Search

Stochastic resonance of an asymmetric tristable system driven by a periodic forcing and cross-correlated Ornstein-Uhlenbeck (O-U) noise is investigated. An approximate Fokker-Planck Equation for an asymmetric tristable system driven by a periodic forcing and cross-correlated O-U noise is derived in the adiabatic limit by using the unified colored-noise approximation method and the stochastic equivalent rules. Afterwards, the expression for spectral amplification (SA), the stationary probability density function (PDF) and the mean first passage time (MFPT) are derived. By considering the influence of asymmetric potential function and cross-correlated noise, the phenomenon of noise enhancing stability (NES) and stochastic resonance (SR) can be found. It can also be found that the response of the tristable system can be effectively improved by properly selecting the relevant noise intensity and the asymmetric constant. This proves the excellent performance of the system, and provides a good theoretical support for practical engineering application.

In this paper, we present a stochastic hepatitis B virus (HBV) infection model and the dynamic behaviors of the model are investigated. When the fraction of vertical transmission μ ω ν C is not considered to be new infections, the existence and ergodicity of the stationary distribution of the model are obtained by constructing a suitable Lyapunov function, which determines a critical value ρ 0 s corresponding to the basic reproduction number of ODE system. This implies the persistence of the diseases when ρ 0 s > 1 . Meanwhile, the sufficient conditions for the extinction of the diseases are derived when ρ 0 T 0 . What is more, we give the specific expression of the probability density function of the stochastic model around the unique endemic quasi-equilibrium by solving the Fokker–Planck equation. Finally, the numerical simulations are illustrated to verify the theoretical results and match the HBV epidemic data in China.

We examine the influence of using a restart mechanism on the stationary distributions of a particular class of Markov chains. Namely, we consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We show that the restart mechanism may change significantly the limiting distribution. We obtain a limit theorem with a novel type of limiting distribution, a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case without restart mechanism. The main theorem is accompanied by a number of examples and auxiliary results of their own interest.

In this paper, we study the mean square exponential stability of discrete-time stochastic neural networks with partially unstable subsystems and mixed delays. The mixed delays under consideration involve discrete delay and distributed delay. Moreover, the discrete delay term satisfies the Bernoulli distribution . Different from the deterministic switching, we consider Markov switching and our system has partially unstable subsystems. By constructing a novel Lyapunov–Krasovskii functional and using the stationary distribution of Markov chain , we give sufficient conditions for the mean square exponential stability of the suggested system. Finally, two numerical examples are given to check the theory results.

This paper proposes a penalty strategy with differentiated service rate based on the free riding phenomenon in P2P networks, and establishes an M/M/c+d queueing model. Based on this model, a sleep/wakeup mechanism is introduced for the peers at the service end, and a single asynchronous vacation strategy is adopted to reduce the energy consumption of the system. In addition, the energy consumption of peers in each state is quantified, and the relationship between the energy consumption and parameters of the system is analyzed. In order to avoid excessive requests for unnecessary services from requesting nodes and increasing energy consumption of the system, this paper analyzes the Nash equilibrium between the arrival rate and the net profit of a single node, and then studies the optimization of social profit. The stationary distribution of queueing model is obtained by the method of matrix geometric solution, the performance indicators of the system are constructed, and the system performance is analyzed by numerical experiments. Experimental results show that the model developed in this paper has a significant penalty effect on free riding behavior, and that the single asynchronous vacation strategy not only saves more than 10% of the total energy consumption compared with the single synchronous vacation strategy, but also makes the hybrid P2P networks more flexible and efficient.

This paper studies the heavy-traffic limit of the moments of the stationary distribution in GI/G/1-type Markov chains. For these Markov chains, several researchers have derived heavy-traffic-limit formulas for the stationary distribution itself. However, for its moments, no such formulas have been reported in the literature. This paper presents a heavy-traffic-limit formula for the moments of the stationary distribution and a sufficient condition for the formula to hold, by using a characteristic function approach.

For two independent Levy processes ξ and η and an exponentially distributed random variable τ with parameter q > 0 , independent of ξ and η , the killed exponential functional is given by V q , ξ , η ≔ ∫ 0 τ e − ξ s − d η s . Interpreting the case q = 0 as τ = ∞ , the random variable V q , ξ , η is a natural generalisation of the exponential functional ∫ 0 ∞ e − ξ s − d η s , the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein–Uhlenbeck process. In this paper we show that also the law of the killed exponential functional V q , ξ , η arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein–Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of ∫ 0 t e − ξ s − d η s for fixed t ≥ 0 , as well as for integrals of the form ∫ 0 ∞ f ( s ) d η s for deterministic functions f . Furthermore, applying the same techniques to the case q = 0 , new results on the absolute continuity of the improper integral ∫ 0 ∞ e − ξ s − d η s are derived.

