Stability criteria for the consumption and exchange of essential resources
Authors: Gibbs Theo , Zhang Yifan , Miller Zachary R. , O’Dwyer James P. 
Published: November 2021

Models of pairwise interactions have informed our understanding of when ecological communities will have stable equilibria. However, these models do not explicitly include the effect of the resource environment, which has the potential to refine or modify our understanding of when a group of interacting species will coexist. Recent consumer-resource models incorporating the exchange of resources alongside competition exemplify this: such models can lead to either stable or unstable equilibria, depending on the resource supply. On the other hand, 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 multiple resources. Consumed resources that do not lead to growth are leaked back into the environment, 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, once 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 we derive analytically apply to a broader class of network structures and resource inflow profiles, including cases in which 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 ecological mechanism is modeled directly.

Author summary

One of the longstanding challenges in community ecology is to understand how diverse ecosystems assemble and stably persist. Microbial communities are 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. In this paper, we incorporate a more realistic understanding of microbial metabolism into classic mathematical models of consumer-resource dynamics. In our model, bacteria deplete multiple abiotic nutrients but only grow on one of these resources. In addition, they recycle some of the nutrients they consume back into the environment as new resources. We analytically derive criteria which, if satisfied, 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.