【作者】 刘国栋；

【导师】 孟新柱；

【作者基本信息】 山东科技大学 ， 计算数学， 2020， 硕士

【摘要】 种群动力学模型是用来研究种群间相互关系的重要工具.随机干扰和空间结构可以显著地影响种群的动力学性质.根据随机微分方程和反应扩散方程相关理论,建立了随机和异质环境中的三类种群系统,分析了不同种群的时空动力学性质,并研究了随机和反应扩散种群系统的最优收获问题.第一章介绍了种群系统的研究背景、现状与意义,简述了关于随机微分方程、马尔科夫半群、反应扩散方程的一些基本理论,并给出了全文的主要工作和创新点.第二章研究了一类具有收获项的随机May型合作系统的渐近稳定性.首先证明了系统正解的全局存在性与唯一性.其次,应用马尔科夫半群理论和Fokker-Planck方程等新的方法,证明了该系统解的分布的密度收敛于一个不变密度,从而证明了系统解的渐近稳定性.最后,利用数值模拟验证了系统解的平稳分布,并探究了收获对种群系统的影响.第三章探究了一类污染环境中具有模式转换和Levy跳的随机合作系统的最优收获策略.首先验证了系统正解的全局存在性与唯一性.其次,根据比较原理和极限确界理论建立了两合作种群平均持久的充分条件.通过遍历、聚合的方法建立了随机环境中的最优收获策略,并给出相应的最大可持续产量.最后,利用数值举例验证了平均持久性和最优收获策略.第四章讨论了一类异质环境中具有避难所效应的捕食者-食饵系统的时空动力学性质以及最优收获问题.首先验证了 Neumann边值条件下常数正平衡态的全局渐近稳定性,并给出获得最大可持续产量的收获策略.其次,建立了非常数正平衡态存在与不存在的条件,并给出获得最大经济产量的收获策略.最后,利用数值模拟探究了避难所对种群动力学性质和最优收获策略的影响.第五章总结了全文研究的主要内容,解释了相应结论的生物学意义,并对以后的工作作了展望.更多还原

【Abstract】 Population dynamics model is an important tool to investigate the interaction among different populations.Stochastic disturbance and spatial structure can significantly affect the population dynamics.According to the theories of stochastic differential equations and reaction-diffusion equations,we establish three population systems in stochastic and heterogeneous environment,analyze the spatiotemporal dynamics of different populations,and explore the optimal harvesting problems of stochastic and reaction-diffusion population systems.The first chapter introduces the research background,current situation and significance of population system,and briefly narrates some preliminary theories about stochastic differential equations,Markov semigroup and reaction-diffusion equations,and presents the main work and innovation of the whole paper.The second chapter studies the asymptotic stability of a stochastic May mutualism system with harvesting term.Firstly,the global existence and uniqueness of positive solutions are proved.Secondly,applying the new method of Markov semigroup theory and Fokker-Planck equation,it is verified that the density of the distribution of solutions converges to an invariant density,which implies the asymptotic stability of the solutions of the system.Finally,the theoretical results are verified by numerical simulations,and the effects of harvesting on the system are also considered.The third chapter explores the optimal harvesting strategy for a stochastic mutualism system with regime switching and Levy jump in polluted environment.Firstly,the global existence and uniqueness of positive solution are proved.Secondly,sufficient conditions for persistence in mean of two cooperative populations are established by virtue of comparison principle and limit superior theory.By the method of aggregation and ergodicity,the optimal harvesting strategy in stochastic environment is established.Finally,numerical examples are carried out to verify the persistence in mean and optimal harvesting policy.The fourth chapter discusses the spatiotemporal dynamics and optimality for a predator-prey system system in a heterogeneous environment incorporating a prey refuge.First of all,the global stability of positive constant steady state under Neumann boundary conditions is proved,and the optimal harvesting strategy to get the maximum sustainable yield is established.Secondly,the conditions for existence and nonexistence of nonconstant positive steady state are proved,and the optimal harvesting strategy to get maximum economic yield is given.Finally,some numerical examples are presented to explore the effects of prey refuge on population dynamics and optimal harvesting strategy.The fifth chapter summarizes the main contents of the work,explains the biological significance of the corresponding conclusions and looks forward to the future work.更多还原