具时滞和食物补贴的捕食者—食饵模型的分支研究

发布时间：2018-05-19 21:02

本文选题：捕食者-食饵 + 时滞　；参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：为了保护物种的多样性,维护生态平衡,需要对种群动力学模型进行深入研究,揭示出种群之间的相互作用关系。在种群动力学中,捕食者-食饵模型因其重要性一直受到各界学者的关注。在描述种群数量变化时,需要考虑到物种的成熟期和能量的转化时间,因此有必要在系统中引入时滞,以便更好地反应实际情况。所以本文讨论了一类具时滞的捕食者-食饵模型,并在模型中引入了食物补贴项的影响。首先,讨论了系统正平衡点的存在唯一性,在此基础上利用特征方程根的分布分析方法分析其稳定性,得到了在正平衡点处存在局部Hopf分支的充分条件。又由中心流形定理和规范型理论,分析了正平衡点处Hopf分支的性质,包括分支的方向、分支周期解的稳定性以及周期解周期的变化等。其次,在局部Hopf分支的基础上进一步研究系统周期解的大范围存在性问题。由全局Hopf分支定理可以得到每个连通分枝是无界的,接着证明了系统的解具有正性,又利用常微分方程高维Bendixson定理证明系统没有非常值?-周期解,进而得到了周期解的全局存在性结论。最后,本文分为两部分进行数值模拟。第一部分以时滞为参数,观察系统在不同时滞处的稳定性和全局Hopf分支的存在性,对之前的理论结果给予了算例支撑;第二部分分别以食物补贴投放率、环境承载量、捕食者消耗食饵的最大速率和转换因子为参数。通过模拟观察其对第一个分支值的影响,从而得到各参数对系统稳定区间的影响,同时解释了各种情况下的生物学意义。
[Abstract]:In order to protect species diversity and maintain ecological balance, it is necessary to deeply study the population dynamics model and reveal the interaction between populations. In population dynamics, predator-prey model has been concerned by scholars because of its importance. In order to better reflect the actual situation, it is necessary to take into account the mature period of species and the time of energy conversion when describing the population quantity change, so it is necessary to introduce time delay in the system. In this paper, we discuss a kind of predator-prey model with time delay, and introduce the effect of food subsidy term into the model. Firstly, the existence and uniqueness of the positive equilibrium point of the system are discussed. On this basis, the stability of the system is analyzed by using the distribution analysis method of the root of the characteristic equation, and the sufficient conditions for the existence of local Hopf bifurcation at the positive equilibrium point are obtained. Based on the center manifold theorem and normal form theory, the properties of Hopf bifurcation at positive equilibrium point are analyzed, including the direction of bifurcation, the stability of bifurcation periodic solution and the variation of periodic solution. Secondly, on the basis of local Hopf bifurcation, the existence of periodic solutions in a large scale is studied. By using the global Hopf bifurcation theorem, we can obtain that every connected branch is unbounded, then we prove that the solution of the system is positive, and by using the high dimensional Bendixson theorem of ordinary differential equation, we prove that the system has no nonconstant value or periodic solution. The global existence of periodic solutions is obtained. Finally, this paper is divided into two parts for numerical simulation. In the first part, the stability of the system at different time delays and the existence of global Hopf bifurcation are observed, and the previous theoretical results are supported by examples. The maximum rate of predator consumption and the conversion factor are parameters. The influence of each parameter on the stability interval of the system is obtained by observing the influence of the first branch value by simulation, and the biological significance under various conditions is explained.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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