Stochastic Permanence Research Articles

Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations

In this article, an impulsive stochastic tumor-immune model with regime switching is formulated and explored. Firstly, it is proven that the model has a unique global positive solution. Then sufficient criteria for extinction, non-persistence in the mean, weak persistence and stochastic permanence are provided. The threshold value between extinction and weak persistence is gained. In addition, the lower- and the upper-growth rates of tumor cells are estimated. The results demonstrate that the dynamics of the model are intimately associated with the random perturbations and impulsive perturbations. Finally, biological implications of the results are addressed with the help of real data and numerical simulations.

Asymptotic Properties of Multi-species Lotka–Volterra Models with Regime Switching Involving Weak and Strong Interactions

This work focuses on multi-species Lotka–Volterra models with regime switching modulated by a continuous-time Markov chain involving a small parameter. The small parameter is used to reflect different rates of the switching among a large number of states representing the discrete events. Using perturbed Lyapunov function methods and the structure of the limit system as a bridge, stochastic permanence and extinction are obtained. Sufficient conditions under which the measures of the original system converge to the invariant measure of that of the limit system are provided. A couple of examples and numerical simulations are given to illustrate our results.

Analysis of a stochastic predator–prey system with modified Leslie–Gower and Holling-type IV schemes

In this paper, we investigate the dynamics of a stochastic predator–prey system with modified Leslie–Gower and Holling-type IV schemes. We first show the existence and uniqueness of the global positive solution to the system with positive initial values. In some case, the stochastic boundedness and stochastic permanence are obtained. Then, under some conditions, we prove the persistence in mean and extinction of the stochastic system. Moreover, under certain parametric restrictions, we obtain that the system has a stationary distribution which is ergodic. Finally, some numerical simulations are carried out to support our results.

Analysis of a stochastic hybrid population model with Allee effect

This research formulates and studies a stochastic population model with Allee effect and Markovian switching. Sufficient criteria for extinction, stochastic permanence and the existence of a unique ergodic invariant measure of the model are established. The result indicates that these long term behaviors of the model are intimately associated with the random perturbations.

Dynamics analysis of a stochastic non-autonomous one-predator\u2013two-prey system with Beddington\u2013DeAngelis functional response and impulsive perturbations

In this paper, we explore a stochastic non-autonomous one-predator–two-prey system with Beddington–DeAngelis functional response and impulsive perturbations. First, by using Itô’s formula, exponential martingale inequality, Chebyshev’s inequality and other mathematical skills, we establish some sufficient conditions for extinction, non-persistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the solution of the stochastic system. Then the limit of the average in time of the sample path of the solution is estimated by two constants. Afterwards, the lower-growth rate and the upper-growth rate of the positive solution are estimated. In addition, sufficient conditions for global attractivity of the system are established. Finally, we carry out some simulations to verify our main results and explain the biological implications: the large stochastic interference is disadvantageous for the persistence of the population and the strong impulsive harvesting can lead to extinct of the population.

Analysis of a budworm growth model with jump–diffusion

This article formulates and researches a single-species stochastic population model with Allee effect and Lévy jumps. Firstly, it is shown that the model has a unique global positive solution. Then sufficient conditions for extinction and stochastic permanence are obtained. Finally, the growth rate of the solution is estimated. The results demonstrate that the Lévy jumps could change the properties of the model significantly. Several numerical figures are given to explain the theoretical results.

Dynamics of a stochastic population model with Allee effect and Lévy jumps

This article formulates and researches a single-species stochastic population model with Allee effect and Lévy jumps. Firstly, it is shown that the model has a unique global positive solution. Then sufficient conditions for extinction and stochastic permanence are obtained. Finally, the growth rate of the solution is estimated. The results demonstrate that the Lévy jumps could change the properties of the model significantly. Several numerical figures are given to explain the theoretical results.

