-}h{\mathbf{\sum }}_{\mathrm{j}=3}^{\mathrm{i}-2}{x}_j\right), \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{i}=5,6,\dots {\mathrm{S}}_{\mathrm{T}},\hfill \end{array}$$ [1e]

where x_i is the population density (or biomass) of the i^th species in competition, and a is the main competition coefficient relating the change in per capita growth rate per density of competitor (classically these usually have subscripts indicating which species is interfering with which other species – here, such extra notation would be cumbersome). The parameter h is the competitive effect of species i on species with higher indices for i > 3 (Fig. 2), specifying the degree to which the transitive part of the model is a chain or hierarchy (Fig. 1). The system {x₁, x₂, x₃} admits of either a heteroclinic orbit oscillating among the three attractors, {1,0,0}, {0,1,0}, and {0,0,1} (when a > 2), or oscillates to a stable focal point on a triangular-shaped manifold (when a < 2) (21, 43). If h = 0 (i.e., a transitive chain; Fig. 1), the system is identical to that reported in a previous work (21), where the relationship to niche differentiation and other factors is elaborated more fully. Here, we explore the connection between an intransitive loop and a transitive hierarchy. This system formally represents the connection between Fig. 1 B and C, which is presented more completely in Fig. 2, where the specific intransitive species studied are indicated, and the three transitive species (species 4, 5, and 6) are left undefined.

Rather than a complete matrix of competition coefficients, our construct includes only those competition coefficients that specify the intransitivity (i.e., x₃ dominating in competition with x₁, x₁ dominating in competition with x₂, x₂ dominating in competition with x₃) and presumes, as has been done elsewhere (41), that they are all equal. This makes the intransitive loop part of the model a one-parameter model (a). Then, the “trailing” species have that same competition coefficient for the basic structure of the transitive chain, but also involve the additional competition coefficient, h, which stipulates variable levels of deviation from a transitive chain toward a transitive hierarchy, and can be regarded as a measure of the hierarchical nature of the trailing species. Given that, the overall model has only two parameters, the principal competition coefficients between all species (all equal to a) and the hierarchical competition coefficients between the species in the transitive sequence, starting with species 3 and ending with species 6 (all equal to h; Fig. 2).

Extensive simulation reveals, as expected, a tendency of the trailing species (the transitive component of the system) to be maintained by the continued oscillation of the three main species in the intransitive loop (21). Species x₄, x₅, …, are set to be transitive, with a ≥ h > 0 stipulating a transitive hierarchy. A simple time series and associated qualitative plots of the relevant manifolds illustrate the main qualitative outcomes of the model (Fig. 3). The behavior of the simple three-species model is well known (21, 40, 43) (heteroclinic cycle for a > 2, neutral oscillations for a = 2, and stable focal point for a < 2; Fig. 3 A–C). When a > 2 and the intransitive loop is connected to a transitive hierarchy, the system converges into a 6-dimension heteroclinic cycle (Fig. 3D), but when a < 2, it converges into a 6-dimension focal point attractor (Fig. 3E). Both extreme situations can result in the maintenance of the entire community.

Fig. 3.

Basic qualitative results of an intransitive loop (a, b, and c, which are the solutions to Eqs. 1 for the three intransitive species and represent a “control” of sorts in that the transitive part of the system is not included), and the intransitive loop plus the attached transitive chain (d and e, where all the six species are in the system) from equation set 1. (A) Time series for each species with a qualitative diagram of the manifold in the triangles above when the competition coefficient (a) = 2.1. (B) The same as a but when the competition coefficient (a) = 2.0. (C) Same as A, but with the competition coefficient (a)= 1.9. (D) Adding the three species in the transitive chain with high competition (a = 2.1). The manifolds above are diagrammatically expanded to indicate the behavior of the transitive component of the system. (E) Same as D but with a = 1.9. h = 0.7 for all cases.

