|
|
Bazykin捕食系统的平衡点和余维2 Bogdanov-Takens分支：全局渐近稳定性
|
Abstract:
本文从一个全局渐近稳定性定理出发讨论了Bazykin捕食系统，并在特定参数条件下给出若干内平衡点和一个余维2 BT分支，包括重数为1的多重焦点、余维2尖点和余维3 Bogdanov-Takens奇点(焦点或中心)。最后，结合数值类比，系统经历相应的余维2 Bogdanov-Takens分支。
This paper discusses the Bazykin’s predator-prey system from a globally asymptotically stable theorem, and presents several interior equilibria and a Bogdanov-Takens bifurcation of codimension 2 under certain parameter conditions, including multiple focus of multiplicity one, cusp of codimension 2 and Bogdanov-Takens singularity of codimension 3 (focus or center). Finally, combining numerical simulations, this system undergoes the corresponding codimension 2 Bogdanov-Takens bifurcation.
| [1] | Bazykin, A.D. (1998) Nonlinear Dynamics of Interacting Populations. World Scienti？c, Singapore.
https://doi.org/10.1142/2284 |
| [2] | Bazykin, A.D. (1976) Structural and Dynamic Stability of Model Predator-Prey Systems. RM-76-8 International Institute for Applied Systems Analysis, Laxenburg. |
| [3] | Pei, Y.Z., Chen, L.S., Zhang, Q.R. and Li, C.G. (2005) Extinction and Performance of One-Prey Multi-Predators of Holling Type II Function Response System with Impulsive Biological Control. Journal of Theoretical Biology, 235, 495-503. https://doi.org/10.1016/j.jtbi.2005.02.003 |
| [4] | Holling, C.S. (1965) The Functional Response of Predators to Prey Density and its Role in Mimicry and Population Regulation. Memoirs of the Entomological Society of Canada, 97, 3-60. https://doi.org/10.4039/entm9745fv |
| [5] | Misha, P. and Raw, S.N. (2019) Dynamical Complexities in a Predator-Prey System Involving Teams of Two Prey and One Predator. Journal of Applied Mathematics and Computing, 61, 1-24. https://doi.org/10.1007/s12190-018-01236-9 |
| [6] | Huang, J.C., Ruan, S.G. and Song, J. (2014) Bifurcation in a Predator-Prey System of Leslie Type with Generalized Holling Type III Functional Response. Journal of Differential Equations, 257, 1721-1752.
https://doi.org/10.1016/j.jde.2014.04.024 |
| [7] | Rosenzweig, M.L. and MacArthur, R. (1963) Graphical Representation and Stability Conditions of Predator-Prey Interactions. The American Society of Naturalists, 97, 209-223. https://doi.org/10.1086/282272 |
| [8] | Hsu, S.B. and Huang, T.W. (1995) Global Stability for a Class of Predator-Prey Systems. SIAM Journal on Applied Mathematics, 55, 763-783. https://doi.org/10.1137/S0036139993253201 |
| [9] | 陈兰荪, 井竹君. 捕食者-食饵相互作用中微分方程的极限环存在性和唯一性[J]. 科学通报, 1984, 9(3): 521-523. |
| [10] | Metzler, W. and Wischniewsky, W. (1985) Bifurcations of Equilibria in Bazykin’s Predator-Prey Model. Mathematical Modelling, 6, 111-123. https://doi.org/10.1016/0270-0255(85)90003-X |
| [11] | Wang, Y.Q., Jing, Z.J. and Chan, K.Y. (1999) Multiple Limit Cycles and Global Stability in Predator Prey Model. Acta Mathematicae Applicatae Sinica, 15, 206-219. https://doi.org/10.1007/BF02720497 |
| [12] | Freedman, H.I. (1979) Stability Analysis of a Predator Prey System with Mutual Interference and Density Dependent Death Rate. Bulletin of Mathematical Biology, 41, 67-78. https://doi.org/10.1016/S0092-8240(79)80054-3 |
| [13] | Hainzl, J. (1988) Stability and Hopf Bifurcation in a Predator-Prey System with Several Parameters. SIAM Journal on Applied Mathematics, 48, 170-190. https://doi.org/10.1137/0148008 |
| [14] | Kazarinoff, N.D. and Driessche, P. Van Den. (1978) A Model Predator-Prey System with Functional Response. Mathematical Biosciences, 39, 125-134. https://doi.org/10.1016/0025-5564(78)90031-7 |
| [15] | Qiu, X.X. and Xiao, H.B. (2013) Qualitative Analysis of Holling Type II Predator-Prey Systems with Prey Refuges and Predator Restricts. Nonlinear Analysis: Real World Applications, 14, 1896-1906.
https://doi.org/10.1016/j.nonrwa.2013.01.001 |
| [16] | Wang, S.T., Yu, H.G., Dai, C.J. and Zhao, M. (2020) The Dynamical Behavior of a Certain Predator-Prey System with Holling Type II Functional Response. Journal of Applied Mathematics and Physics, 8, 527-547.
https://doi.org/10.4236/jamp.2020.83042 |
| [17] | Perko, L. (2001) Differential Equations and Dynamical Systems. Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4613-0003-8 |
| [18] | Kuznetsov, Y. (1998) Elements of Applied Bifurcation Theory. 2nd Edition, Springer-Verlag, Berlin. |
| [19] | Wang, S.T. and Yu, H.G. (2021) Complexity Analysis of a Modified Predator-Prey System with Beddington-DeAngelis Functional Response and Allee-Like Effect on Predator. Discrete Dynamics in Nature and Society, 2021, Article ID: 5618190. https://doi.org/10.1155/2021/5618190 |
| [20] | Huang, J.C., Gong, Y.J. and Chen, J. (2013) Multiple Bifurcations in a Predator-Prey System of Holling and Leslie Type with Constant-Yield Prey Harvesting. International Journal of Bifurcation and Chaos, 23, Article ID: 1350164.
https://doi.org/10.1142/S0218127413501642 |
| [21] | Cai, L.L., Chen, G.T. and Xiao, D.M. (2013) Multiparametric Bifurcations of an Epidemicological Model with Strong Allee Effect. Journal of Mathematical Biology, 67, 185-215. https://doi.org/10.1007/s00285-012-0546-5 |
| [22] | Tang, B. and Xiao, Y.N. (2015) Bifurcation Analysis of a Predator-Prey Model with Anti-Predator Behavior. Chaos, Solitons and Fractals, 70, 58-68. https://doi.org/10.1016/j.chaos.2014.11.008 |
| [23] | Huang, J.C., Xia, X.J., Zhang, X.N. and Ruan, S.G. (2016) Bifurcation of Codimension 3 in a Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response. International Journal of Bifurcation and Chaos, 26, Article ID: 1650034. https://doi.org/10.1142/S0218127416500346 |