In this paper, we consider the Israeli queue which consists of a main queue with at most N groups and an infinite capacity retrial orbit. The retrial customers may become non-persistent before receiving service. This model was considered before and the decay rate function of the stationary distribution was obtained. To strengthen the result, we characterize the exact tail asymptotics by calculating the coefficient before the decay rate function.

Different from the existing methods, a new method is introduced to analyze the stochastic permanence and extinction of a stochastic predator–prey model with a general functional response and the random factors acting on both the intrinsic growth rates and the intra-specific interaction rates. In particular, the existence of a stationary distribution and weak convergence to a boundary process are investigated as well. Some numerical simulations are performed to illustrate our theoretical results and to show that the stochastic noises play an essential role in determining the permanence and extinction. To be more specific, appropriate intensities of white noises may make the predator and prey population fluctuate around their deterministic steady-state values; but too large intensities of white noises may make the predator and/or prey population go to extinction.

Let ( U , d ) be a complete separable metric space and ( F n ) n ≥ 0 a sequence of random functions from U to U . Motivated by studying the stability property for Markovian dynamic models, in this paper, we assume that the random function ( F n ) n ≥ 0 is driven by a Markov chain X = { X n , n ≥ 0 } . Under some regularity conditions on the driving Markov chain and the mean contraction assumption, we show that the forward iterations M n u = F n ∘ ⋯ ∘ F 1 ( u ) , n ≥ 0 , converge weakly to a unique stationary distribution Π for each u ∈ U , where ∘ denotes composition of two maps. The associated backward iterations M n u = F 1 ∘ ⋯ ∘ F n ( u ) are almost surely convergent to a random variable M ∞ which does not depend on u and has distribution Π . Moreover, under suitable moment conditions, we provide estimates and rate of convergence for d ( M ∞ , M n u ) and d ( M n u , M n v ) , u , v ∈ U . The results are applied to the examples that have been discussed in the literature, including random coefficient autoregression models and recurrent neural network.

The stationary probability density and stochastic resonance phenomenon of a tumor growth model under the excitation of Levy noise and Gaussian white noise are investigated in this paper. The fourth-order Runge-Kutta method and the Janick-Weron algorithm are used to simulate the stationary probability density. Meanwhile, the signal-to-noise ratio(SNR) is studied as a function of Levy noise intensity and Gaussian white noise intensity by numerical simulation respectively. The results indicate that: (i)both Levy and Gaussian noise sources give rise to noise-induced transitions for the system, with a peculiarity that smaller stability indexα and Levy noise intensity D enhance the likelihood of tumor cell death; (ii)both noise parameters and system parameters can induce the occurrence of stochastic resonance. However, the effect of Gaussian white noise on SNR is different from that of Levy noise.

While facultative sex is common in sexually reproducing species, for reasons of tractability most mathematical models assume that such sex is asynchronous in the population. In this paper, we develop a model of switching environments to instead capture the effect of an entire population transitioning synchronously between sexual and asexual modes of reproduction. We use this model to investigate the evolution of the number of self-incompatible mating types in finite populations, which empirically can range from two to thousands. When environmental switching is fast, we recover the results of earlier studies that implicitly assumed populations were engaged in asynchronous sexual reproduction. However when the environment switches slowly, we see deviations from previous asynchronous theory, including a lower number of mating types at equilibrium and bimodality in the stationary distribution of mating types. We provide analytic approximations for both the fast and slow switching regimes, as well as a numerical scheme based on the Kolmogorov equations for the system to quickly evaluate the model dynamics at intermediate parameters. Our approach exploits properties of integer partitions in number theory. We also demonstrate how additional biological processes such as selective sweeps can be accounted for in this switching environment framework, showing that beneficial mutations can further erode mating type diversity in synchronous facultatively sexual populations., EBC acknowledges a President’s Doctoral Scholarship (The University of Manchester). TG is grateful for partial financial support by the Maria de Maeztu Program for Units of Excellence in R&D, Spain (MDM-2017-0711). GWAC thanks the Leverhulme Trust for funding through a Leverhulme Early Career Fellowship.

A class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion is investigated. By using the Lyapunov function method, the existence of global positive solutions and the ultimate boundedness with probability one are obtained. By using the Markov semigroups theory, Fokker–Planck equation and Khasminskiǐ functions, the existence of unique stationary distribution for the model is established. That is, when the stochastic basic reproduction number R 0 S > 1 and some extra conditions are satisfied then the distribution density of any positive solutions of the model converges to a unique invariant density as t → + ∞ . Finally, the main conclusions and open problems are illustrated and verified by the numerical simulations.

