Complexity Analysis Of Several Kinds Of Stochastic Predator-prey Bioeconomic Systems

In recent years,the research on population ecology has become an important branch of biology mathematics.Almost scholars use deterministic differential equations to study populations initially,whereas with the development of deterministic differential equations,which can't describe the real population situations.Therefore,stochastic differential equations are utilized by more and more researchers,which make a great progress and have profound theoretical and practical significance.For the further development of stochastic differential equations,the stochastic differential algebraic equations are considered.However,there are few studies on stochastic differential algebraic equations in the population ecosystem so far.Based on the fundamental theory of stochastic system dynamics,the fundamental theory of nonlinear systems,constructing appropriate Lyapunov function,the dynamical behavior of several kinds of stochastic predator-prey bioeconomic systems is proposed in this paper.The main research works are as follows:1.A class of stochastic Lotka-Volterra predator-prey bioeconomic model with Brown motion is proposed.By utilizing the theory of stochastic dynamical system,the dynamical behavior is discussed.Firstly,under some conditions,the existence and uniqueness of a positive solution is given using Lyapunov function and some inequalities.Then the local stability analysis are obtained and the stability of this system in time average is also gained.Finally,some numerical results are provided,which illustrate that the solution of stochastic model is relative to disturbance intensity.2.A class of stochastic Lotka-Volterra predator-prey bioeconomic model with double noise is proposed.Based on stochastic averaging method,singular boundary theory and invariant measure theory,the stability and bifurcation of this model are considered.Firstly,the stability of stochastic system is provided in terms of Lyapunov exponent and singular boundary theory,then it is analyzed that the location and probability which the system occurred stochastic Hopf bifurcation by invariant measure theory.3.A class of stochastic Lotka-Volterra predator-prey bioeconomic model with Levy jumps is proposed.According to Lyapunov functionality and martingale,the dynamical behavior of this system is studied.Firstly,the existence and uniqueness of a global positive solution are discussed,then we can get persistence and extinction of the population under some conditions,at the same time,the results illustrates that Levy jumps can change the dynamical behavior of the system,and the stability in time average of the model is also proven.Lastly,some numerical simulations are given.