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A stochastic two-species mutualism model that incorporates nonlinear perturbations and distributed delays is proposed and analyzed in this paper, in where saturation effects and distributed delays with weak kernel are contained in the interspecies mutualism term. We first investigate that a nonlinear stochastic mutualism model possesses a unique global positive solution regarding arbitrarily given initial value. Therewith the sufficient conditions for two-species extinction and the existence of a unique stationary distribution (USD) are obtained. Whereafter, it is derived that the stationary solution near the quasi-positive equilibrium adheres a unique probability density function with the aid of tackling Fokker–Planck (FP) equation corresponding to the linearized system and applying the theory of algebraic equations we have developed. At last, several numerical examples are presented to validate the validity of our theoretical findings. • A stochastic mutualism model with distributed delays is developed. • The sufficient criteria for two-species extinction and the existence of a unique staionary distribution are obtained. • We gain the specific expression of probability density around the quasi-positive equilibrium of the system. [ABSTRACT FROM AUTHOR]

Abstract: An impulsive SI model with Monod-Haldane functional response for pest control is proposed and investigated. First, we have proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the above system can be permanent. Then, influences of impulsive perturbation including impulse period, the time of spraying pesticide and the quantity of releasing infective pests on the above system have been studied. Moreover, numerical simulations show that the system has rich dynamical behaviors. Finally, it is concluded that the approach of combining impulsive infective releasing with impulsive pesticide spraying is more effective than the classical one if the chemical control is adopted rationally. [Copyright &y& Elsevier]

Abstract: A Monod–Haldane functional response predator–prey system with impulsive harvesting and stocking is proposed, where the Monod–Haldane functional response involves group defense theory. Conditions for the system to be extinct are given and permanence conditions are established via the method of comparison involving multiple Lyapunov functions. Further influences of the impulsive harvesting and stocking on the system are studied, and numerical simulations show that the system has rich dynamical behaviors. [Copyright &y& Elsevier]

Abstract: In most population models, diffusion between patches is assumed to be continuous or discrete, but in practice many species diffuse only during a single period. In this paper we propose a single species model with impulsive diffusion between two patches, which provides a more natural description of population dynamics. By using the discrete dynamical system generated by a monotone, concave map for the population and a ϵ 1 − ϵ 2 variation we prove that the map always has a globally stable positive fixed point. This means that a single species system with impulsive diffusion always has a globally stable positive periodic solution. This result is further substantiated by numerical simulation. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of positive periodic solution of two-species competitive system with toxicant and birth pulse. A set of easily verifiable sufficient conditions are derived for the existence of at least one positive periodic solution of the above system by using the method of coincidence degree. Numerical simulations are also presented to illustrate the feasibility of our main results. [Copyright &y& Elsevier]