A Bayesian approach to the ecosystem inverse problem

This study investigates a probabilistic approach for the inverse problem associated with blending time-dependent ecosystem models and observations. The goal is to combine prior information, in the form of ecological dynamics and substantive knowledge about uncertain parameters, with available measurements. Posterior estimates of both the time-varying ecological state variables and the model parameters are obtained, along with their uncertainty. Ecological models of interacting populations are considered in the context of a nonlinear, non-Gaussian state space model. This comprises a nonlinear stochastic difference equation for the ecological dynamics, and an observation equation which relates the model state to the measurements. Complex error processes are readily incorporated. The posterior probability density function provides a complete solution to the inverse problem. Bayes’ theorem allows one to obtain this posterior density through synthesis of the prior information and the observations. To illustrate this Bayesian inverse method, these ideas are applied to a simple ecosystem box model concerned with predicting the seasonal co-evolution of a population of grazing shellfish and its two food sources: plankton and detritus. Observations of shellfish biomass over time are available. Lognormal system noise was incorporated into the ecosystem equations at all time steps. Ingestion and respiration parameters for shellfish growth are considered as uncertain quantities described by beta distributions. Stochastic simulation was carried out and provided predictions of the model state with uncertainty estimates. The Bayesian inverse method was then used to assimilate the additional information contained in the observations. Posterior probability density functions for the parameters and time-varying ecological state were computed using Markov Chain Monte Carlo methods. The ecological dynamics spread the measurement information to all state variables and parameters, even those not directly observed. Probabilistic state estimates are refined in comparison to those from the stochastic simulation. It is concluded that this Bayesian approach appears promising as a framework for ecosystem inverse problems, but requires careful control of the dimensionality for practical applications.