Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/17559
標題: 利用Liapunov-Schmidt化簡法與數值延續法來探討捕食者-被捕者的數學模式
Liapunov-Schmidt Reduction and Numerical Continuation for Predator-Prey Models
作者: 蕭竣中
Hsiao, Chun-Chung
關鍵字: Predator-prey models
捕食者-被捕者的數學模型
Liapunov-Schmidt reduction
bifurcation
numerical continuation
two-gird method
Liapunov-Schmidt化簡法
分支,延續法
雙重網格法
出版社: 應用數學系所
引用: [1] Allgower, E. L. & Georg, K. [2003] it Introduction to Numerical Continuation Methods (SIAM, Philadelphia). [2] Bank, R. E. & Chan, T. F. [1986] ``PLTMGC: A multigrid continuation package for solving parametrized nonlinear elliptic systems,'' SIAM J. Sci. Stat. Comput. 7, 540--559. [3] Brown, K. J. [1987] ``Nontrivial solutions of predator-prey systems with small diffusion,'' it Nonlinear Anal.11, 685--689. [4] Chien, C.-S. & Jeng, B.-W. [2005] ``A two--grid discretization scheme for semilinear elliptic eigenvalue problems,'' SIAM J. Sci. Comput., to appear. [5] Choi, Y.-S., Lui, R. & Yamada, Y. [2003] ``Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion,'' Discrete Contin. Dyn. Syst. 9, 1193--1200. [6] Choi, Y.-S., Lui, R. & Yamada, Y. [2004] ``Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,'' Discrete Contin. Dyn. Syst. 10, 719--730. [7] Conway, E. D., Gardner, R. & Smoller, J. [1982] ``Stability and bifurcation of steady-state solutions for predator-prey equations,'' Adv. Appl. Math. 3, 288--334. [8] Dancer, E. N. [1984] ``On positive solutions of some pairs of differential equations,'' Trans. Amer. Math. Soc. 284, 729--743. [8] Dancer, E. N. [1985] ``On positive solutions of some pairs of differential equations II,'' J. Diff. Eqs. 60, 236--258. [9] Du, Y.-H. & Lou, Y. [1997] ``Some uniqueness and exact multiplicity results for a predator-prey model,'' Trans. Amer. Math. Soc. 349, 2443--2475. [10] Golubitsky, M. & Schaeffer, D. G. [1985] Singularities and Groups in Bifurcation Theory, Vol. I (Springer-Verlag, New York). [11] Keller, H. B. [1987] Lectures on Numerical Methods in Bifurcation Problems (Springer-Verlag, Berlin). [12] Kuto, K. & Yamada, Y. [2004] ``Multiple coexistence states for a prey-predator system with cross-diffusion,'' J. Diff. Eqs. 197, 315--348. [13] Li, K.-T. & Wang, L.-H. [2001] ``Global bifurcation and long time behavior of the Volterra-Lotka ecological model,'' Int. J. Bifurcation and Chaos 11, 133--143. [14] Lotka, A. J. [1920a] ``Undamped oscillations derived from the law of mass action,'' J. Amer. Chem. Soc. 42, 1595--1599. [15] Lotka, A. J. [1920b] ``Analytic note on certain rhythmic relations in organic systems,'' Proc. Nat. Acad. Sci. U.S.A. 6, 410--415. [16] Shih, S.-D. & Chow, Y.-S. [2004] ``A power series in small energy for the period of the Lotka-Volterra system,'' Taiwanese J. Math. 8, 569--591. [17] van der Vorst, H. A. [1992] ``Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,'' SIAM J. Sci. Statist. Comput. 13, 631--644. [18] Volterra, V. [1926] ``Variazioni e fluttuazioni del numero d''individui in specie animali conviventi,'' Mem. R. Acad. Naz. dei Lincei 2, 31--113.
摘要: 我們探討三個捕食者-被捕者之數學模型的分支解局部性質。首先應用Liapunov-Schmidt化簡法來計算出對應於這三個數學模式的標準式,我們證明出這些分支解皆為transcritial,我們也探討具有雙重迴圈延續法的雙重網格中央差分離散法來追蹤反應-擴散系統的解曲線,一次與二次逼近的方程可被用來推導出雙重網格法。接著利用延續法來追蹤這些數學模式的解曲線,並且藉由數值結果可以驗證出我們的理論證明。
We study bifurcation scenario of three predator-prey models. The Liapunov-Schmidt reduction is applied to compute normal forms of these models. We show that the bifurcations are all transcritical. We also study a two-grid centered difference discretization scheme with two-loop continuation algorithms for tracing solution curves of reaction-diffusion systems. Both linear and quadratic approximations of the operator equations are exploited to derive the scheme. A numerical continuation algorithm is exploited to trace solution curves of the models. The theoretical proofs are verified by our numerical results.
URI: http://hdl.handle.net/11455/17559
其他識別: U0005-1707200612452300
文章連結: http://www.airitilibrary.com/Publication/alDetailedMesh1?DocID=U0005-1707200612452300
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