具有時滯和間接控制的捕食-被捕食模型的分支分析

徐昌進,張千宏

(貴州財經大學經濟系統仿真重點實驗室,貴州 貴陽 550004)

具有時滯和間接控制的捕食-被捕食模型的分支分析

徐昌進,張千宏

(貴州財經大學經濟系統仿真重點實驗室,貴州 貴陽 550004)

研究了一類具有時滯和間接控制的捕食-被捕食模型.選擇時滯τ為分支參數,證實了系統在一定的時滯范圍內是漸近穩定的.當時滯τ通過一系列的臨界值時,Hopf分支產生,即當時滯τ通過某些臨界值時,從平衡點處產生一簇周期解.最后,用數值模擬驗證了理論分析結果的正確性.

捕食-被捕食;Hopf分支;穩定性;間接控制

1 引言

近年來,具有時滯的種群模型的動力學行為(包括穩定性、不穩定性、周期性和混沌等)已經成為生物學和數學界研究的焦點問題.特別是因時滯引起的Hopf分支周期解吸引了諸多學者的興趣.自從文獻[1]發現了時滯會破壞Logistic模型的正平衡點的穩定性并引起周期振蕩以來,已有大量的文獻研究時滯,導致了生態模型的Hopf分支的出現,并得到了諸多很有指導意義和現實價值的結果[2-8].文獻[9]研究了下列具有變時滯和間接控制的捕食-被捕食模型的全局漸近穩定性:

本文的主要目的是研究模型(2)的Hopf分支.具體地說,就是選擇時滯τ為參數和運用Hopf分支定理,分析系統對應的特征方程,得到了系統漸近穩定和Hopf分支產生的條件.證實存在一系列的臨界值,使得系統在平衡點附近產生Hopf分支.

2 Hopf分支的存在性

3 數值模擬

圖1 當τ=2.1＜τ0≈2.2時,系統(10)的軌線圖和相圖.正平衡點E(1.9152,0.1311,0.1915,0.0197)是漸進穩定的,初值為(2,0.13,0.2,0.015).

圖2 當τ=0.85＞τ0≈0.82時,系統(10)的軌線圖和相圖.正平衡點E(1.9152,0.1311,0.1915,0.0197)附近Hopf分支產生,初值為(2,0.13,0.2,0.015).

參考文獻

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[4]Yan Xiangping,Li Wangtong.Bifurcation and global periodic solutions in a delayed facultative mutualism system[J].Physica D,2007,227(1):51-69.

[5]Xu Changjin,Tang Xianhua,Liao Maoxin.Stability and bifurcation analysis of a delayed predator-prey model of prey dispersal in two-patch environments[J].Appl.Math.Comput.,2010,216(10):2920-2936.

[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,竇霽虹,張金龍.一類帶時滯的Watt型功能性反應的捕食系統的Hopf分支[J].純粹數學與應用數學, 2010,26(5):798-803.

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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.

國家自然科學基金(11261010);貴州省優秀科技教育人才省長基金([2012]53);貴州省科學技術基金(黔科合J字[2012]2100號);貴州財經大學博士科研啟動項目(2010);貴州省軟科學研究項目(黔科合體R字[2011]LKC2030號).

徐昌進(1970-),博士,副教授,研究方向：泛函微分方程理論及其應用.

2010 MSC:34K20,34C25