离散Lotka-Volterra系统的持久性及周期解的存在性与全局吸引性

【作者】 于跃华；

【导师】 周展；

【作者基本信息】 湖南大学 ， 应用数学， 2003， 硕士

【摘要】 在第一章，我们陈述了研究生物模型的现实意义，介绍了一些已知的Lotka-Volterra类型的竞争和食饵-捕食者系统模型的研究结果；另外，还给出了一些最基本定义相关的初始条件。 在第二章，利用李雅普诺夫函数研究了离散时滞Lotka-Volterra竞争系统的持久性。早在1974年，May首先提出了二维离散Lotka-Volterra竞争系统模型。从那以后，二维离散Lotka-Volterra竞争系统模型已引起了许多学者的注意和兴趣，而且发现此系统有相当复杂的动力行为。在现实中，时滞总是存在的，而且会对模型的模拟产生影响。因此，最近几篇文献都考虑了二维自治时滞Lotka-Volterra竞争系统。进一步，由于系数可能会改变，张勤勤和周展考虑了二维非自治时滞Lotka-Volterra竞争系统的持久性，而且获得了此系统持久性的充分必要条件。本章所研究的模型包含了前面涉及到所有二维系统模型。王稳地研究了具有时滞和振动环境的离散人口模型的全局稳定性，虽然此模型包含了我们在本章中所研究的模型，但是在我们的证明过程中去掉条件(i=1,2)。因而我们的结论改进了相关的结果。 在第三章，我们讨论了多维Lotka-Volterra竞争系统。首先获得此系统的持久性，再假定所有系数均是周期的，得到了系统周期解的存在性；而且，增加一些适当的条件后，我们证明了周期解的全局吸引性。我们的结果可以推得二维系统的相应的结果；同时，我们的结果可以推得当耦合消失时标量方程的结果，而且改进和弥补了其不足。 在第四章，通过构造李雅普诺夫函数，我们研究了在有限滞量背景下的二维离散食饵-捕食者系统，得到了与连续情形类似的结论，即我们证明了时滞对所研究系统解的持久性没有影响。更多还原

【Abstract】 In the first chapter, we firstly state the necessity for the study of biological models. Then, we introduce some known competition and predator-prey models of Lotka-Volterra type. Finally, some basic definitions and the related initial conditions of those studied system are given.In the second chapter, the permanence for an autonomous delaycompetition model of Lotka-Volterra type is considered by means of Liapunovfunctionals. In 1974, May first proposed the discrete two-species competitionmodel of Lotka-Volterra type. Since then, the discrete two-species competitionmodel of Lotka-Volterra type has attracted great attention and interest of manyauthors, and it has been found that the system can demonstrate quite rich andcomplicated dynamics. In real situations, however, time delays always existand they should be taken into account in modeling. Therefore, several recentpapers considered the discrete delay two-species competition model ofLotka-Volterra type. Due to the possible change of the coefficients in realworld, Qinqin Zhang and Zhan Zhou considered the permanence of thenonautonomous two-species competition model of Lotka-Volterra type withdelays, and the necessary and sufficient conditions for the permanence wereobtained. Our model includes all those models mentioned earlier. Wendi Wangstudied the global stability of discrete population models with time delays andfluctuating environment. Though those models studied by Wcndi Wang includeour model in this chapter, during our proof, we dismiss the conditionsinf a(k)>0 for i=1,2. Therefore, our conclusion improves thecorresponding ones.In the third chapter, we discuss a discrete Lotka-Volterra competition system with m-species. We first obtain the persistence of the system. Assuming that the coefficients in the system are periodic, we obtain the existence of a periodic solution. Moreover, under some additional conditions, the periodic solution is globally attractive. Our results can be reduced to those for two-species system. At the same time, our results not only can bereduced to those for the scalar equation when there is no coupling, but also improve and complement some in the literature.Finally, in the fourth chapter, by constructing Liapunov functional, we study a discrete two-dimensional predator prey-system with a finite number of delay, the similar conclusions to the continuous case are obtained. That is, the time delays are harmless for permanence of the solution of the system.更多还原