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Stability criteria for the consumption and exchange of essential resources [1]

['Theo Gibbs', 'Lewis-Sigler Institute For Integrative Genomics', 'Princeton University', 'Princeton', 'New Jersey', 'United States Of America', 'Yifan Zhang', 'Department Of Plant Biology', 'University Of Illinois', 'Urbana']

Date: 2022-09 

Abstract Models of consumer effects on a shared resource environment have helped clarify how the interplay of consumer traits and resource supply impact stable coexistence. Recent models generalize this picture to include the exchange of resources alongside resource competition. These models exemplify the fact that although consumers shape the resource environment, the outcome of consumer interactions is context-dependent: such models can have either stable or unstable equilibria, depending on the resource supply. However, these recent models focus on a simplified version of microbial metabolism where the depletion of resources always leads to consumer growth. Here, we model an arbitrarily large system of consumers governed by Liebig’s law, where species require and deplete multiple resources, but each consumer’s growth rate is only limited by a single one of these resources. Resources that are taken up but not incorporated into new biomass are leaked back into the environment, possibly transformed by intracellular reactions, thereby tying the mismatch between depletion and growth to cross-feeding. For this set of dynamics, we show that feasible equilibria can be either stable or unstable, again depending on the resource environment. We identify special consumption and production networks which protect the community from instability when resources are scarce. Using simulations, we demonstrate that the qualitative stability patterns derived analytically apply to a broader class of network structures and resource inflow profiles, including cases where multiple species coexist on only one externally supplied resource. Our stability criteria bear some resemblance to classic stability results for pairwise interactions, but also demonstrate how environmental context can shape coexistence patterns when resource limitation and exchange are modeled directly.

Author summary One longstanding challenge in community ecology is to understand how diverse ecosystems assemble and stably persist. Microbial communities pose a particularly acute example of this open problem, because thousands of different bacterial species can coexist in the same environment. Interactions between bacteria are of central importance across a wide variety of systems, from the dynamics of the human gut microbiome to the functioning of industrial bioreactors. As a result, a predictive understanding of which microbes can coexist together, and how they do it, will have far-reaching applications. Here, we incorporate a more realistic understanding of microbial metabolism into a classic mathematical model of consumer-resource dynamics. In our model, bacteria deplete multiple abiotic nutrients but their growth rates are only sensitive to one of these resources at a time. In addition, they recycle some of the nutrients they consume back into the environment as new (transformed) resources. We analytically derive criteria which guarantee that any number of microbes will coexist. We find that there are special types of interaction networks which remain stable even when resources are scarce. Our theory can be used in conjunction with experimentally determined interaction networks to predict which species assemblages are likely to stably coexist in a specified resource environment.

