Nonlinear Statistical Modeling and Model Discovery for Ecological Data

The search for dynamical models (dynamical inference) underlying time-varying phenomena is of fundamental importance for understanding and controlling complex systems in science and technology. Often, however, only part of the system's dynamics can be measured and the state of the dynamical system remains invisible (or hidden). Furthermore, the measurements are usually corrupted by noise and the dynamics is complicated by the interplay of nonlinearity and random perturbations. The problem of dynamical inference in these general settings is challenging researchers for decades. We demonstrate here a path-integral approach to this problem, in which measured data act effectively as a control force driving algorithm towards the most probable solution. The approach is semi-analytical; consequently, the resulting algorithm does not require an extensive global search for the model parameters, provides optimal compensation for the effects of dynamical noise, and is robust for a broad range of dynamical models [1,2]. The strengths of the algorithm are illustrated by inferring the parameters of the stochastic Lorenz system and comparing the results with those of earlier research. The efficiency of the algorithm is further demonstrated by solving an intensively studied problem from the population dynamics of predator-prey system [3] where the prey populations may be observed while the number of predators is difficult or impossible to estimate. We emphasize that the predator-prey dynamics is fully nonlinear, perturbed stochastically by environmental factors and is not known beforehand. 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 The presented method can be further extended to encompass cases of colored noise and specially distributed systems. . It is hoped that techniques such as developed here may be very useful in the future for inferring important ecological variables from limited observational data, as well as in many other fields where similar practical problems are faced. In particular, it can be applied to the inference of a climate forcing mechanisms from reconstructed from the measurements of carbon dioxide in ocean sediment [4]. [1] V. N. Smelyanskiy, D. G. Luchinsky, A. Stefanovska, P. V. E. McClintock, Physical Review Letters, 94, 098101 (2005). [2] V. N. Smelyanskiy, D. G. Luchinsky, D. A. Timucin, A. Bandrivskyy, Physical Review E, 72, 026202 (2005). [3] I. Hanski, H. Henttonen, E. Korpimanaki, L. Oksanen, P. Turchin, Ecology 82, 1505 (2001). [4] S. Rahmstorf, Nature 419, 207 (2002)