bioRxiv preprint doi: https://doi.org/10.1101/694257; this version posted July 5, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Spatial of Territorial Populations

Benjamin G. Weiner Department of Physics, Princeton University, Princeton, NJ 08544

Anna Posfai Cold Spring Harbor Laboratory, Cold Spring Harbor, NY 11724

Ned S. Wingreen∗ Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544 (Dated: July 5, 2019) Many , from to biofilms, are composed of territorial populations that com- pete for both nutrients and physical space. What are the implications of such spatial organization for ? To address this question, we developed and analyzed a model of territorial re- source . In the model, all obey trade-offs inspired by biophysical constraints on metabolism; the species occupy non-overlapping territories while nutrients diffuse in space. We find that the nutrient diffusion time is an important control parameter for both biodiversity and the timescale of . Interestingly, fast nutrient diffusion allows the populations of some species to fluctuate to zero, leading to extinctions. Moreover, territorial competition spon- taneously gives rise to both multistability and the Allee effect (in which a minimum population is required for survival), so that small perturbations can have major ecological effects. While the as- sumption of trade-offs allows for the coexistence of more species than the number of nutrients – thus violating the principle of competitive exclusion – overall biodiversity is curbed by the domination of “” species. Importantly, in contrast to well-mixed models, spatial structure renders di- versity robust to inequalities in metabolic trade-offs. Our results suggest that territorial ecosystems can display high biodiversity and rich dynamics simply due to competition for resources in a spatial . Keywords: spatial ecology | biodiversity | trade-offs | | modeling

Living things exist not in isolation but in commu- tial structure, along with realistic metabolic constraints, nities, many of which are strikingly diverse. Tropical impact diversity? rainforests can have more than 300 tree species in a Various studies have clarified how intrinsic environ- single hectare [1], and it has been estimated that one mental heterogeneity (e.g. an external gradi- gram of soil contains 2,000-30,000+ distinct microbial ent) fosters biodiversity by creating spatial niches [9– genomes [2, 3]. Understanding the relationship between 13]. Others have demonstrated that migration between biodiversity and the environment remains a major chal- low-diversity local environments can lead to “metacom- lenge, particularly in light of the competitive exclusion munities” with high global diversity [14–19]. But how principle: in simple models of resource competition, no is diversity impacted by local spatial structure? Re- more species can coexist indefinitely than the number of cent models suggests that spatial environments with- limiting resources [4, 5]. In modern niche theory, com- out intrinsic heterogeneity can support higher diversity petitive exclusion is circumvented by mechanisms which than the well-mixed case [20–25], although the effect reduce niche overlaps and/or intrinsic fitness differences depends on the interactions and details of spatial struc- [6, 7], suggesting that trade-offs may play an important ture [26, 27]. In these models, competition follows phe- role in the maintenance of biodiversity. Intriguingly, di- nomenological interaction rules. In some cases, trade- versity beyond the competitive-exclusion limit was re- offs have been invoked to limit fitness differences [21] cently demonstrated in a resource-competition model and penalize niche overlap [25], but did not otherwise with a well-mixed environment and exact metabolic structure the spatial interactions. All these models al- trade-offs [8]. However, many ecosystems are spatially low coexistence when the combination of spatial segre- structured, and metabolic trade-offs are unlikely to be gation and local interactions weakens interspecific com- exact. While some spatial structure is externally im- petition relative to intraspecifc competition. However, posed, it also arises from the capacity of to it remains unclear how such interactions relate to con- shape their environment. How does self-generated spa- crete biophysical processes. Here, we study biodiversity in a model where species interact through spatial resource competition. We specifically consider surface-associated populations ∗ Correspondence email address: [email protected] which exclude each other as they compete for territory. bioRxiv preprint doi: https://doi.org/10.1101/694257; this version posted July 5, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