Dynamics Analysis of a Stochastic Delay Gilpin-Ayala Model with Markovian Switching

This paper considers a stochastic delay Gilpin-Ayala model with Markovian switching. Using Lyapunov method, we show existence and uniqueness of global positive solution. Then, by using Chebyshev’s inequality, M-matrix method, and BDG’s inequality, stochastic permanence and asymptotic estimations of solutions are studied. Finally, numerical simulations illustrate the theoretical results. Our results generalize and improve the existing results.

Permanence of Stochastic Biological Systems

Area of research related to prey-predator systems is an important topic. The concept of permanence is an important issues related to biological systems. In general permanence is considered as a combination of persistence and boundedness. Following this, this paper reviews few existing definitions of stochastic permanence. Following the existing definition, in this paper a new definition of permanence for stochastic biological systems has been proposed, which modifies the existing ones. The permanence of a general Lotka-Volterra stochastic two species cooperation, competition and predation systems is proved by new definition.

Note on the permanence of stochastic population models

Abstract The concept of permanence of any system is an important technical issue. This concept is very significant to all kind of systems, e.g., social, medical, biological, population, mechanical, or electrical. It is desirable by scientists and investigators that any system under consideration must be long time survival. For example, if we consider any ecosystem, it is always pre-requisite that this system is permanent. In general language, permanence is just the persistent and bounded system in a particular surface time frame. But the meaning may vary with the type of systems. For example, deterministic and stochastic biological systems have different concepts of permanence in an abstract mathematical platform. The reason is simple: it is due to the mathematical nature of parameters, methods of derivations of the model, biological assumptions, details of the study, etc. In this short note, we consider the stochastic models for their permanence. To address stochastic permanence of biological systems, many different approaches have been proposed in the literature. In this note, we propose a new definition of permanence for stochastic population models (SPM). The proposed definition is applied to the well-known Lotka–Volterra two species stochastic population model. The note is closed with the open ended discussion on the topic.

A regime switching model for species subject to environmental noises and additive Allee effect

Taking into account the environmental noises and Allee effect, we develop a regime switching model for a single species, and then we consider dynamics of species population, such as persistence, ergodicity and global attractivity. More precisely, firstly we prove the global existence of the positive solution, and then study the extinction of species, non-persistence in mean, persistence in mean and stochastic permanence. It is followed by the study of positive recurrence, ergodicity and the global attractivity. Finally, we conclude our study by a numerical example and discussions.

The threshold of stochastic Gilpin–Ayala model subject to general Lévy jumps

This paper investigates a stochastic Gilpin–Ayala model with general Levy jumps and stochastic perturbation to around the positive equilibrium of corresponding deterministic model. Sufficient conditions for extinction are established as well as nonpersistence in the mean, weak persistence and stochastic permanence. The threshold between weak persistence and extinction is obtained. Asymptotic behavior around the positive equilibrium of corresponding deterministic model is discussed. Our results imply the general Levy jumps is propitious to population survival when its intensity is more than 0, and some changes profoundly if not. Numerical simulink graphics are introduced to support the analytical findings.

Periodic Solution and Ergodic Stationary Distribution of Stochastic SIRI Epidemic Systems with Nonlinear Perturbations

This paper formulates two stochastic nonautonomous SIRI epidemic systems with nonlinear perturbations. The main aim of this study is to investigate stochastic dynamics of the two SIRI epidemic systems and obtain their thresholds. For the nonautonomous stochastic SIRI epidemic system with white noise, the authors provide analytic results regarding the stochastic boundedness, stochastic permanence and persistence in mean. Moreover, the authors prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii’s theory. For the system with Markov conversion, the authors establish sufficient conditions for positive recurrence and existence of ergodic stationary distribution. In addition, sufficient conditions for the extinction of disease are obtained. Finally, numerical simulations are introduced to illustrate the main results.

Stochastic permanence of solution to stochastic non-autonomous logistic equation with jumps

It is investigated the non-autonomous logistic differential equation with disturbance of coeffcients by white noise, centered and non-centered Poisson noises. The coeffcients of equation are locally Lipschitz continuous but do not satisfy the linear growth condition. This equation describes the dynamics of population in the Verhulst model which takes into account the logistic eect: an increase of the population size produces a fertility decrease and a mortality increase; since resources are limited, if the population size exceeds some threshold level, the habitat cannot support the growth. The property of stochastic permanence is desirable since it means the long time survival in a population dynamics. The suffcient conditions for the stochastic permanence of population in the considered model is obtained.

