Stability in a system subject to noise with regulated periodicity

Abstract

The stability of a simple dynamical system subject to multiplicative one-side pulse noise with hidden periodicity is investigated both analytically and numerically. The stability analysis is based on the exact result for the characteristic functional of the renewal pulse process. The influence of the memory effects on the stability condition is analyzed for two cases: (i) the dead-time-distorted Poissonian process, and (ii) the renewal process with Pareto distribution. We show that, for fixed noise intensity, the system can be stable when the noise is characterized by high periodicity and unstable at low periodicity.

Stochastic models for phytoplankton dynamics in Mediterranean Sea

“…This new approach is used to better analyze the real dynamics of phytoplankton populations, which are continuously exposed to random and deterministic changes in environmental variables. Indeed, it is worth recalling that marine ecosystems are complex systems, that is open systems characterized by nonlinear interactions between their parts and external perturbations (Goryachev et al, 2005;Maye et al, 2007), both deterministic and random, due to environmental variables (Grenfell et al, 1998;Zimmer, 1999;Bjornstad and Grenfell, 2001;Spagnolo et al, , 2003Spagnolo et al, , 2004Spagnolo et al, , 2005La Barbera and Spagnolo, 2002;Valenti et al, 2004aValenti et al, , 2006Caruso et al, 2005;Chichigina et al, 2005Chichigina et al, , 2011Fiasconaro et al, 2006;Chichigina, 2008;La Cognata et al, 2010). As a consequence, the study of a marine ecosystem has to be performed by considering also the effects of random perturbations, which can be treated as environmental noise sources.…”

Dynamics of Two Picophytoplankton Groups in Mediterranean Sea: Analysis of the Deep Chlorophyll Maximum by a Stochastic Advection-Reaction-Diffusion Model

“…Two novelties are present in this work: i) the use of a stochastic approach to model the dynamics of more phytoplankton populations; ii) the comparison between theoretical and experimental distributions of chlorophyll concentration; this is performed by using, for each phytoplankton population, a conversion curve to obtain from the biomass concentrations the equivalent chlorophyll content. It is important to stress that marine ecosystems, because of the presence as well of non-linear interactions among their parts as deterministic and random perturbations due to environmental variables, are complex systems [7]–[23]. Therefore, in order to better reproduce this non-linear and noisy dynamics, it is necessary that the model takes into account the presence of external random fluctuations [24], [25] including, in the equations of our model, terms of multiplicative noise [14], [26]–[28].…”

Effect of time delay on pattern dynamics in a spatial epidemic model

“…Usually, one needs taking care of the joint effect of diffusion and time delay [54] to obtain stability conditions which depends on transcendental equation associating characteristic eigenvalue λ with a function e λτ . In addition, the presence of noise [55][56][57][58] or other terms may give rise to a rich variety of dynamical effects [59], including noise-enhanced stability [60], noise-delayed extinction [61,62] and noise-induced transitions [63]. For example, pattern formation induced by the noise in two competing species was analyzed by Valenti et al [55].…”