Study On Several Kinds Of Population Dynamical Models With Uncertain Factors In A Polluted Environment

The environmental pollution has great impacts on the survival of the biological population.There are many uncertain factors in the interactive porcess between pollutant and population.Therefore,it is meaningful to analyze the effect mechanism of the uncertain factors and pollutant on the survival of population,and propose the effective approaches and strategies for controlling the pollution.In this paper,we consider several kinds of population models with uncertain factors in a polluted environment.By choosing suitable stochastic processes(e.g.,Lévy noise,Brownian motion,Ornstein-Uhlenbeck process,Markov switching,Fuzzy initial value)to describe these uncertain factors,we establish some reasonable nonlinear stochastic dynamic models.Then,by use of the techniques of stochastic analysis,we estimate the extinction risk of the population and discuss the effects of uncertain factors in the pollution process.We prove the persistence,stability,the survival threshold for each population,the existence of the stationary distribution,positive recurrent,ergodicity,stability in distribution,etc.Moreover,we obtain the existence and convergence of the fuzzy numerical solutions of stochastic population dynamic model.More precisely,the outline of this paper is as follows:In the first chapter,we give a brief summary of researching background and significance of the toxicant-population models,and introduce some newly developments in this filed.Then,we state the main results obtained in this paper.In the second chapter,we introduce some fundational concepts of stochastic process,related theories of stochastic differential equation driven by different kinds of the noises,and frequently-used stochastic inequalities.In the third chapter,we develop and study a stochastic model for the competition of three species with a generalized dose-response function in a polluted environment.We first carry out the survival analysis,and obtain sufficient conditions for the extinction,non-persistence,weak persistence in the mean,strong persistence in the mean,and stochastic permanence.The threshold between weak persistence in the mean and extinction is established for each species.Then,using Hasminskii's methods and a Lyapunov function,we derive sufficient conditions for the existence of stationary distribution for each population.Moreover,we further investigate a stochastic competitive model of two species with Lévy noise in an impulsive polluted environment.The threshold between weak persistence in the mean and extinction is established for each species.Sufficient conditions for strong persistence in themean,stable in the mean and stochastic permanence are obtained by using the Ito formula with jumps and constructing Lyapunov function.Numerical simulations are carried out to support our theoretical results and some biological significance is presented.In the fourth chapter,a fuzzy stochastic single-species age-structure model in a polluted environment is presented.Both the fuzziness of the initial condition and the stochastic disturbance of the environment are incorporated into the model.By using the theory of fuzzy stochastic differential equation(FSDE)and the successive approximation,the global existence and uniqueness of solutions of the model are proved.In addition,the error estimation and stability of the numerical solutions are obtained.Furthermore,making use of Euler-Maruyama(EM)method,the convergence of the EM numerical approximation is established.Numerical simulations are carried out to verify the theoretical results.Our results show that the technique of numerical solution of FSDE can be used to estimate the evolution tendency of the population density in a polluted environment.In the fifth chapter,we develop and analyze a stochastic phytoplankton allelopathy model,which takes both white and colored noises into consideration.We first prove the existence of the global positive solution of the model.By use of the stochastic Lyapunov functions,we investigate the positive recurrence and ergodic property of the model,which implies the existence of a stationary distribution of the solution.Moreover,we obtain the mean and variance of the stationary distribution.Our results illustrate that both the two kinds of environmental noises and toxic substances have great impacts on the evolution of the phytoplankton populations.At the same time,we also analyze a non-autonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation.First,we obtain the existence of the global positive solution and the boundary periodic solutions of the model.Then,by using Khasminskii's method and constructing Lyapunov function,we derive the sufficient conditions of the existence of the nontrivial positive stochastic periodic solution of the model.Our results show that: the allelopathic effect plays an important role of the existence of the stochastic periodic solution,and can reduce the peaks of the cyclic outbreaks of the harmful algal blooms.Finally,numerical simulations are carried out to support our results.In the sixth chapter,we consider a n-species competitive Lotka-Volterra model with Lévy jumps under regime-switching.First,we prove the existence of the global positive solution,obtain the upper and lower boundedness.Then,asymptotic stability in distribution as the main result is derived under some sufficient conditions.Finally,numerical simulations are carried out to support our theoretical results and a brief discussion is given.In the last chapter,we give a summary of this paper and provide some future direction of research on this field.