Effects Of Stochastic Perturbations Research Articles

Effects of stochastic perturbations on a phytoplankton allelopathy competitive system

In this paper, we investigate stochastic models of two species competitive phytoplankton system with allelopathic interaction. We construct stochastic models from deterministic models by introducing three different stochastic perturbations to the equations. For the first model, we prove that this model is asymptotically stable in probability at three different equilibrium points. It is also shown that the system is positively recurrent under some conditions. For the second model, we study the extinction, existence of positive recurrence and global asymptotic stability under some sufficient conditions. For the last system, almost sufficient and necessary conditions for global asymptotic stability, the existence of positive recurrence and extinction of each population are established. In addition, using a suitable stochastic Lyapunov method, we prove that there is a unique solution to three models, we also discuss the ultimate boundedness of the three stochastic systems. Moreover, some numerical simulations are introduced to illustrate our mathematical results. We show that stochastic models still retain the desirable stability of their deterministic counterparts if stochastic perturbations are relatively small.

Effects of stochastic perturbations on the tree–grass coexistence in savannas

Effects of stochastic perturbations on the tree–grass coexistence in savannas

Impact of incubation delay and stochastic perturbation on Dengue fever virus transmission model

Consider the fluctuation of environmental noise and the incubation period of virus, a stochastic vector-host model with incubation delay is proposed to describe the transmission of Dengue fever virus (DFVs) between humans and mosquitoes. By constructing suitable stochastic Lyapunov functional, the existence and uniqueness of global positive solutions of this model are proved. Furthermore, some sufficient conditions are obtained for the asymptotic pathwise estimation of solutions. In addition, we also consider the asymptotic pathwise estimation of solutions, the existence and uniqueness of stationary distribution of this model without delay, which imply that disease is stochastic persistent. Finally, numerical simulations explain the theoretical results and discuss the effects of stochastic perturbations and delays on DFVs transmission. Our results imply that the spread of DFVs presents many unpredictability under the influence of stochastic perturbation, and the reproduction number can no longer be used as a threshold condition for the extinction or persistence of disease. Moreover, ignoring the incubation delay will not only overestimate the risk of disease outbreaks but also underestimate the duration of diseases. Rational application of stochastic perturbations on disease transmission and incubation periods will play a very important role in vector-borne disease control.

Most probable transition paths in eutrophicated lake ecosystem under Gaussian white noise and periodic force

The effects of stochastic perturbations and periodic excitations on the eutrophicated lake ecosystem are explored. Unlike the existing work in detecting early warning signals, this paper presents the most probable transition paths to characterize the regime shifts. The most probable transition paths are obtained by minimizing the Freidlin–Wentzell (FW) action functional and Onsager–Machlup (OM) action functional, respectively. The most probable path shows the movement trend of the lake eutrophication system under noise excitation, and describes the global transition behavior of the system. Under the excitation of Gaussian noise, the results show that the stability of the eutrophic state and the oligotrophic state has different results from two perspectives of potential well and the most probable transition paths. Under the excitation of Gaussian white noise and periodic force, we find that the transition occurs near the nearest distance between the stable periodic solution and the unstable periodic solution.

An HIV latent infection model with cell-to-cell transmission and stochastic perturbation

During the HIV infection process in vitro cell cultures, recent experimental and mathematical investigations have revealed that there are two fundamentally distinct viral transmission modes: cell-free infection and cell-to-cell transmission. A stochastic HIV model with latent infection, cell-free infection and cell-to-cell transmission is formulated in this paper. The existence of the unique ergodic stationary distribution is established by constructing appropriate Lyapunov function. Moreover, virus can be eradicated under certain sufficient conditions. Numerical simulations are carried out to investigate three aspects: (i) the effect of stochastic perturbation; (ii) the influence of latency; (iii) the relative contributions of cell-to-cell and cell-free infections.

Effects of stochastic perturbation and vaccinated age on a vector-borne epidemic model with saturation incidence rate

For reasons that the universality of stochastic perturbation and heterogeneity in the spread of vector-borne epidemic diseases, we formulate a stochastic vector-borne epidemic model with age-structure to discuss the effects of these factors. By constructing appropriate Lyapunov functions, the existence and uniqueness of global positive solutions of this model are derived. Further, we obtained some sufficient conditions for the extinction of the disease. In addition, the existence of a unique stationary distribution is studied which leads to the persistence of disease. Some numerical simulations are carried to explain our theoretical results. This implicates that, under the effects of these factors, the intensity and timing of outbreaks of the disease are unpredictable.

