A stochastic predator-prey system with time-dependent delays is considered. Firstly, we show the existence of a global positive solution and stochastically ultimate boundedness. Secondly, the critical value between weak persistence and extinction of the prey is obtained and we also give the asymptotic pathwise estimation. Finally, we simulate the model to illustrate our results. MSC: 92B05; 92D25; 93E03

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Abstract-In this paper, we consider an autonomous LotkaVolterra competitive system with stochastic perturbation and feedback controls. Firstly, we show the existence, the uniqueness and the positivity of the solution. Secondly, under a simple assumption, sufficient conditions for stability in the mean and extinction of each population are established.Keywords-feedback controls; competitive system; stochastic perturbation; extinction; stability in the mean I INTRODUCTIONFor the last decades, the classical Lotka-Volterra competition system has been studied extensively. Many excellent results are obtained (see [13,14]).In [1], the authors argued that in a situation where the equilibrium is not the desirable one (or affordable) and a smaller value is required, we are required to alter the system structurally by introducing a feedback control variable [2]. This can be implemented by means of a biological control or some harvesting procedure so as to make the population stabilize at a lower value. In 1931 V. Volterra explained the balance between two populations of fish in a closed pond using the theory of feedback. Later, a series of mathematical models have been established to describe the dynamics of feedback control systems. dx t x t r a x t a x t c u t dt dx t x t r a x t a x t c u t dt du t e u t d x t dt du t e u t d x t dtThey obtained sufficient conditions for the globally asymptotically stable of system (1).But, in the real world population systems often subject to environmental perturbations. In many cases, these perturbations should not be neglected, and there are many authors have introduced stochastic population models in order to investigate the effect of environmental noises; see (e.g. [4,[6][7][8][9][10]12] x t x t r a x t a x t dt x t dB t x t x t r a x t a x t dt x t dB tWhere ,,However, to this day, no scholar has investigated the dynamic behaviors of the stochastic Lotka-Volterra competitive system with feedback controls. In this paper, we consider a stochastic Lotka-Volterra competitive system with feedback controls. Suppose that the environmental noises mainly affect the growth rate i r , t x t r a x t a x t c u t dt x t dB t x t x t r a x t a y t c u t dt x t dB t du t e u t d x t dt du t e u t d x t dt

Considering the influence of sublethal concentration of pesticides on pests and natural enemies, we propose a pest-management model with impulsive effect on chemical control and biological control strategies–periodic spraying pesticide and releasing predatory natural enemies. By using the Floquet theory and the comparison theorem of impulsive differential equations, a sufficient condition for the global asymptotic stability of the pest-eradication periodic solution is obtained. The persistence of the system is further studied, and a sufficient condition for the persistence of the system is obtained. Finally, some numerical simulations are shown to verify our theoretical works. Our works indicate that the sublethal effects of insecticides and the release of predatory natural enemies play significant roles in pest control in agricultural production.