scholarly journals Stochastic Lotka-Volterra systems under regime switching with jumps

Filomat ◽  
2014 ◽  
Vol 28 (9) ◽  
pp. 1907-1928 ◽  
Author(s):  
Ruihua Wu ◽  
Xiaoling Zou ◽  
Ke Wang ◽  
Meng Liu
Keyword(s):  
Markov Chain ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
Volterra Model ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Stochastic Permanence ◽  
Color Noise ◽  
Volterra Systems ◽  
The Moment

A stochastic Lotka-Volterra model with Markovian switching driven by jumps is proposed and investigated. In the model, the white noise, color noise and jumping noise are taken into account at the same time. This model is more feasible and applicable. Firstly, sufficient conditions for stochastic permanence and extinction are presented. Then the moment average in time and the asymptotic pathwise properties are estimated. Our results show that these properties have close relations with the jumps and the stationary probability distribution of the Markov chain. Finally, several numerical simulations are provided to illustrate the effectiveness of the results.

scholarly journals Dynamical Behavior of a Stochastic Delayed One-Predator and Two-Mutualistic-Prey Model with Markovian Switching and Different Functional Responses

2016 ◽  
Vol 2016 ◽  
pp. 1-16
Author(s):  
Rensheng He ◽  
Zuoliang Xiong ◽  
Desheng Hong
Keyword(s):  
Lyapunov Functions ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Matrix Analysis ◽  
Stationary Probability ◽  
Functional Responses ◽  
Stationary Probability Distribution ◽  
Stochastic Permanence ◽  
Prey Model ◽  
Telegraph Noise

We propose a stochastic delayed one-predator and two-mutualistic-prey model perturbed by white noise and telegraph noise. By theM-matrix analysis and Lyapunov functions, sufficient conditions of stochastic permanence and extinction are established, respectively. These conditions are all dependent on the subsystems’ parameters and the stationary probability distribution of the Markov chain. We also investigate another asymptotic property and finally give two examples and numerical simulations to illustrate main results.

scholarly journals Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

2021 ◽  
Vol 58 (2) ◽  
pp. 372-393
Author(s):  
H. M. Jansen
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
The State ◽  
Stochastic Integrals ◽  
Occupation Measure ◽  
Generator Matrix ◽  
Markov Modulated ◽  
Erlang Loss

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

The censored Markov chain and the best augmentation

1996 ◽  
Vol 33 (03) ◽  
pp. 623-629 ◽  
Author(s):  
Y. Quennel Zhao ◽  
Danielle Liu
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
The Best Approximation ◽  
Finite Matrix ◽  
Countable State ◽  
Stationary Probabilities

Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.

scholarly journals Stochastic Delay Logistic Model under Regime Switching

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Zheng Wu ◽  
Hao Huang ◽  
Lianglong Wang
Keyword(s):  
Regime Switching ◽  
Logistic Model ◽  
Sufficient Conditions ◽  
Function Technique ◽  
Stochastic Permanence ◽  
Stochastic Delay ◽  
Gronwall’S Inequality ◽  
Diffusion In Random Environment ◽  
Stochastically Ultimate Boundedness ◽  
Generalized Itô Formula

This paper is concerned with a delay logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall's inequality, and Young's inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue ofV-function technique,M-matrix method, and Chebyshev's inequality. Finally, an example is given to illustrate the main results.

Mixing properties of harris chains and autoregressive processes

1986 ◽  
Vol 23 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Krishna B. Athreya ◽  
Sastry G. Pantula
Keyword(s):  
Sufficient Conditions ◽  
Autoregressive Process ◽  
Strong Mixing ◽  
Stationary Probability ◽  
Autoregressive Processes ◽  
Stationary Probability Distribution ◽  
Mixing Properties ◽  
Recurrent Markov Chain ◽  
General State Space ◽  
Necessary And Sufficient

Let {Yn: n ≧ 1} be a Harris-recurrent Markov chain on a general state space. It is shown that {Yn} is strong mixing, provided there exists a stationary probability distribution π (·) for {Yn}. Necessary and sufficient conditions for an autoregressive process to be uniform mixing are given.

