Neural ordinary differential equations for ecological and evolutionary time-series analysis

Inferring the functional shape of ecological and evolutionary processes from time-series data can be challenging because processes are often not describable with simple equations. The dynamical coupling between variables in time series further complicates the identification of equations through model selection as the inference of a given process is contingent on the accurate depiction of all other processes. We present a novel method, neural ordinary differential equations (NODEs), for learning ecological and evolutionary processes from time-series data by modelling dynamical systems as ordinary differential equations and dynamical functions with artificial neural networks (ANNs). Upon successful training, the ANNs converge to functional shapes that best describe the biological processes underlying the dynamics observed, in a way that is robust to mathematical misspecifications of the dynamical model. We demonstrate NODEs in a population dynamic context and show how they can be used to infer ecological interactions, dynamical causation and equilibrium points. We tested NODEs by analysing well-understood hare and lynx time-series data, which revealed that prey-predator oscillations were mainly driven by the interspecific interaction, as well as intraspecific densitydependence, and characterised by a single equilibrium point at the centre of the oscillation. Our approach is applicable to any system that can be modelled with differential equations, and particularly suitable for linking ecological, evolutionary and environmental dynamics where parametric approaches are too challenging to implement, opening new avenues for theoretical and empirical investigations.

Automated summary

The NODEs method is a novel approach for learning ecological and evolutionary processes from time-series data by modelling dynamical systems as ordinary differential equations and dynamical functions with artificial neural networks. It not only describes the functional shapes behind biological processes, but also robustly handles mathematical misspecifications of the dynamical model.