Analysis Of The Dynamic Behaviors And Optimal Harvesting Of A Stochastic Mutualism Model With Time-varying Coefficients

The application of differential equations in the field of biological mathematics is mainly shown in describing the law of population change rate with time flying.The rapid development of stochastic biological mathematical models provides a practical and solid theoretical basis for accurately studying the changes of ecosystems.This paper starts from the two perspectives of stochastic properties and optimal harvest,and takes a type of general mutualism models proposed by May as the object to carry out in-depth discussion.The article is arranged as follows.In chapter 1,the background,development status and evolution process of the three main models are introduced,and the preliminary knowledge associated with them is given.In chapter 2,the stochastic nature of the general mutualism model with variable coefficient is discussed.The comparison principle of stochastic equations is used to prove the existence and uniqueness of the global positive solution,then the sufficient conditions for the persistence and extinction of the system are obtained,along with the dynamic nature such as stochastically ultimate boundedness,uniformly Holder-continuous.Finally,numerical simulation is employed to analyze the impact of different intensities of disturbances?_i(t)on population growth dynamics,laying a theoretical foundation for the subsequent optimal harvesting problem.In chapter 3,the proportional harvest strategy is considered.Firstly,the existence of the unique positive solution is proved,then the sufficient conditions for the system to persist and extinct are obtained.Secondly,the invariant measure theory is used to prove that the system is ergodic and,by constantizing the harvesting effectiveness the optimal ranges of harvesting output and effectiveness are determined.Therefore,the eligible harvesting mode _iq(t)is simulated according to the change law of r_i(t),?_i(t),and the dynamic harvesting in a non-autonomous environment is verified superiority.Finally,the conclusions obtained are used to analyze the optimal capture situation of the corresponding autonomous system,and combined with the images it is shown that the reduced value of biological density is consistent with the theoretical value of optimal harvest.In chapter 4,the disturbance of complex noise on mutualism model is involved,and the sufficient conditions of persistence and extinction are mainly established.The method is to construct the upper and lower bounded equations by comparison principle and then four parameters reflected by related probability density function which satisfies Fokker-Planck equation are obtained,and the signs of four numbers represent the persistence and extinction of the model.Finally,the frequency distribution of species under different intensities of noise is analyzed by growth graphs,and it is speculated that compared to simple noise,the complex noise with the same intensity has a low degree of interference to the model.