Consider first, the case of strong competition (i.e., a > 2.0 – the generally heteroclinic case). The topological position of x₄ in this system is evidently of extreme importance. If, for example, the competition effect of x₃ on x₄ (in Fig. 2, species 3 on species 4) is large (>2.0), then x₄ (the dominant species in the transitive part of the community) may be the first of the 6D heteroclinic cycle to be eliminated, thus disconnecting the transitive from the intransitive parts, leaving the transitive component to be dominated by x₅. This could result in the loss of the rest of the species within the transitive chain, depending, to some extent, on the idiosyncrasies of the timing of competitive exclusion. Extensive simulations show that if competition is strong (experiments herein done with a = 2.1), all the six species in the whole system enter into a 6D heteroclinic cycle (Fig. 3D), eventually extinguishing the species whose population transgresses the lower limit of population persistence and resulting in the long-term persistence of either one or two species. Clearly, which species is the first to be extinguished in this way sets the path of extinction for the rest of the community. If, for example, x₃ (species 3 in Fig. 2) is the first to go extinct, the intransitive loop ceases to exist, and the only species left in the system will be x₁ and x₄. However, if species x₅ is first to go extinct, that leaves x₆ with no competitor and it will survive in perpetuity, but the other four species will remain in a new 4D heteroclinic cycle. If the first species to go extinct is x₂ (species 2 in Fig. 2), x₃ will go unchecked, and the competitive pressure on x₄ from x₃ will increase, causing x₄ to go extinct, along with x₁ (for the same reason), leaving x₃ and x₅ as the only two surviving species. A moment’s reflection reveals that there are nine alternative stable (populations persist in perpetuity) outcomes involving more than one species of this heteroclinic cycle: 1) x₂ and x₄, 2) x₂ and x₅, 3) x₂ and x₆, 4) x₁ and x₄, 5) x₁ and x₅, 6) x₁ and x₆, and, of course, each of the species in the original three species loop also, all depending on which species first transcends the limit at which it goes extinct.

More generally, if more species are involved in the transitive part, for every two species added to the transitive sequence, one will tend to survive in perpetuity (depending on the value of h), reflecting the elementary result of a pure transitive chain (alternating extinctions along the chain or complete takeover by the dominant species in a hierarchy). This mathematical result can be made intuitive if we begin with an intransitive/transitive system (e.g., Fig. 2) where the transitive part is a chain. Uncoupling the chain from the intransitive loop, we see that the first species in the chain beats the second, the third beats the fourth, the fifth beats the sixth, and so forth. The result is that every other species in the chain (beginning with the second one) goes extinct, while every other one (beginning with the first) survives in perpetuity. Such a pattern holds true, qualitatively, for a transitive hierarchy also, as long as h is not too large, at which point the order of approaching the heteroclinic limit determines which species will survive, as described above.

Considering now a lower competition coefficient (a < 2), the theoretical structure of the intransitive/transitive system suggests an additional complication. At the extreme of h = 0 (a transitive chain, Fig. 2), all species survive as a focal point (assuming a < 2.0; Fig. 3E). As h increases, this pattern remains until a critical value of h, at which point species 5, the penultimate species in the chain, goes extinct. Further increases in h eventually extinguish species 6 also. This basic process operates in a parallel fashion for much larger transitive chains and hierarchies (e.g., Fig. 4), wherein the penultimate species (e.g., species 11 in Fig. 4), if odd, begins to reduce its population density, while the ultimate species (species 12 in Fig. 4) increases its density, due to the reduced pressure from its main competitor in the hierarchy. However, the penultimate species declines to zero as h increases, at which point the only pressure on the ultimate species is from the other species in the hierarchy, and only from the collection of competitive pressures brought on by the “h” competition coefficients, eventually resulting in the extinction of the ultimate species (see the sequence of species 11 and 12 in Fig. 4, as h increases). This sequence repeats itself as the parameter h is increased, first driving an odd-numbered species to extinction, followed by the decline and eventual extinction of the associated even-numbered species (see Fig. 4, for a concrete example). The final conclusion is that, although it is complicated, the stable intransitive loop (a < 2) provides a safety net for the species in a transitive competitive hierarchy, meaning that the intransitive/transitive structure in general may promote species diversity, when the transitive nature of the system is strong (i.e., when the parameter a is large relative to h).

