一类捕食者-食饵系统的分歧问题和Turing不稳定性 Alternative Title Bifurcation Problems and Turing Instability in a class of Predator-Prey Systems 张嘉防 Thesis Advisor 李万同 2008-05-23 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 稳定性 Hopf分歧 鞍-结点分歧 Bogdanov-Takens分歧 Turing不稳定性 规范形 中心流形约化 周期解 Abstract 众所周知，捕食系统是描述客观世界的重要工具之一，对其进行研究对于理解 现实世界具有重要的指导意义。基于此，本文主要研究一类捕食系统的分歧问题 和Turing不稳定性。 首先，在没有扩散影响时，讨论了捕食系统的稳定性和Hopf分歧周期解。获得 了正平衡点渐近稳定的充分条件以及在它周围分歧出周期解的条件。另外，还获得 了确定周期解Hopf分歧的方向、分歧周期解稳定性的显式算法。同时，通过一些数 值模拟验证了文中所得结论的正确性。 其次，考虑了扩散系统的分歧和Turing不稳定性。研究了正常数平衡解的渐近 稳定性和系统存在Hopf分歧的条件。更进一步，通过使用偏微分方程的规范形理论 和中心流形约化，获得了决定周期解Hopf分歧的方向、分歧周期解稳定性的显式公 式，给出了分歧周期解在中心流形上轨道渐近稳定和不稳定的充分条件。特别地， 我们注意了扩散对捕食系统稳定性的影响。通过分析系统的分歧现象，对比捕食系 统和对应的扩散系统，首次发现在Neumann边界条件下扩散系统能出现更多的分歧 现象，得到了新的、更多的分歧现象和Turing不稳定性出现的条件和结论。同时也 通过一些数值模拟验证了文中所得结论的正确性。 Other Abstract It is well known that predator-prey systems are one of the important tools describing the real world and studying these systems has important to understand the real world. Based on this fact, in this thesis we mainly consider bifurcation problems and Turing instability of a class of predator-prey systems. Firstly, in the absence of diffusion effects, the stability and Hopf bifurcation of periodic solutions of the predator-prey system are discussed. The suffcient conditions ensuring that the positive equilibrium is asymptotically stable and the conditions guaranteeing that the system can bifurcate periodic solutions from the positive equilibrium are established. In addition, an explicit algorithm determining the direction of Hopf bifurcation, the stability of bifurcated periodic solutions is given. Meanwhile, some numerical simulations are also included. Secondly, in the presence of diffusion effects, the bifurcations and Turing instability of the corresponding diffusion system are studied. The stability of positive equilibrium is studied and the conditions under which the system undergoes a Hopf bifurcation of periodic solutions are obtained. Furthermore, by using the normal form theory and the center manifold reduction for partial differential equations, an explicit algorithm determining the direction of Hopf bifurcation of periodic solutions, the stability and period of bifurcated periodic solutions is given and the sufficient conditions ensuring that the bifurcated periodic solutions are orbitally asymptotically stable and unstable on the center manifold are also obtained. Es pecially, we noticed that the diffusion had a notable impact on the stability of the predator-prey system. By analyzing the bifurcation phenomena of the diffusion system and comparing with the corresponding system without diffusion, we find that under Neumann boundary condition the corresponding diffusion system can appear more bifurcation phenomena and obtain some more new conditions and conclusions regarding bifurcation phenomena and Turing instability. Meanwhile, to verify the theoretical conclusions obtained in this part, some numerical simulations are also included. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224522 Collection 数学与统计学院 Recommended CitationGB/T 7714 张嘉防. 一类捕食者-食饵系统的分歧问题和Turing不稳定性[D]. 兰州. 兰州大学,2008.
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