Environmental Noise and Nonlinear Relaxation in Biological Systems

We analyse the effects of environmental noise in three different biological systems: (i) mating behaviour of individuals of \emph{Nezara viridula} (L.) (Heteroptera Pentatomidae); (ii) polymer translocation in crowded solution; (iii) an ecosystem described by a Verhulst model with a multiplicative L\'{e}vy noise.Comment: 32 pages; In "Ecological Modeling" by Ed. Wen-Jun Zhang. ISBN: 978-1-61324-567-5. - Nova Science Publishers, New York, 201

Stationary states for underdamped anharmonic oscillators driven by Cauchy noise

Using methods of stochastic dynamics, we have studied stationary states in the underdamped anharmonic stochastic oscillators driven by Cauchy noise. Shape of stationary states depend both on the potential type and the damping. If the damping is strong enough, for potential wells which in the overdamped regime produce multimodal stationary states, stationary states in the underdamped regime can be multimodal with the same number of modes like in the overdamped regime. For the parabolic potential, the stationary density is always unimodal and it is given by the two dimensional $\alpha$-stable density. For the mixture of quartic and parabolic single-well potentials the stationary density can be bimodal. Nevertheless, the parabolic addition, which is strong enough, can destroy bimodlity of the stationary state.Comment: 9 page

Probability distribution function for systems driven by superheavy-tailed noise

We develop a general approach for studying the cumulative probability distribution function of localized objects (particles) whose dynamics is governed by the first-order Langevin equation driven by superheavy-tailed noise. Solving the corresponding Fokker-Planck equation, we show that due to this noise the distribution function can be divided into two different parts describing the surviving and absorbing states of particles. These states and the role of superheavy-tailed noise are studied in detail using the theory of slowly varying functions.Comment: 9 page

Bimodality and hysteresis in systems driven by confined L\'evy flights

We demonstrate occurrence of bimodality and dynamical hysteresis in a system describing an overdamped quartic oscillator perturbed by additive white and asymmetric L\'evy noise. Investigated estimators of the stationary probability density profiles display not only a turnover from unimodal to bimodal character but also a change in a relative stability of stationary states that depends on the asymmetry parameter of the underlying noise term. When varying the asymmetry parameter cyclically, the system exhibits a hysteresis in the occupation of a chosen stationary state.Comment: 4 pages, 5 figures, 30 reference

Verhulst model with Levy white noise excitation

The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Levy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise, (ii) noise with a probability density of increments expressed in terms of Gamma function, and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Levy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.Comment: 9 pages, 12 figures, to appear in EPJ B (2008

Ecological Complex Systems

Main aim of this topical issue is to report recent advances in noisy nonequilibrium processes useful to describe the dynamics of ecological systems and to address the mechanisms of spatio-temporal pattern formation in ecology both from the experimental and theoretical points of view. This is in order to understand the dynamical behaviour of ecological complex systems through the interplay between nonlinearity, noise, random and periodic environmental interactions. Discovering the microscopic rules and the local interactions which lead to the emergence of specific global patterns or global dynamical behaviour and the noises role in the nonlinear dynamics is an important, key aspect to understand and then to model ecological complex systems.Comment: 13 pages, Editorial of a topical issue on Ecological Complex System to appear in EPJ B, Vol. 65 (2008