Fig. 4.

Final value of x₅, x₆, … x₁₂, as a function of the competition coefficient of the transitive sequence (h), for this parameter instantiation (a = 1.9). x₄ remains in perpetuity, and x_i, when odd, is driven to extinction at a value of h lower than that which drives x_i+1 to extinction, effectively reproducing the basic pattern well known for the case of a strict transitive chain (i.e., h = 0) (8).

An Empirical Example.

This theoretical framing seems to describe a real community of ants on a coffee farm where we have collected spatial data for several years (Methods). As discussed above, the three dominant species, S. invicta, W. auropunctata, and M. floricola, form an intransitive loop that is connected to the other species in the system, the latter of which represent the putative transitive portion of the community. We have encountered 13 other species (Table 1) at the site, species that we presume to be competitors with one another as well as with one or another of the species in the intransitive triad. Although interactions among these species have not been studied in detail, we assume that they are likely to form transitive competitive hierarchies as reported in the ant literature for many ant communities (23–26).

In Fig. 5, we show the spatial pattern of the three dominant ant species for six sampling periods. The original expectation (from the first two sampling periods; Fig. 5 A and B) that W. auropunctata would completely take over the farm (31), seemed plausible. However, on the third sampling date, although W. auropunctata did indeed become quite dominant, hints of more complexity were evident in the new appearance of S. invicta in a part of the field that had not been occupied by this species before (Fig. 5C). Furthermore, the strictly arboreal species, M. floricola, had been relatively abundant in the eastern third of the plot for the first two time periods, only to be overwhelmed by W. auropunctata by January 2020 (Fig. 5C). The overall pattern conforms to expectations from other studies examining the competition between these species (31, 44). W. auropunctata takes over from M. floricola, clearly visible in the change from January 2019 to January 2020 (Fig. 5 A–C), and S. invicta seems to be in the process of taking over from W. auropunctata from January 2020, through July 2022 (Fig. 5 C–F), as expected (35). The competitive dominance of M. floricola over S. invicta is less clear, but nevertheless could be inferred from the patterns seen from July 2021 to July 2022 (Fig. 5 D–F) and amply reinforced by field observations of worker-to-worker aggression, as described above. More fine-tuned qualitative evidence of the intransitive structure is presented in SI Appendix, Supplementary Material S2.

Fig. 5.

Spatial distribution of the three dominant ant species on coffee plants on six sampling times over a three-and-a-half-year period (Panels A–F). The size of each bubble represents the ant abundance/activity (from 1 to 5 baits occupied) of that species on a coffee bush. The time gap between January 2020 and July 2021 was caused by the COVID epidemic, which disabled our field operation for that year. Parts A–C, are from the same data reported in Perfecto and Vandermeer, 2020 (31).

Although the patterns displayed in Fig. 5 are all from arboreal foraging (on coffee plants), the well-known habitat or niche differences among these species was observed regularly in all of our sampling. In extensive sampling on the ground (not reported here), M. floricola was never encountered, while S. invicta and W. auropunctata were regularly encountered both on the ground and on coffee bushes. In a related study, all the three species were encountered regularly in the citrus trees that cover the area (44), with S. invicta occasionally foraging all the way to the upper canopy of the citrus, but mainly concentrated in the mid to bottom trunk area. Initial ground sampling of S. invicta and W. auropunctata also suggested that the advance of S. invicta into W. auropunctata territory is mainly terrestrial (see evidence below), with regular foraging of S. invicta on the ground extending well into territory dominated on the coffee trees by W. auropunctata.

