Logistic models with regime switching: Permanence and ergodicity

This paper is concerned with a stochastic Leslie–Gower Holling-type II predator–prey system with regime switching and prey harvesting. Firstly, sharp sufficient criteria for the extinction and the existence of a unique ergodic stationary distribution of the system are established, and the convergence of transition probability of the solution to the stationary distribution is estimated. Then, sharp sufficient criteria for the existence of optimal harvesting strategy are obtained. Some critical influences of random perturbations on the existence of a unique ergodic stationary distribution, extinction and optimal harvesting strategy are revealed. Finally, several numerical simulations are presented to illustrate the theoretical results.

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 of V‐function technique, M‐matrix method, and Chebyshev′s inequality. Finally, an example is given to illustrate the main results.

Stationary distribution of stochastic population systems under regime switching

In this paper, the dynamical behaviors of a stochastic competition system with a saturation effect are analyzed. The existence and uniqueness of globally positive solution are proved in detail, and the sufficient conditions for stochastic permanence, strong persistence in the mean, weak persistence, and extinction are obtained respectively. Then the existence and uniqueness of stationary distribution are also obtained under some appropriate assumptions. Finally, several numerical simulations are provided to justify the analytical results.

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

Stochastic mutualism model with Lévy jumps

Asymptotic properties and simulations of a stochastic logistic model under regime switching II

Taking both white noise and colored environmental noise into account, a single-species logistic model with population’s nonlinear diffusion among two patches is proposed and investigated. The sufficient conditions of the existence of positive solutions, stochastic permanence, persistence in mean and extinction are established. Moreover, we use an example and simulation figures to illustrate our main results.

Stability of a budworm growth model with random perturbations

Dynamical behavior of stochastic multigroup S-DI-A epidemic models for the transmission of HIV

In this paper, we consider a stochastic SI epidemic model with regime switching. The Markov semigroup theory is employed to obtain the existence of a unique stable stationary distribution. We prove that if mathcal{R}^{s}<0, then the disease becomes extinct exponentially; whereas if mathcal{R}^{s}>0 and beta(i)>alpha(i), iinmathbb{S}, then the densities of the distributions of the solution can converge in L^{1} to an invariant density.

A remark on stationary distribution of a stochastic SIR epidemic model with double saturated rates

We study a stochastic SIS (susceptible-infected-susceptible) epidemic dynamics on network, under the effect of a Markovian regime-switching. We first prove the existence of a unique global positive solution, and find a positive invariant set for the system. Then, we find sufficient conditions for a.s. extinction and stochastic permanence, showing also their relation with the stationary probability distribution of the Markov chain that governs the switching and with the network topology. We provide an asymptotic lower bound for the time average of the sample-path solution under the conditions ensuring stochastic permanence. From this bound, we are able to prove the existence of an invariant probability measure if the condition of stochastic permanence holds. Under a different condition, we prove the positive recurrence and the ergodicity of the regime-switching diffusion.

A stochastic SIS-type epidemic model with general nonlinear incidence and disease-induced mortality is investigated. It is proved that the dynamical behaviors of the model are determined by a certain threshold value . That is, when and together with an additional condition, the disease is extinct with probability one, and when , the disease is permanent in the mean in probability, and when there is not disease-related death, the disease oscillates stochastically about a positive number. Furthermore, when , the model admits positive recurrence and a unique stationary distribution. Particularly, the effects of the intensities of stochastic perturbation for the dynamical behaviors of the model are discussed in detail, and the dynamical behaviors for the stochastic SIS epidemic model with standard incidence are established. Finally, the numerical simulations are presented to illustrate the proposed open problems.

Asymptotic properties of stochastic nutrient-plankton food chain models with nutrient recycling