Citation: Gibbs T, Zhang Y, Miller ZR, O’Dwyer JP (2022) Stability criteria for the consumption and exchange of essential resources. PLoS Comput Biol 18(9): e1010521. https://doi.org/10.1371/journal.pcbi.1010521 Editor: Jacopo Grilli, Abdus Salam International Centre for Theoretical Physics, ITALY Received: December 6, 2021; Accepted: August 29, 2022; Published: September 8, 2022 Copyright: © 2022 Gibbs et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All relevant data are within the manuscript and the Supporting information files. The code for generating the data is available on GitHub at https://github.com/theogibbs/essential-stability-criteria. Funding: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-2039656 and Grant No. DGE-1746045. T.G. was supported by Grant No. DGE-2039656 and Z.R.M. was supported by Grant No. DGE-1746045. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. J.P.O. acknowledges funding from Simons Foundation Grant No. 376199 (www.simonsfoundation.org) and McDonnell Foundation Grant No. 220020439 (www.jsmf.org). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction Pairwise interaction models have informed our understanding of when competitive interactions will lead to stable equilibria. For example, these classic models imply the coexistence of two competing species when the strength of interspecific competition is less than the strength of intraspecific competition, as well as more general stability criteria for large, multi-species systems with randomly distributed interaction strengths [1–4]. On the other hand, models of pairwise interactions do not explicitly include the effect of environmental context, and this context has the potential to refine or modify our understanding of when a group of interacting species will coexist. For example, one species may exclude another if both compete for and rely on a given resource, but the same two species may coexist if that resource is replaced by two alternative resources, each of which is consumed by only one of the two species. Recent consumer-resource models incorporating the exchange of resources alongside resource competition have shed light on stable coexistence in systems where interactions are mediated by abiotic resources [5–11]. In these open systems, the environmental context is specified by resource inflow from outside, and stability turns out to depend both on the structure of which species consume and produce specific resources, and on the resource inflow rates. However, this recent theory has focused on a simplified version of microbial metabolism where the depletion of resources always leads to consumer growth. Specifically, the models studied in [5] assumed either that the impact of consumers on resources was proportional to the growth rate resulting from consumption of these substitutable resources (ie. the impact of resources on consumers), or else a generalization of this type of assumption [6, 12, 13] for multiplicative colimitation by essential resources. In general, the rates at which a consumer depletes resources are not always proportional to the benefits that it derives from their consumption. This kind of mismatch between impact and growth can arise for many reasons. One example is ‘waste’ by consumers with large uptake rates [14], where otherwise usable resources are degraded and made unavailable for other consumers. An even more basic origin for this mismatch arises when internal metabolism requires multiple resources, but is only limited by the availability of a single resource—known as Liebig’s law [15–19]. In this case, a consumer will deplete and make use of multiple resources, but its growth rate will only be sensitive to one of those resources at a time. Thus, a given consumer can strongly affect the growth rate of others that are limited by one of its own non-limiting resources. The consumption of non-limiting resources does not result in biomass production or cell division. These non-limiting resources could instead be used for cellular maintenance, or transformed via cellular metabolism into byproducts and then leaked back into the environment [20–22]. The depletion of non-limiting resources is one way to generate a mismatch between consumption and microbial growth. Because of the conservation of resource biomass, this mismatch gives rise to the production of resource byproducts, and hence the potential for cross-feeding. For example, a recent reconstruction of the metabolic evolution of the marine cyanobacteria Prochlorococcus suggests that Prochlorococcus is nitrogen limited and leaks organic carbon, forming a mutualism with the heterotrophic bacterium SAR11 [14]. Both the mismatch between depletion and growth as well as the cross-feeding of nutrients may have important ecological consequences because they shape the resource environment for competing species. Yet, we do not know whether this more realistic picture of microbial metabolism changes our theoretical understanding of coexistence in diverse microbial communities. In this paper, we model an arbitrarily large system of consumers undergoing growth governed by Liebig’s law. We consider dynamics near a positive equilibrium where each consumer is limited by a single, distinct resource, but can potentially deplete additional resources. Each consumer leaks the resources it does not use for growth back into the environment in other forms, thereby tying the mismatch between growth and depletion directly to cross-feeding. Although we are primarily modeling microbial communities, the mismatch between growth and depletion is a general ecological phenomenon, and our results apply equally well to non-microbial systems. We find that with certain additional assumptions it is possible to analytically derive sufficient stability criteria in terms of resource inflow rates and ecological network structure which guarantee that a feasible equilibrium is stable. These criteria mirror those found earlier for a different form of positive interactions [5], and show that the structure of consumption and production networks, as well as the environmental context, affect the stability of this equilibrium. Our theory generalizes well-known results for low diversity consumer-resource dynamics [23], but also identifies stabilizing interaction network structures which do not have a clear low-dimensional analog. Using simulations, we show that our stability criteria apply more broadly to network structures and parameter regimes which do not precisely satisfy our mathematical assumptions, including situations where many microbial species coexist on only one externally supplied resource [9, 10]. As a result, our theory could be used to select species assemblages whose consumption preferences and nutrient production networks permit coexistence in a specified resource environment.