Ground sampling, employed along 10 m transects (Methods) during January of 2022, revealed a regular pattern of S. invicta emerging from foraging holes as part of the basic nest and foraging structure of this species, sometimes quite far from the nearest evident above-ground nest mound (Fig. 6). M. floricola does not appear in any of the 312 ground-foraging baits illustrated (within the 10 m transects, Fig. 6). There are, however, significant swarms of S. invicta within the area dominated by W. auropunctata on the coffee bushes (Fig. 6), a consequence of the underground tunnels with periodic foraging exit holes, from where these swarms emerged (39). Furthermore, the actual ground foraging of S. invicta goes far further than any active nest mounds might indicate. The penetration of S. invicta into the area dominated by W. auropunctata, well beyond any evident arboreal foraging of the former, suggests that the active site of competitive exclusion is on or below the ground for the interaction between these two species.

Fig. 6.

Relative abundance of W. auropunctata (green) and S. invicta (red) in 10 m ground transects at 12 strategic positions in a main plot within a coffee farm (same as in Fig. 4). Abundance/activity of the three dominant species on the coffee plants presented as a background heat map (based on the data in Fig. 4E—including M. floricola) so as to locate each of the transects relative to the position of ant dominance in the coffee bushes. Left bottom Inset indicates the basic structure of the baiting transects: two rows indicating discovery density (Methods) of each species (green for W. auropunctata and red for S. invicta) and 26 columns indicating each baiting site on the transect (note that M. floricola does not appear on any of the transects because it does not forage on the ground). Red circular symbols are the locations of all active S. invicta nest mounds at the site at the time of sampling.

Contrarily, there is no possible direct interaction between M. floricola and either of the other two species on the ground, since M. floricola does not forage on the ground. Nevertheless, as evident in Fig. 6, there is little ground-foraging activity of S. invicta in the areas dominated on the coffee bushes by M. floricola. Aggressive attacks of M. floricola against S. invicta are observed regularly when the two species temporarily occupy the same arboreal baiting site, and the response is always the same—S. invicta retreats. It is evident that the abundance of S. invicta in the M. floricola-dominated areas is low, meaning that the ground-foraging gaps (areas between the foraging exit holes of S. invicta) are extensive, simply due to the lower population density of the species. These are precisely the areas where one would expect a significant number of other species, suggesting what we propose theoretically, that intransitive competition provides recurrent opportunities for other species to be maintained in the system. Competitive pressure on S. invicta from M. floricola must be felt mainly arboreally, but is reflected in the lower activity of the former on the ground.

We propose that the additional species in the system (Table 1), the “transitive” subcomponents of the community (Fig. 2), can be maintained (21) through this structure. Although competitive interactions among ants generally (as other organisms) remain largely unknown, or perhaps oversimplified (26), for most ant communities, there is a vast literature reporting on competitive hierarchies, which are fundamentally transitive (22–26). Given the basic natural history facts of the collection of species in this system, it is fair to suggest that various subcomponents of the “other species” will form transitive hierarchies. Certainly, not all species at any given point in space will be so organized, but various subcomponents vying for local resources that may be in short supply are likely to form transitive hierarchies.

A snapshot of the field system (i.e., Fig. 6) suggests that, in accordance with the theory, much area is left open for other groups of species to be maintained, through this intransitive loop. S. invicta alone is expected to create “no ants’ lands” between foraging territories of individual nests (45). However, we expect that the entire area of S. invicta in this study is a single polydomous megacolony and the question of where one foraging territory begins and ends seems mute. Nevertheless, the basic theoretical community structure is dependent on the recurrent cycle of the three elements of this intransitive loop, wherein most opportunities for other species in the system occur either arboreally when M. floricola is not present, or terrestrially when S. invicta leaves foraging gaps that other species can exploit, and less when both S. invicta and W. auropunctata dominate all terrestrial surfaces. Given the relatively complete occupancy of baits when both S. invicta and W. auropunctata are present (see transects in the W. auropunctata territory in Fig. 6), the opportunities for other species are limited. The 13 other species encountered in these 12 terrestrial transects (Fig. 6) are thus, not surprisingly, related to the foraging activity of both S. invicta and W. auropunctata (Fig. 7), corresponding to the general theoretical result that the intransitive/transitive structure is a generator of species diversity.