The purpose of this study is to analyse the distribution pattern of centrifugal fertiliser distributor with two shutter holes and obtain the optimal position of the shutter holes for improved distribution uniformity. A simulation model of the centrifugal fertiliser distributor was generated by discrete element method (DEM). The behaviour of fertiliser particles on the spinner was analysed by the simulation and the simulation model was verified through a stationary distribution pattern test. The distributor design was modified using the verified simulation model to improve the distribution uniformity. The simulation results of the existing fertiliser distributor showed the amount of fertiliser applied on the right side of the machine was 1.59 times higher than the amount applied on the left (R/L ratio) and the coefficient of variation (CV) was 34.88% at stationary condition indicated non-uniform fertiliser distribution. As the position of the shutter holes was moved in the direction of rotation of the spinner (i.e. clockwise) the amount of fertiliser distributed towards the right side decreased. The shutter hole position accordingly adjusted to improve fertiliser distribution uniformity. When the optimal position of the shutter hole was used under stationary condition, the R/L ratio was close to 1.0, whilst the CV value decreased by approximately 50%. The optimal solution was also suitable during continuous application when applying at a width of 8 m since CV values were

Incorporating the environmental perturbations and available resources of the public health system, we construct both deterministic and stochastic models of SIRS type. The deterministic model exhibits very rich dynamics, such as Hopf bifurcation and backward bifurcation which leads to the co-existence of the stable disease-free state and a stable endemic equilibrium. For the stochastic model, we show that under mild extra conditions, if the basic reproduction number is less than one, then the disease will be eradicated almost surely, and if the basic reproduction number is greater than one, the stochastic model will admit a unique ergodic stationary distribution, which implies that the disease persists almost surely. Part of our numerical simulations indicate that: (i) The introduction of environmental perturbations may drift the endemic equilibrium to the disease-free equilibrium, or vice versa; (ii) Increasing available resources is necessary in order to mitigate the infections.

This paper investigates a stochastic Markovian switching predator–prey model with infinite memory and general Levy jumps. Firstly, we transfer a classic infinite memory predator–prey model with weak kernel case into an equivalent model through integral transform. Then, for the corresponding stochastic Markovian switching model, we establish the sufficient conditions for permanence in time average and the threshold between stability in time average and extinction. Finally, sufficient criteria for a unique ergodic stationary distribution of the model are derived. Our results show that, firstly, both white noise and infinite memory are unfavorable to the existence of the stationary distribution; secondly, the general Levy jumps could make the stationary distribution vanish as well as happen; finally, the Markovian switching could make the stationary distribution appear.

This paper presents a simulation study on turnpike phenomena in stochastic optimal control problems. We employ the framework of Polynomial Chaos Expansions (PCE) to investigate the presence of turnpikes in stochastic LQ problems. Our findings indicate that turnpikes can be observed in the evolution of PCE coefficients as well as in the evolution of statistical moments. Moreover, the turnpike phenomenon can be observed for optimal realization trajectories and with respect to the optimal stationary distribution. Finally, while adding variance penalization to the objective alters the turnpike, it does not destroy the phenomenon.

Deterministic mathematical models are frequently used to represent the predator-prey interaction for a deeper understanding of population dynamics. This approach, however, always has some limitations as it does not capture the environmental noise or demographic stochasticity which are important features of any natural system. In this paper, we study a minimal deterministic model of phytoplankton-zooplankton (prey and predator, respectively) interaction and compare its dynamics with its stochastic version formulated by two different stochastic perturbation techniques. In the first method, two parameters of the deterministic system are replaced by its average value plus an error term. In the second method, the stochastic perturbation is considered proportional to the distance of state variables from their deterministic equilibrium value. We analyze both the stochastic models and explore their dynamics. In particular, we determine sufficient conditions for extinction probability, stochastic persistence of populations, and the existence of stationary distribution of the first stochastic system. For the other system, we determine sufficient conditions for the asymptotic mean square stability by defining a suitable Lyapunov function. Different analytical results are illustrated numerically and interpreted biologically. The stochastic behaviours of the system are compared with its deterministic counterpart. A case study has also been done considering the 24 months data of phytoplankton-zooplankton interaction in the Lake Trasimeno, Italy.

