【作者】 李艳；

【导师】 刘萍；

【作者基本信息】 云南大学 ， 应用数学， 2022， 硕士

【摘要】 对种群模型进行定性分析是种群生态学的重要研究内容,在种群系统中引入反馈控制变量,研究反馈控制变量对种群动力学行为的影响,这有益于维持生态系统的动态平衡.本文对三类具有反馈控制的种群生态系统进行研究,通过利用微分方程定性和稳定性理论、随机微分方程理论、概周期函数理论、时标理论及方法,得到了一些新的结果:第一,研究了一类具单反馈控制和非线性抑制项的随机竞争系统.利用随机微分方程比较原理、广义积分中值定理、反证法和伊藤公式,分别获得了系统全局正解存在唯一性的充分条件,系统中两种群都灭绝的充分条件,以及某种群灭绝,另一个种群均值稳定的充分条件.第二,研究了一类具高次非线性反馈控制函数的非自治四种群混合系统.利用微分不等式和分析技巧得到了三个种群持久,一个种群灭绝的充分条件.在此基础上,构造合适的Lyapunov函数,利用Barbalat引理获得了系统的解全局吸引的充分条件.在某种群灭绝时,得到了该系统对应子系统的正概周期解的存在唯一性及一致渐近稳定性的充分条件.最后,给出一个例子说明结果的可行性.第三,研究了一类时标上具反馈控制和毒素影响的Schoner竞争系统.利用微分不等式分别得到了系统一个种群灭绝,另一种群持久,以及两种群都持久的充分条件.进一步,在两种群都持久时,构造恰当的Lyapunov函数并利用微分中值定理,得到了正概周期解的存在唯一性和一致渐近稳定的充分条件.更多还原

【Abstract】 Qualitative analysis of population model is an important research content of population ecology.Introducing feedback control variables into the population system to study the effects of feedback control variables on population dynamics behavior,which is beneficial to maintain the dynamic balance of the ecosystem.We mainly study three classes of population ecosystems with feedback controls,by using the qualitative and stability theory of differential equations,the theory of stochastic differential equations,the theory of almost periodic functions,the theory of time scale and methods.Some new results have been obtained:Firstly,a class of stochastic competitive system with single feedback control and nonlinear suppression term is studied.Using the comparative principle of stochastic differential equations,the generalized integral mean value theorem,the method of contradictory evidence and the It(?) formula,we get the sufficient conditions for the existence and uniqueness of the global positive solution of the system and the extinction of two groups and the extinction of a certain population,stability in the mean of another population,respectively.Secondly,a class of non-autonomous four species mixed system with higher-order nonlinear feedback control is studied.Applying differential inequalities and analytical skills obtains sufficient conditions for the persistence of three populations and the extinction of one population.Based on this,constructing a suitable Lyapunov function and applying Barbalat lemma,the sufficient condition of the global attractivity of the system is obtained.When one group extincts,we establish sufficient conditions for the uniformly asymptotically stability of a unique positive almost solutions of the corresponding subsystem of the system.Finally,an example is given to illustrate the feasibility of the result.Thirdly,a class of Schoner competitive system with feedback control and toxin on time scale is studied.Using differential inequalities,we can obtain the sufficient conditions for extinction of one population and the persistence of the another population of the system and the persistence of two populations,respectively.Moreover,when two populations are persistent,by constructing a suitable Lyapunov function and applying differential mean value theorem,the sufficient conditions for the existence and uniqueness and uniform asymptotic stability of positive almost periodic solutions are obtained.更多还原