Fig. 7.

Relationship between the total foraging number at time 10 min (discovery-density) of the dominant terrestrial species on a 10 m transect (Fig. 6) and the total number of transitive species on that transect (after 30 min) for each of the 10 transects. (A) W. auropunctata; (B) S. invicta; the dotted black regression line represents transects that have large W. auropunctata discovery-densities; the dashed black regression line represents transects that have low W. auropunctata discovery-densities. (C) Relationship between the combined total foraging number (discovery-density of both dominant species) and the total number of transitive species (exponential fit, R² = 0.66). Positions of transects are shown in Fig. 6.

The relationship between W. auropunctata “discovery-density” and the number of putatively transitive species is dramatically nonlinear, with four transects with zero or close to zero discovery-density of W. auropunctata and a substantial number of other transitive species and another four transects with larger W. auropunctata discovery-density and almost no transitive species (Fig. 7A). The relationship between S. invicta discovery-density and the number of transitive species is only very weakly negative. The natural history of these two species makes this apparent contradiction clear, in that at low S. invicta discovery-density, there are both a small number of transitive species and a large number of transitive species, the former associated with transects that have large W. auropunctata discovery-densities (the dotted black regression line in Fig. 7B) and the latter associated with transects that have small W. auropunctata discovery-densities (the dashed black regression line in Fig. 7B). Consequently, the combined discovery-densities (i.e., taking W. auropunctata and S. invicta as if they were a single species), shows a much clearer relationship between the discovery-density of these two dominant intransitive species and the abundance of the so-called transitive species (Fig. 7C). The obvious difference between the effect of W. auropunctata and S. invicta underscores the basic natural history of the two species and suggests that it is the idiosyncratic nest and foraging structure of the S. invicta that makes it the main species that allows the presumed transitive part of the community to persist over the long run.

Discussion

In recent years, intransitivity has been increasingly recognized as a potential force in community assembly and species persistence. For example, in a statistical analysis of two large datasets, Soliveres et al. (12) note that intransitive competition is widespread in plant communities. From a more theoretical perspective, there is now a vast literature on the subject. Allesina and Levine (6) have shown that intransitive competitive networks are the most likely outcome in ecological systems with multiple limiting factors and suggested intransitivity as a general mechanism for maintaining species diversity. Levine et al. (7) (p. 63) also note that there is limited empirical evidence because of “… the intractability of empirically evaluating competition between many species and the technical difficulties that are inherent in tightly coupling theory to data.” We speak specifically to such concerns with our concrete empirical example.

Here, we elaborate on the recent appreciation of intransitive structures in the process of community assembly (6–10). Most of the literature on this topic focuses on statistical properties of empirical data (11, 12), or the intransitivities themselves (14–16), acknowledging that concrete examples from nature are rare. In the spirit of searching for mechanistic structures that result in intransitivities, we propose a generalized theoretical structure that explicitly connects an intransitive loop to a transitive hierarchy, showing how this combination may promote the coexistence of otherwise incompatible competitors. These theoretical structures are stimulated, first from a more survey-styled study published elsewhere (31), where we speculated on the likelihood of an intransitive structure, and finally from a detailed study of our field system, where three of the species create an intransitive loop. The consequent theoretical structure is mathematically evident and leads to an intuitive understanding of how intransitive structures may stabilize otherwise unstable transitive structures, and the empirical data from our field system broadly correspond to this organizing metaphor. The empirical structure (31) (and its intransitive nature) of these three species can be deduced from previous literature, laboratory experiments, and/or extensive natural history observations. Here, we elaborate on the generality of connecting intransitive with transitive structures in a competitive community, and the association with species diversity, offering a specific example that appears to follow that theory.