International audience; In this work, we study the exponential stability of the stationary distribution of a McKean-Vlasov equation, of nonlinear hyperbolic type which was recently derived in \cite{de_masi_hydrodynamic_2015,fournier_toy_2016}. We complement the convergence result proved in \cite{fournier_toy_2016} using tools from dynamical systems theory. Our proof relies on two principal arguments in addition to a Picard-like iteration method. First, the linearized semigroup is positive which allows to precisely pinpoint the spectrum of the infinitesimal generator. Second, we use a time rescaling argument to transform the original quasilinear equation into another one for which the nonlinear flow is differentiable. Interestingly, this convergence result can be interpreted as the existence of a locally exponentially attracting center manifold for a hyperbolic equation.

This paper investigates the synchronized stationary distribution of stochastic coupled networks with Markovian switching (CNMS) and the exponential synchronization of CNMS. Except for the Lyapunov method, the periodically intermittent control strategy and the technique of graph theory are applied to the study of synchronized stationary distribution and exponential synchronization. Some sufficient conditions to achieve synchronized stationary distribution and exponential synchronization are presented. The results show that the existing domain of synchronized stationary distribution is affected by control rate and the realization of exponential synchronization is also influenced by intensity of control, control rate. In addition, the theoretical results are applied to stochastic coupled oscillators and coupled Chua’s circuits with Markovian switching. Finally, two examples with numerical simulations are provided to illustrate the effectiveness of the results developed.

Wind farms are subject to unavoidable energy-production stoppage periods because of the stochastic behaviour of wind velocity; however, these periods provide additional opportunities for maintenance activities. In this research, we investigated a novel joint opportunistic maintenance strategy for wind turbines considering the stochastic wind velocity. Our strategy entailed considering two types of opportunities, namely, internal opportunities provided by the general degradation of wind turbines, and external opportunities generated by the weak wind-speed periods. An expected cost-rate model was formulated by utilising the semi-regenerative process theory based on all possible maintenance scenarios. In this model, all the maintenance activities at each decision point were determined in accordance with the maintenance requirement that is dictated by the state of deterioration of the wind turbines. Furthermore, a stationary probability density function was developed by derivation of the stationary law of the wind turbine condition, influenced by opportunistic maintenance. The maintenance activities and their corresponding probabilities were all derived on this basis. Finally, the applicability and correctness of the proposed approach were demonstrated by a case study and sensitivity analysis. The results of our study indicated that the maintenance policy we proposed could be of economic benefit to wind farms.

A stochastic compartmental model for cholera is investigated, which incorporates the effect of two transmission pathways on the diseases via contaminated water. Multiple stages of infection and multiple states of pathogen and environmental white noises are considered, which include or extend the cholera models in some existing articles. The existence of a unique stationary distribution and ergodicity of the model are obtained by using a suitable Lyapunov function, which determines a critical value R 0 * corresponding to a basic reproduction number R0 of the ordinary differential equation. The results mean if R 0 * > 1 , the system still retains the stability and all individuals are persistent in a long term. In addition, the sufficient conditions for extinction of diseases are derived. What’s more, we provide a new method in constructing a Lyapunov function, which can be successfully applied to other complex high-dimensional systems.

To screen for virus carriers and for infected individuals to actively seek treatment are the key factors to curb the spread of HIV/AIDS. We propose a stochastic HIV/AIDS model to evaluate their effect. First, we theoretically prove that the solution of the stochastic model is positive and global. Second, we obtain the existence of an ergodic stationary distribution if R 0 S > 1 , and we establish sufficient conditions R 0 ′ 1 for disease extinction. We provide examples and numerical simulations to verify our theoretical results. Finally, we analyze the impact of the above two factors on the dynamical behavior of the stochastic system, and draw conclusions regrading disease prevention and elimination.

In this paper, we investigate the fluctuation impact of human mobility on influenza transmissions through a stochastic differential equation (SDE) model with environmental driving forces through both analytical and numerical analysis. We define a stochastic basic reproduction number R 0 s through a detailed process of calculations for the SDE model. Through R 0 s , the conditions for the stochastic extinction and persistence of the influenza disease are identified. We show that an epidemic stationary distribution exists in the case of persistence of influenza disease by constructing an appropriate Lyapunov function. We find that: (1) large environment fluctuations can suppress the outbreak of the influenza; (2) R 0 s determines stochastic distributions; (3) proper noise perturbations can be beneficial to control the spread of the influenza; (4) high economic capability can prohibit the influenza outbreaks; (5) seemingly irregular but appropriate human mobility response can significantly reduce the epidemic prevalence; (6) the positivity of the minimum intrinsic growth rate of human mobility density is the key factor of the stochastic persistence of the influenza disease.

