Minimum stationary values of sparse random directed graphs

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.

A stochastic SIRS epidemic model with nonlinear incidence rate

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

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.

Dynamics of a stochastic Markovian switching predator–prey model with infinite memory and general Lévy jumps

Stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion

We consider the subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are change of measure and some conditional limit theorems for random walks.

In this paper we consider a mathematical model of a transaction oriented computing system (airline reservation system, file system, data-base system, etc.) operating in the presence of intermittent failures, and with a checkpoint and roll-back recovery scheme built-in to preserve system integrity. An arbitrary distribution of inter-checkpoint times is assumed. A condition on the traffic intensity of transaction requests is established which insures that the system will not saturate. This condition is essential if meaningful simulation studies of these complex systems are to be conducted. In probabilistic terms, this is the necessary and sufficient condition for the existence of the stationary probability distribution associated with the non-Markovian queueing process representing the backlog of requets to the computer system.

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.

Stationary distribution of stochastic population systems under regime switching

A stochastic epidemic model with infectivity rate in incubation period and homestead–isolation on the susceptible is developed with the aim of revealing the effect of stochastic white noise on the long time behavior. A good understanding of extinction and strong persistence in the mean of the disease is obtained. Also, we derive sufficient criteria for the existence of a unique ergodic stationary distribution of the model. Our theoretical results show that the suitably large noise can make the disease extinct while the relatively small noise is advantageous for persistence of the disease and stationary distribution.

In this paper, we study the dynamical behavior of a stochastic two-compartment model of [Formula: see text]-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.

A general stochastic compartment model for cholera with higher-order perturbation is proposed, which incorporates direct and indirect transmission by contaminated water. Nonlinear incidence, multiple stages of infection, multiple states of pathogen, and second-order white-noises perturbation are introduced into the model, which includes and extends the existing cholera model. The existence and ergodicity of the stationary distribution for the cholera system are obtained by constructing a suitable Lyapunov function, which determines a sharp critical value R0s corresponding to the basic productive number R0 of the ordinary differential equation. The results show that, if R0s>11$\\end{document}]]>, the system has a unique and ergodic stationary distribution, which implies the persistence of the diseases. Our general results are applied to a cholera system with a Holling type-II functional response.

This paper deals with the stochastic Zika virus model within the human and mosquito population. Firstly, we prove that there exists a global positive solution. Further, we found the condition for a viral infection to be extinct. Besides that, we discuss the existence of a unique ergodic stationary distribution through a suitable Lyapunov function. The stationary distribution validates the occurrence of infection in the population. From that, we obtain the threshold value for prevail and disappear of disease within the population. Through the numerical simulations, we have verified the reproduction ratio R 0 S ${R}_{0}^{S}$ as stated in our theoretical findings.

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.