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This is an appropriate description for biofilms, vegeta- and C show an example of the time evolution of one tion, and marine ecosystems like mussels [28] or coral such spatial community consisting of 11 species com- [29], in contrast with models that represent populations peting for three nutrients. as overlapping densities and better describe motile or While the supply of nutrients is spatially uniform, planktonic populations [9, 30]. The well-mixed envi- the local rate of nutrient consumption depends on the ronment is an explicit limit of our model, so we are able metabolic strategy of the local species, and nutrients to isolate the unique effects of spatial structure. diffuse in space. We study the regime where popula- We find that, contrary to expectations, introduc- tion growth is nutrient-limited, so the rate of uptake of ing population territories into a model with metabolic each nutrient is linear in its concentration. Thus, within trade-offs reduces biodiversity relative to the well-mixed each region occupied by a single species σ, the nutrient case. Extinctions occur over a new timescale inversely concentrations cσi obey related to the nutrient mixing time. Spatial structure ∂c ∂2c also leads to the of multiple steady states σi = S − α c + D σi , (1) ∂t i σi σi ∂x2 and the Allee effect, so that small perturbations may have drastic consequences. Finally, we find that overall where D is the diffusion coefficient for all nutrients. biodiversity is curbed by the domination of “oligotroph” As nutrient processing is generally much faster than species but is robust to inequalities in metabolic trade- growth, we assume a separation of timescales, such that offs. nutrient concentrations equilibrate before populations ∂c change. Then ∂t = 0, and r r Si  ασi   ασi  RESULTS cσi(x) = +Aσi exp x +Bσi exp −x . ασi D D (2) Model The constants of integration Aσi and Bσi are fixed by the physical requirement that ci(x) be continuous and We developed a model of territorial populations com- differentiable at the population boundaries. Figure 1D peting for diffusing resources to clarify the relationship shows the concentrations of the three nutrients after between spatial structure, metabolic trade-offs, and bio- the populations shown in Fig. 1B and C have reached diversity. The model is spatially explicit and relates steady state. The competitors transform the uniform the mechanistic dynamics of competition to parame- nutrient supply into a complex spatial environment by ters with clear biological meaning. Crucially, compet- depleting their preferred nutrients while allowing other ing populations are not interpenetrating, so populations nutrients diffuse to their neighbors. are competing for both nutrients and territory. The populations change in time according to ! Specifically, we consider m species competing for p Z nσ dnσ X = α v c (x) dx − δn , (3) nutrients in a one-dimensional space of size L with dt σi i σ periodic boundary conditions (a ring). The rate of i 0 supply of nutrients is specified by the supply vector where δ is the death rate and the integral is taken over ~ P S = (S1,S2...Sp) such that i Si = S, where S is the territory occupied by species σ. v is a length that the total nutrient supply rate in units of concentra- converts nutrients to territory growth. The total pop- tion/time. The nutrient supply is spatially uniform, so ulation remains fixed at L, corresponding to competi- there is no external environmental heterogeneity. Each tion for a share of a fixed total territory. This implies P species σ ∈ [1...m] is defined by its metabolic strategy σ n˙ σ = 0, which requires δ = vS, i.e. the death rate ~ασ = (ασ1, ασ2...ασp), which specifies the proportion of matches the nutrient value of the total supply rate. We its metabolic resources (e.g. enzymes) it allocates to choose units of time and concentration such that v = 1 the consumption of each nutrient. Metabolic trade-offs and S = 1, without loss of generality. In the exam- are implemented via a constraint on the enzyme budget, ple shown in Fig. 1B and C, nine species coexist, far P namely i ασi = E for all species (except where noted). exceeding the three-species limit set by competitive ex- Metabolic strategies and the supply can be represented clusion. as points on a simplex of dimension p − 1 (Fig. 1A and The spatial nutrient environment influences the pop- Fig. 2A inset, for example). Each species occupies a ulation dynamics via the dimensionless diffusion time 2 segment of the ring corresponding to its population nσ, τD ≡ L E/D, which is the time for nutrients to diffuse a so that nσ is a length and σ = 1...m specifies a spatial distance L relative to the uptake time. Competitors in- ordering. For example, the population with strategy ~α2 teract only through the nutrient environment, so when occupies the segment of the ring between populations nutrients diffuse instantaneously (τD = 0), the spatial with strategies ~α1 and ~α3. Populations never overlap, so dynamics reduce to the well-mixed dynamics. (See SI P the total population satisfies σ nσ = L. Figures 1B Appendix for the τD → 0 expansion.) bioRxiv preprint doi: https://doi.org/10.1101/694257; this version posted July 5, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

