A Research On Dynamical Of Several Stochastic Population Models

Biological mathematical models are used to explain all kinds of population dynamic behavior from the angle of mathematics,which allow people to understand population dynamics more scientifically,so that the population can be intend to control more sci-entifically and purposefully.Especially,the random biological mathematical models can better describe the actual situation of dynamic population systems,because the popula-tions are always affected by environment in the real word.In recent years,the stochastic systems are widely used in the management of the ecological environment,wildlife protec-tion,infectious disease and pest management and so on.In this dissertation,population dynamic models are established to consider several problems in population controls and the corresponding dynamic behavior,including the existence and uniqueness of positive solution,the boundedness of expectation,the extinction and persistence of the system are investigated by means of the theory of method of stochastic differential equations.The main results of this paper are as follows:In chapter 2,a stochastic three populations model with parital-advantage for prey is studied.By constructing the suitable Liapunov functions,we show the system exist a unique positive global solution,and prove the boundedness of the solution,we also got the sufficient conditions for the solution of population extinction and persistence,furthermore,we show that the system has a unique stationary distribution and it has ergodic property under certain conditions.Finally,we give the main numerical simulations to verify our conclusion.In chapter 3,a stochastic predator-prey model with pulse input in a polluted envi-ronment is considered.We show the system exist a unique positive global solution,and prove the boundedness of the solution,then,the sufficient conditions for the existence boundedness periodical solution are obtained and we proved that it is globally attractiv-ity with probability 1,and the threshold P~*of population extinction and persistence in the mean are obtained too.If P>P~*,then the population will be extinct,If P<P~*,then the population will be exist,we also find that the larger pulse value,the faster the population will tend to destroy,and when the white noise,toxic pulse are weaker enough,the population will be persistence in the mean.Finally,numerical simulations about d-ifferent noise disturbance coefficients and pulse value are carried to illustrate the main obtained conclusions.In chapter 4,a stochastic toxin-producing phytoplankton-zooplankton system is in-vestigated.For the autonomous system,we first establish the sufficient conditions for the existence of the globally positive solution.Secondly,the conditions of extinction for system and persistence in mean for species are established.Furthermore,by construct-ing some suitable Lyapunov function,we also prove that there exists unique stationary distribution and they are ergodic,what is important is that Lyapunov function does not depend on existence and stability of equilibrium.For the non-autonomous periodic sys-tem,we prove that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii,and for the periodic system without stochastic distribution,there still exists a periodic solution.Finally,we illustrate our conclusions through several examples.