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In this paper, we investigate a classical Lotka–Volterra predator–prey model with telephone noise and a higher order perturbation of white noise. The existence of a unique positive solution is discussed and sufficient conditions for the existence of an ergodic stationary distribution is established. Some simulation figures are presented to illustrate the analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and analyze a stochastic predator–prey model with additional food and nonlinear perturbation. Firstly, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution of the positive solution to the model. Then we obtain sufficient conditions for extinction of the predator species in two cases, one is the prey species surviving and the predator species extinction; the other is both the prey and predator species are extinct. Finally, some examples together with numerical simulations are provided to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dynamical behavior of a stochastic SIQR epidemic model with standard incidence which is disturbed by both white and telegraph noises. Firstly, we obtain sufficient conditions for persistence in the mean of the disease. Then we establish sufficient conditions for extinction of the disease. In addition, in the case of persistence, we get sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic Lyapunov function with regime switching. Meanwhile, the threshold between persistence in the mean and extinction of the stochastic system is also obtained. Finally, some numerical simulations are introduced to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

A stochastic HIV infection model with T-cell proliferation and CTL immune response is formulated to investigate the effect of environmental fluctuations on the HIV viral dynamics. We obtain that the model solution is positive and global, and analyze the extinction of the model. We also derive a critical condition R 0 s , when R 0 s is greater than one, the existence of ergodic stationary distribution of the model solution is established by constructing suitable Lyapunov functions. Numerical simulations are performed to investigate the effect of white noises on model behavior, we investigate that the small intensities of white noise can maintain the irregular recurrence of HIV virus and CTL immune response, while the larger ones may be help to the elimination of the virus and CTL immune response, and the medium intensities of white noises may cause both the persistence and extinction on model dynamics behavior. [ABSTRACT FROM AUTHOR]

• A stochastic SEIRS rabies epidemic models with two patches is studied. • We prove the existence and uniqueness of global positive solution for the stochastic model. • Sufficient conditions for the existence of an ergodic stationary distribution. • For no dog travel model, we obtain the exact expression of probability density function around. • Numerical results are carried out to illustrate the theoretical results. In this paper, we propose a stochastic SEIRS rabies epidemic models with two patches to investigate the spatial spread of rabies under the influence of environmental noise. Adopting a new technique to construct the stochastic Lyapunov functions, we obtain the sufficient conditions for the existence of an ergodic stationary distribution. In addition, if the movement of dogs is forbidden, we obtain the exact expression of probability density function around a quasi-equilibrium point. Finally, numerical simulations are employed to verify our analytical results and reveal the impact of stochastic perturbations on rabies transmission. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic Lotka–Volterra model with two predators competing for one prey. We give the sufficient conditions which guarantee that the principle of coexistence holds for this perturbed model via Markov semigroup theory. Furthermore, we prove that the densities of the solution can converge in L 1 to an invariant density under appropriate conditions. Finally we make simulations to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

The paper characterizes some qualitative dynamic properties of a stochastic non-autonomous multi-species mutualism model, with continuous periodic parameters. Using Khasminskii theory of stability with suitable Lyapunov functions, and M -Matrices, sufficient conditions are established to guarantee existence of positive periodic solutions to the system. We also provide conditions for the global attractiveness of the latter, or extinction of all species for sufficiently high volatility levels. Results are finally supported by numerical computations. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a predator–prey model with disease in the prey by stochastic perturbation. Using Lyapunov analysis methods we discuss the asymptotic behavior of this solution. Furthermore, we investigate whether that there exists a stationary distribution for this system and if it has the ergodic property. Finally we make simulations to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we analyze long-time behavior of densities of the distributions of the solution for a stochastic SIR epidemic model. We prove that the densities can converge in to an invariant density or can converge weakly to a singular measure. [Copyright &y& Elsevier]

Abstract: Using the mixed monotone method we establish the existence and uniqueness of positive solutions for some fourth-order nonlinear singular continuous and discrete boundary value problem. The theorems obtained are very general and complement previously known results. [Copyright &y& Elsevier]

Abstract: This paper is devoted to study the existence of multiple positive solutions for the second order periodic boundary value problem with impulse effects. The main results here are the generalization of Jiang [Daqing Jiang, On the existence of positive solutions to second order periodic BVPs, Acta Math. Sci. 18 (1998) 31–35] for ordinary differential equations. Existence is established via the theory of fixed point index in cones. [Copyright &y& Elsevier]

Abstract: In this paper we present some new existence results for singular semipositone Dirichlet boundary value problem for second-order impulsive differential equations using the upper and lower solutions method and a fixed point theorem in cones. [Copyright &y& Elsevier]

This paper presents a new optimal existence theory for single and multiple positive periodic solutions to functional difference equations. Existence is established using a fixed point theorem in cones. We illustrate our theory be examining several biomathematical models. [Copyright &y& Elsevier]

In this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problemu(4)(t)=f(t,u(t),u″(t)), t∈[0,2π],u(0)=u(2π), u′(0)=u′(2π), u″(0)=u″(2π), u‴(0)=u‴(2π)in the presence of an upper solution β and a lower solution α with β⩽α. [Copyright &y& Elsevier]

