Seeing beyond invisible with noise: application to populational biology

The problem of determining dynamical models and trajectories that describe observed time­series data (dynamical inference) allowing for the understanding, prediction and possibly control of complex systems in nature is of great interest in a variety of fields. Often, however, in multidimensional systems only part of the system’s dynamical variables can be measured directly. The measurements are usually corrupted by noise and the dynamics is complicated by interplay of nonlinearity and random perturbations. We solve the problem of dynamical inference in these general settings by applying a path-­integral approach to fuctuational dynamics, and show that, given the measurements, the most probable 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 algorithm towards the most probable solution. We illustrate the efficiency of the approach by solving an intensively studied problem from the population dynamics of a predator-­prey system where the prey populations may be observed while the number of predators 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 real data measurements of the prey dynam­ics.