具有时滞和间接控制的捕食-被捕食模型的分支分析

徐昌进,张千宏

(贵州财经大学经济系统仿真重点实验室,贵州 贵阳 550004)

具有时滞和间接控制的捕食-被捕食模型的分支分析

徐昌进,张千宏

(贵州财经大学经济系统仿真重点实验室,贵州 贵阳 550004)

研究了一类具有时滞和间接控制的捕食-被捕食模型.选择时滞τ为分支参数,证实了系统在一定的时滞范围内是渐近稳定的.当时滞τ通过一系列的临界值时,Hopf分支产生,即当时滞τ通过某些临界值时,从平衡点处产生一簇周期解.最后,用数值模拟验证了理论分析结果的正确性.

捕食-被捕食;Hopf分支;稳定性;间接控制

1 引言

近年来,具有时滞的种群模型的动力学行为(包括稳定性、不稳定性、周期性和混沌等)已经成为生物学和数学界研究的焦点问题.特别是因时滞引起的Hopf分支周期解吸引了诸多学者的兴趣.自从文献[1]发现了时滞会破坏Logistic模型的正平衡点的稳定性并引起周期振荡以来,已有大量的文献研究时滞,导致了生态模型的Hopf分支的出现,并得到了诸多很有指导意义和现实价值的结果[2-8].文献[9]研究了下列具有变时滞和间接控制的捕食-被捕食模型的全局渐近稳定性:

本文的主要目的是研究模型(2)的Hopf分支.具体地说,就是选择时滞τ为参数和运用Hopf分支定理,分析系统对应的特征方程,得到了系统渐近稳定和Hopf分支产生的条件.证实存在一系列的临界值,使得系统在平衡点附近产生Hopf分支.

2 Hopf分支的存在性

3 数值模拟

图1 当τ=2.1＜τ0≈2.2时,系统(10)的轨线图和相图.正平衡点E(1.9152,0.1311,0.1915,0.0197)是渐进稳定的,初值为(2,0.13,0.2,0.015).

图2 当τ=0.85＞τ0≈0.82时,系统(10)的轨线图和相图.正平衡点E(1.9152,0.1311,0.1915,0.0197)附近Hopf分支产生,初值为(2,0.13,0.2,0.015).

参考文献

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[6]Xu Changjin,Tang Xianhua,Liao Maoxin.Bifurcation analysis in a delayed Lokta-Volterra predator-prey model with two delays[J].Nonlinear Dynamics,2011,66(1/2):169-183.

[7]石雁,窦霁虹,张金龙.一类带时滞的Watt型功能性反应的捕食系统的Hopf分支[J].纯粹数学与应用数学, 2010,26(5):798-803.

[8]戎晓剑,赵晓青,徐瑞.一个具有变时滞和间接控制的捕食模型的全局渐近稳定性[J].河北师范大学学报:自然科学版,2007,31(2):149-154.

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Bifurcation analysis in a delayed predator-prey model with indirect control

Xu Changjin,Zhang Qianhong
(Guizhou Key Laboratory of Economics System Simulation,Guizhou University of Finance and Economics, Guiyang 550004,China)

In this paper,a delayed predator-prey model with indirect control is investigated.By choosing the delay τ as a bifurcation parameter,we prove that system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as τ passes a sequence of critical values.This means that a family of periodic solutions bifurcate from the equilibrium when the bifurcation parameter exceeds a critical value.Some numerical simulations are given to justify the theoretical analysis results.

predator-prey,Hopf bifurcation,stability,indirect control

O175.13

A

1008-5513(2012)05-0573-07

2011-12-15.

国家自然科学基金(11261010);贵州省优秀科技教育人才省长基金([2012]53);贵州省科学技术基金(黔科合J字[2012]2100号);贵州财经大学博士科研启动项目(2010);贵州省软科学研究项目(黔科合体R字[2011]LKC2030号).

徐昌进(1970-),博士,副教授,研究方向：泛函微分方程理论及其应用.

2010 MSC:34K20,34C25