分数阶捕食者—食饵系统的动力学研究

发布时间：2018-06-08 08:23

本文选题：分数阶 + 捕食者-食饵系统　；参考：《北京交通大学》2015年硕士论文

【摘要】：近年来,分数阶动力学系统因其在生物数学、社会科学、统计力学等领域中存在巨大的应用价值而成为当前国内外的热点研究课题.作为非线性科学的一个新的研究方向,分数阶种群系统的动力学研究具有重要的理论与实际意义.目前,对于分数阶种群系统的动力学研究,主要是针对分数阶单种群和两种群生物系统,探讨其平衡点的稳定性和数值解.然而,对于分数阶三种群生物系统的研究还十分欠缺.鉴于此,本文以三维分数阶捕食者-食饵系统为研究对象,建立了四类分数阶捕食者-食饵系统,并对其进行了深入细致的研究. 一方面,建立了两类分数阶三种群捕食者-食饵系统,分别研究了这两类系统的稳定性和分岔性质.其一,研究了一类具有种间竞争的分数阶捕食者-食饵系统的稳定性,给出了系统各个平衡点局部渐近稳定的充分性条件,并通过数值仿真验证了所给条件的正确性；其二,研究了另一类具有种内竞争的分数阶捕食者-食饵系统的分岔行为,将分数阶阶数作为分岔参数,得到了系统发生分岔的临界值,并在数值仿真中发现了Hopf分岔现象. 另一方面,建立了两类分数阶具有时滞和阶段结构的捕食者-食饵系统,研究了这两类系统的稳定性.根据分数阶时滞系统的稳定性定理,得到了系统各个平衡点局部渐近稳定的充分性条件,并通过数值仿真验证了理论分析的有效性.此外,在数值仿真中讨论了时滞参数对于系统收敛速率的影响.
[Abstract]:In recent years, fractional order dynamics system has become a hot research topic at home and abroad because of its great application value in the fields of biology mathematics, social science, statistical mechanics and so on. As a new research direction of nonlinear science, the dynamics of fractional population system has important theoretical and practical significance. At present, the stability and numerical solution of the equilibrium point of the fractional population system are studied mainly for the fractional order single population and two species biological systems. However, the study of fractional three species biological system is still very lacking. In view of this, four kinds of fractional predator-prey systems are established and studied in detail, taking the three-dimensional fractional predator-prey system as the research object. In this paper, two classes of fractional order three species predator-prey systems are established, and their stability and bifurcation properties are studied respectively. First, the stability of a class of fractional predator-prey systems with interspecific competition is studied. The sufficient conditions for the local asymptotic stability of each equilibrium point of the system are given, and the correctness of the conditions is verified by numerical simulation. The bifurcation behavior of another kind of fractional predator-prey system with intraspecific competition is studied. The fractional order is taken as the bifurcation parameter and the critical value of bifurcation is obtained. Hopf bifurcation is found in numerical simulation. On the other hand, two classes of fractional order predator-prey systems with time delay and stage structure are established, and the stability of these two systems is studied. According to the stability theorem of fractional delay systems, the sufficient conditions for the local asymptotic stability of each equilibrium point of the system are obtained, and the validity of the theoretical analysis is verified by numerical simulation. In addition, the effect of delay parameters on the convergence rate of the system is discussed in the numerical simulation.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

【引证文献】