种群竞争系统分形动力学与控制

发布时间：2018-12-27 12:16

【摘要】：研究生物种群的数量增长模型对于人类社会的发展有着重要的意义,其在控制人口、调配社会资源、监控和改善生态环境、保护物种和开发养殖业等方面都有重要应用。近年来,生物数学得到不断发展,生物数学模型也得到了很好的运用。人们在研究种群时,最为关心问题是种群是否具有一个正的平衡态,以及这个平衡态是否能够保持稳定。而在数学上,种群平衡态就是关于种群竞争模型解的稳定性问题。本文对Lotka-Volterra竞争模型、连续的种群竞争模型和复数域上的种群竞争模型分别进行了分析。其中,Lotka-Volterra种群竞争模型奠定了竞争模型的基础。本文首先把分形几何中Julia集的思想和方法应用于Lotka-Volterra种群竞争模型,并建立了该竞争模型的Julia集,采用反馈控制方法对其加以控制,并且考虑了不同参数下竞争模型的同步,使得其中一个Julia集同步到另一个Julia集。其次,对连续的种群竞争模型进行了分析,离散化并建立了该竞争模型的Julia集,采用反馈控制法以及最优控制法对其加以控制,同时计算每一Julia集所对应的分形盒维数,利用分形盒维数的值刻划Julia集和吸引域的复杂性。以上研究内容是把Julia集的思想和方法放在实数系统范围内考虑的,但Julia集自身是定义在复数域上的,因此文章最后把Julia集推广到复系统种群竞争模型中。将种群竞争模型拓展到复数域,并把分形Julia集思想应用于离散化的种群竞争模型,利用Jury准则判断系统的稳定性,并建立复种群竞争模型的Julia集,从种群初始数量考虑对模型的影响。并通过反馈控制和追踪控制对模型Julia集加以合理的控制;在实际生态系统中,就是对种群数量加入人为的干扰,从而达到保护生物种群,保持生态平衡的目的。为方便,本文将讨论的范围仅限于两种群之间的竞争关系上。
[Abstract]:The model of population growth of graduate students is of great significance to the development of human society. It has important applications in controlling population, allocating social resources, monitoring and improving ecological environment, protecting species and developing breeding industry. In recent years, the biological mathematics has been continuously developed, and the biological mathematical model has been well used. In the study of population, the most important question is whether the population has a positive equilibrium state and whether the equilibrium state can be kept stable. In mathematics, population equilibrium is about the stability of the solution of population competition model. In this paper, the Lotka-Volterra competition model, the continuous population competition model and the population competition model in the complex number domain are analyzed respectively. Among them, Lotka-Volterra population competition model laid the foundation of competition model. In this paper, the idea and method of Julia set in fractal geometry are first applied to the Lotka-Volterra population competition model, and the Julia set of the competition model is established, and the feedback control method is used to control it, and the synchronization of the competition model under different parameters is considered. Synchronizes one of the Julia sets to the other Julia set. Secondly, the continuous population competition model is analyzed, the Julia set of the competition model is discretized, the feedback control method and the optimal control method are used to control it, and the fractal box dimension corresponding to each Julia set is calculated. The complexity of the Julia set and the domain of attraction is characterized by the value of fractal box dimension. The ideas and methods of Julia set are considered in the real system, but the Julia set itself is defined in the complex number field, so the Julia set is extended to the complex system population competition model at the end of this paper. The population competition model is extended to the complex number domain, and the fractal Julia set is applied to the discrete population competition model. The stability of the system is judged by using the Jury criterion, and the Julia set of the complex population competition model is established. The effect of the initial population number on the model is considered. The model Julia set is controlled reasonably by feedback control and tracking control, and in the actual ecosystem, the artificial disturbance is added to the population quantity to protect the biological population and maintain the ecological balance. For convenience, the scope of this paper is limited to the competitive relationship between two species.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175;O189

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