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A B 푛1 푛2 푛3 … 푛휎

훼3 ←Time

훼1 훼2

Space 퐿 →

C D

Populationsize Nutrientconcentration

Time (1/훿) Space (퐿)

Figure 1. A model with spatial structure and metabolic trade-offs supports more species than expected from the Figure 2. Spatial structure reduces diversity compared to principle of competitive exclusion. Example with three nu- the well-mixed limit of instantaneous nutrient diffusion. (A) trients and eleven species starting with equal populations. Top: A well-mixed population of ten species with equal ini- (A) Each species uptakes nutrients according to its enzyme- tial populations competing for two nutrients. All ten coex- ist at steady state. Bottom: Same as Top, but with dif- allocation strategy (ασ1, ασ2, ασ3). Because strategies sat- P ferent initial populations. The community reaches a new isfy the budget constraint i ασi = E, each can be repre- sented as a point on a triangle in strategy space. The nu- steady state. (Inset) Strategies ~ασ and resource supply trient supply ~s = (E/S)S~ = (0.25, 0.45, 0.3) is represented ~s = (0.4, 0.6). (B) Top: Same species and nutrient supply as a black diamond. Colors correspond to strategies and are as A, but in a spatial environment with nonzero nutrient dif- consistent throughout the figure. (B) Each species occupies fusion time. Only three species survive. Bottom: Same as a fraction of a one-dimensional space (a ring) and has a cor- Top, but with different initial populations. The community reaches the same steady state. (C ) Fraction of initial species responding time-dependent nσ(t). Here, the coexisting at steady state with ~s = (0.4, 0.6); a population nutrient diffusion time τD is 400. (C ) Population dynamics −6 from A. Nine species coexist on three nutrients. (D) Con- is considered extinct if nσ/L < 10 (mean ± SD for 400 centrations of the three nutrients at steady state (vertical random sets of ten strategies). The well-mixed model has black lines denote boundaries between populations). survival fraction 0.99±0.08. (D) Effective number of species M at steady state (mean ± SD for same strategies as C ).

Biodiversity tions shrink until the nutrient fluxes are balanced or the How does territorial spatial structure influence bio- species goes extinct. This contrasts with the well-mixed diversity? As an illustrative example, we consider ten case, where at steady state all the nutrient concentra- species competing for two resources. The simplex in the tions are equal so every strategy can coexist [8]. Thus inset of Fig. 2A shows how each of the strategies (col- territorial spatial structure heightens competitive differ- ored dots) and the nutrient supply (diamond) divide ences between strategies, even when all obey the same between the two nutrients. In Fig. 2A, the nutrients trade-offs. are well-mixed (τD = 0), and all ten species coexist How representative is the behavior seen in Fig. 2A at steady state. The steady state of the spatial case and B? In Fig. 2C and D we show results for many shown in Fig. 2B still exceeds competitive exclusion, randomly generated territorial communities, confirming with three species coexisting on two resources, but much that the loss of biodiversity is a generic feature of spa- of the biodiversity is lost. This behavior is striking, as it tial structure, and that the nutrient diffusion time τD contrasts with many competition models where spatial acts as a control parameter for biodiversity. In the well- structure increases diversity relative to the well-mixed mixed model, all ten species typically coexist. (See [8] case [20–26]. In those models, diversity increases be- for a discussion of the “convex hull condition” for coex- cause spatial segregation, combined with local interac- istence.) Figure 2C shows the mean fraction of species tions, weakens interspecific competition. Here, however, coexisting at steady state for nonzero τD. Spatial com- the resource environment is uniformly coupled via dif- munities still violate competitive exclusion, but a large fusion, so competition remains strong. Strategies that fraction of species go extinct. Even among those that are poorly matched to the nutrient supply allow unused survive, spatial structure reduces biodiversity by ren- nutrients to diffuse away to competitors; such popula- dering abundances highly unequal. Using the same data bioRxiv preprint doi: https://doi.org/10.1101/694257; this version posted July 5, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