In this paper, we propose and analyze a deterministic model along with its stochastic version to address the problem of scanty rainfall by means of forestry resources. For deterministic model, boundedness of the system, feasibility of equilibria and their stability behavior are discussed. For stochastic model, boundedness, existence, uniqueness of global positive solution and sufficient conditions for the existence of unique stationary distribution are obtained. Model analysis reveals that the stability of the forest cover equilibrium state depends only on the model parameters in the deterministic case, however it also depends on the magnitude of the intensities of white noise terms in the stochastic case. To validate analytically obtained results and see the effect of key parameters, we have simulated proposed models using Indian annual rainfall data. The proposed model suggests that for the parameter values given in Table 2 , the plantation of trees with slight higher intrinsic growth rate is beneficial to increase the rainfall.

A correction is noted to the previously published paper (Zou and Lv, 2017). Regularity, positive recurrence and stationary distribution are considered in this paper. A complete threshold analysis of strong stochastic persistence and extinction is investigated. Stochastic Hopf bifurcation is considered from the viewpoint of numerical simulations.

We investigate the global asymptotic stabilities of disease-free equilibrium and endemic equilibrium of the deterministic susceptible–exposed–infected–recovered epidemic model (short for SEIR model). The basic reproduction number R 0 , depends on constant contact rate β and natural death rate d and other parameters as well, indicates the critical value of stability, and completely determines the dynamical behavior of the deterministic model. After taking the perturbations of the environments into account, the corresponding stochastic SEIR model with generalized nonlinear incidence is discussed in existence and uniqueness, the extinction in the mean, and the existence of the unique stationary distribution as well. As a consequence, we carry out several numerical simulations to support the main theoretical results of this paper.

This paper investigates dynamic properties of a class of stochastic Markovian jump linear systems. Firstly, we introduce quasi-alternative Markovian jump linear switching systems with both stable and unstable subsystems. Then, we provide sufficient conditions of finite-time boundedness by using Lyapunov methods and system future analysis. These presented conditions guarantee the existence of boundary within a given time interval. Next, we further analyse an useful dynamic property, named as finite-time chaos-like property, based on boundedness and stationary distribution characteristics. Finally, a detailed example is presented to illustrate our main results.

The significant cost required for implementation of WEC sites and the uncertainty associated with their performance, due to the randomness of the marine environment, can bring critical challenges to the industry. This paper presents a probabilistic methodology for predicting the long-term power generation of WECs. The developed method can be used by the operators and designers to optimize the performance of WECs by improving the design or in selecting optimum site locations. A Markov Chain model is constructed to estimate the stationary distribution of output power based on the results of hydrodynamic analyses on a point absorber WEC. To illustrate the application of the method, the performance of a point absorber is assessed in three locations in the south of Tasmania by considering their actual long-term sea state data. It is observed that location 3 provides the highest potential for energy extraction with a mean value for absorbed power of approximately 0.54MW, while the value for locations 1 and 2 is 0.33MW and 0.43MW respectively. The model estimated that location 3 has the capacity to satisfy industry requirement with probability 0.72, assuming that the production goal is to generate at least 0.5MW power.

This paper deals with a mutualism system in random environments, in which the cooperation of two species is guaranteed depending on increasing the carrying capacity of each other. Mathematically, we obtain the existence and uniqueness of a stable stationary distribution by means of Markov semigroup theory and Fokker–Planck equation. In addition, we prove that densities of the distributions for the solutions converge in L 1 to an invariant density under appropriate conditions. Numerical simulations are also presented to illustrate the theoretical results.

Consider a fleet of N autonomous electric vehicles (EVs), where EV n has a battery state βn. Customers arrive at rate λ and each customer requests a trip that costs some battery charge b. The goal is to coordinate the pick-ups and charging of the fleet in order to minimize the discounted number of unpicked customers over an infinite time horizon. Solving the dynamic program for this scheduling problem is infeasible since the state space has an exponential size in N. We propose a distributed approach instead where each EV models the fleet as a "field" that picks the customer i.i.d. at random with some pick-up probability α (b). Based on this model, each EV computes its optimal charging and pick-up strategy, and reports which customer they are available to pick-up (as a function of b) or if they decided to charge. The server computes a maximal matching between customers and EVs, taking into account their reported availability. The process continues iteratively as each EV recomputes its optimal response to the new empirical α (b). We prove that the optimal local strategy is a threshold policy with respect to βn, where the threshold depends on α (b) and b. We also prove that the empirical pick-up probability α (b) converges to the true stationary probability, which explains why our algorithm converges to an equilibrium. We then study our algorithm in simulations, that show that at equilibrium, its performance is significantly better than that of an uncoordinated charging strategy. Therefore, our solution has low-complexity but is still sophisticated enough to coordinate the charging of the fleet.

