Noise-Induced Phenomena in the Environmental Sciences

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Sieber, Eur.

He, M. Mobilia, U. Valenti et al. Kraut, U. Feudel, Phys. Nowakowski, Phys.

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Bashkirtseva, A. Ryashko, Phys.

1. Introduction

Alexandrov, I. Bashkirtseva, L. Ryashko, Eur. Hemerik, M. Boer, B. Kooi, Math Biosci. Lidicker Jr. Drake, D. Lodge, Biological Invasions 8 , CrossRef Google Scholar. Sun, Z. Jin, L. Li, Q. Liu, J. Hart, L. Kurrer, K. Freidlin, A. Bashkirtseva, G. The differences of the jump sizes in the two examples can be observed in Fig.

The derivation of the master equation and the steady-state probability distribution in Box 2. The master equation corresponding to Langevin equation 2. Unfortunately, no analytical solution is known of Eq. The domain of the steady-state pdf under the Stratonovich interpretation can be inferred from the general rule presented for the case of DMN, i. We can now refer to the examples presented in Subsection 2. We first transform generic multiplicative Langevin equation 2. Now, if expression B2.

Backtransforming Eq. Moreover, the expressions of master equation B2. Thus, to compare with 2. Following the same steps as in B2.

Noise-induced stabilization and fixation in fluctuating environment

In fact, in the case of mechanistic use of DMN, the underlying dynamics are intrinsically dichotomous [i. We can obtain a state-dependent form of WSN from state-dependent dichotomous noise with functional usage discussed in Subsection 2. It is interesting to observe that when such limits are taken in the case of state-dependent dichotomous noise see also Laio et al.

In other words, by taking the limits in Eq. Example of a typical realization of the Wiener process. An enlargement of a small portion of the path is shown in the inset to underline the fractal nature of Brownian motion. The first two properties are valid for any white-noise process; the third one is peculiar to white Gaussian noise. As for WSN, the problem with the Gaussian white noise is that, even though the notion of white noise is commonly used, it refers to a singular mathematical object.

A typical path of the Wiener process is represented in Fig. The average and the covariance function of the Wiener process are Parzen, , p. We can also obtain Gaussian white noise from WSN by letting the frequency of the shots go to infinity and the average intensity go to zero. This justifies the use of a white-noise memoryless representation of the external forcing in a number of environmental systems. Moreover, in many situations, fluctuations in the external forcing are the result of several factors simultaneously acting on the environmental system.

When the number of these factors is large enough, the central-limit theorem ensures that the fluctuations in the external parameter have approximatively a Gaussian distribution. The quality of the approximation depends on the number of concurring factors, the shape of their probability distributions, and the cross correlation among the factors. The frequent simultaneous presence of these two conditions memoryless external forcing and numerous environmental factors explains the common use of Gaussian white noise to represent the external forcing in models of biological and geophysical processes.

We refer again, as in Subsection 2. The intermittent nature of a rainfall time series is lost when the temporal resolution or aggregation scale of rainfall is much larger than the diurnal one.

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For example, if we consider rainfall at an annual resolution, we typically find a sequence of random uncorrelated precipitation amounts, approximatively Gaussian distributed. If the response time of the dynamical system subjected to precipitation is larger than the year e.

Other examples of environmental processes in which the forcing can be represented as white Gaussian noise are given in Chapters 4 and 6. We follow a similar reasoning as in Subsection 2. In fact, in this case only the terms up to h 2 remain in Eqs. As a consequence, Eqs. We remark again that this is valid only for systems driven by Gaussian white noise. Example of a realizations of processes driven by Gaussian white noise: a Eq.

From the comparison of the two panels it is clear that in this case the presence of the multiplicative term has an even stronger influence than in the case of the shot-noise process see also Fig. In fact, the presence of the multiplicative term changes the domain of the process and the shape of the corresponding probability distribution, as explained in the following subsections.


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We obtain e. We obtain the same solution by considering the master equation for the evolution of the probability density in time, which is reported in Box 2. The domain of steady-state pdf 2. For example, if the noise is additive, i. This is the case for Example 2.

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For Example 2. The interested reader is referred to these books for details. The Fokker—Planck equation corresponding to Langevin equation 2. The steady-state solution of Eq. Transient solutions in the form of time-dependent probability distributions exist in particular cases, including the Ornstein—Uhlenbeck O-U process [represented by Eq.

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To exemplify the application of Eq. Examples of steady-state pdfs for processes driven by Gaussian white noise, corresponding to a Eq. The pdfs in Eqs. In spite of the existence of a plethora of different types of correlated noises, exact analytical results are known only for some classes of systems driven by non-Markovian Gaussian colored noises and for two types of Markovian colored noise, namely the dichotomous noise and the Markovian Gaussian colored noise, i.

The case of dichotomous noise was presented at the beginning of this chapter Section 2. In Eq.

The master equation for the evolution in time of the pdf, corresponding to Langevin equation 2. In this case some important approximated results can be obtained. However, some approximated solutions were proposed in the literature. Essentially, two approaches were followed.

The other approach treats the process as driven by non-Markovian Gaussian noise and tries to obtain from exact master equation 2. The differences between these two approaches translate into different ranges of validity of the approximated expressions, particularly with respect to the correlation scale of the noise. Here we summarize the main results along with their range of validity.

An often-adopted approach consists of dealing with colored noises close to the white-noise limit, i. Thus, starting from exact master equation 2.