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nutrients does not explain the difference in outcomes. However, diverse steady states proliferate as the supply becomes more balanced between nutrients. What dis- tinguishes high diversity outcomes? Figures 3B and D show every strategy present in every community with high diversity. Diverse communities have one thing in common: they lack species in the region of strategy Pp space where Rσ ≡ i Rσi < p. Here, Rσi ≡ Si/ασi is the uniform concentration of Nutrient i an isolated population with uptake ασi would produce given a sup- ply rate Si. Thus Rσ is the total nutrient concentration maintained by and sustaining an isolated species σ at steady state. For comparison, Rσ diverges for special- ists (ασi = 0), while a perfect generalist (ασi = 1/p) has Rσ = p, as does a strategy that perfectly matches the supply (ασi = Si). Strategies satisfying Rσ < p sur- vive on even lower total nutrient concentrations, so we Figure 3. Steady-state diversity is governed by a simple christen them “.” Their ability to create and condition: diversity crashes if there is an “oligotroph” whose survive on the minimum total nutrient concentration al- strategy satisfies Rσ < p. (A) Probability of effective num- ber of species M at steady state. For each nutrient sup- lows them to drive competitors extinct, thus reducing ply, we simulated 2000 sets of 20 strategies. Strategies were diversity. This recalls Tilman’s famous result that the chosen uniformly at random except the case shown in red species with the lowest equilibrium concentration of its ∗ (~s = (0.2, 0.8)), where oligotrophs were excluded. τD = 10 limiting resource (the lowest R ) can displace all oth- here and below. (B) For ~s = (0.4, 0.6) (orange in A), we ers competing for that resource [15]. However, the R∗ plot all strategies that appear in the most diverse 10% of rule is due to a species’ innate superiority in consum- simulations (90th percentile and above of M). No strategies ing a single resource, whereas the oligotroph condition appear in the oligotroph region, demarcated by the blue arises in a competition for multiple resources between dashed lines. (C ) Same as A but for three nutrients. (D) intrinsically equal species. Strategies that appear in the most diverse 10% of simula- tions for ~s = (0.2, 0.4, 0.4) (teal in C ). The oligotroph region To test whether it is simply the presence/absence of is nearly empty. oligotrophs that controls overall biodiversity, we gen- erated random communities in an environment with an asymmetric nutrient supply (s1 = 0.2), but ex- cluded oligotrophs. The resulting steady-state commu- as C, Fig. 2D quantifies this via the effective number nities are much more diverse (Fig. 3A, red) than the P of species M = exp{(H)}, where H = − σ pσ log pσ case where oligotrophs are allowed (Fig. 3A, purple). is the Shannon entropy and pσ = nσ/L. (Intuitively, Hence an asymmetric nutrient supply reduces diversity M is the number of equal populations yielding H. See by increasing the probability that an oligotroph will be SI Appendix for full rank curves.) The aver- present. The oligotroph condition captures the intuition age community loses ≈ 1/3 of its steady-state diversity of the R∗ rule – species with low resource requirements as τD grows from 0.01 − 1600. In well-mixed commu- dominate – but does not require biological superiority nities, all ten species are typically present in compara- or preclude coexistence beyond competitive exclusion. ble proportions, but in the spatial model, one species dominates. Once this population is large compared to p E/D, increasing τD adds to the “bulk” population in Alternative Steady States and Slow Dynamics its interior, decreasing overall diversity (see SI Appendix for details). However, aggregate measures of diversity How does the outcome of spatial competition depend in 2C and D belie a wide distribution of outcomes. For on initial conditions? Consider the well-mixed system example, only three species survive in Fig. 2B, whereas in Fig. 2A. All that differs between the top and bottom nine coexist in Fig. 1. Why are some steady-state com- subplots are the initial populations, but the same set of munities so much more diverse than others? species has two very different steady states; not even the In order to identify which features of the initial set hierarchy of populations is preserved. In fact, there is an of species determine steady-state diversity, we gener- m − p dimensional degenerate manifold of fixed points ated many random communities with species drawn uni- corresponding to the communities that construct the ∗ formly from strategy space. Figures 3A and C show the same steady-state nutrient environment ci = S/E ∀ i. distributions of the steady-state diversity M as a func- The final population may lie anywhere on this manifold. tion of s1, the supply of Nutrient 1. The number of By contrast, in the spatial of Fig. 2B, both bioRxiv preprint doi: https://doi.org/10.1101/694257; this version posted July 5, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

