Modeling and Analysis of Switching Diffusion Systems: Past-Dependent Switching with a Countable State Space

Stochastic Analysis and Applications

2019

Self Cite

This work is concerned with competitive Lotka-Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective of the paper is to reduce the computational complexity by using the two-time-scale systems. Because existence and uniqueness as well as continuity of solutions for Lotka-Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the generator of the Markov chain. When the small parameter goes to 0, there is a limit system or reduced system. It is established in this paper that if the reduced system possesses certain properties such as permanence and extinction, etc., then the complex system also has the same properties when the parameter is sufficiently small. These results are obtained by using the perturbed Lyapunov function methods. Lotka [17] and Volterra [29], the well-known Lotka-Volterra models have been investigated extensively in the literature and used widely in ecological and population dynamics, among others. When two or more species live in close proximity and share the same basic requirements, they usually compete for resources, food, habitat, or territory. Initially being posed as a deterministic model, subsequent study has taken randomness into consideration; see [6,10,24] and references therein. Recent effort on the so-called hybrid systems has much enlarged the applicability of Lotka-Volterra systems. One class of such hybrid systems uses a continuous-time Markov chain to model environmental changes and other random factors not represented in the usual stochastic differential equations; see [33] for a comprehensive study of switching diffusions. Introduced byDeterministic Lotka-Volterra systems have been studied by many people. A number of important results were obtained. A set of sufficient conditions for the existence of a globally stable equilibrium point in various models of the n-dimensional Lotka-Volterra system was obtained in [16]; limit cycles for some deterministic competitive three-dimensional Lotka-Volterra systems were treated in [30]. Random perturbations to the Lotka-Volterra model were considered in the literature; see for example [3,12] and many references therein, and also [11,13] for up-to-dated progress on stochastic replicator dynamics. Recently, much effort has been devoted to studying the stochastic Lotka-Volterra with regime-switching; see [7,18,27,34,35]. While most recent works focus on Markov chain with a finite state space, to take into consideration of various factors, it is also natural to consider the Markov chain with a countable state space, which is the effort of the current paper.One of our main aims here is to reduce the computational complexity. Assuming that the Markov chain has a two-time-scale structure so that it has a fast changing part and a slowly varying part, we show that the systems under consi...

Section: A Proofs Of Technical Resultsmentioning

confidence: 99%

“…Our paper establishes these properties. Although general regime-switching diffusions were considered in [23], the spatial variable x there lives in the whole space R n , whereas for the Lotka-Volterra systems considered here, x ∈ R n + . It needs to be established that the solution is in R n + as well.…”

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confidence: 99%

Hybrid competitive Lotka–Volterra ecosystems: countable switching states and two-time-scale models

Bui

Stochastic Analysis and Applications

2019

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“…In another direction, Xi and Zhu (2017) dealt with regime-switching jump diffusions with countable number of switching values. Nguyen and Yin (2016) considered switching diffusions in which the switching process depends on the past information of the continuous state and takes values in a countable state space; the corresponding recurrence and ergodicity was considered in Nguyen and Yin (2018).…”

Section: Introductionmentioning

confidence: 99%

Regime-Switching Jump Diffusions with Non-Lipschitz Coefficients and Countably Many Switching States: Existence and Uniqueness, Feller, and Strong Feller Properties

Modeling, Stochastic Control, Optimization, and Applications

Zhu

2019

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This work focuses on a class of regime-switching jump diffusion processes, which is a two component Markov processes (X(t),Λ (t)), where Λ (t) is a component representing discrete events taking values in a countably infinite set.Considering the corresponding stochastic differential equations, our main focus is on treating those with non-Lipschitz coefficients. We first show that there exists a unique strong solution to the corresponding stochastic differential equation. Then Feller and strong Feller properties are investigated.

“…To be able to treat more complex models and to broaden the applicability, we have undertaken the task of investigating the dynamics of (X(t), α(t)) in which α(t) has a countable state space and its switching intensities depend on the history of the continuous component X(t). As a first attempt, this type of switching diffusion was considered in [16], which was motivated by queueing and control systems applications. In particular, the evolution of two interacting species was considered in the aforementioned reference.…”

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confidence: 99%

“…The reproduction process of α(t) is non-instantaneous, resulting in past-dependent switching. In [16], we gave precise formulation of the process (X(t), α(t)) and established the existence and uniqueness of solutions together with such properties as Markov-Feller property and Feller property of function-valued stochastic processes associated with our processes under suitable conditions. Many real-world systems are in operation for a long period of time.…”

mentioning

confidence: 99%

Recurrence and Ergodicity of Switching Diffusions with Past-Dependent Switching Having a Countable State Space

Nguyen

2017

Potential Anal

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This work focuses on recurrence and ergodicity of switching diffusions consisting of continuous and discrete components, in which the discrete component takes values in a countably infinite set and the rates of switching at current time depend on the value of the continuous component over an interval including certain past history. Sufficient conditions for recurrence and ergodicity are given. Moreover, the relationship between systems of partial differential equations and recurrence when the switching is pastindependent is established under suitable conditions. Emerging and existing applications in wireless communications, queueing networks, biological models, ecological systems, financial engineering, and social networks demand the mathematical modeling, analysis, and computation of hybrid systems in which continuous dynamics and discrete events coexist. Switching diffusions are one of such hybrid models. A switching diffusion is a two-component process (X(t), α(t)), a continuous component and a discrete component taking values in a set consisting of isolated points. When the discrete component takes a value α (i.e., α(t) = α), the continuous component X(t) evolves according to the diffusion process whose drift and diffusion coefficients depend on α. Such processes have received growing attention recently because of their ability to a wide range of applications; see [11,13,20,26] and the references therein.In the comprehensive treatment of hybrid switching diffusions in [14], it was assumed that α(t) is a continuous-time and homogeneous Markov chain independent of the Brownian motion and that the generator of the Markov chain is a constant matrix. For broader impact on applications, considering the two components jointly, the work [25] extended the study to the Markov process (X(t), α(t)) by allowing the generator of α(t) to depend on the current state X(t). Until very recently, most of the works treat α(t) as a process taking values in a finite set. Even when α(t) is allowed to take values in a countable state space, almost all works required the systems being memoryless. That is, the switching depends on the continuous state, with the dependence on the current continuous state only, no delays are involved; see, for example, [14,19,20,25] and references therein. To be able to treat more complex models and to broaden the applicability, we have undertaken the task of investigating the dynamics of (X(t), α(t)) in which α(t) has a countable state space and its switching intensities depend on the history of the continuous component X(t). As a first attempt, this type of switching diffusion was considered in [16], which was motivated by queueing and control systems applications. In particular, the evolution of two interacting species was considered in the aforementioned reference. One of the species is micro described by a logistic differential equation perturbed by a white noise, and the other is macro. Let X(t) be the density of the micro species and α(t) the population of the macro species. The reproduction proce...