Consensus of delayed multi-agent dynamical systems with stochastic perturbation via dual-stage impulsive approach

This paper investigates the problem of consensus for delayed multi-agent systems with stochastic perturbation via dual-stage impulsive approach. A novel dual-stage impulsive control technique is adopted in the field of consensus of multi-agent systems. Furthermore, the effects of stochastic perturbation have been taken into consideration simultaneously in the model, which provide a more piratical framework for the consensus of delayed multi-agent systems. Parameter uncertainties are also discussed in the research. Finally, we present the simulation results to verify the correctness of the proposed control mechanism.

Permanence and extinction in a stochastic service–resource mutualism model

In this paper, a stochastic service–resource mutualism model is proposed and studied. For each species, the threshold between stochastic permanence and extinction is established. The effects of stochastic perturbations on the permanence and extinction of the resource species and the service species are revealed.

Effects of stochastic perturbation on the SIS epidemic system

In this paper we extend the classical SIS epidemic model from a deterministic framework to a stochastic one. We also study the long time behavior of the stochastic system. We mainly establish conditions for the extinction of disease from the population as well as the persistence of disease under different conditions. In the case of persistence, we show the existence of a stationary distribution. we found that the introduction of stochastic noise changes the basic reproduction number. The presented results are demonstrated by numerical simulations.

Noisy homoclinic pulse dynamics.

The effect of stochastic perturbations on nearly homoclinic pulse trains is considered for three model systems: a Duffing oscillator, the Lorenz-like Shimizu-Morioka model, and a co-dimension-three normal form. Using the Duffing model as an example, it is demonstrated that the main effect of noise does not originate from the neighbourhood of the fixed point, as is commonly assumed, but due to the perturbation of the trajectory outside that region. Singular perturbation theory is used to quantify this noise effect and is applied to construct maps of pulse spacing for the Shimizu-Morioka and normal form models. The dynamics of these stochastic maps is then explored to examine how noise influences the sequence of bifurcations that take place adjacent to homoclinic connections in Lorenz-like and Shilnikov-type flows.

Natural Resource Management for Nonlinear Stochastic Biotic-Abiotic Ecosystems: Robust Reference Tracking Control Strategy Using Limited Set of Controllers

The management of natural resources has become an important subject because of the increasing population. To meet human requirements for resources such as food and water, many control strategies have been proposed to improve biomass production and nutrient supply. However, implementing these strategies may be limited by the expense of large numbers of control devices and by the effects of stochastic perturbations. In this study, a robust reference tracking control strategy for the natural resource management of nonlinear stochastic biotic-abiotic ecosystems is proposed, using a limited set of controllers, to regulate the systematic dynamics achieving a desired reference trajectory under the influence of intrinsic fluctuations and environmental disturbances. To simplify the design procedure and make the robust reference tracking control strategy more feasible, we propose a fuzzy stochastic partial differrential equations system to represent the ecosystem, itself approximated by a fuzzy stochastic spatial state space model, based on a finite difference scheme. This allows replacement of a complex Hamilton-Jacobi integral inequality by an equivalent set of local linear matrix inequalities which can be easily solved. We verify the efficiency of the proposed approach using a nonlinear stochastic biomass-nutrient control example and compare the robust tracking control performance for different arrangements of control devices. Managers may select appropriate tracking control schemes based on this comparison. The robust reference tracking control strategy can be applied not only to agricultural systems but also to biophysical systems in ecological conservation, ecosystem restoration or engineering.

Measuring the Effect of Stochastic Perturbation Component in Cellular Automata Urban Growth Model

Urban environments are complex dynamic systems whose prediction of the future states cannot exclusively rely on deterministic rules. Although several studies on urban growth were carried out using different modelling approaches, the measurement of uncertainties was commonly neglected in these studies. This paper investigates the effect of uncertainty in urban growth models by introducing a stochastic perturbation method. A cellular automaton is used to simulate predicted urban growth. The effect of stochastic perturbation is addressed by comparing series of urban growth simulations based on different degree of stochastic perturbation randomness with the original urban growth simulation, obtained with the sole cellular automata neighbouring effects. These simulations are evaluated using cell-to-cell location agreement and a number of spatial metrics. The model framework has been applied to the Ourthe river basin in Belgium. The results show that the accuracy of the model is increased by introducing a stochastic perturbation component with a limited degree of randomness, in the cellular automata urban growth model.

A Multi-Scale Non-Deterministic Approach to Composite Curing Process Simulation

Thermosetting matrix composite materials are often subject to a curing process to enhance the mechanical properties of the final product. In recent years computational models of the curing process allowed for a remarkable time and cost compression with respect to trial and error procedures. Most of the proposed models, however, rely on deterministic hypothesis. In this paper a multi-scale non deterministic approach to cure process simulation has been proposed, evidencing the effect of stochastic perturbations of fibers distribution on simulative results on macro-scale.

