一类具有阶段结构的时滞捕食系统的周期解
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摘要
研究一类捕食者具有阶段结构和Crowley-Martin功能性反应的时滞捕食系统.通过分析特征方程根的分布,得到系统正平衡点的局部稳定性和局部Hopf分叉的存在性的充分条件.进一步,利用中心流形定理和规范型理论,给出确定Hopf分叉方向和分叉周期解稳定性的计算公式.最后,利用仿真实例证明了理论分析结果的正确性.
In this paper,we analyze a delayed and stage- structured predator- prey system with Crowley-Martin functional response. By analyzing the distribution of the roots of the associated characteristic equation,sufficient conditions for the local asymptotic stability of the positive equilibrium and the existence of the local Hopf bifurcation are obtained. Further,explicit formulae for determining the direction of the Hopf bifurcation and the stability of the periodic solutions are derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are presented to support the theoretical results.
In this paper,we analyze a delayed and stage- structured predator- prey system with Crowley-Martin functional response. By analyzing the distribution of the roots of the associated characteristic equation,sufficient conditions for the local asymptotic stability of the positive equilibrium and the existence of the local Hopf bifurcation are obtained. Further,explicit formulae for determining the direction of the Hopf bifurcation and the stability of the periodic solutions are derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are presented to support the theoretical results.
引文
[1]宋永利,韩茂安,魏俊杰.多时滞捕食-食饵系统正平衡点的稳定性及全局Hopf分叉[J].数学年刊,2004,25A(6):783-790.
[2]Yang Yu.Hopf bifurcation in a two-competitor,one-prey system with time delay[J].Applied Mathematics and Computation,2009,214(1):228-235.
[3]Liu Juan,Li Yimin.Hopf bifurcation analysis of a predator-prey system with delays[J].浙江大学学报(理学版),2013,40(6):618-626.
[4]刘娟,李医民.具有功能性反应的微分生态模型的极限环分析[J].四川师范大学学报(自然科学版),2012,35(4):500-504.
[5]鲁铁军,王美娟.一类基于比率的捕食-食饵系统的全局稳定性分析[J].应用数学和力学,2008,29(4):447-452.
[6]Shi Xiangyun,Zhou Xueyong,Song Xinyu.Analysis of a stage-structured predator-prey model with Crowley-Martin function[J].Journal of Applied Mathematics andComputing,2011,36(1-2):459-472.
[7]Li Feng,Li Hongwei.Hopf bifurcation of a predator-prey modelwith time delay and stage structure for the prey[J].Mathematical and Computer Modelling,2012,55(3-4):672–679.
[8]Xu Rui.Global stability and Hopf bifurcation of a predator-prey model with stage structure and delayed predator response[J].Nonlinear Dynamics,2012,67(2):1683-1693.
[9]Xu Rui.Global dynamics of a predator-prey model with time delay and stage structure for the prey[J].Nonlinear Analysis:Real World Applications,2011,12(4):2151-2162.
[10]Hu Guangping,Li Xiaoling.Stability and Hopf bifurcation for a delayed predator-prey model with disease in the prey[J].Chaos,Solitons and Fractals,2012,45(3):229-237.
[11]B.D.Hassard,N.D.Kazarinoff,Y.H.Wan.Theory and Applications of Hopf Bifurcation[M].Cambridge University Press,Cambridge,1981.
[2]Yang Yu.Hopf bifurcation in a two-competitor,one-prey system with time delay[J].Applied Mathematics and Computation,2009,214(1):228-235.
[3]Liu Juan,Li Yimin.Hopf bifurcation analysis of a predator-prey system with delays[J].浙江大学学报(理学版),2013,40(6):618-626.
[4]刘娟,李医民.具有功能性反应的微分生态模型的极限环分析[J].四川师范大学学报(自然科学版),2012,35(4):500-504.
[5]鲁铁军,王美娟.一类基于比率的捕食-食饵系统的全局稳定性分析[J].应用数学和力学,2008,29(4):447-452.
[6]Shi Xiangyun,Zhou Xueyong,Song Xinyu.Analysis of a stage-structured predator-prey model with Crowley-Martin function[J].Journal of Applied Mathematics andComputing,2011,36(1-2):459-472.
[7]Li Feng,Li Hongwei.Hopf bifurcation of a predator-prey modelwith time delay and stage structure for the prey[J].Mathematical and Computer Modelling,2012,55(3-4):672–679.
[8]Xu Rui.Global stability and Hopf bifurcation of a predator-prey model with stage structure and delayed predator response[J].Nonlinear Dynamics,2012,67(2):1683-1693.
[9]Xu Rui.Global dynamics of a predator-prey model with time delay and stage structure for the prey[J].Nonlinear Analysis:Real World Applications,2011,12(4):2151-2162.
[10]Hu Guangping,Li Xiaoling.Stability and Hopf bifurcation for a delayed predator-prey model with disease in the prey[J].Chaos,Solitons and Fractals,2012,45(3):229-237.
[11]B.D.Hassard,N.D.Kazarinoff,Y.H.Wan.Theory and Applications of Hopf Bifurcation[M].Cambridge University Press,Cambridge,1981.