【摘要】：Gurney等人1990年在Nature上提出的時(shí)滯果蠅模型可以很好的擬合Nicholson的果蠅實(shí)驗(yàn)數(shù)據(jù),所以文中的模型被稱為Nicholson果蠅模型,并得到了學(xué)者廣泛的研究�？紤]擴(kuò)散的影響,學(xué)者們也研究了具擴(kuò)散的Nicholson果蠅模型,包括無窮空間上行波解,漸近傳播速度;有限空間上Neumann邊界條件和Dirichlet邊界條件下模型解的存在性、穩(wěn)定性及分支問題等。Dirichlet邊界條件下正解附近的Hopf分支問題是個(gè)困難的問題。應(yīng)用隱函數(shù)定理和Liapunov-Schmidt方法學(xué)者們對某些模型得到了一些結(jié)果,但是此方法對Nicholson果蠅模型不適用。為了考察Dirichlet邊界條件下果蠅模型正穩(wěn)態(tài)解附近的Hopf分支存在性問題,本文研究Dirichlet邊界條件下兩個(gè)格點(diǎn)上的果蠅模型。特別地,證明了正平衡點(diǎn)的存在唯一性;利用特征值分析的方法分析正平衡點(diǎn)的穩(wěn)定性,以及Hopf分支存在的條件,然后利用中心流形定理和規(guī)范型的方法,分析周期解的穩(wěn)定性。對于選定的參數(shù),文章中證明了Hopf分支的存在性以及周期解的穩(wěn)定性,且數(shù)值模擬的結(jié)果與理論相一致。
[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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

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