The general natural history of the three intransitive species reflects and reinforces the grounding of this community, at least at the site of the current study and perhaps on other coffee farms and other habitats in Puerto Rico (31, 34, 35). W. auropunctata is one of the most invasive ants in the world and is well known to be an aggressive competitor of other ants, at least in its nonnative range (46, 47), known to use chemical repellants (48), and, in addition to its strong competitive effect, preys upon other ants (36). It nests and forages both arboreally and terrestrially. S. invicta, another extremely noxious invasive species, is also well known to displace other ants in its invasive range (49, 50). Our results demonstrate the importance of its nest structure of underground tunnels with periodic foraging exits (39), wherein it clearly penetrates areas dominated by W. auropunctata, far from any evident nest mounds of its own species, and slowly takes them over, reflecting clear results from laboratory studies (SI Appendix, Supplementary Material S3). It nests exclusively on the ground, although, in Puerto Rico, regularly forages arboreally. Its nest structure, consisting of underground tunnels with foraging exit holes, creates a patchwork on the ground wherein its competitive effect is limited to a local area surrounding the exit holes (Fig. 5), enabling (as neither of the other species do) foraging opportunities for other ground-foraging ants. M. floricola is the third exotic species involved in the intransitive triad, nesting and foraging only arboreally and exerting its competitive effect against S. invicta through easily observed aggressive encounters, directly spraying a chemical at individual foragers. Its displacement by W. auropunctata is presumably due to physical attacks on its colonies, as in a similar situation with a related species of Monomorium (36). These natural history accounts, coupled with abundant related literature (38, 46–52), clearly support the idea that this system of three species of nonnative ants (W. auropunctata, S. invicta, and M. floricola) found on coffee farms in Puerto Rico are expected to form an intransitive loop. From a four-year census, the dynamic spatial pattern formed by the three species (Fig. 5) is consistent with the intransitive nature of their interactions, as gleaned from the literature and natural history observations. Additional fine-tuned observations of the patterns over the four-year period are presented in SI Appendix, Supplementary Material S2.

Regarding the transitive portion of our argument, the ant species in this study (Table 1) are not necessarily strictly arranged in a transitive way. Some forage only arboreally, some only terrestrially, sometimes reflecting the choice of nesting site. While our theoretical formulation is a qualitative idealization meant to encourage intuition about community structure, our empirical contribution is less “idealized” since it involves real animals. Yet the basic story of competitive release in the gaps left by one species in the intransitive loop and our observations of various species foraging in those gaps, and only when the one species, S. invicta, is the local resident, fit well, qualitatively, with the theoretical perspective. Whatever ants do show up in the gaps are clearly the result of the competitive (and perhaps other) interactions that were underway before S. invicta arrived and certainly may have been more complicated than a simple hierarchy or chain.

The periodic release of foraging area as S. invicta takes over territory from W. auropunctata creates opportunities for other ants in the ground area between the foraging exits from the underground tunnels of S. invicta. The further depression of the general population density of S. invicta through competition from M. floricola creates yet more foraging areas terrestrially for other species. The final conclusion, then, is that the basic intransitive/transitive structure theoretically provides for a distinct form of competitive coexistence despite strong competition, certainly not Lyapunov stable, but persistent through time. The point made elsewhere (7) that these structures may be common elements of community dynamics is partially what encouraged us to do the study in the first place, thus speaking to the generalization that there is a “… glaring absence of evidence to evaluate the degree to which observed intransitivity stabilizes coexistence in nature,” (7) providing a concrete example of the more general phenomenon of intransitivity as a stabilizing force. Furthermore, the unique theoretical contribution here, elaborating on previous theory (21), is that transitive structures may be stabilized by intransitive structures, speaking to the general phenomena of intransitivity as a force in species coexistence.

