Stochastic model corrections for reduced Lotka–Volterra models exhibiting mutual, commensal, competitive, and predatory interactions

Abstract

We explore model-form error and how to correct it in systems of ordinary differential equations. In particular, we focus on the Lotka–Volterra equations, which are used broadly in fields such as ecology, biology, economics, chemistry, and physics. Accounting for every object and their complex interactions with a complete model often becomes infeasible, thereby requiring reduced models. However, reduced models may omit vital relationships, resulting in discrepancies between reduced model predictions and observations from the true system. In this work, we propose a model correction framework for decreasing such discrepancies. Specifically, we embed a stochastic enrichment operator into the reduced model’s system of equations. The enrichment operator is theory-informed, calibrated with observations from the complete model, and extended to extrapolative combinations of parameters and initial conditions. The complete model involves N species, while the reduced and enriched models only track M<N species. Numerical results show the enriched models significantly decrease discrepancies, consistently predict equilibria, and improve the species’ transient behavior.