Two Stochastic Predator-prey Systems With Bedding-deangelis Functional Response

Any population dynamics in the real biological systems is inevitably affected by environmental noises.In order to adapt to the actual needs,it is necessary to take stochastic perturbations fully into consideration during systems modeling.In addition,time delays are commonly encountered in practice,adding time delays factors into the random predator system with Beddington-DeAngelis functional response will be more accepted.Therefore,further studying with Beddington-DeAngelis functional response of the random delayed predator system and its some vital properties is overwhelmingly significant.Besides,because of the existence of environmental noise and delay factors,there are a variety of the random delayed system.Inspired by this,this paper adds time delays factors into the random predator system with Beddington-DeAngelis functional response,and random item also contains time delay factor,which are one of the point of innovation of this paper.Besides,population systems may suffer sudden environmental perturbations,such as epidemics,earthquakes,hurricanes,etc.These phenomena cannot be modeled by stochastic system with environmental noises.Many authors suggested that these phe-nomena can be described by a L(?)vy jump process and they considered stochastic Lotka-Volterra population systems with jumps for the first time.To the best of our knowledge,no results related to a stochastic Bedding-DeAngelis predator-prey system with L(?)vy jumps has been reported.Motivated by these,in this paper we consider stochastic sys-tem with jumps.Therefore,a stochastic Beddington-DeAngelis predator-prey system with L(?)vy jumps is investigated.In this page,based on the Lyapunov stability theory,stochastic functional differ-ential equations theory,employing Lyapunov functions,stopping time technique,and inequalities technique,we systematically study the random delayed predator system with Beddington-DeAngelis functional response and a stochastic Bedding-DeAngelis predator-prey system with L(?)vy jumps.The main contributions are summarized as follows:In the first part,we introduce the background,research significance and current situation of the predator system with Beddington-DeAngelis functional response and a stochastic Bedding-DeAngelis predator-prey system with L(?)vy jumps axe reviewed.Then,some preliminaries are presented.In the second part,we simply introduce the modeling of the predator system with Beddington-DeAngelis functional response.Under the condition of a given hypothesis the random delayed predator system with Beddington-DeAngelis functional response exist global positive solutions.This paper shows that,although the coefficients of the system neither satisfy the linear growth condition,nor local Lipschitz continuous,the model still has a globally positive solution.Using Lyapunov function and to formula,we can verify that system exists the uniqueness of positive solutions,and the sufficient conditions for the positive solution of the system are obtained.On this basis,fur-ther prove that the solution is Stochastically ultimate boundedness,and stochastically asymptotic stability in the large are obtained.In the third part,a stochastic Bedding-DeAngelis predator-prey system with L(?)vy jumps is investigated.Using the construction of Lyapunov functions and stopping time technique,the existence of global unique positive solution is obtained.Based on that,by constructing function the solution to the system is stochastically ultimate bound-edness.Finally,some sufficient conditions of extinction are established,In the four part,Using Milstein method to verify the main conclusions of the paper.