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Figure 5. (A, B) Two species competing for two nutrients, with supply ~s = (0.3, 0.7) and τD = 400. Arrows indicate flow away from unstable fixed points. (A) Bistability for α11 = 0.29. (B) Allee effect for α11 = 0.31. Species 2 goes Figure 4. Spatial structure replaces the steady-state degen- extinct if its initial population is too low. (C ) The Allee eracy of the well-mixed case with slow modes in population effect in a competition with ten species and two nutrients, space. (A) Trajectories in population space for a three-way with ~s = (0.4, 0.6) and τD = 100. The blue and brown competition at different values of τD (Inset: strategies and species displace each other, depending on initial conditions. supply). The direction and color of the arrows show the di- rection and magnitude of dnσ/dt, respectively. (B) Same as A, but with stochastic dynamics due to random births and nities with m − p  1 could have tens or hundreds of deaths; see SI Appendix for details. (Inset: Trajectory color as a function of time.) Left: Species 3 drifts to extinction. slow modes for population changes. These modes shape the response to perturbations: a microbial community Right: nσ(t) for the population trajectory at Left. might recover from one antibiotic very rapidly and an- other very slowly, depending on the shift in popula- tion space. Even without an intervention, real popula- sets of initial populations converge to the same unique tions will have stochastic fluctuations around the steady steady state. state. Figure 4B shows trajectories through population The relationship between the steady states in the space for the same species and nutrient supply as in Fig. well-mixed and spatial regimes can be visualized in a 4A, but with demographic noise due to stochastic births simple example. Figure 4A shows the phase behav- and deaths. Ecological drift is confined to the slow ior of three species competing for two resources. Here, manifold, and fluctuations primarily excite the popula- m − p = 1, so the well-mixed case (left) has a one- tion’s “soft mode” of the balance between Species 2 and dimensional degeneracy of steady states. In the spatial Species 3. These two have similar strategies, and either community (middle/right), the degenerate manifold col- can drift to extinction, whereas Species 1 always sur- lapses to a single fixed point. (Here the fixed point is vives. In the absence of noise, increasing τD decreases unique, but this is not always the case; see Fig. 5.) fixed-point diversity (Fig. 2D). With noise, however, This discontinuous change in the steady states is re- steady states in the well-mixed limit are unstable to flected in Fig. 2C and D, where the diversity for any fluctuations along the degenerate manifold. Increasing τD 6= 0 is substantially lower than for the well-mixed the “restoring force” (∼ τD) can prevent species from limit τD = 0. Figure 4A also clarifies another striking fluctuating to extinction, and so spatial structure can difference between Fig. 2A, in which the well-mixed stabilize diversity. community approaches steady state at approximately Although the well-mixed case has degenerate steady the individual death rate δ, and Fig. 2B, in which the states, the steady-state nutrient environment is unique, spatial community approaches steady state orders of and small initial population differences lead to small dif- magnitude more slowly. This emergent slow timescale ferences in the steady state (see Fig. 4A). By contrast, and the breaking of degeneracy are intimately related: spatial communities can have multiple steady-state nu- for any nonzero diffusion time τD, the degenerate man- trient environments, and similar populations may end ifold becomes a corresponding slow manifold, which the in very different steady states. For example, in a com- population rapidly reaches and then crawls to a fixed petition of two species for two resources, Fig. 5A and point. Linear stability analysis around this fixed point B show the steady states as a function of α21, with α11 reveals a relaxation time t ∼ 1/τ , which diverges slow D held fixed. (ασ1 is the enzyme allocation of Species σ as τ → 0. (See SI Appendix for details.) D to Nutrient 1. Due to trade-offs, this also fixes ασ2.) What are the ecological implications of this slow re- In Fig. 5A, there are two alternative steady states laxation to steady state? In general, diverse commu- with both species coexisting. The unstable fixed point bioRxiv preprint doi: https://doi.org/10.1101/694257; this version posted July 5, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

