章培軍, 王震, 楊穎惠
(1. 西京學(xué)院 理學(xué)院, 陜西 西安 710123;2. 西京學(xué)院 智能控制技術(shù)研發(fā)中心, 陜西 西安 710123;3. 西南交通大學(xué) 數(shù)學(xué)學(xué)院, 四川 成都 611756)
具有收獲和Beddington-DeAngelis功能反應(yīng)的捕食-食餌模型
章培軍1,2, 王震1,2, 楊穎惠3
(1. 西京學(xué)院 理學(xué)院, 陜西 西安 710123;2. 西京學(xué)院 智能控制技術(shù)研發(fā)中心, 陜西 西安 710123;3. 西南交通大學(xué) 數(shù)學(xué)學(xué)院, 四川 成都 611756)
研究食餌具有階段結(jié)構(gòu),捕食者具有收獲和時滯的Beddington-DeAngelis功能反應(yīng)的捕食-食餌模型.選取合適的收獲率,通過分析相應(yīng)平衡點處的特征方程,得到各平衡點局部漸近穩(wěn)定的條件.以時滯τ為分支參數(shù),運用Hopf分支理論,得到當(dāng)τ經(jīng)過臨界值τ0時系統(tǒng)出現(xiàn)Hopf分支.最后,用Matlab軟件進行數(shù)值仿真,并驗證結(jié)論的正確性. 關(guān)鍵詞: Beddington-DeAngelis功能反應(yīng); 捕食-食餌模型; 時滯; 階段結(jié)構(gòu); Hopf分支
在實驗的基礎(chǔ)上,Holling[1]對不同類型的物種提出了3種不同的功能反應(yīng)函數(shù),這些功能反應(yīng)函數(shù)只與食餌的密度有關(guān),稱為食餌依賴功能反應(yīng).Jost等[2]對其進行了系統(tǒng)的研究,并證明功能性反應(yīng)項與捕食者有關(guān),即捕食者依賴,如Hassel-Varley功能反應(yīng)[3],Crowley-Martin功能反應(yīng)[4]等.由于Beddington-DeAngelis功能反應(yīng)[5-6]在描述捕食關(guān)系時具有食餌依賴型和捕食者依賴型的雙重特點,建立在Beddington-DeAngelis功能反應(yīng)之下的捕食-食餌模型,受到學(xué)者的廣泛關(guān)注[7-8].近幾十年來,對時滯微分方程的穩(wěn)定性和分支的研究引起了許多學(xué)者的關(guān)注[9-12],尤其是時滯引起模型產(chǎn)生分支從而誘發(fā)周期解.由于種群的存活率、增長率和繁殖力受年齡和種群發(fā)展階段的影響,很多學(xué)者考慮了捕食者和食餌的年齡因素對生物系統(tǒng)的影響[12-15].鑒于此,本文在文獻[8,15-16]的基礎(chǔ)上,研究食餌具有階段結(jié)構(gòu),捕食者具有收獲和時滯的 Beddington-DeAngelis 功能反應(yīng)的捕食-食餌模型.
食餌具有階段結(jié)構(gòu),捕食者具有收獲和時滯的Beddington-DeAngelis功能反應(yīng)的捕食-食餌模型為
(1)
2.1 平衡點及存在性
令模型(1)的右端為零,易得系統(tǒng)的平衡點及存在條件.
定理1 1) 模型(1)總存在平衡點E1(0,0,0).
2.2 平衡點的穩(wěn)定性
定理2 1) 當(dāng)H1)成立時,E1不穩(wěn)定;當(dāng)H2):br-d2(b+d1)<0成立時,E1局部漸近穩(wěn)定.
3) 對任意τ≥0,若條件H5):M2+N2>0,(M2+N2)(M1+N1)-(M0+N0)>0,M0+N0>0,且M0>N0成立,則E3局部漸近穩(wěn)定.
其特征方程為F1(λ,τ)=|λE-Ji|=0,i=1,2,3.
1) 在E1(0,0,0)處,有
(2)
當(dāng)H1)成立時,方程(2)有一個根的實部為正,從而平衡點E1(0,0,0)不穩(wěn)定;當(dāng)H2)成立時,方程(2)的所有根具有負(fù)實部,從而平衡點E1(0,0,0)局部漸近穩(wěn)定.
2) 在E2處,有
(3)
當(dāng)H1)和H4)成立時,得f(0)>0,且f(λ)→-∞ (λ→-∞),從而f(λ)=0有負(fù)實部的根.根據(jù)Kuang[17]中的定理4.1可得,對任意τ≥0,E2局部漸近穩(wěn)定.
(4)
(5)
當(dāng)τ=0時,式(5)變?yōu)棣?+(M2+N2)λ2+(M1+N1)λ+M0+N0=0.
2.3Hopf分支及其周期解
設(shè)λ=iω是方程(5)的根,并分離實部與虛部,則可得到
(6)
平方后,可得到(M0-M2ω2)2+(M1ω-ω3)2=(N0-N2ω2)2+(N1ω)2.即
(7)
(8)
其中:k=1,2,3;j=0,1,2,….
即有
所以有
所以,若條件H7):f′(z0)≠0成立,那么橫截性條件滿足.根據(jù)Hopf分支存在定理[20],得到下列結(jié)論.
