Bistability driven by white shot noise

Exponential increase of transition rates in metastable systems driven by non-Gaussian noise

Noise-induced escape from metastable states governs a plethora of transition phenomena in physics, chemistry, and biology. While the escape problem in the presence of thermal Gaussian noise has been well understood since the seminal works of Arrhenius and Kramers, many systems, in particular living ones, are effectively driven by non-Gaussian noise for which the conventional theory does not apply. Here we present a theoretical framework based on path integrals that allows the calculation of both escape rates and optimal escape paths for a generic class of non-Gaussian noises. We find that non-Gaussian noise always leads to more efficient escape and can enhance escape rates by many orders of magnitude compared with thermal noise, highlighting that away from equilibrium escape rates cannot be reliably modelled based on the traditional Arrhenius-Kramers result. Our analysis also identifies a new universality class of non-Gaussian noises, for which escape paths are dominated by large jumps.

Probability characteristics of nonlinear dynamical systems driven by δ-pulse noise

For a nonlinear dynamical system described by the first-order differential equation with Poisson white noise having exponentially distributed amplitudes of δ pulses, some exact results for the stationary probability density function are derived from the Kolmogorov-Feller equation using the inverse differential operator. Specifically, we examine the "effect of normalization" of non-Gaussian noise by a linear system and the steady-state probability density function of particle velocity in the medium with Coulomb friction. Next, the general formulas for the probability distribution of the system perturbed by a non-Poisson δ-pulse train are derived using an analysis of system trajectories between stimuli. As an example, overdamped particle motion in the bistable quadratic-cubic potential under the action of the periodic δ-pulse train is analyzed in detail. The probability density function and the mean value of the particle position together with average characteristics of the first switching time from one stable state to another are found in the framework of the fast relaxation approximation.

Multiplicative jump processes and applications to leaching of salt and contaminants in the soil

We consider simple systems driven multiplicatively by white shot noise, which appear in the modeling of the dynamics of soil nutrients and contaminants. The dynamics of these systems is analyzed in two ways: solving a hierarchy of linear ordinary differential equations for the moments, which gives a time scale of convergence of the stationary probability density function; and characterizing the crossing properties, such as the mean first-passage time and the mean frequency of level crossing. These results are readily applicable to the study of geophysical systems, such as the problem of accumulation of salt in the root zone, i.e., soil salinization.

Numerical method for solving stochastic differential equations with Poissonian white shot noise

We propose a numerical integration scheme to solve stochastic differential equations driven by Poissonian white shot noise. Our formula, which is based on an integral equation, which is equivalent to the stochastic differential equation, utilizes a discrete time approximation with fixed integration time step. We show that our integration formula approaches the Euler formula if the Poissonian noise approaches the Gaussian white noise. The accuracy and efficiency of the proposed algorithm are examined by studying the dynamics of an overdamped particle driven by Poissonian white shot noise in a spatially periodic potential. We find that the accuracy of the proposed algorithm only weakly depends on the parameters characterizing the Poissonian white shot noise; this holds true even if the limit of Gaussian white noise is approached.

A Probabilistic Analysis of Fire‐Induced Tree‐Grass Coexistence in Savannas

Fires play an important role in determining the composition and structure of vegetation in semiarid ecosystems. The study of the interactions between fire and vegetation requires a stochastic approach because of the random and unpredictable nature of fire occurrences. To this end, this article develops a minimalist probabilistic framework to investigate the impact of intermittent fire occurrences on the temporal dynamics of vegetation. This framework is used to analyze the emergence of statistically stable conditions favorable to tree-grass coexistence in savannas. It is found that these conditions can be induced and stabilized by the stochastic fire regime. A decrease in fire frequency leads to bush encroachment, while more frequent and intense fires favor savanna-to-grassland conversions. The positive feedback between fires and vegetation can convert states of tree-grass coexistence in semiarid savannas into bistable conditions, with both woodland and grassland as possible, though mutually exclusive, stable states of the system.

Mean first passage times of processes driven by white shot noise

We consider mean first passage times in systems driven by white shot noise with exponentially distributed jump heights. Simple interpretable results are obtained and the linkage between those results and the steady-state probability density function of the process is presented. The virtual waiting-time or Takács process (constant losses) and the shot noise process with linear losses are analyzed in depth, along with a more complex process with useful implications for the modeling of the soil moisture dynamics in hydrology.