More generally, competition among arboreal ant species is generally thought to be the main element generating community structure (4, 53), sometimes a contentious issue (25, 54), but frequently cited as a mechanism for the formation of the commonly observed ant mosaics (55, 56). Some past studies have hypothesized that dominant and subdominant species are maintained through such spatial mosaics (57, 58). We offer a potentially different lens. For example, in a particular cacao system, there are three dominant arboreal species (Crematogaster sp., Oecophylla longinoda, and Pheidole sp.) (59), and a variety of other ant species, thought to be retained in the system because they form a spatial mosaic. The unstudied prospect that they might be maintained through intransitive competition was not a point of interrogation since the subject had not really entered the ecological literature when this study was published. Other similar studies (60, 61) frequently cite cases of three species apparently sharing dominance, again through assumed fixed spatial mosaics, and associated with other, likely subdominant, species. In coconut plantations in the Solomon Islands, for example, four species are dominant on coconut trees (62), although on any given plantation, only three tend to dominate. Yet there always seem to be “minor species” (subdominant species) associated with the dominant ones. A twenty-year time series suggests, for several of the plantations, a sequence that strongly resembles a three-species intransitive loop, along with minor species inserted. We suggest that some of these ant studies, although treating issues such as dominance, spatial mosaics, and pairwise competition, reveal patterns that are at least consistent with the mechanism of community organization we propose to sometimes operate.

Other sessile or spatially restrained organisms, such as corals, plants, bacteria, and others, share with ants a natural history that might suggest this basic structure. The classic case of bacteria that form a spatial mosaic driven by a three-species intransitive loop when unmixed (14) is a case in point, where transitivity is likely, if realized on bacterial films and thus subject to spatial patterns. Coral reefs with mixed strategies of overgrowth involving algae, sponges, and corals have been described as potentially harboring intransitive structures (19, 63). Competition between stony corals and gorgonian corals is “likely intransitive” (64) in an environment with more than 100 species, an arrangement that could be investigated as a possible candidate for the process here described. The classic case of the now well-appreciated intransitivity associated with plant communities (12) presents ample opportunity to ask whether the reported massive intransitivities are somehow coupled with transitive structures to produce the fundamental mechanism of community structure that we propose herein.

Methods

Field Surveys.

We repeated the methods as described elsewhere (31). In that study of three visits over a 2-y period to 25 farms, we encountered relatively high variability from farm to farm, although about half the farms contained all the three focal species, W. auropunctata, S. invicta, and M. floricola. The basic methodology is to place small pieces of tuna fish (canned in vegetable oil), directly on the bark of coffee bushes, at five distinct points, attempting to sample the entire coffee bush. In the present study, the basic methodology was repeated for one specific farm (UTUA2 from ref. 31) on all coffee bushes over an area of approximately 110 m × 30 m, for six distinct sampling dates (January 2019, July 2019, January 2020, July 2021, January 2022, and July 2022).

Ground-Foraging Transects.

Ground-foraging ants were sampled along strategically located transects in January 2022. Small thin plastic discs (8 cm diameter) were placed directly on the ground at 0.4 m intervals for a total of 26 baits along a 10-m transect. The total number of foraging ants (from 0 to >20) was counted at each bait, for each species present, at intervals of 10, 30, and 60 min. Twelve such transects were employed along the range of all coffee trees in the plot (see map in Fig. 5). The number encountered at 10 min was labeled “discovery” density, and the number encountered at 60 min was labeled “dominance” density.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Numerical data have been deposited in figshare and are available at https://figshare.com/articles/dataset/Intransitive_competition_among_three_dominant_ants_in_coffee_farms_in_Puerto_Rico/22299052 (65).