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In this paper, we investigate the dynamics of a high‐dimensional human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome (AIDS) model with treatment and standard incidence, which is perturbed by telegraph noise. The switching is formulated by a continuous time Markov chain. Firstly, we show the existence and uniqueness of the global positive solution. Then, conditions for extinction of the disease are established. Moreover, through constructing suitable Lyapunov functions, we derive sufficient conditions for the existence of positive recurrence of the solutions. Positive recurrence implies all the individuals can coexist and persist in the long term. Finally, numerical experiments are performed for supporting the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on the spread dynamics of an HIV/AIDS model with multiple stages of infection and treatment, which is disturbed by both white noise and telegraph noise. Switching between different environmental states is governed by Markov chain. Firstly, we prove the existence and uniqueness of the global positive solution. Then we investigate the existence of a unique ergodic stationary distribution by constructing suitable Lyapunov functions with regime switching. Furthermore, sufficient conditions for extinction of the disease are derived. The conditions presented for the existence of stationary distribution improve and generalize the previous results. Finally, numerical examples are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behavior of a stochastic two-compartment model of B -cell chronic lymphocytic leukemia, which is perturbed by white noise. Firstly, by constructing suitable Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Then, conditions for extinction of the disease are derived. Furthermore, numerical simulations are presented for supporting the theoretical results. Our results show that large noise intensity may contribute to extinction of the disease. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behavior of a stochastic food chain chemostat model, in which the white noise is proportional to the variables. Firstly, we prove the existence and uniqueness of the global positive solution. Then by constructing suitable Lyapunov functions, we show the system has a unique ergodic stationary distribution. Furthermore, the extinction of microorganisms is discussed in two cases. In one case, both the prey and the predator species are extinct, and in the other case, the prey species is surviving and the predator species is extinct. Finally, numerical experiments are performed for supporting the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider two chemostat models with random perturbation, in which single species depends on two perfectly substitutable resources for growth. For the autonomous system, we first prove that the solution of the system is positive and global. Then we establish sufficient conditions for the existence of an ergodic stationary distribution by constructing appropriate Lyapunov functions. For the non-autonomous system, by using Has'minskii theory on periodic Markov processes, we derive it admits a nontrivial positive periodic solution. Finally, numerical simulations are carried out to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic multimolecular biochemical reaction model with lévy jumps is investigated. Firstly, we prove the existence and uniqueness of the global positive solution. Then we derive the conditions when the reaction will end and when the reaction will proceed. Moreover, the existence of positive recurrence to the solutions is studied by constructing suitable Lyapunov functions. Results show that the end and persistence of the reaction are closely related to the intensity of lévy noise. Finally, numerical simulations are carried out to illustrate the theoretical results. • A stochastic multimolecular biochemical reaction model with lévy jumps is studied. • We establish sufficient conditions for the end and persistence of the reaction. • The existence of positive recurrence to the solutions is investigated. • Numerical simulations are carried out to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic chemostat model with single-species growth on two perfectly substitutable nutrients under telegraph noise is investigated. Firstly, we prove the solution of the system is positive and global. Then we establish sufficient conditions for the existence of ergodic stationary distribution of the model by constructing suitable Lyapunov functions with regime switching. Finally, numerical simulations are carried out to demonstrate our theoretical results. • A stochastic chemostat model with two nutrients under telegraph noise is considered. • We establish sufficient conditions for the existence of stationary distribution. • The existence of stationary distribution implies stochastic weak stability. • Numerical simulations are carried out to illustrate the results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Lotka–Volterra food chain chemostat model that incorporates both distributed delay and stochastic perturbations. We obtain sufficient conditions for the existence of stationary distribution by constructing suitable Lyapunov functions. Stationary distribution indicates the two species in the chemostat can coexist in the long term. [ABSTRACT FROM AUTHOR]

This article focuses on a stochastic viral model with distributed delay and CTL responsiveness. It is shown that the viral disease will be extinct if the stochastic reproductive ratio is less than one. However, when the stochastic reproductive ratio is more than one, the viral infection system consists of an ergodic stationary distribution. Furthermore, we obtain the existence and uniqueness of the global positive solution by constructing a suitable Lyapunov function. Finally, we illustrate our results by numerical simulation. [ABSTRACT FROM AUTHOR]