The Threshold For Persistence Of The Population Models In The Noises Environment

In this dissertation,three stochastic population models are established,which are periodic intraguild predation model with white noises,periodic predatorprey model with Holling type II functional response and L(?)vy noises and Lotka–Volterra type predator-prey model with delay and white noises.We investigate the persistence,periodicity,extinction and the existence and uniqueness of stationary distribution,the threshold conditions for the persistence and extinction of each population are established.These results can provide a theoretical strategy for biological population control.The main results in this thesis are as follows:In Chapter 1,we mainly introduce the background of models,the status of the research,some mathematical methods,notations,definitions and usefule lemmas.In Chapter 2,a hybrid competition-predation model driven by white noise is proposed: in which the relationship between two groups(competition or predation),is related to time period.In the real world,there are some miscellaneous food species that play different roles throughout their lives.They are sometimes predators and sometimes competitors.For example,blackmouth bass is a native fish species in North America.Young juveniles compete with each other and they compete for common food.However,adult salmon,which uses fish as the main food,will eat smaller young squid in summer when there is not enough food.Therefore,in different stages of growth,blackmouth bass plays a different role(predator or competitor).First,we assume that the density of a population is zero,The existence of stochastic periodic solutions for the model subsystem is carried out,and necessary and sufficient conditions for the global asymptotic stability are established.Then,using the stochastic comparison principle,small parameter perturbation scaling time,stopping time theory,and the law of large numbers of martingale,the threshold condition between stochastic persistence and extinction is given by analyzing the stability of the stochastic periodic solution of the boundary model.Finally,we also prove the existence of stochastic periodic solutions for the stochastic boundary model.Our results show that,environmental white noise is unfavorable to the persistence of the population,and large noise intensity will cause the extinction of the population.In Chapter 3,a stochastic predator-prey model with L(?)vy noises and time periodic coefficients is considered.First,the existence and uniqueness and stochastically ultimate boundedness of the positive solutions are proved.When assuming that the predator population density is zero,the threshold condition between persistence and extinction of subsystem is obtained,then we discuss the existence and global stability of periodic solutions of the model.As a result of the solution for this model is not continuous,so most methods of the second chapter cannot be applied to this model.By using the stochastic comparison theory with jump,the Lyapunov function,the solution of the Markov property,laws of large numbers of martingale method,threshold conditions between persistence and extinction are obtained by the analysis stability of periodic solutions of the boundary model.Finally,we prove the persistence implies the existence of stochastic periodic solutions.Our results show that if the jump part of model is composed of compensation of Poisson measure,then the environmental Levy noise is unfavorable for population persistence,big Levy noise intensity would lead to population tends to extinct.If the model jump part is composed of Poisson measure,so the environmental L(?)vy noise sometimes is good for population persistence and sometimes is unprofitable.In Chapter 4,a delay Lotka-Volterra Predator-Prey model with white noises is established.First,the existence and uniqueness and stochastically ultimate boundedness of the positive solutions are proved.By constructing a suitable persistence function and time transformation,we give the threshold conditions between stochastic permanence and extinction of the model.According to Markov property of fragment solution,we define Markov semigroup and a KrylovBogoliubov measure sequence.Under the condition of stochastic permanence,we proved that the semigroup has Feller property and Krylov-Bogoliubov measure sequence is tight,then we obtain the existence of a stationary distribution.Finally,if noise intensity matrix of the model is non-degenerate,we will show that the semigroup is strongly Feller and irreducible by the truncation method,then it is easy to obtain the stationary distribution is unique by Doob theorem.