# Whence Lotka-Volterra?

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## Abstract

Competition in ecology is often modeled in terms of direct, negative effects of one individual on another. An example is logistic growth, modeling the effects of intraspecific competition, while the Lotka-Volterra equations for competition extend this to systems of multiple species, with varying strengths of intra- and interspecific competition. These equations are a classic and well-used staple of quantitative ecology, providing a framework to understand species interactions, species coexistence, and community assembly. They can be derived from an assumption of random mixing of organisms, and an outcome of each interaction that removes one or more individuals. However, this framing is somewhat unsatisfactory, and ecologists may prefer to think of phenomenological equations for competition as deriving from competition for a set of resources required for growth, which in turn may undergo their own complex dynamics. While it is intuitive that these frameworks are connected, and the connection is well-understood near to equilibria, here, we ask the question: when can consumer dynamics alone become an exact description of a full system of consumers and resources? We identify that consumer-resource systems with this property must have some kind of redundancy in the original description, or equivalently there is one or more conservation laws for quantities that do not change with time. Such systems are known in mathematics as integrable systems. We suggest that integrability in consumer-resource dynamics can only arise in cases where each species in an assemblage requires a distinct and unique combination of resources, and even in these cases, it is not clear that the resulting dynamics will lead to Lotka-Volterra competition.

## Keywords

Lotka-Volterra Competition Consumer-resource dynamics Integrability## Notes

### Acknowledgments

I thank Mario Muscarella, Nick Laracuente, and Alice Doucet Beaupré for constructive criticism on an earlier draft of this manuscript, and Rafael D’Andrea for productive conversations on the history of modeling colimitation. I thank two anonymous reviewers for extremely helpful comments and constructive criticism. I acknowledge Peter Abrams for sending comments and references relating to the separation of timescales approximation, and Sally Otto and Troy Day for commenting on the history of reducing the number of degrees of freedom in SIS and related models.

### Funding information

This works was supported by the Simons Foundation Grant #376199 and the McDonnell Foundation Grant #220020439.

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