6 separates the relatively equal community of the lower branch from the upper branch where Species 1 dom- inates. This bistability leads to discontinuous transi- tions, where small changes (the populations crossing the separatrix, or the strategy exiting the bistable phase) can have dramatic consequences. Figure 5B shows an- other region of strategy space where the outcomes are bistable, but now the alternatives are coexistence and exclusion. Above the unstable fixed point in Fig. 5B, Species 1 drives Species 2 to extinction. Otherwise, they coexist with Species 2 having the larger popula- tion. This is an example of the Allee effect: Species 2 can only survive if its population exceeds a thresh- old. (See SI Appendix for a phase diagram of the full strategy space.) The Allee effect persists in more complex communi- ties. Figure 5C shows a ten-species competition where the brown and blue species can displace each other de- pending on the initial conditions, modifying the eight other species’ fates in the process. Thus multistabil- Figure 6. Territorial spatial structure renders diversity ro- ity and the Allee effect emerge naturally in our territo- bust to variation in enzyme budgets. (A) In a community rial model, even though the species interact exclusively with equal enzyme budgets and ~s = (0.4, 0.6) (Left), ten through competition for resources. species coexist in the well-mixed model (Center), whereas only three coexist in the spatial model (Right). (B) In a community with the same strategies as A but unequal en- Unequal Enzyme Budgets zyme budgets (Left), two species coexist in the well-mixed model (Center), whereas seven coexist in the spatial model (Right). (C ) Probability of effective number of species M at Metabolic trade-offs are plausible because all mi- steady state for random enzyme budgets, in the well-mixed crobes face the same biophysical constraints on model with ~s = (0.4, 0.6). For 2000 sets of strategies, each metabolism and protein production, but trade-offs are of 20 initial species’ enzyme budgets Eσ was drawn from unlikely to be exact in real ecosystems. How does this N (1, δE). (D) Same as C but for spatial model. impact biodiversity in our model? Figure 6A shows re- P sults for ten species with exact trade-offs ( i ασi = E for all species) competing for two resources. The well- DISCUSSION mixed community is very diverse, while the spatial com- munity is not. In Fig. 6B, each species allocates the same fraction of its enzyme budget to each nutrient as We analyzed a model of spatial resource compe- in 6A, but each with its own total enzyme budget Eσ. tition among territorial surface communities such as Diversity collapses in the well-mixed system, but the biofilms, vegetation, or coral. Each species has a con- spatial community actually becomes more diverse. Fig- crete metabolic strategy subject to biophysical trade- ures 6C and D show that this behavior is typical via offs. The nutrient environment has no intrinsic hetero- comparison of the steady-state diversity of communi- geneity but is globally coupled via diffusion, so com- ties where each species’ enzyme budget is drawn from a petitors shape it via consumption. We found that the Normal Distribution with mean 1 and standard devia- resulting spatial structure restricts biodiversity, in stark tion δE. Well-mixed communities (C ) are very diverse contrast to previous models where spatial segregation if trade-offs are exact (δE = 0), but any disparity in increases diversity by weakening competition. In the the enzyme budgets causes diversity to collapse; only simplest of these cases, different resources are parti- one or two species survives at steady state. The spa- tioned into different regions, providing spatial niches [9– tial communities (D) are less diverse for δE = 0, but 13]. Alternatively, competitors may self-organize into their diversity hMi actually increases with δE. This patches linked by migration [14–16, 18, 19] or into ag- is due to an asymmetric effect: oligotrophs lose their gregates with local interactions [20–26]. External re- with a very small decrease in Eσ, but other source gradients can also increase diversity, because dif- species require a large increase in Eσ to dominate (see fusion of dense motile populations prevents any species SI Appendix for details). As a result, spatial commu- from monopolizing resource-rich regions [9, 10]. In our nities with imperfect trade-offs can display biodiversity model, the situation is very different. Because the ex- well beyond the competitive-exclusion limit. ternal nutrient supply is uniform and the entire space is bioRxiv preprint doi: https://doi.org/10.1101/694257; this version posted July 5, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

