Identification of nonlinear noisy dynamics of an ecosystem from observations of one of its trajectory components

The problem of determining dynamical models and trajectories that describe observed time-series data allowing for the understanding, prediction and possibly control of complex systems in nature is of a great interest in a wide variety of fields. Often, however, only part of the system's dynamical variables can be measured, the measurements are corrupted by noise and the dynamics is complicated by an interplay of nonlinearity and random perturbations. The problem of dynamical inference in these general settings is challenging researchers for decades. We solve this problem by applying a path-integral approach to fluctuational dynamics, and show that, given the measurements, the system trajectory can be obtained from the solution of the certain auxiliary Hamiltonian problem in which measured data act effectively as a control force driving the estimated trajectory toward the most probable one that provides a minimum to certain mechanical action. The dependance of the minimum action on the model parameters determines the statistical distribution in the model space consistent with the measurements. We illustrate the efficiency of the approach by solving an intensively studied problem from the population dynamics of predator-prey system where the prey populations may be observed while the predator populations or even their number is difficult or impossible to estimate. We apply our approach to recover both the unknown dynamics of predators and model parameters (including parameters that are traditionally very difficult to estimate) directly from measurements of the prey dynamics. We provide a comparison of our method with the Markov Chain Monte Carlo technique.