• A stochastic cholera model between communities linked by migration is studied. • We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution. • Our result shows that the migration and the small white noise are not beneficial for eliminating cholera. • Some examples together with numerical simulations are introduced to support the theoretical result. In this paper, we develop and study a stochastic cholera model between communities linked by migration. Firstly, we show that there exists a unique global positive solution of the considered system with any positive initial value. Then by using the stochastic Lyapunov analysis method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Our result shows that the migration and the small white noise are not beneficial for eliminating cholera. Finally, some examples together with numerical simulations are introduced to support the theoretical result. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an avian influenza epidemic model with half saturated incidence is investigated. Firstly we study the local asymptotical stability of equilibria of the deterministic model. Then for the stochastic model, the existence of the global positive solution is proved and different thresholds are obtained. In addition, we get sufficient conditions for the existence of an ergodic stationary distribution by using Has'minskii theory and Lyapunov function. Finally, some numerical simulations illustrate the analytical results. [ABSTRACT FROM AUTHOR]

This paper introduces the DS-I-A model with periodic parameter function and Markovian switching. First, we will prove that the solution of the system is positive and global. Furthermore, we draw a conclusion that there exists nontrivial positive periodic solution for the stochastic system and we establish sufficient conditions for extinction of system. Moreover, we construct stochastic Lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to DS-I-A model perturbed by white and telephone noises and we also establish sufficient conditions for extinction of system with regime switching. Finally, we test our theory conclusion by simulations. [ABSTRACT FROM AUTHOR]

• A stochastic HIV model with cell-free and cell-to-cell infections is formulated. • The positive and global solution of the model is rigorously verified. • Existence of the unique ergodic stationary distribution of the system is studied. • Probability density function is derived by applying asymptotic analysis. • The effects of noise and cell-to-cell infection on model behavior are investigated. In this paper, a stochastic HIV model with CD 4 + T-cell proliferation, cell-free infection and cell-to-cell transmission is proposed. By constructing suitable Lyapunov function, we establish the existence of unique and ergodic stationary distribution of the model. Moreover, by using asymptotic analysis and employing the Fokker-Planck equation, we derive the probability density function around the quasi-steady state of the system. Through numerical simulations, the effects of the stochastic perturbation and cell-to-cell infection on model dynamic behavior are investigated, thus the probability density function of the system is also given under the realistic parameter values. [ABSTRACT FROM AUTHOR]

• A hybrid stochastic SEQIHR model is studied for disease dynamics with media effect. • We obtain the sufficient conditions for disease extinction of stochastic model. • We obtain the criteria for the existence of an ergodic stationary distribution. • The developed θ -stochastic method can be greatly applied to other hybrid models. Combining a deterministic SEQIHRS model proposed by Sahu et al. (2015) and the present mathematical modelling of media coverage effect (2020), a hybrid stochastic SEQIHR model, perturbed by both nonlinear white noise and colored noise, is formulated and studied for the transmission dynamics of an infectious disease with media coverage, quarantine strategies and pre-existing immunity in a community. First, we prove that the stochastic model possesses a unique global positive solution. Second, by means of the basic reproduction number R 0 of the corresponding deterministic model, two relevant critical values which include R ¯ 0 and R 0 C are derived. Next, we obtain the disease extinction under R 0 < 1 and R ¯ 0 < 1. Moreover, we further establish the sufficient condition R 0 C > 1 for the existence and uniqueness of an ergodic stationary distribution of the stochastic model, which means the infectious disease will be prevailing and persistent in a community. Finally, several numerical simulations are performed to validate the above theoretical results. Besides, the impact of media coverage and nonlinear hybrid noises on the dynamical behavior of the stochastic model are studied at the end of this paper. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with existence theorems for single and multiple positive solutions to a kind of Neumann boundary value problems with impulse actions. The proof of the main results relies on a nonlinear alternative principle of Leray–Schauder type and a well-known fixed point theorem in cones. The impulses are not restricted to be monotonous in the results obtained by us. [Copyright &y& Elsevier]

• A stochastic SEI epidemic model with general distributed delay is established. • The existence of a unique global positive solution for the stochastic model are proved. • We prove that there is a stationary solution to the stochastic system. • Two probability density functions with respect to the stationary solution are obtained. • Some sufficient conditions are established for the disease extinction of stochastic system. A stochastic SEI (Susceptible-Exposed-Infected) epidemic model with general distributed delay is studied in this paper. First, we prove the existence and uniqueness of a global positive solution to the stochastic system. By means of the Lyapunov method, we verify the existence of a stationary distribution of the positive solution under a stochastic criterion R 0 p > 1 , which is known as stationary solution. Moreover, if R 0 p > 1 , two exact probability density functions around the quasi-stable equilibrium are obtained by solving the corresponding Fokker-Planck equation. Notably, both the explicit expression of density function and the existence of stationary distribution suggest the disease persistence in biological sense. For completeness, some sufficient conditions for disease extinction are established. At last, several numerical simulations are provided to verify our analytical results and reveal the impact of stochastic perturbations on disease transmission. [ABSTRACT FROM AUTHOR]