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linked via diffusion, no spatial niches emerge; competi- to persist with imprecise metabolic trade-offs. In the tors have nowhere to hide. This is reminiscent of ecolog- well-mixed system without noise, any deviation from ex- ical reaction-diffusion models without external sources, actly equal enzyme budgets leads to ecosystem collapse where global coupling reduces diversity [26] and nonuni- [8]. Spatial communities, however, remain diverse with form steady states only become possible for unequal dif- only approximate trade-offs. In fact, variation in en- fusion coefficients or complex geometries [27]. Our com- zyme budgets actually increases mean diversity by im- munities are fundamentally different, however, as they pairing oligotrophs. The persistence of diversity beyond occupy exclusive territories and exceed the competitive- competitive exclusion with inexact trade-offs makes it exclusion limit in spite of a simple geometry and uni- more credible that trade-offs play a role in maintaining form diffusion coefficients. the surprising diversity of real ecosystems. What controls diversity in our model? The degree of Our results suggest several future research directions. nutrient mixing τD controls the evenness of abundances A two-dimensional extension of the model exhibits the by setting the population of the dominant species, same loss of biodiversity due to oligotrophs and uneven while the presence of oligotrophs distinguishes steady abundances (see SI Appendix), and it will be interest- states of high coexistence from those with many extinc- ing to explore 2D in more depth. tions. Oligotrophs drive competitors extinct because One might also consider resources that diffuse at dif- they have the lowest total nutrient requirements, in ferent rates. This can lead to nonuniform steady states ∗ rough analogy with the lowest R rule for well-mixed in reaction-diffusion systems [27]. Finally, in microbial systems [15]. However, oligotrophs obey the same communities, gene regulation and evolution are often trade-offs as every other species, and their dominance relevant on ecological timescales, so it would be natural arises from the relationship between their strategies and to allow species to modify their strategies. the nutrient supply rather than any innate superiority. In summary, we find that spatial structure engenders The composition of the nutrient supply sets the strategy more realistic communities: it curtails the unlimited range of oligotrophs, so it is effectively another control diversity of the well-mixed model, but allows for coex- parameter for diversity. In spite of highly nonlinear dy- istence beyond the competitive exclusion principle even namics and many parameters, the oligotroph condition in the absence of exact metabolic trade-offs. Our results provides a simple criterion for diversity. demonstrate that mechanistic interactions, arising from Spatial structure also provides a novel mechanism biophysical constraints such as space and metabolism, for discontinuous transitions between alternative steady can allow even simple models to capture some of the states. Such sudden shifts, or “catastrophes”, attract rich behaviors of real ecosystems. significant attention due to their implications for ecosys- tem resilience [31]. The Allee effect occurs in a large variety of ecosystems [32], and is particularly relevant to the conservation of rare species. It is usually un- derstood as the result of transparently cooperative pro- METHODS cesses, such as production of a public good [32], and modeled via an explicit cooperative term. In resource- The population ODEs (Eq. 3) were solved numeri- competition models, multistability has been observed cally using Mathematica’s “NDSolve”. The cσ,i depend when species consume nutrients one at a time [33] or on nσ through the coefficients {Aσi,Bσi}, which are with unequal stoichiometries [34]. Here, both the Allee fixed by requiring that ci(x) be continuous and dif- effect and multistability emerge naturally from the abil- ferentiable at the population boundaries. The nutri- ity of a population to render its resource environment ent equations are simpler under the change of variables P more favorable to itself. Interestingly, the Allee-effect x → x − nσ0 . Then cσ,i(x) runs from 0 to nσ, yield- species are oligotrophs, underscoring the special ability σ0<σ these strategies have to impact their ecosystems. ing the system It has been observed that spatial structure increases the time to reach equilibrium [35]. Here we showed cσ,i(nσ) = cσ+1,i(0) (4) precisely how a new dynamical timescale emerges from c0 (n ) = c0 (0) spatial structure. We found that the slow dynamics are σ,i σ σ+1,i confined to a manifold in population space. These slow modes of the population are subject to large fluctua- which was solved using Mathematica’s “LinearSolve” tions due to noise (e.g. demographics). Slow relaxation with periodic boundary conditions (cmi(nm) = c1i(0), also means that for a rapidly changing nutrient supply, corresponding to a ring). the population might never reach steady state, poten- Details on the well-mixed model, stochastic dynam- tially saving some species from extinction. ics, and figure parameters can be found in SI Appendix. Finally, we find that spatial structure allows diversity Code will be available on GitHub. bioRxiv preprint doi: https://doi.org/10.1101/694257; this version posted July 5, 2019. 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ACKNOWLEDGEMENTS ter for the Physics of Biological Function (PHY- 1734030) (B.G.W.), the National Institutes of Health We thank S. Levin for valuable conversations and (www.nih.gov), Grant R01 GM082938 (N.S.W.), and suggestions. This work was supported in part by the Princeton University Diekman Genomics Fund the National Science Foundation, through the Cen- (A.P.).

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