【作者】 蒋利群；

【导师】 周展；

【作者基本信息】 湖南大学 ， 应用数学， 2004， 硕士

【摘要】 本文在已有的Lotka-Volterra模型的基础上，考虑多个物种并加入常时滞或变时滞，得到了更符合现实的几类离散时滞人口模型。我们主要对这几类模型的持久性和周期解的存在性进行了较深入的研究。本篇论文由四章构成。 第一章概述了数学生态学的发展历史和前人所做的一些相关工作以及本文问题的产生。另外还简单介绍了本文的主要工作。 第二章讨论了具有变时滞的一般n-种群周期Lotka-Volterra互惠差分系统。利用拓扑度理论中的连续定理以及M-矩阵的性质获得了该系统正周期解存在的充分条件。由此结果推得的二维系统正周期解存在的充分条件与相应微分系统的已知结果相同。 第三章考虑了具有时滞的n-种群非自治的Lotka-Volterra竞争差分系统。通过构造一个新的拟李雅谱诺夫函数，得到了判定该系统持久的一些易于检验的准则。这些结果较好地推广了一些已知结论。另外，尽管有的模型覆盖面比该模型要广，但它要求的条件过于苛刻，而我们大大地减弱了这些条件。在此条件下，我们较大程度上改进了这个结果。 第四章研究的仍然是n-种群的Lotka-Volterra竞争差分系统，但是是具有不同变时滞的周期系统。利用重合度理论得到了该系统正周期解存在的充分条件。更多还原

【Abstract】 Based on the existed Lotka-Volterra population models, we consider multi-species models with constant or varied delays instead of two species ones and obtain several classes of discrete delayed population models which seem more practical than those existed. We mainly make much investigation for the existence of periodic solution and the permanence of these new models. This thesis is composed of four chapters.In the first chapter, we state the history of Mathematical ecology’s development, the existed related work and the origin of the problems we discussed. And the main work of this paper is also simply introduced.In the second chapter, we consider a periodic Lotka-Volterra cooperative difference system with varied delays. By using the continuation theorem of topology degree theory and properties of nonsingular M-matrix, we obtain sufficient conditions for the existence of positive periodic solutions of this system. As the special case, the results for the two species model are obtained, which are same to the known results of the corresponding differential system.The purpose of the third chapter is to study the permanence of a nonau-tonomous n-species Lotka-Volterra competitive discrete system with delays. By constructing a new qusi-Liapunov function, we obtain some new criteria which are easily checked. The known results of several two-species competitive models are well generalized. On the other hand, although some models cover this model, the required conditions are too rigorous to fit, and the conditions are weakened in our theorems. In this sense, we improve these results.Finally, in the fourth chapter, we still consider a n-species Lotka-Volterra competitive difference system, but this model is a periodic system with different varied delays. The existence of positive periodic solutions of this system is established by using coincidence degree theory.更多还原