Research On Dynamical Properties Of Several Stochastic Predator-Prey Systems With Time Delay

With the development of society and the progress of science and technology,the ability of human beings to exploit natural resources is constantly improving,and the ecological environment is facing more and more challenges and problems.In practice,environmental fluctuations inevitably affect the dynamic behavior of the population.At the same time,the biological process of population generally exists time delay phenomenon.Therefore,it is of great practical significance to study the dynamic properties of ecosystem affected by stochastic environmental fluctuations and time delay.In this paper,based on the actual survival status of the population,considering the time delay of biological processes,the predator’s predation ability,the fear effect on prey,and the influence of environmental white noise on the ecological behavior of the population,three types of stochastic predatorprey models with time delay are established.In view of the importance of the properties of population dynamics,this paper is based on the theory of delay differential equation and stochastic differential equation,using Lyapunov stability theory,Hopf-bifurcation theory,Routh-Hurwitz criterion,stochastic differential comparison theorem,It^o formula and other classical inequalities,combined with integral transformation method and ergodic method.The existence,positive and boundedness of the solution,the existence and stability of the equilibrium point,and the bifurcation behavior of the coexistence equilibrium point are studied.The existence and uniqueness of the global positive solution,the moment boundedness and stochastic ultimate boundedness,the average persistence and extinction of predator and prey populations in the sense of time are discussed under the influence of random factors,and the stability in distribution of the system are obtained.Then the theoretical results are verified from the perspective of numerical analysis by examples,and the specific effects of factors such as time delay,fear effect and stochastic perturbation on the system dynamics are further discussed.The results show that the environmental white noise has a significant effect on the persistence and extinction of the population,and a large white noise intensity will destroy the stability of the system.At the same time,time delay will affect the stability of coexistence equilibrium point of the system.With the increase of time delay,the system will have oscillation behavior.When the delay strength increases to a certain value,Hopf-bifurcation will occur near the equilibrium point,and the system will switch from a stable state to an unstable state.In addition,the fear effect of predators does not change the dynamics of the system within a certain range,but the population density of the system decreases as the level of fear increases.The results enrich the dynamic properties of stochastic population system,reveal the effects of time delay and white noise on system stability,and provide a theoretical basis for strengthening ecosystem protection,promoting biodiversity and maintaining ecological balance.