Dynamical Analysis Of Two Kind Of Stochastic Models With Markovian Switching

In this paper,two kind of stochastic predator-prey models with Markovian switching and mutualism are established,which are a stochastic delayed one-predator and two-mutualistic-preys model perturbed by white and telegraph noise(referred to as "Model One")and its functional response are Holling II and Beddington-DeAngelis types,and a stochastic ratio-dependent one-predator and two-mutualistic-preys model perturbed by white and telegraph noise(referred to as "Model Two")and its functional response is Holling III type.Dynamical behaviors of the above models are mainly investigated in the paper.By constracting Lyapunov functions,based on stochastic differential equation theory and M-matrix theory,sufficient conditions of stochastic permanence and extinction are established.Another asymptotic property is also studyed,and furthermore,asymptotic estimates of the boundary of limit superior and inferior of the average in time of the solution of Model Two are obtained under stochastical permanence.These estimates are only dependent on the subsystems' parameters and the stationary probability distribution of the Markov chain.Finally examples and its numerical simulations are given to illustrate main results.Results of Model One and Two show that strong white noise can cause the system to be extinct.And when perturbed by white and telegraph noise,stochastic permanence and extinction of the system are only dependent on the parameters of subsystems and the stationary probability distribution of the Markov chain.