两个格点上具扩散和Dirichlet边界条件的果蝇模型的Hopf分支研究
发布时间：2018-10-20 10:15
【摘要】：Gurney等人1990年在Nature上提出的时滞果蝇模型可以很好的拟合Nicholson的果蝇实验数据,所以文中的模型被称为Nicholson果蝇模型,并得到了学者广泛的研究。考虑扩散的影响,学者们也研究了具扩散的Nicholson果蝇模型,包括无穷空间上行波解,渐近传播速度;有限空间上Neumann边界条件和Dirichlet边界条件下模型解的存在性、稳定性及分支问题等。Dirichlet边界条件下正解附近的Hopf分支问题是个困难的问题。应用隐函数定理和Liapunov-Schmidt方法学者们对某些模型得到了一些结果,但是此方法对Nicholson果蝇模型不适用。为了考察Dirichlet边界条件下果蝇模型正稳态解附近的Hopf分支存在性问题,本文研究Dirichlet边界条件下两个格点上的果蝇模型。特别地,证明了正平衡点的存在唯一性;利用特征值分析的方法分析正平衡点的稳定性,以及Hopf分支存在的条件,然后利用中心流形定理和规范型的方法,分析周期解的稳定性。对于选定的参数,文章中证明了Hopf分支的存在性以及周期解的稳定性,且数值模拟的结果与理论相一致。
[Abstract]:The delayed Drosophila model proposed by Gurney et al in 1990 on Nature can well fit the experimental data of Drosophila from Nicholson, so the model in this paper is called Nicholson Drosophila Model, and has been widely studied by scholars. Considering the effect of diffusion, scholars have also studied the existence of solutions for Nicholson flies with diffusion, including travelling wave solutions on infinite spaces, asymptotic propagation rates, Neumann boundary conditions and Dirichlet boundary conditions in finite space. Stability and bifurcation problems. The Hopf bifurcation problem near positive solutions under Dirichlet boundary conditions is a difficult problem. Some results are obtained by using implicit function theorem and Liapunov-Schmidt method for some models, but this method is not suitable for Nicholson Drosophila model. In order to investigate the existence of Hopf bifurcation near the positive steady-state solution of the Drosophila model under the Dirichlet boundary condition, this paper studies the Drosophila model on two lattice points under the Dirichlet boundary condition. In particular, the existence and uniqueness of positive equilibrium point are proved, the stability of positive equilibrium point is analyzed by the method of eigenvalue analysis, and the condition of existence of Hopf bifurcation is analyzed, then the method of center manifold theorem and normal form is used. The stability of periodic solutions is analyzed. For the selected parameters, the existence of Hopf bifurcation and the stability of periodic solution are proved, and the numerical simulation results are in agreement with the theory.
【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
本文编号：2282842
[Abstract]:The delayed Drosophila model proposed by Gurney et al in 1990 on Nature can well fit the experimental data of Drosophila from Nicholson, so the model in this paper is called Nicholson Drosophila Model, and has been widely studied by scholars. Considering the effect of diffusion, scholars have also studied the existence of solutions for Nicholson flies with diffusion, including travelling wave solutions on infinite spaces, asymptotic propagation rates, Neumann boundary conditions and Dirichlet boundary conditions in finite space. Stability and bifurcation problems. The Hopf bifurcation problem near positive solutions under Dirichlet boundary conditions is a difficult problem. Some results are obtained by using implicit function theorem and Liapunov-Schmidt method for some models, but this method is not suitable for Nicholson Drosophila model. In order to investigate the existence of Hopf bifurcation near the positive steady-state solution of the Drosophila model under the Dirichlet boundary condition, this paper studies the Drosophila model on two lattice points under the Dirichlet boundary condition. In particular, the existence and uniqueness of positive equilibrium point are proved, the stability of positive equilibrium point is analyzed by the method of eigenvalue analysis, and the condition of existence of Hopf bifurcation is analyzed, then the method of center manifold theorem and normal form is used. The stability of periodic solutions is analyzed. For the selected parameters, the existence of Hopf bifurcation and the stability of periodic solution are proved, and the numerical simulation results are in agreement with the theory.
【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
【参考文献】
相关期刊论文 前3条
1 刘畅;张莉;秦爽;;一个超混沌Lorenz系统的Hopf分岔分析[J];重庆文理学院学报(社会科学版);2016年02期
2 胡万紫;郭谦;;二阶时滞微分方程的Hopf分歧分析及近似其周期解的一种约化方法(英文)[J];上海师范大学学报(自然科学版);2010年02期
3 魏俊杰,黄启昌;泛函微分方程分支理论发展概况[J];科学通报;1997年24期
，本文编号：2282842
本文链接：https://www.wllwen.com/kejilunwen/yysx/2282842.html