In this paper, we consider stationarity of a class of second-order stochastic evolution equations with memory, driven by Wiener processes or Levy jump processes, in Hilbert spaces. The strategy is to formulate by reduction some first-order systems in connection with the stochastic equations under investigation. We develop asymptotic behavior of dissipative second-order equations and then apply them to time delay systems through Gearhart–Pruss–Greiner’s theorem. The stationary distribution of the system under consideration is the projection on the first coordinate of the corresponding stationary results of a lift-up stochastic system without delay on some product Hilbert space. Last, two examples of stochastic damped wave equations with memory are presented to illustrate our theory.

A stochastic resonance (SR) tristable system based on a high-order time-delayed feedback is investigated and the feasibility of the system for weak fault signature extraction is discussed. The potential function, the mean first-passage time (MFPT) and the signal-to-noise ratio (SNR) are used to evaluate the model. Firstly, the potential function and stationary probability function (PDF) of the system are derived, and then the influence of the time delay parameters on the MFPT of the particles is analyzed. Secondly, the influences of time-delyed strength e and delyed length τ on the SR system from the perspective of the transition of the particles in the potential wells are discussed, and then the SNR and the effect of the parameters on the SNR are derived. In addition, the high-order time-delayed feedback tristable stochastic resonance (HTFTSR) system is used to deal with faulty bearing data and is compared with traditional tristable stochastic resonance (TSR). The result shows that the nonlinear system model can accurately identify the fault frequency and improve the energy of the characteristic signal under the appropriate system parameters.

To investigate the effects of stochasticity and seasonality on the dynamics of aquatic ecosystems, in this paper, we propose two stochastic nutrient-plankton food chain models. For the stochastic model with regime switching, we establish sufficient conditions for the persistence and extinction of plankton. Under mild extra assumptions, the existence of an ergodic stationary distribution is studied. For the stochastic model with seasonal fluctuation, we first perform the survival analysis. Then, based on Khasminskii’s theory for periodic Markov processes, we give sufficient criteria for the existence of a positive stochastic periodic solution. Numerical simulations are carried out to illustrate the analytical results, showing that stochasticity and seasonality may cause the periodic bloom of phytoplankton, which partly explains the formation of algal blooms.

In the present paper, considering the significant nonsmooth factors, friction and impact, the responses and asymptotic stability with probability one of a system are investigated for the cases of additive and multiplicative Gaussian white noise random excitations. First, the original system which contains impact and viscoelastic terms can be approximated as an equivalent system by using nonsmooth transformation and converting the viscoelastic force to the sum of stiffness and damping terms. Then, the stationary probability density functions are analytically obtained by utilizing the stochastic averaging method. And the equation of the Top Lyapunov Exponent is also obtained with the help of combining the stochastic averaging method and Khasminskii method, which is used to discuss the effects of various parameters on the system’s stability. The validity of the analytical results derived from the proposed procedures is verified by comparing with directly numerical simulation. In particular, under certain conditions, there are examples about how to control the change of stability of the system by adjusting the variation of the restitution coefficient and the friction from the mathematical standpoint.

The dynamical evolution of a vegetation-water system subjected to Gaussian noise disturbance is explored. With regard to the regime shift of the stationary response of the system, the early warning signal can be detected by the maximum of the stationary probability density function. And for the transient response of the bistable system, the basin stability is investigated by two different deterministic quantities, the first escape probability and the mean first exit time. Firstly, the stochastic system with two variable is translated into the Ito stochastic differential equation to derive the analytical expression of the first escape probability and the mean first exit time with different Balayage-Dirichlet exterior boundary condition. Then, the two indexes are calculated by the finite difference method, respectively. The results, compared with the ones through Monte Carlo simulations, shows that the proposed method works very well. It turns out that the increase of noise intensity induces advance of the early warning and decreases basin stability of the vegetation-water system.

We consider n independent p -dimensional Gaussian vectors with covariance matrix having Toeplitz structure. The aim is two-fold: to test that these vectors have independent components against a stationary distribution with sparse Toeplitz covariance matrix, and also to select the support of non-zero entries under the alternative hypothesis. Our model assumes that the non-zero values occur in the recent past (time-lag less than p / 2 ). We build test procedures that combine a sum and a scan-type procedure, but are computationally fast, and show their non-asymptotic behaviour in both one-sided (only positive correlations) and two-sided alternatives, respectively. We also exhibit a selector of significant lags and bound the Hamming-loss risk of the estimated support. These results can be extended to the case of nearly Toeplitz covariance structure and to sub-Gaussian vectors. Numerical results illustrate the excellent behaviour of both test procedures and support selectors - larger the dimension p , faster are the rates.

