一類食物鏈模型的Fold-Hopf分支現(xiàn)象分析

郭 爽,張 玲

(大慶師范學(xué)院 數(shù)學(xué)科學(xué)學(xué)院,黑龍江 大慶 163712)

0 引 言

目前,關(guān)于捕食者及其食餌之間的關(guān)系研究已有許多結(jié)果[1-6].Freedman等[1]提出了一類Gause型食物鏈模型：

(1)

其中:x,y和z分別表示t時(shí)刻食餌、捕食者和頂層捕食者的數(shù)量;g(x)是食餌的內(nèi)部增長函數(shù);p(x)和q(y)分別是捕食者和頂層捕食者的功能反應(yīng)函數(shù);h,s>0分別是捕食者和頂層捕食者的死亡率;e,m>0分別是食餌和捕食者的轉(zhuǎn)換率.由于種群會(huì)出現(xiàn)振蕩現(xiàn)象,所以合理的解釋是引入一個(gè)單一的時(shí)滯到捕食者的功能反應(yīng)函數(shù)中,如: Hastings等[3]討論了種群的滅絕、邊界行為以及共存平衡點(diǎn)的全局漸近穩(wěn)定性;郭爽等[6]強(qiáng)調(diào)了該模型隨著時(shí)滯的增加會(huì)出現(xiàn)穩(wěn)定的Hopf分支,并模擬出Hopf分支的全局存在性等.本文將時(shí)滯引入頂層捕食者方程的功能反應(yīng)函數(shù)中,討論該模型復(fù)雜的動(dòng)力學(xué)現(xiàn)象.

1 Fold-Hopf分支

選擇g(x)=a(1-x/k),p(x)=βx/(1+px),q(y)=ry,將時(shí)滯引入q(y)中,可得下列時(shí)滯微分方程：

(2)

這里a,β,k,p,h,e,r,s,m都是正參數(shù).

為方便,將式(2)非量綱化,可得下列方程：

(3)

其中:m11=1-2x*-by*/(x*+b)2;m12=-x*/(x*+b)<0;m21=eby*/(x*+b)2>0;m23=-dy*<0;n32=uz*>0.特征值λ滿足如下特征方程：

D(λ,τ)=λ3+a2λ2+a1λ+(b1λ+b0)e-λτ=0,

(4)

其中:a2=-m11;a1=-m12m21>0;b1=-m23n32>0;b0=m11m23n32.

如果m11=0,則a2=0,b0=0,易知λ=0是特征方程(4)的根.將式(4)對(duì)λ求導(dǎo),有

表明λ=0是式(4)的單根.將λ=iω代入式(4)并分離實(shí)虛部,有

b1ωsinωτ=0, -ω3+a1ω+b1ωcosωτ=0,

(5)

將式(5)化簡(jiǎn)得

(6)

(7)

所以(±ω0,τ0)是式(4)的解,即±iω0是τ=τ0時(shí)式(4)的純虛特征根.記λ(τ)=α(τ)+iω(τ)是方程(3)滿足α(τ0)=0,ω(τ0)=ω0的根,定義p*=(1-x*)2/(1-2x*) ?m11=0,則當(dāng)p=p*,τ=τ0時(shí),有下述定理.

定理1假設(shè)2l/e

證明：根據(jù)前面的討論知,當(dāng)p=p*,τ=τ0時(shí),方程(4)有一個(gè)單零特征根和一對(duì)純虛根.假設(shè)方程(4)有一個(gè)正實(shí)部的根λ=α0+iβ0,令λ=α(τ)+iβ(τ)是p=p*時(shí)方程(4)滿足α(τ0)=α0>0和β(τ0)=β0的解,則當(dāng)τ∈(τ0-δ,τ0)時(shí),存在正數(shù)0<δ<τ0,使得α(τ)>0.

矛盾.證畢.

2 數(shù)值模擬

圖1 當(dāng)b=0.3,d=0.62,e=0.532,l=0.22,c=0.15, u=0.55時(shí),系統(tǒng)(2)在參數(shù)平面上的分支圖Fig.1 Bifurcation diagram of system (2) on the parameter plane for b=0.3,d=0.62, e=0.532,l=0.22,c=0.15,u=0.55

圖2 當(dāng)p=0.213 5,τ=0.746 7時(shí)平衡點(diǎn)的波動(dòng)曲線Fig.2 Fluctuation curves of (0.737 3,0.272 7,0.716 2) when p=0.213 5,τ=0.746 7

圖3 當(dāng)p=1.157,τ=2.408 7時(shí)平衡點(diǎn)附近的周期波動(dòng)曲線Fig.3 Fluctuation cycle curves near the equilibrium when p=1.157,τ=2.408 7

圖4 當(dāng)p=5.514,τ=5.801 7時(shí)平衡點(diǎn)附近的擬周期波動(dòng)曲線Fig.4 Quasi-periodic motion curves near the equilibrium when p=5.514,τ=5.801 7

圖5 當(dāng)p=2.133,τ=15.807時(shí)平衡點(diǎn)附近的爆發(fā)行為Fig.5 Bursting behavior near the equilibrium when p=2.133,τ=15.807

[1] Freedman H I,Waltman P.Mathematical Analysis of Some Three-Species Food-Chain Models [J].Mathematical Biosciences,1977,33(3): 257-276.

[2] Ginoux J M,Rossetto B,Jamet J L.Chaos in a Three-Dimensional Volterra-Gause Model of Predator-Prey Type [J].International Journal of Bifurcation and Chaos,2005,15(5): 1689-1708.

[3] Hastings A,Powell T.Chaos in a Three-Species Food Chain [J].Ecology,1991,72(3): 896-903.

[4] LIU Gui-rong,YAN Wei-ping,YAN Ju-rang.Positive Periodic Solutions for a Class of Neutral Delay Gause-Type Predator-Prey System [J].Nonlinear Analysis: Theory,Methods &Applications,2009,71(10): 4438-4447.

[5] WANG Hong-bin,JIANG Wei-hua.Hopf-Pitchfork Bifurcation in Van Der Pol’s Oscillator with Nonlinear Delayed Feedback [J].Journal of Mathematical Analysis Applications,2010,368(1): 9-18.

[6] GUO Shuang,LIU Yang,SHA Yuan-xia,et al.Stability and Bifurcation Analysis on Gause-Type Predator-Prey Model [J].Journal of Jilin University：Science Edition,2012,50(5): 940-944.(郭爽,劉洋,沙元霞,等.Gause型捕食模型的穩(wěn)定性與分支分析 [J].吉林大學(xué)學(xué)報(bào)：理學(xué)版,2012,50(5): 940-944.)

[7] Hale J K.Theory of Functional Differential Equations [M].New York: Springer,1977.

[8] Faria T,Magalhves L T.Restrictions on the Possible Flows of Scalar Retarded Functional Differential Equations in Neighborhoods of Singularities [J].Journal Dynamics and Differential Equations,1996,8(1): 35-70.

[9] JIANG Wei-hua,WANG Hong-bin.Hopf-Transcritical Bifurcation in Retarded Functional Differential Equations [J].Nonlinear Analysis: Theory,Methods &Applications,2010,73(11): 3626-3640.