Fitting deterministic population models tostochastic data: from trajectory matching tostate-space models

Abstract

Population and community ecology traditionally has a very strong theoretical foundation with well-known dynamical models, such as the logistic and its variations, and many modification of the classical Lotka-Volterra predator-prey and interspecific competition models. More and more, these classical models are being confronted with data via fitting to empirical time series for purposes of projections or for estimating model parameters of interest. However, using statistical models to fit theoretical models to data is far from trivial, especially for time series data where subsequent measurements are not independent. This raises the question of whether statistical inferences using pure observation error models, such as simple (non-)linear regressions, are biased, and whether more elaborate process error models or state-space models have to be used to address this complexity. In order to help empiricists, especially researchers working with experimental laboratory populations in micro- and mesocosms, make informed decisions about the statistical formalism to use, we here compare different error structures one could use when fitting classical deterministic ODE models to empirical data. We consider a large range of biological scenarios and theoretical models, from single species to community dynamics and trophic interactions. In order to compare the performance of different error structure models, we use both realistically simulated data and empirical data from microcosms in a Bayesian framework. We find that many model parameters can be estimated precisely with an appropriate choice of error structure using pure observation error or state-space models, if observation errors are not too high. However, Allee effect models are typically hard to identify and state-space models should be preferred when model complexity increases. Our work shows that, at least in the context of low environmental stochasticity and high quality observations, deterministic models can be used to describe stochastic population dynamics that include process variability and observation error. We discuss when more complex state-space model formulations may be required for obtaining accurate parameter estimates. Finally, we provide a comprehensive tutorial for fitting these models in R. Open researchCode for stochastic individual-based simulations is available from https://doi.org/10.5281/zenodo.5500442. A tutorial for fitting ODE models to time series data in R is presented in the Supplementary Information and is also available online https://github.com/benjamin-rosenbaum/fitting deterministic population models. Data (Fronhofer et al., 2020) will be provided via GitHub and Zenodo.