In this paper, we study a stochastic SIR epidemic model with distributed delay and degenerate diffusion. Firstly, we transform the stochastic model into an equivalent system which contains three equations. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We verify the densities of the distributions of the solutions can converge in L1 to an invariant density. Then we establish sufficient conditions for extinction of the disease. Some examples and numerical simulations are introduced to illustrate our analytical results.

In this paper, the issue about the stationary distribution for hybrid multi-stochastic-weight coupled networks (HMSWCN) via aperiodically intermittent control is investigated. Specially, when stochastic disturbance gets to zero, the exponential stability in pth moment for hybrid multi-weight coupled networks (HMWCN) is considered. Under the framework of the Lyapunov method, M-matrix and Kirchhoff’s Matrix Tree Theorem in the graph theory, several sufficient conditions are derived to guarantee the existence of a stationary distribution and exponential stability. Different from previous work, the existing area of a stationary distribution is not only related to the topological structure of coupled networks, but also aperiodically intermittent control (the rate of control width and control duration). Subsequently, as an application to theoretical results, a class of hybrid multi-stochastic-weight coupled oscillators is studied. Ultimately, numerical examples are carried out to demonstrate the effectiveness of theoretical results and effects of the control schemes.

In this paper, a stochastic epidemic model for cholera is proposed and investigated. Firstly, we establish sufficient conditions for extinction of the disease. Then we establish sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. The existence of an ergodic stationary distribution implies that all the individuals can be coexistent in the long run. Finally, some examples together with numerical simulations are introduced to illustrate our theoretical results.

The linkage disequilibrium coefficient r(2) is a measure of statistical dependence of the alleles possessed by an individual at different genetic loci. It is widely used in association studies to search for the locations of disease-causing genes on chromosomes. Most studies to date treat r(2) as a fixed property of two loci in a finite population, and investigate the sampling distribution of estimators due to the statistical sampling of individuals from the population. Here, we instead consider the distribution of r(2) itself under a process of genetic sampling through the generations. Using a classical two-locus model for genetic drift, mutation, and recombination, we investigate the probability density function of r(2) at stationarity. This density function provides a tool for inference on evolutionary parameters such as mutation and recombination rates. We reconstruct the approximate stationary density of r(2) by calculating a finite sequence of the distribution’s moments and applying the maximum entropy principle. Our approach is based on the diffusion approximation, under which we demonstrate that for certain models in population genetics, moments of the stationary distribution can be obtained without knowing the probability distribution itself. To illustrate our approach, we show how the stationary probability density of r(2) can be used in a maximum likelihood framework to estimate mutation and recombination rates from sample data of r(2).

In this paper we consider a batch matching system with impatient servers and boundedly rational customers, which is essentially a double-ended batch service system. Each server is responsible for a batch of customers and serves them instantaneously. Especially, we assume that the servers have the characteristic of ”impatience”, which will lead to incomplete batch processing of each service. To derive the stationary distribution of system states, we combine two methods (called probability generating function method and G-matrix method), and some main performance measures are captured. In addition, we assume the customers are boundedly rational, that is, they are unable to estimate their sojourn times and utilities accurately. To deal, a logit model is employed to model this characteristic of the customers, and by which, we obtain the customers’ equilibrium joining fraction which is compared with the equilibrium joining probability of fully rational customers afterwards. Depending on whether the service is visible or invisible, we conduct some numerical analyses to show main parameters’ impact on equilibrium fraction, along with some intuitive explanations. Finally, to make the system viable economically, a function of service provider’s utility is developed and several numerical plots are presented to study the impact of several main parameters on the utility. Through the PSO (Particle Swarm Optimization) algorithm, we numerically obtain the maximum utility point within a certain range of multiple parameters.

We present a stochastic HIV infection model with logistic target cell growth, general nonlinear incidence rate, CTL immune response and parameter perturbations. Through a rigorous analysis of the model, we obtain that the solution of the model is positive and global. A critical condition R 0 s of the model is derived, which depends not only on the general incidence function but also on the noise intensities. Under certain assumptions, by constructing suitable Lyapunov functions, we find that the system has a unique ergodic stationary distribution when R 0 s > 1 . We further explore the effect of the noise intensity on model behavior. Our conclusion improves and generalizes the results of the existing HIV stochastic models.

In this paper, an avian influenza epidemic model with half saturated incidence is investigated. Firstly we study the local asymptotical stability of equilibria of the deterministic model. Then for the stochastic model, the existence of the global positive solution is proved and different thresholds are obtained. In addition, we get sufficient conditions for the existence of an ergodic stationary distribution by using Has’minskii theory and Lyapunov function. Finally, some numerical simulations illustrate the analytical results.

