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This paper considers a three species eco-epidemiological model perturbed by white noise. Stochastic stability and long time behavior around the equilibrium of deterministic eco-epidemiological model are analyzed by Lyapunov methods. Numerical simulations for a set of parameter values are presented to illustrate the analytical findings. [ABSTRACT FROM AUTHOR]

This paper studies a stochastic non-autonomous Lotka–Volterra predator–prey model. We prove that there exists at least one positive periodic solution under some simple and reasonable conditions. In addition we obtain sufficient conditions for persistence in mean and extinction for the stochastic non-autonomous system. Our method and results are new and the ideas could be used to study other types of non-autonomous stochastic predator–prey systems. [ABSTRACT FROM AUTHOR]

Abstract: This paper discusses a randomized n-species Lotka–Volterra competition system. We show that this system is stable in time average under certain conditions. Furthermore, there is a stationary distribution of this system, if extra conditions are satisfied. Also we give the extinction condition of this system. Finally, numerical simulations are carried out to support our results. [Copyright &y& Elsevier]

Abstract: In this paper, we include stochastic perturbations into SIR and SEIR epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number . The long time behavior of the two stochastic systems is studied. Mainly, we utilize stochastic Lyapunov functions to show under some conditions, the solution has the ergodic property as , while exponential stability as . At last, we make simulations to conform our analytical results. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss a stochastic density dependent predator–prey system with Beddington–DeAngelis functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we investigate the asymptotic behavior of this system. When the white noise is small, the stochastic system imitates the corresponding deterministic system. Either there is a stationary distribution, or the predator population will die out. While if the white noise is large, besides the extinction of the predator population, both species in the system may also die out, which does not happen in the deterministic system. Finally, simulations are carried out to conform to our results. [Copyright &y& Elsevier]

Abstract: In this paper, we show there is a stationary distribution of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation and it has ergodic property. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a DI SIR epidemic model with two categories stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of deterministic system under different conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the two-group SIR model introduced by Guo, Li and Shuai [H.B. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q. 14 (2006) 259–284], allowing random fluctuation around the endemic equilibrium. We prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large. In addition, the stability condition is obtained by the construction of Lyapunov function. Finally, numerical simulations are presented to illustrate our mathematical findings. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. We show there is a unique positive solution to the system with positive initial value, and mainly investigate the long time behavior of the system. Condition for the system to be extinct is given and persistent condition is established. At last, numerical simulations are carried out to support our results. [Copyright &y& Elsevier]

Abstract: This paper discusses a randomized non-autonomous logistic equation , where is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164–172], the authors show that has a unique positive T-periodic solution provided , and are continuous T-periodic functions, , and . We show that this equation is stochastically permanent and the solution is globally attractive provided , and are continuous T-periodic functions, , and . By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded. [Copyright &y& Elsevier]

Abstract: This paper is devoted to study the existence of multiple positive solutions for the second order Dirichlet boundary value problem with impulse effects. The main results here is the generalization of Liu and Li [L. Liu, F.Y. Li, Multiple positive solution of nonlinear two-point boundary value problems, J. Math. Anal. Appl. 203 (1996) 610–625] for ordinary differential equations. Existence is established via the theory of fixed point index in cones. [Copyright &y& Elsevier]

Abstract: In this paper, we characterize the eigenvalues and show existence of positive solutions to discrete boundary value problem (here , and is a parameter) [Copyright &y& Elsevier]

Abstract: In this paper, we study positive periodic solutions to the repulsive singular perturbations of the Hill equations. It is proved that such a perturbation problem has at least two positive periodic solutions when the anti-maximum principle holds for the Hill operator and the perturbation is superlinear at infinity. The proof relies on a nonlinear alternative of Leray–Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones. [Copyright &y& Elsevier]

Abstract: This paper discusses a randomized nonautonomous logistic equation where is 1-dimensional standard Brownian motion. We show that has a unique positive T-periodic solution provided , , and are continuous T-periodic functions, , and . [Copyright &y& Elsevier]

In this paper, we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations. We obtain some new results relative to the lower solution α and upper solution β with α⩽β. [Copyright &y& Elsevier]

The existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper. Our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of Leray–Schauder type and on a fixed point theorem in cones. [Copyright &y& Elsevier]

The singular boundary value problem [φ(u′)]′+g(t,u,u′)=0,for t∈(0,1),u(0)=u(1)=0, where φ(s)=&z.sfnc;s&z.sfnc;p−2s, p>1, is studied in this paper. The singularity may appear at u=0, t=0 and t=1, and the function g may change sign. The existence of solutions is obtained via an upper and lower solution method. [Copyright &y& Elsevier]

This paper presents Part I of a two-part series on studying the long-term coexistence states of stochastic generalized Kolmogorov systems with small diffusion. Part I establishes a mathematical framework for approximating the invariant probability measures (IPMs) and density functions (IPDFs) of these systems, while Part II will focus on analyzing their non-autonomous periodic counterparts. Compared with the existing approximation methods available only for systems with non-degenerate linear diffusion, this paper introduces two new and easily implementable approximation methods, the log-normal approximation (LNA) and updated normal approximation (uNA), which can be used for systems with not only non-degenerate but also degenerate diffusion. Moreover, we utilize the Kolmogorov-Fokker-Planck (KFP) operator and matrix algebra to develop algorithms for calculating the associated covariance matrix and verifying its positive definiteness. Our new approximation methods exhibit good accuracy in approximating the IPM and IPDF at both local and global levels, and significantly relaxes the minimal criteria for positive definiteness of the solution of the continuous-type Lyapunov equation. We demonstrate the utility of our methods in several application examples from biology and ecology. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove that a stochastic logistic population under regime switching controlled by a Markov chain is either stochastically permanent or extinctive, and we obtain the sufficient and necessary conditions for stochastic permanence and extinction under some assumptions. In the case of stochastic permanence we estimate the limit of the average in time of the sample path of the solution by two constants related to the stationary probability distribution of the Markov chain and the parameters of the subsystems of the population model. Finally, we illustrate our conclusions through two examples. [Copyright &y& Elsevier]