1. It is well known that fluctuations of species abundances observed in ecological time series emerge from an interplay between deterministic nonlinear dynamics and stochastic forces. Importantly, nonlinearity and stochasticity introduce significant challenges to the analysis of ecological time series, such as the inference of the effect of species interactions on community dynamics and forecasting of species abundances.2. Local linear fits with state-space-dependent kernel functions, known as S-maps, provide an efficient method to infer Jacobian coefficients (a proxy for the local effect of species interactions) and to make reliable forecasts from nonlinear time series. Yet, while it has been shown that the S-map outperforms existing methods for nonparametric inference and forecasting, the methodology is sensitive to process noise. To overcome this limitation, we integrate the S-map with different regularization schemes.3. To validate our approach, we test our methodology against different levels of noise and nonlinearity using three standard population dynamics models. We show that an appropriate choice of the regularization scheme, alongside an accurate choice of the kernel functions, can significantly improve the in-sample inference of Jacobian coefficients and the out-of-sample forecast of species abundances in the presence of process noise. We further validate our methodology using two empirical time series of marine microbial communities. 4. Our results illustrate that the regularized S-map is an efficient method for nonparametric inference and forecasting from noisy, nonlinear, ecological time series.Yet, attention must be paid on the regularization scheme and the structure of the kernel for whether inference or forecasting is the ultimate goal of a research study. K E Y W O R D Snonlinear time series, out-of-sample forecast, parameter inference, process noise, regularization, S-map
In population dynamics, the concept of structural stability has been used to quantify the tolerance of a system to environmental perturbations. Yet, measuring the structural stability of nonlinear dynamical systems remains a challenging task. Focusing on the classic Lotka-Volterra dynamics, because of the linearity of the functional response, it has been possible to measure the conditions compatible with a structurally stable system. However, the functional response of biological communities is not always well approximated by deterministic linear functions. Thus, it is unclear the extent to which this linear approach can be generalized to other population dynamics models. Here, we show that the same approach used to investigate the classic Lotka-Volterra dynamics, which is called the structural approach, can be applied to a much larger class of nonlinear models. This class covers a large number of nonlinear functional responses that have been intensively investigated both theoretically and experimentally. We also investigate the applicability of the structural approach to stochastic dynamical systems and we provide a measure of structural stability for finite populations. Overall, we show that the structural approach can provide reliable and tractable information about the qualitative behavior of many nonlinear dynamical systems.
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