This paper formulates a new switched two species chemostat model and discusses the coexistence behavior in the chemostat. A complete classification on the single-species chemostat is carried out firstly, where the stationary distribution with ergodicity is derived to exist and be unique. Then, based on the obtained stationary distribution and the comparison theorem, we put forward some sufficient conditions for the coexistence of microorganisms in the two species chemostat with Markov switchings. Moreover, when the species coexist in the deterministic chemostat for each state and have the same break-even concentrations for all states, they are proved to coexist still in the switched chemostat, which randomized the results of the classical deterministic chemostat. Results in this paper show that Markov switchings can contribute to coexistence of the two species.

Graph-theoretic approach as a new technique is applied to analyze the issue of stationary distribution for stochastic multi-group models (MGMs) with Markovian switching and dispersal. The feature of our model is combining the white noise, the dispersal with Markovian switching, which can model systems in practice reasonably. By constructing a global Lyapunov function for stochastic MGMs with Markovian switching and dispersal, sufficient criteria, which guarantee the existence of a stationary distribution, are presented via the combination of the graph theory, the Lyapunov method and the M-matrix method. Moreover, stochastic coupled oscillators with Markovian switching and dispersal are presented as a practical application to illustrate our results in the end.

A model of series expansion with random amplitudes is proposed to generate two correlated Gaussian stochastic processes. The spectral density of each process can be adjusted to fit that of any practical process, and the correlation coefficient of the two processes can be specified arbitrarily. The model is then used to simulate two correlated random ocean wave forces of Pierson–Moskowitz spectrum. Based on the model, ship rolling motion under two correlated wave forces is investigated. The focus is on the effect of the correlation of the two random wave forces. Two different situations are considered. When the wave forces are weak, capsizing will not occur, and the motion will reach the stationary state after a period of time. The stationary probability densities of the rolling angle are calculated by simulation. It is found that the effect of the correlation on the long-term behavior of the ship rolling motion may be neglected. Another situation is with strong wave forces, in which case capsizing may happen as a random event. The ship rolling motion is non-stationary transient process before capsizing occurs. The average capsizing time is calculated also by simulation. It is shown that the correlation may have significant effect on the capsizing process, depending on system and excitation parameters.

In this paper, the combined effects of delayed feedbacks and noise on the enzyme–substrate reaction in the brain are examined. By means of the approximate methods, we analyze the stationary probability density function of the amplitude and the joint of displacement and velocity, finding that stochastic bifurcations can be induced by noise correlation and time delay. In Particular, the system undergoes a succession of two transitions as time delay is varied monotonically, namely, the reentrance phenomenon appears. The analytical results appear to be in reasonable agreement with the direct numerical simulations of the original system. From the biological point of view, the reaction rate of the enzymatic substrate reaction can be enhanced or diminished by Gaussian colored noise. More intriguingly, an increase in displacement feedback will retard the reaction rate, nevertheless, whether an increasing velocity feedback will benefit the reaction rate depends on time delay. It is worth noting that the chemical oscillations can be controlled to a low level when the appropriate time delay is chosen.

In this paper, based on the existing literature, we further study an important statistical character of a stochastic delayed chemostat model. By constructing suitable Lyapunov functional and using the stochastic Lyapunov analysis method, we investigate the existence of stationary distribution and the ergodicity of a stochastic delayed chemostat model, which can help us better understand the dynamic behavior and statistical characteristics of stochastic delayed biological models.

A single-degree-of-freedom random oscillator with a piecewise linear restoring force (experiencing softening after a certain point value of the response, called a “knuckle” point) is studied with the goal of understanding the structure of the distribution tail of its response or (local) maximum. A theoretical analysis is carried out by two approaches: first, by focusing on the maximum and response after crossing the “knuckle” point, where explicit calculations can be performed assuming standard distributions for the derivative at the crossing, and second, by considering the white noise random external excitation, where the stationary distribution of the response is readily available from the literature. Both approaches reveal the structure of the distribution tails where a Gaussian core is followed by a heavier tail, possibly having a power-law form, which ultimately turns into a tail having a finite upper bound (referred to as a light tail). The extent of the light tail region is also investigated, and shown to be the result of the conditioning for the system not to reach the unstable equilibrium. The study is motivated by applications to ship motions, where the considered random oscillator serves as a prototypical model for ship roll motion in beam seas, and estimation of the probabilities of these motions exceeding large critical angles is of interest. Standard statistical methodology for such estimation is based on the peaks-over-threshold approach, for which several lessons are drawn from the analysis of the tail structure of the considered random oscillator.