Population Dynamical Behavior Of A Stochastic Predator-Prey System With Beddington-DeAngelis Functional Response

There are many different kinds of predator-prey systems in the mathematical ecologyliterature. It is well known that the traditional predator-prey systems with prey-dependentfunctional response fail to model the interference among predators. Some biologists haveargued that in many cases, especially when predators have to search for food and there-fore, have to share or compete for food, the functional response in a prey-predator mod-el should be predator-dependent. There are many significant evidences to suggest thatpredator dependence in the functional response occurs quite frequently in natural system-s and laboratory . Especially, by comparing the statistical evidence from 19 predator-prey systems with the three classical predator-dependent functional responses (Hassell-Varley , Beddington-DeAngelis and Crowley-Martin ), Skalski and Gilliam claimed thatthe predator-dependent can provide better descriptions of predator feeding over a range ofpredator-prey abundances, and in some cases the Beddington-DeAngelis type functionalresponse preformed even better. On the one hand, in the real world, population systemsare inevitably affected by stochastic noises. In addition, the density-dependence of thepredator population should be taken into account. However, both the stochastic factor andthe density-dependence of the predator were neglected in almost all existing studies.In this paper author shall propose a stochastic non-autonomous predator-prey systemwith Beddington-DeAngelis functional response by taking into account the stochastic fac-tor and the density-dependence of the predator. The population system is non-autonomousand therefore more complicated and the mathematics presented is more diffcult, for ex-ample, the boundedness of x(t) and y(t) in deterministic system is destroyed by stochasticnoises in (SBD). Author overcomes these problems, and studies the population dynamicalproperties. They are the innovations of this paper.Firstly , author introduces the research background of this paper. Since the sys-tem denotes a population system, then author need to show that the model has a positiveand global solution. Author uses It(?) formula and substitution technique to show theexistence, the uniqueness and the positivity of the solution. Then by using It(?) formulaand constructing some appropriate Lyapunov functions, author studys the boundednessof moments and the upper-growth rate of the solution. After that author discusses thepersistence and extinction of the model. Author establishes the su?cient conditions for persistence and extinction of each population by using It(?) formula and Lyapunov func-tions. Then author uses It(?) formula and Lyapunov functions to investigate the globalattractivity of the system. At the end , author introduces some numerical simulationsto confirm the results by using Milstein method. Finally, author closes the paper withconclusions and discussions.