AAM  >> Vol. 3 No. 4 (November 2014)

    A Delayed Predator-Prey Model with Migration Rate and Holling-II Type Functional Response

  • 全文下载: PDF(823KB) HTML    PP.231-244   DOI: 10.12677/AAM.2014.34033  
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时滞微分方程系统Holling-II型功能性反应迁移率稳定性Hopf分支Delayed Differential System Holling-II Functional Response Migration Rate Stability Hopf Bifurcation



In this paper, we study global dynamic properties of a predator-prey model with Holling-II func- tional response and predator migration, which reflect manual control. Because of the delay effect of predations on the variation of predator’s quantity, this model is a system of delayed differential equations. Firstly, we study the existence and stability of equilibria. Then, sufficient conditions are obtained which ensures that Hopf bifurcation occurs when the delay is regarded as a bifurcation parameter. We also derive computational formula for direction and stability of Hopf bifurcation by applying the center manifold theorem and norm form theory. Some numerical simulations illu-strate the theoretical results.

段全恒, 郭志明. 一类具有迁移率和Holling-II型功能性反应的时滞捕食–食饵模型[J]. 应用数学进展, 2014, 3(4): 231-244. http://dx.doi.org/10.12677/AAM.2014.34033


[1] 马世骏 (1976) 谈农业害虫的综合防治. 昆虫学报, 2, 129-140.
[2] 陈兰荪, 井竹君 (1984) 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. 科学通报, 9, 521-523.
[3] DeBach, P. and Rosen, D. (1991) Biological control by natural enemies. CUP Archive.
[4] Barclay, H.J. (1982) Models for pest control using predator release, habitat management and pesticide release in combination. Journal of Applied Ecology, 19, 337-348.
[5] Tang, S.Y., Tang, G.Y. and Cheke, R.A. (2010) Optimum timing for integrated pest management: Modelling rates of pesticide ap-plication and natural enemy releases. Journal of Theoretical Biology, 264, 623-638.
[6] 焦建军, 陈兰荪 (2007) 具非线性传染率与生物化学控制的害虫管理SOI模型. 应用数学和力学, 4, 487-496.
[7] Wang, X., Guo, Z. and Song, X.Y. (2011) Dynamical behavior of a pest management model with impulsive effect and nonlinear incidence rate. Computational & Applied Mathematics, 30, 381-398.
[8] 傅金波, 陈兰荪 (2011) 无公害害虫治理策略的数学研究. 数学实践与认识, 2, 144-150.
[9] Hsu, S.B., Hwang, T.W. and Kuang, Y. (2003) A ratio-dependent food chain model and its applications to biological control. Mathematical Biosciences, 181, 55-83.
[10] Zhu, G.H. and Chen, L.S. (2008) Pest management about omnivora with continuous biological control. International Journal of Pure and Applied Mathematics, 44, 41-49.
[11] 成定平 (2003) 鼠类–天敌系统渐近稳定性的数学分析. 生物数学学报, 3, 283-286.
[12] Chen, Y.M. and Zhang, F.Q. (2013) Dynamics of a delayed predator-prey model with predator migration. Applied Mathematical Modelling, 37, 1400-1412.
[13] Smith, H. (2011) An introduction to delay differential equations with sciences applications to the life. Springer, New York.
[14] Kuang, Y. (1993) Delay differential equations with applications in population dynamics. Academic Press, New York.
[15] Wang, Z.H. (2012) A very simple criterion for characterizing the crossing direction of time-delay systems with delay-dependent parameters. International Journal of Bifurcation and Chaos, 22.
[16] Hassard, B.D., Kazarinoff, N.D. and Wan, Y.H. (1981) Theory and applications of Hopf bifurcation. Cambridge University Press, Cambridge.
[17] 魏俊杰, 王洪滨, 蒋卫华 (2012) 时滞微分方程的分支理论及应用. 科学出版社, 北京.