Noise-induced shifts in the ecological model with delay

A Hassell-type mathematical model of population dynamics with delay and stochastic disturbances is considered. In this bistable model, one of the attractors corresponds to the extinction, and the other one describes non-trivial stable modes of dynamics. These modes can be both regular and chaotic. Structural stability zones are separated by local and global bifurcations. We study how noise shifts these bifurcation points and contracts the persistence zone. Abilities of the theoretical analysis of these phenomena with the help of the stochastic sensitivity function technique is discussed. © 2019 Author(s).Russian Science Foundation, RSF: N 16-11-10098The work was supported by Russian Science Foundation (N 16-11-10098)

Modality analysis of patterns in reaction-diffusion systems with random perturbations

In this paper, a distributed Brusselator model with diffusion is investigated. It is well known that this model undergoes both Andronov–Hopf and Turing bifurcations. It is shown that in the parametric zone of diffusion instability the model generates a variety of stable spatially nonhomogeneous structures (patterns). This system exhibits a phenomenon of the multistability with the diversity of stable spatial structures. At the same time, each pattern has its unique parametric range, on which it may be observed. The focus is on analysis of stochastic phenomena of pattern formation and transitions induced by small random perturbations. Stochastic effects are studied by the spatial modality analysis. It is shown that the structures possess different degrees of stochastic sensitivity. © 2019 Udmurt State University. All right reserved.Russian Science Foundation, RSF: 16–11–10098Funding. This research was supported by the Russian Science Foundation (project no. 16–11–10098)

Nonlinear dynamics of mushy layers induced by external stochastic fluctuations

The time-dependent process of directional crystallization in the presence of a mushy layer is considered with allowance for arbitrary fluctuations in the atmospheric temperature and friction velocity. A nonlinear set of mushy layer equations and boundary conditions is solved analytically when the heat and mass fluxes at the boundary between the mushy layer and liquid phase are induced by turbulent motion in the liquid and, as a result, have the corresponding convective form. Namely, the ‘solid phase–mushy layer’ and ‘mushy layer–liquid phase’ phase transition boundaries as well as the solid fraction, temperature and concentration (salinity) distributions are found. If the atmospheric temperature and friction velocity are constant, the analytical solution takes a parametric form. In the more common case when they represent arbitrary functions of time, the analytical solution is given by means of the standard Cauchy problem. The deterministic and stochastic behaviour of the phase transition process is analysed on the basis of the obtained analytical solutions. In the case of stochastic fluctuations in the atmospheric temperature and friction velocity, the phase transition interfaces (mushy layer boundaries) move faster than in the deterministic case. A cumulative effect of these noise contributions is revealed as well. In other words, when the atmospheric temperature and friction velocity fluctuate simultaneously due to the influence of different external processes and phenomena, the phase transition boundaries move even faster. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’. 10.1098/rsta.2017.0216Ministry of Education and Science of the Russian Federation, Minobrnauka: 1.9527.2017/8.9Data accessibility. This article has no additional data. Authors’ contributions. All authors contributed equally to the present research article. Competing interests. We declare we have no competing interests. Funding. This work was supported by the Ministry of Education and Science of the Russian Federation (project no. 1.9527.2017/8.9)

On controlling stochastic sensitivity of oscillatory systems

For a nonlinear oscillatory stochastic system, we study the control problem for the variance of random trajectories around a deterministic cycle. To describe the range of random trajectories, we use the method of stochastic sensitivity functions. We consider the problem of designing a given stochastic sensitivity function, discuss problems of controllability and reachability. Complete stochastic controllability is only possible when the control's dimension coincides with the system's dimension. Otherwise, the design problem becomes ill-posed. To solve it, we propose a regularization method that lets us produce a given stochastic sensitivity function with any given precision. The efficiency of the proposed approach is demonstrated with the example of controlling stochastic oscillations in a brusselator model. © 2013 Pleiades Publishing, Ltd

Sea Ice Dynamics Induced by External Stochastic Fluctuations

The influence of stochastic fluctuations in the atmosphere and in the ocean caused by different occasional phenomena (noises) on dynamic processes of sea ice growth with a mushy layer is studied. It is shown that atmospheric temperature variances substantially increase the sea ice thickness, whereas dispersion variations of turbulent flows in the ocean to a great extent decrease the ice content produced by false bottom evolution. © 2013 Springer Basel

Solidification dynamics under random external-temperature fluctuations

The nonlinear dynamic mechanisms of solid-phase formation with a phase transition region are studied under periodic and random fluctuations of the cooling-boundary temperature. It is theoretically shown that a mushy zone can form even at close liquid and cooling-boundary temperatures due to random temperature field fluctuations. The growth of a solid phase with the mushy zone is investigated as a function of the autocovariance characteristics of random noises. © 2013 Pleiades Publishing, Ltd

