徐昌進,張千宏

(貴州財經(jīng)大學(xué)經(jīng)濟系統(tǒng)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(jīng)大學(xué)經(jīng)濟系統(tǒng)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(shù),證實了系統(tǒng)在一定的時滯范圍內(nèi)是漸近穩(wěn)定的.當(dāng)時滯τ通過一系列的臨界值時,Hopf分支產(chǎn)生,即當(dāng)時滯τ通過某些臨界值時,從平衡點處產(chǎn)生一簇周期解.最后,用數(shù)值模擬驗證了理論分析結(jié)果的正確性.

捕食-被捕食;Hopf分支;穩(wěn)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(xué)行為(包括穩(wěn)定性、不穩(wěn)定性、周期性和混沌等)已經(jīng)成為生物學(xué)和數(shù)學(xué)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(xué)者的興趣.自從文獻[1]發(fā)現(xiàn)了時滯會破壞Logistic模型的正平衡點的穩(wěn)定性并引起周期振蕩以來,已有大量的文獻研究時滯,導(dǎo)致了生態(tài)模型的Hopf分支的出現(xiàn),并得到了諸多很有指導(dǎo)意義和現(xiàn)實價值的結(jié)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(wěn)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(shù)和運用Hopf分支定理,分析系統(tǒng)對應(yīng)的特征方程,得到了系統(tǒng)漸近穩(wěn)定和Hopf分支產(chǎn)生的條件.證實存在一系列的臨界值,使得系統(tǒng)在平衡點附近產(chǎn)生Hopf分支.

2 Hopf分支的存在性

3 數(shù)值模擬

圖1 當(dāng)τ=2.1＜τ0≈2.2時,系統(tǒng)(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(wěn)定的,初值為(2,0.13,0.2,0.015).

圖2 當(dāng)τ=0.85＞τ0≈0.82時,系統(tǒng)(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(chǎn)生,初值為(2,0.13,0.2,0.015).

參考文獻

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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(xué)基金(11261010);貴州省優(yōu)秀科技教育人才省長基金([2012]53);貴州省科學(xué)技術(shù)基金(黔科合J字[2012]2100號);貴州財經(jīng)大學(xué)博士科研啟動項目(2010);貴州省軟科學(xué)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(yīng)用.

2010 MSC:34K20,34C25