Discussion Unlike many classic ecological models in which species interactions are completely determined by the abundances of the competing species [38, 39], our model explicitly tracks the dynamics of resources [8, 40–42]. As the resource abundances vary over time, the per-capita effect of one consumer on another can change, in contrast to most pairwise models, where these per-capita interactions are fixed in time. Additionally, consumers grow according to Liebig’s law, meaning that each consumer is limited by only a single resource at a time, even though they can deplete other resources. Liebig’s law has been found to accurately describe microbial growth rates, as well as the resource limitation properties of plants, across many different ecosystems [12, 13, 18, 43–45]. In our model of Liebig’s law, there is a mismatch between microbial growth and resource depletion that directly gives rise to the exchange of resources. In this paper, we investigated how these two linked processes—microbial growth governed by Liebig’s law and the subsequent cross-feeding of nutrients—jointly affect stability in arbitrarily large microbial communities. We analytically determined two stability criteria that are together sufficient for stability of the entire community. Our first stability criterion ensures that the interactions between the microbial species are reciprocal. Under our first condition, each consumer must affect the limiting resource of other consumers in exactly the same way that every other consumer affects its limiting resource. In our second stability criterion, we identified an effective interaction matrix for the community. When our first stability condition is satisfied, the stability of this effective interaction matrix implies that the equilibrium where all consumers coexist is also stable. Guided by these stability criteria, we found four main results. First, our two stability criteria were not only sufficient for, but in fact appear exactly predictive of, stability in simulated communities. Second, reciprocal interaction structures protected the community from instability at low consumer abundances, but communities with non-symmetric interactions generically became unstable when resources were scarce. Third, our analytical results applied qualitatively to situations where microbes compete for a single external resource and therefore have highly variable consumer abundances at equilibrium. Fourth, interaction networks that were not exactly symmetric still promoted stability, as long as the spectra of these networks had small imaginary parts. Our model incorporates a more mechanistic picture of microbial interactions by considering how essential resources and cross-feeding alter ecological dynamics. At the same time, it is still a highly coarse-grained version of true microbial metabolism. We have focused on a particular form of Liebig’s law with explicit stochiometric requirements, but other models without these requirements have been investigated as well [46–48]. In our consumption matrix parameterizations, we have also implicitly assumed that each consumer can deplete many of the available resources. Similarly, we have assumed that any resource can be produced from any other. In other words, although we have conserved total biomass in our model, we have not ensured that the basic biochemical building blocks are conserved when consumed resources are being converted into other nutrients. In reality, there are stochiometric rules that these matrices must obey [49–51]. Intriguingly, recent experiments [10] have shown that resource production networks are approximately, though not precisely, hierarchical because of biochemical constraints. Understanding how the chemical properties of abiotic nutrients and the metabolic strategies of specific bacterial strains constrain the consumption and production networks is therefore an important direction for future work. It would be particularly interesting to evaluate whether these biochemical constraints on resource exchange create production networks with spectral properties similar to those we have shown to promote stability. Even though our consumer-resource model does not capture the full complexity of microbial metabolism, it still produced stability criteria that refine our understanding of microbial coexistence. Recent theory [52, 53] has shown that population abundances do not affect stability in a randomly parameterized Lotka-Volterra model. In the present work, we find the opposite result—given consumption and production networks which do not satisfy our symmetry conditions, some choices of equilibrium consumer abundances generate stable systems, while others do not. By contrast, our theoretical results do align with other analyses of consumer-resource dynamics. The resilience of a food web (defined as the speed at which the abundances return to equilibrium after a perturbation) has been shown to increase as the residence time of nutrients in the system decreases [54]. One possible interpretation of our results is that, when resource supply is low, resources remain in the community for longer times, inducing instability, but further work is needed to understand this connection more completely. Recent simulations of model microbial communities with cross-feeding showed that, when consumers are resource limited, the constituent species interact in a characteristic pattern at equilibrium [7]. These results mirror our simulations, where ecosystems with specific symmetric interaction structures are protected from instability at low resource inflow. In [7], the characteristic interaction patterns emerge from community assembly, while in our theory, we impose them from the outset. This connection is particularly interesting because it suggests that special interaction structures may emerge from assembly processes in specific resource environments. Similarly, a recent mathematical analysis of consumer-resource models with multiple forms of consumption also showed that resource inflow mediates a transition to instability [6]. This recent theory, however, treats resource exchange as coming directly from consumer biomass, as though resource production were an additional source of mortality, rather than as an explicit transformation of resources. As a result, the overall strength of production for each species is a tunable parameter and, if it exceeds the total consumption of a single species, then the feasible equilibrium can be unstable [6]. In our model, the strength of production is determined by the resource consumption that is not used for growth, and so cannot exceed the total consumption for each species. Nevertheless, we find that unstable equilibria are still possible due to the mismatch between resource depletion and consumer growth, rather than from the strength of resource production overwhelming resource