On hybrid stochastic population models with impulsive perturbations

This paper considers the dynamic behaviours of a hybrid stochastic population model with impulsive perturbations. The existence of the global positive solution is studied in this paper. Moreover, under some conditions on the noises and impulsive perturbations, the properties of the persistence and extinction, stochastic permanence, global attractivity and stability in distribution are presented. Our results illustrate that impulsive perturbations play a crucial role in these properties. The bounded impulse term will not affect these properties, however, when the impulse term is unbounded, some of the properties, such as the persistence and extinction may be changed significantly. As a part of this paper, a couple of examples and numerical simulations are provided to illustrate our results.

Survival analysis of an impulsive stochastic delay logistic model with Lévy jumps.

This paper studies a stochastic delay logistic model with Lévy jumps and impulsive perturbations. We show that the model has a unique global positive solution. Suffcient conditions for extinction, non-persistence in the mean, weak persistence, stochastic permanence and global asymptotic stability are established. The threshold between weak persistence and extinction is obtained. The results demonstrate that impulsive perturbations which may represent human factor play an important role in protecting the population even if it suffers sudden environmental shocks that can be discribed by Lévy jumps.

ANALYSIS OF A PREDATOR-PREY MODEL WITH CROWLEY-MARTIN AND MODIFIED LESLIE-GOWER SCHEMES WITH STOCHASTIC PERTURBATION

In this paper, we study a nonautonomous predator-prey model with Crowley-Martin and modified Leslie-Gower schemes with stochastic perturbation. The existence of a global positive solution and stochastically ultimate boundedness are obtained. Sufficient conditions are established for extinction, persistence in the mean, and stochastic permanence of the system. Finally, simulations are carried out to verify our results.

Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

This paper investigates dynamic behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte (CTL) to the growth of an immunogenic tumour. The main methods are stochastic Lyapunov analysis, comparison theorem for stochastic differential equations (SDEs) and strong ergodicity theorem. Firstly, we prove the existence and uniqueness of the global positive solution for the tumor-immune system. Then we go a further step to study the boundaries of moments for tumor cells and effector cells and the asymptotic behavior in the boundary equilibrium points. Furthermore, we discuss the existence and uniqueness of stationary distribution and stochastic permanence of the tumor-immune system. Finally, we give several examples and numerical simulations to verify our results.

Dynamical behavior in a hybrid stochastic triple delayed prey predator bioeconomic system with Lévy jumps

In this paper, a hybrid triple delayed prey predator bioeconomic system with prey refuge and Lévy jumps is established, where both maturation delay for prey and predator population and gestation delay for predator population are considered. For deterministic system, positivity and uniform permanence of solution are discussed. Local stability of deterministic system around interior equilibrium is investigated due to variations of triple time delays. For stochastic system without time delay, sufficient conditions for stochastically ultimate boundedness and stochastic permanence are discussed. Existence of stochastic Hopf bifurcation and stochastic stability are investigated. For stochastic system with triple time delays, existence and uniqueness of global positive solution are studied. Finally, combined dynamic effects of triple time delays and Lévy jumps on the hybrid stochastic system are discussed by constructing appropriate Lyapunov functions. Numerical simulations are supported to illustrate theoretical analysis.

Asymptotic properties of a stochastic Gilpin–Ayala model under regime switching

In this paper, a stochastic Gilpin–Ayala population model with regime switching and white noise is considered. All parameters are influenced by stochastic perturbations. The existence of global positive solution, asymptotic stability in probability, pth moment exponential stability, extinction, weak persistence, stochastic permanence and stationary distribution of the model are investigated, which generalize some results in the literatures. Moreover, the conditions presented for the stochastic permanence and the existence of stationary distribution improve the previous results.