Analysis of additive and parametric noise effects on Morris - Lecar neuron model

This paper is devoted to the analysis of the effect of additive and parametric noise on the processes occurring in the nerve cell. This study is carried out on the example of the well-known Morris - Lecar model described by the two-dimensional system of ordinary differential equations. One of the main properties of the neuron is the excitability, i.e., the ability to respond to external stimuli with an abrupt change of the electric potential on the cell membrane. This article considers a set of parameters, wherein the model exhibits the class 2 excitability. The dynamics of the system is studied under variation of the external current parameter. We consider two parametric zones: the monostability zone, where a stable equilibrium is the only attractor of the deterministic system, and the bistability zone, characterized by the coexistence of a stable equilibrium and a limit cycle. We show that in both cases random disturbances result in the phenomenon of the stochastic generation of mixed-mode oscillations (i. e., alternating oscillations of small and large amplitudes). In the monostability zone this phenomenon is associated with a high excitability of the system, while in the bistability zone, it occurs due to noise-induced transitions between attractors. This phenomenon is confirmed by changes of probability density functions for distribution of random trajectories, power spectral densities and interspike intervals statistics. The action of additive and parametric noise is compared. We show that under the parametric noise, the stochastic generation of mixed-mode oscillations is observed at lower intensities than under the additive noise. For the quantitative analysis of these stochastic phenomena we propose and apply an approach based on the stochastic sensitivity function technique and the method of confidence domains. In the case of a stable equilibrium, this confidence domain is an ellipse. For the stable limit cycle, this domain is a confidence band. The study of the mutual location of confidence bands and the boundary separating the basins of attraction for different noise intensities allows us to predict the emergence of noise-induced transitions. The effectiveness of this analytical approach is confirmed by the good agreement of theoretical estimations with results of direct numerical simulations. © 2017 Lev B. Ryashko, Evdokia S. Slepukhina.The work was supported by the Government of the Russian Federation (Act 211, contract No. Russian Foundation for Basic Research (project No. 16-31-00317 mol_a)

Stochastic phenomena in pattern formation for distributed nonlinear systems

We study a stochastic spatially extended population model with diffusion, where we find the coexistence of multiple non-homogeneous spatial structures in the areas of Turing instability. Transient processes of pattern generation are studied in detail. We also investigate the influence of random perturbations on the pattern formation. Scenarios of noise-induced pattern generation and stochastic transformations are studied using numerical simulations and modality analysis. © 2020 The Author(s) Published by the Royal Society. All rights reserved.Russian Science Foundation, RSF: 16-11-10098Data accessibility. This article has no additional data. Authors’ contributions. All authors contributed equally. Competing interests. We declare we have no competing interests. Funding. The work was supported by the Russian Science Foundation (grant no. 16-11-10098)

Analysis of stochastic model for nonlinear volcanic dynamics

Motivated by important geophysical applications we consider a dynamic model of the magma-plug system previously derived by Iverson et al.~(2006) under the influence of stochastic forcing. Due to strong nonlinearity of the friction force for a solid plug along its margins, the initial deterministic system exhibits impulsive oscillations. Two types of dynamic behavior of the system under the influence of the parametric stochastic forcing have been found: random trajectories are scattered on both sides of the deterministic cycle or grouped on its internal side only. It is shown that dispersions are highly inhomogeneous along cycles in the presence of noises. The effects of noise-induced shifts, pressure stabilization and localization of random trajectories have been revealed by increasing the noise intensity. The plug velocity, pressure and displacement are highly dependent of noise intensity as well. These new stochastic phenomena are related to the nonlinear peculiarities of the deterministic phase portrait. It is demonstrated that the repetitive stick–slip motions of the magma-plug system in the case of stochastic forcing can be connected with drumbeat earthquakes

ANALYSIS OF SPATIAL STRUCTURES IN THE STOCHASTICALLY FORCED REACTION-DIFFUSION MODEL

In this paper we consider a stochastically perturbed Brusselator model with two spatial variables. For this reaction-diffusion model, the dependence of possible regimes of dynamics is investigated depending on the parameters of the system, including the intensity of random perturbations. The possibility of forming spatial structures caused by even small random perturbations is shown. A description of the shape of these structures and their characteristics are given. The phenomenon of stochastic transitions between different structures is investigated.Исследование выполнено при поддержке Российского научного фонда (проект 16-11-10098)