選擇參數(shù)值r=2,b=0.2,d1=0.2,β=1.5,m=0.5,n=1,a=0.1,d2=0.2,c=0.5,d3=0.1,當(dāng)收獲率和時滯不同時,可能會出現(xiàn)捕食者滅絕、捕食者與幼成年食餌共存、Hopf分支及周期解的情況.
3.1E2的穩(wěn)定性
經(jīng)計算得到捕食者滅絕的平衡點為E2(40,8,0),分別對時滯τ=0,τ=2進行數(shù)值仿真,得到了系統(tǒng)的軌線趨向于E2,如圖1所示.
(a) τ=0 (b) τ=2圖1 E2穩(wěn)定性的數(shù)值仿真Fig.1 E2 stability of numerical simulation
3.2 E3的穩(wěn)定性
根據(jù)以上的參數(shù)值,取qE=0.6,可得捕食者與幼成年食餌共存的平衡點E3(18.708 3, 3.741 7, 1.138 1),且條件H5):M2+N2=1.774 2>0,M0+N0=0.059 5>0,(M2+N2)(M1+N1)-(M0+N0)=0.822 5>0,M0-N0=0.038 8>0成立,根據(jù)定理2,對任意τ≥0,E3漸近穩(wěn)定.分別對時滯τ=0,τ=2進行數(shù)值仿真,得到了系統(tǒng)的軌線趨向于E3,如圖2所示.
(a) τ=0 (b) τ=2圖2 E3穩(wěn)定性的數(shù)值仿真Fig.2 E3 stability of numerical simulation
3.3 E3的漸近穩(wěn)定性與Hopf分支及周期解
根據(jù)以上的參數(shù)值,取qE=0.1,可得到捕食者與幼成年食餌共存的平衡點E3(3.507 8,0.701 6,1.280 1),z0=0.008 9,τ0=7.742 7且H6):M0-N0=-0.029 3;H7):f′(z0)≠0成立,根據(jù)定理3,當(dāng)τ∈[0,7.742 7)時,正平衡點 E3局部漸近穩(wěn)定;當(dāng)τ∈(7.742 7,+∞)時,E3不穩(wěn)定;而當(dāng)τ=7.742 7時,系統(tǒng)在正平衡點E3產(chǎn)生Hopf分支,即系統(tǒng)在τ=7.742 7附近產(chǎn)生一簇分支周期解.取τ=7.5∈(0,7.742 7),系統(tǒng)的正平衡點E3是局部漸近穩(wěn)定的,而取τ=8.0∈(7.742 7,+∞),系統(tǒng)的正平衡點E3將失去穩(wěn)定性并產(chǎn)生Hopf分支,在正平衡點E3處分支出一簇周期解,如圖3所示.
(a) τ=7.5∈(0,7.742 7) (b) τ=8.0∈(7.742 7,+∞)圖3 E3漸近穩(wěn)定性與Hopf分支及周期解的數(shù)值仿真Fig.3 E3 asymptotic stability and numerical simulation of Hopf bifurcation and periodic solution
研究收獲和時滯對模型性態(tài)的影響,用 Matlab 軟件進行數(shù)值仿真驗證了結(jié)論的正確性.當(dāng)進行不同的收獲時,可能會出現(xiàn)捕食者滅絕、捕食者與幼成年食餌共存等情況;而當(dāng)進行適當(dāng)收獲,不同的時滯也可能導(dǎo)致模型出現(xiàn)不同的形態(tài),并最終趨向正平衡點和產(chǎn)生Hopf分支及周期解.該研究可為今后對生物資源的開發(fā)和種群數(shù)量的收獲提供寶貴的理論依據(jù).
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(責(zé)任編輯: 黃曉楠 英文審校: 黃心中)
Predator-Prey Model With Beddington-DeAngelis Functional Response and Harvesting
ZHANG Peijun1,2, WANG Zhen1,2, YANG Yinghui3
(1. School of Science, Xijing University, Xi′an 710123, China;2.Intelligent Control Technology Research and Development Center, Xijing University, Xi′an 710123, China;3. School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China)
A predator-prey model with Beddington-DeAngelis functional response of predator with harvesting and time delay and the stage structure for prey are investigated in this paper. Select the appropriate harvest rate, the conditions for the local asymptotic stability of the equilibrium point are obtained by analyzing the characteristic equation of the corresponding equilibrium point; by means of the Hopf bifurcation theorem and considering the delayτas a bifurcation parameter, Hopf bifurcation occurs whenτpasses through the critical valueτ0. Finally, Matlab is employed to carry out numerical simulation to verify our results.
Beddington-deAngelis functional response; predator-prey model; time delay; stage structure; Hopf bifurcation
10.11830/ISSN.1000-5013.201704025
2017-01-11
章培軍(1984-),男,講師,主要從事生物數(shù)學(xué)與計算機模擬、常微分方程與動力系統(tǒng)的研究.E-mail:zhangpj2006@126.com.
國家自然科學(xué)基金資助項目(61473237); 陜西省自然科學(xué)基礎(chǔ)研究計劃資助項目(2016JM1024); 陜西省教育廳科研計劃項目(15JK2181); 西京學(xué)院科研基金資助項目(XJ160143)
O 175; Q 141
A
1000-5013(2017)04-0579-06