Mixing properties of harris chains and autoregressive processes

1986 ◽  
Vol 23 (04) ◽  
pp. 880-892 ◽  
Author(s):  
Krishna B. Athreya ◽  
Sastry G. Pantula
Keyword(s):  
Sufficient Conditions ◽  
Autoregressive Process ◽  
Strong Mixing ◽  
Stationary Probability ◽  
Autoregressive Processes ◽  
Stationary Probability Distribution ◽  
Mixing Properties ◽  
Recurrent Markov Chain ◽  
General State Space ◽  
Necessary And Sufficient

Let {Yn:n≧ 1} be a Harris-recurrent Markov chain on a general state space. It is shown that {Yn} is strong mixing, provided there exists a stationary probability distributionπ(·) for {Yn}. Necessary and sufficient conditions for an autoregressive process to be uniform mixing are given.

Mixing properties of harris chains and autoregressive processes

1986 ◽  
Vol 23 (04) ◽  
pp. 880-892 ◽  
Author(s):  
Krishna B. Athreya ◽  
Sastry G. Pantula
Keyword(s):  
Sufficient Conditions ◽  
Autoregressive Process ◽  
Strong Mixing ◽  
Stationary Probability ◽  
Autoregressive Processes ◽  
Stationary Probability Distribution ◽  
Mixing Properties ◽  
Recurrent Markov Chain ◽  
General State Space ◽  
Necessary And Sufficient

Let {Yn: n ≧ 1} be a Harris-recurrent Markov chain on a general state space. It is shown that {Yn } is strong mixing, provided there exists a stationary probability distribution π (·) for {Yn }. Necessary and sufficient conditions for an autoregressive process to be uniform mixing are given.

scholarly journals Dynamics of a stochastic Gilpin–Ayala population model with Markovian switching and impulsive perturbations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yuan Jiang ◽  
Zijian Liu ◽  
Jin Yang ◽  
Yuanshun Tan
Keyword(s):  
Regime Switching ◽  
Population Model ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Markovian Switching ◽  
Critical Number ◽  
Stochastic Permanence ◽  
The Mean ◽  
Impulsive Perturbations

Abstract In this paper, we consider the dynamics of a stochastic Gilpin–Ayala model with regime switching and impulsive perturbations. The Gilpin–Ayala parameter is also allowed to switch. Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are provided. The critical number among the extinction, nonpersistence in the mean, and weak persistence is obtained. Our results demonstrate that the dynamics of the model have close relations with the impulses and the Markov switching.

scholarly journals Dynamics of a General Stochastic Nonautonomous Lotka-Volterra Model with Delays and Impulsive Perturbations

2015 ◽  
Vol 2015 ◽  
pp. 1-17
Author(s):  
Qing Wang ◽  
Yongguang Yu ◽  
Shuo Zhang
Keyword(s):  
Theoretical Analysis ◽  
Stochastic Model ◽  
Numerical Simulations ◽  
Sufficient Conditions ◽  
Critical Value ◽  
Volterra Model ◽  
Stochastic Permanence ◽  
Stochastic Noises ◽  
The Mean ◽  
Impulsive Perturbations

A stochastic nonautonomous N-species Lotka-Volterra model with delays and impulsive perturbations is investigated. For this model, sufficient conditions for extinction, nonpersistence in the mean, weak persistence and stochastic permanence are given, respectively. The influences of the stochastic noises, and the impulsive perturbations on the properties of the stochastic model are also discussed. The critical value between weak persistence and extinction is obtained. Finally, numerical simulations are given to support the theoretical analysis results.

scholarly journals High-order Vector Markov Chain with Partial Connections in Data Analysis

2017 ◽  
Vol 46 (3-4) ◽  
pp. 37-45 ◽  
Author(s):  
Yuriy Kharin ◽  
Michail Maltsew
Keyword(s):  
Mathematical Model ◽  
Time Series ◽  
Markov Chain ◽  
Data Analysis ◽  
Probability Distribution ◽  
Discrete Time ◽  
Model Parameters ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Conditional Probability Distribution

A new mathematical model for discrete time series is proposed: homogenous vector Markov chain of the order s with partial connections. Conditional probability distribution for this model is determined only by a few components of previous vector states. Probabilistic properties of the model are given: ergodicity conditions and conditions under which the stationary probability distribution is uniform. Consistent statistical estimators for model parameters are constructed.

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