两类生态模型解的性质及一类二阶非自治微分方程解的振动性研究

【作者】 郎书华；

【导师】 陈斯养；

【作者基本信息】 陕西师范大学 ， 应用数学， 2010， 硕士

【摘要】 生物数学是通过建立数学模型,把复杂的生物学问题转化为数学问题,然后利用数学理论和方法达到对生命现象研究的目的.本文研究了两类生态模型解的性质及一类二阶非自治微分方程解的振动性,其中包括模型正平衡态的存在唯一性、分支周期解的存在性及其近似表达式、解的有界性、正平衡态的全局吸引性以及解的振动性等问题.近年来关于时滞产生周期解的研究得到了迅速发展,大量文献研究了时滞产生的Hopf分支和分支周期解的近似表达式.本文首先研究了一类含时滞和放养项的广义Logistic单种群模型的稳定性和Hopf分支问题.利用特征值理论,讨论了模型正平衡态的局部稳定性和Hopf分支的存在性；通过周期函数正交性方法得到了分支周期解的近似表达式；给出实例验证了定理的可实现性,且运用Matlab绘出了参数取不同数值时的曲线拟合图,并分析了参数对周期解的周期,振幅及正平衡态的影响.在自然界中,任何物种都与其它物种存在着相互制约和相互依赖的关系,其中包括捕食,竞争和互惠共存.本文其次研究了具有反馈控制的多时滞两种群竞争模型正解的全局吸引性.本章利用振动性理论和极限思想,证明了该模型解的有界性；通过构造Lyapunov泛函和不等式估值的方法,得到了该模型解全局吸引性的充分条件.由于微分方程的振动性理论是动力系统研究的重要问题之一.本文最后研究了一类具有多时滞二阶非自治微分方程振动性.通过对一阶系数取不同范围的值进行了讨论,利用Knaster-Tarski不动点原理和微分中值定理得到了方程的线性振动准则,给出了其线性振动的充要条件,使其解的振动性问题得以简化.更多还原

【Abstract】 Ecological mathematics traslates complex biological problems into a mathemat-ical problem by the establishment of mathematical models.Then ecological mathe-matics studies all kinds of natural phenomenon through the mathematical theory and way.The properties of two ecological systems and the oscillation of a second-order differential equation are investigated.Here,the behavior includes the existence and uniqueness of the positive equilibrium state、the existence and approximate expression of the periodic solution、the boundedness of the solutions、the global attractivity of the solutions and the oscillations of solutions.The research of the time delay leads to the periodic solutions’ has developed rapidly in recent years,and the hopf bifurcation produced by the time delay and the periodic solutions similar expressions of the bifurcation have been studied in lots of papers. Frist,the hopf bifurcation of a class of general Logistic model with a discrete time delay and stocking item is investigated.The part stability of the positive equilibrium state and existence of the hopf branch are discussed through the eigenvalue theory. The form of the approximate periodic solution is obtained by orthogonal conditions.The paper points out the example to confirm the theorem’s realizability, and fitted curve figures with different values,in which the influence toperiod, swing, positive equilibrium of period solution are discussed, are achieved by using matlab.In nature,the relationship between one group and another is restriced and de-pendent,such as predator-prey,competition and reciprocity coexistence. Sceond,the global attractivity of the solutions of a two-species competitive system with feedback controls and mutl-delay is investigated.The boundedness of the solutions of this model has been proved through the oscillation theory and the thinking of limit.Sufficient condition of the global attractivity for this model are derived by using the method of constructing Liapunov functional and inequality valuation.Because the differential equation’s oscillation theory is one of the important issues about the power system study. Third,the linearized oscillation for a second-order differential equation with mutl-delay is studied.Through the first-order coefficient s in the different range was discussed.The linearized oscillation criterion of the equation was obtained by using the Knaster-Tarski fixed- point theorem and the value theorem.The whole sufficient and neces-sary condition of it was derived,and so the oscillatin of the equation’s solution was simplified.更多还原