Study on the effect of stochastic perturbations on the dynamics of spiral wave

The cardiac muscle, which is composed of many discrete cells, is a typical excitable medium. In this paper, we study the effect of refractory period with stochastic perturbations on dynamical behaviors of spiral wave using the model of discrete excitable medium. When the perturbations are random in space, the stability of spiral wave is related to the amplitude of the perturbations and the number of perturbation cells. Computer simulation results show that refractory period perturbations can result in meandering, breakup and disappearance of the spiral waves under suitable conditions, and then their mechanisms are analyzed.

Effect of stochastic perturbation on a two species competitive model

In this paper first we study the stability and bifurcation of a two species competitive model with a delay effect. Next we extend the deterministic model system to a stochastic delay differential system by incorporating multiplicative white noise terms in growth equations of both species. We consider the stochastic stability of a co-existing equilibrium point in terms of mean square stability by constructing a suitable Lyapunov functional. We perform a numerical simulation to validate our analytical findings.

Effect of small time delay in a predator-prey model within random environment

We consider a non-Kolmogorov type predator-prey model with and without delay. The local stability and Hopf bifurcation results are stated for both the cases of the deterministic system. The effect of stochastic perturbation in the form of parametric white and colored noise are considered. The exponential mean square stability of the trivial solutions for the stochastic differential equations and stochastic delay differential equations are studied under the Ito interpretation. To study the exponential mean square stability of the interior equilibrium point for the delayed model system in the presence of parametric white and colored noise, an approximation procedure is followed for the system of stochastic delay differential equation model system under the assumption that the magnitude of time delay is small. We have obtained threshold magnitude of discrete time delay. Critical magnitude for noise intensities are obtained for stochastic stability aspects. Numerical simulation results are provided in support of our analytical findings.

Effect of Electrical Uncertainties on Resonant Piezoelectric Shunting

Resonant piezoelectric shunting is one of the most studied techniques for damping structural vibrations by using piezoelectric transducers in conjunction with passive electric networks. It is based on establishing an internal resonance between a mechanical system and an electrical circuit, which is comprised of a shunting impedance and the piezoelectric inherent capacitance. The problem of analyzing the effect of variations of the electrical impedance with respect to its optimal choice is investigated here. Simple closed-form formulas for quantifying the performance loss due to deterministic variations of the electrical elements are derived, and they are validated through numerical tests. The effect of stochastic perturbations is also considered, manageable formulas are provided, and validated by Monte Carlo simulations. The analyses focus on forced vibrations, and the arising H∞ optimization problem is solved by making use of fundamental properties of the system transfer function.

Stochastic dynamo model for subcritical transition

The effects of stochastic perturbations in a nonlinear alpha Omega-dynamo model are investigated. By using transformation of variables we identify a "slow" variable that determines the global evolution of the non-normal alpha Omega-dynamo system in the subcritical case. We apply an adiabatic elimination procedure to derive a closed stochastic differential equation for the slow variable for which the dynamics is determined along one of the eigenvectors of the full system. We derive the corresponding Fokker-Planck equation and show that the generation of a large scale magnetic field can be regarded as a first-order phase transition. We show that the an advantage of the reduced system is that we have explicit expressions for both the stochastic and deterministic potentials. We also obtain the stationary solution of the Fokker-Planck equation and show that an increase in the intensity of the multiplicative noise leads to qualitative changes in the stationary probability density function. The latter can be interpreted as a noise-induced phase transition. By a numerical simulation of the stochastic galactic dynamo model, we show that the qualitative behavior of the "empirical" stationary pdf of the slow variable is accurately predicted by the stationary pdf of the reduced system.

Stochastic analysis of a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition.

The effects of stochastic perturbations on a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition are investigated both analytically and numerically. It is found that a nonlinear dynamical system with non-normal transient linear growth is very sensitive to the presence of weak random perturbations. The effect of non-normality on the exit probability from the zero fixed point is analyzed numerically for small values of the noise intensity parameter. It is found that an increase in the intensity of the noise, or a decrease of the non-normality parameter leads to qualitative changes in the behavior of the trajectories that can be interpreted as noise-induced phase transitions. By using the Itô formula and the adiabatic elimination procedure a stochastic equation governing the slow evolution of the energy of the non-normal system is derived.

Lyapunov Exponents and Moment Lyapunov Exponents of a Two-Dimensional Near-Nilpotent System

The Lyapunov exponents and moment Lyapunov exponents of a near-nilpotent system under stochastic parametric excitation are studied. The system considered is the linearized system of a two-dimensional nonlinear system exhibiting a pitchfork bifurcation. The effect of stochastic perturbation in the vicinity of static pitchfork bifurcation is investigated. Approximate analytical results of Lyapunov exponent are obtained. The eigenvalue problem for the moment Lyapunov exponent is converted to a two-point boundary value problem, which is solved numerically by the method of relaxation.