Qualitative Analysis Of Stochastic Population Models With Time Delays

As an important subject of population dynamics model,the Lotka-Volterra ecological model has attracted extensive attention since it was proposed.After continuous evolution and development,the population structure of Lotka-Volterra has become more complete,and the influencing factors included has become more abundant,which can better describe the growth patterns and interactions of various groups in the ecosystem.Its research can help humans better grasp the living conditions of other species in nature,so as to predict and adjust their populations,and realize the harmonious coexistence between human beings and nature.Compared with the deterministic Lotka-Volterra ecosystem model,the research of multi-groups ecological models with white noise disturbance is still relatively rare.Under this background,this paper improves and generalizes the deterministic Lotka-Volterra ecological models based on previous work,and analyzes the survival status and asymptotic behavior of several types of multi-groups ecological models with time delay and stochastic disturbance.The details are as follows:1.A class of stochastic three-groups competition-predation model with time delay is studied.By using local Lipschitz condition and linear growth condition,the existence of system’s unique global positive solution is proved.By using the comparison theorem of stochastic differential equation and its extension theorem,it is proved that under appropriate conditions,the system has extinction and continuous survivability.By constructing a suitable Lyapunov function,it is proved that under appropriate conditions,the system expectation has globally attractivity.The effects of different initial conditions and different time delays on the stability of the system expectation is also considered.Finally,the correctness of the theoretical analysis is verified by the simulation of the system,and compared with the system that do not have continuous survivability and expectation’s global attractivity,further confirming the correctness of the conclusions.2.A class of stochastic three-groups predator-prey model with time delay and spatial diffusion is studied.On the basis of the first part of study,the flow phenomenon between the prey populations in the communication area is considered,the "space diffusion term" is added to the ecological model.By constructing an auxiliary system and classifying and discussing the diffusion term according to different situations,it is proved that the system has a unique global positive solution.The sufficient conditions for extinction and continuous survivability of system is obtained by generalizing the sufficient conditions for the extinction and continuous survivability of the auxiliary system that derived from the stochastic differential equation comparison theorem and the classification of the diffusion term.By constructing a suitable Lyapunov function,it is proved that under appropriate conditions,the system expectation has globally attractivity.Finally,the correctness of theoretical analysis is verified by using the software of MATLAB and Milstein method.3.A class of stochastic three-groups food-chain predation model with time delay and Beddington-De Angelis functional response is studied.Based on the first part of the study,a "dependency term" is introduced.By using local Lipschitz condition and linear growth condition,the existence of system’s unique global positive solution is proved.The analytical formula of the stochastic system is obtained by using the Ito formula,then,the upper and lower levels of the system solution is calculated by using the comparison theorem of stochastic differential equation to analyse the analytical formula,and the sufficient conditions for uniform persistence of system is obtained.By constructing a suitable Lyapunov function and clever processing of the delay term,it is proved that under appropriate conditions,the system solution has global attractivity.Finally,the correctness of the theoretical analysis is verified by the simulation of the system,and the correctness of the given sufficient conditions is further proved by giving the "counter examples" about the uniform persistence and solution’s global attractivity respectively.4.A class of stochastic three-groups predator-prey model with time delay,Beddington-De Angelis functional response and spatial diffusion is studied.Further improve the model has been discussed.By constructing an auxiliary system,it is proved that the system has a unique global positive solution.By using comparison theorem of stochastic differential and constructing a suitable Lyapunov function and developing some new analytical techniques,it is proved that under appropriate conditions,the system has uniform persistence and the solution has global attractivity.Under the premise that the solution of the system has globally attractivity,it is inferred that the system’s expectation also has globally attractivity.Finally,the correctness of theoretical analysis is verified by using the software of MATLAB and Milstein method.