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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]

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]