depletion, as in [5]. Our results can also be seen as a multi-species generalization of classic stability results for species competing for two resources (termed contemporary niche theory) where instability occurs because of the difference between impact and sensitivity vectors [23, 55–57]. In contemporary niche theory, there are three criteria which must be satisfied for a stable equilibrium to exist. First, the species zero net growth isoclines (ZNGIs) must intersect. In our model, this is always true, since each species is limited by a single resource, so it is straightforward to find resource abundances where every consumers’ growth is zero. Second, each species must impact the resources that it finds most limiting more strongly than it impacts other resources. Our second stability criteria is a direct and quantitative generalization of this result—if each species more strongly regulates the resource it requires for growth than it affects all other resources in the system, then the equilibrium is stable. Third, the supply point must lie above the ZNGIs for the coexisting species. In our theory, we ensure that this criteria is satisfied by requiring the equilibrium to be feasible through Eq (3). We also show in the Feasibility Analysis section of the S1 Appendix that our second stability criterion and the feasibility criteria are closely related—as species more strongly regulate their most limiting resource, the likelihoods of both stability and feasibility are increased. Our theory can be interpreted as an extension of prior work applying contemporary niche theory to competition for essential nutrients to diverse microbial communities [23, 58]. By contrast, there is no clear analog of our first stability criterion for low diversity ecosystems. It can, however, be interpreted as requiring perfectly balanced pairwise competition between the consumers, even though the model itself is not built on pairwise competition coefficients. The phenomenon that reciprocity promotes stability has been found in other theoretical studies of microbial communities [6], but also in a diverse set of other fields, from the evolution of cooperation [59, 60] to the exchange of food in early societies [35, 61]. In addition to our symmetry condition, we showed numerically that other consumption and production networks that generate B matrices whose eigenvalues are purely real also prevent instability at low consumer abundances. Although we lack a precise understanding of how these network structures promote stability, we describe intuitively how the imaginary parts of the eigenvalues of B affect the spectrum of the Jacobian in the Stability Criteria section of the S1 Appendix. A more complete mathematical understanding of the connection between the spectrum of the interaction matrix B and the spectrum of the Jacobian for the entire community would give us a deeper understanding of why certain modes of resource exchange are stabilizing. There are a number of other important directions for future work. Previous simulations of consumer-resource dynamics governed by Liebig’s law have found that large numbers of species can coexist in oscillatory or chaotic dynamics [46–48], so it would be instructive to better understand the behavior of our model away from equilibrium. Similarly, we have only observed a single unique equilibrium in our models, but cross-feeding can generate multiple equilibria [62]. Our stability criteria may still be able to delineate which of these multiple equilibria are attracting. It would also be interesting to rigorously understand the stability properties of an ecosystem where consumers grow on many different substitutable resources at variable efficiencies but still leak resources back into the environment through cross-feeding [7]. Last, the model we have considered here is completely deterministic, and so the mismatch between resource depletion and consumer growth resulted from a specific modeling choice. In a stochastic model, depletion and growth may be decoupled through only the differing fluctuations that the consumers and resources undergo, potentially yielding the same stability transition we have observed in our model. More generally, the combination of stochastic drift and the biological mechanisms we have explored here could produce interesting macroecological patterns [63, 64]. Because our consumer-resource model connects coexistence patterns to empirically accessible quantities, our theoretical results can be tested experimentally. One direct test of our theory would be to design small microbial communities with experimentally-characterized interaction networks [11, 65]. Then, an experimentalist can manipulate, for example, the degree to which the interaction network is reciprocal, and observe whether or not coexistence if favored. Conversely, our theoretical results suggest an interpretation of coexistence outcomes when the exact consumption and production networks are unknown. When an experimentalist reduces the resource inflow rates in a serial dilution experiment, the resulting coexistence (or lack thereof) suggests which types of consumption and production networks may be present. This relationship between the interaction network and the eventual stability properties of the entire community could also potentially be used to constrain the space of possible interactions when inferring these parameters from microbial abundance data [66–69]. Because we showed numerically that our analysis applies to a variety of resource inflow profiles, our theory may also delineate the boundaries of stable coexistence in recent experiments where only one nutrient is externally supplied [9, 10, 70]. For example, recent work has shown experimentally how the resource production network explains the variation in species richness as more resources are externally supplied [10]. Our theory suggests that, if this metabolite production network has eigenvalues with small imaginary parts, the community will be better protected from resource scarcity. Future work should seek to further clarify how the relationship between coexistence outcomes and resource inflow changes depending on network structure. This line of research is especially important because of the difficulty in obtaining well-resolved and quantitative consumer-resource networks for diverse microbial communities.

Supporting information S1 Appendix. Supplementary proofs, calculations and simulation results. We provide proofs of the main results, additional numerical simulations and descriptions of all of the matrix parameterizations. https://doi.org/10.1371/journal.pcbi.1010521.s001 (PDF)

Acknowledgments We thank Seppe Kuehn, Simon Levin and Jonathan Levine for helpful comments and discussion. We also thank three reviewers for their insightful suggestions.

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