于莉琦,賀樹(shù)立,王強(qiáng)
具有Holling-II型功能反應(yīng)函數(shù)的雙時(shí)滯捕食者-食餌系統(tǒng)的Hopf分支
于莉琦1,賀樹(shù)立1,王強(qiáng)2
(黑龍江東方學(xué)院 1. 基礎(chǔ)部,2. 信息工程學(xué)院,黑龍江 哈爾濱 150066)
在具有Holling-Ⅱ型功能反應(yīng)函數(shù)的捕食者-食餌系統(tǒng)中引入2個(gè)時(shí)滯參數(shù),用來(lái)刻畫捕食者和食餌的生長(zhǎng)時(shí)滯,研究了系統(tǒng)平衡點(diǎn)的局部穩(wěn)定性.結(jié)果表明,隨著參數(shù)的變化,系統(tǒng)平衡點(diǎn)發(fā)生了擾動(dòng),進(jìn)而出現(xiàn)了周期解.給出了Hopf分支存在條件的顯示表達(dá)式,并通過(guò)數(shù)值實(shí)驗(yàn)驗(yàn)證了結(jié)論.
Holling-Ⅱ型功能反應(yīng)函數(shù);穩(wěn)定性;時(shí)滯;Hopf分支;捕食者-食餌系統(tǒng)
研究物種間的捕食關(guān)系,有助于預(yù)測(cè)和估計(jì)捕食者與食餌的種群數(shù)量,對(duì)于種群發(fā)展和保護(hù)有著重要意義.捕食者-食餌系統(tǒng)是生物數(shù)學(xué)中的重要研究對(duì)象,長(zhǎng)時(shí)間以來(lái)受到研究者們的廣泛關(guān)注,特別是捕食者-食餌系統(tǒng)的穩(wěn)定性和分支情況[1-9].文獻(xiàn)[8,10]在研究的系統(tǒng)中引入了2個(gè)時(shí)滯量,分析了時(shí)滯變化對(duì)系統(tǒng)穩(wěn)定性的影響.
文獻(xiàn)[6]研究了一個(gè)具有HollingII型功能反應(yīng)函數(shù)的捕食者-食餌系統(tǒng)
鑒于時(shí)滯量在生物系統(tǒng)的廣泛使用,本文在系統(tǒng)(1)中引入2個(gè)時(shí)滯參數(shù)用來(lái)描述捕食者和食餌成長(zhǎng)的滯后量,得到系統(tǒng)
式(3)的特征方程為
式(7)等價(jià)于
當(dāng)條件H2成立時(shí),方程(12)沒(méi)有正根;當(dāng)條件H3成立時(shí),方程(12)至少有一個(gè)正根.
(14)
綜合上述分析可得到定理4.
圖1 當(dāng)時(shí),平衡點(diǎn)是漸近穩(wěn)定的
圖2 當(dāng)時(shí),平衡點(diǎn)不穩(wěn)定
圖4 當(dāng)時(shí),平衡點(diǎn)不穩(wěn)定
在具有Holling-Ⅱ型功能反應(yīng)函數(shù)的捕食者-食餌系統(tǒng)中引入2個(gè)時(shí)滯參數(shù),用來(lái)刻畫捕食者和食餌的生長(zhǎng)時(shí)滯,以2個(gè)時(shí)滯為參數(shù),證明了隨著參數(shù)的變化,系統(tǒng)平衡點(diǎn)的穩(wěn)定性會(huì)發(fā)生改變,出現(xiàn)Hopf分支,經(jīng)計(jì)算給出了分支存在條件的顯示表達(dá)式,數(shù)值實(shí)驗(yàn)驗(yàn)證了所給結(jié)果.
[1]Xiao D M,Ruan S G.Multiple bifurcation in A delayed predator-prey system with nonmonotonic functionalΙresponse[J].DiffEqua,2001,176:494-510.
[2]Xiao Dongmei,Zhu Huaiping.Multiple focus and Hopf bifurcation in a predator-prey system with nonmonotonic functional response[J].SIAM J Appl Math,2006,66:802-819.
[3]范猛,王克.一類具有Holling II型功能性反應(yīng)的捕食者-食餌系統(tǒng)全局周期解的存在性[J].?dāng)?shù)學(xué)物理學(xué)報(bào),2001,21(4):492-497.
[4]王玲書.一類具有時(shí)滯和階段結(jié)構(gòu)的捕食者-食而系統(tǒng)的全局穩(wěn)定性[J].高校應(yīng)用數(shù)學(xué)學(xué)報(bào),2013,28(1):51-62.
[5]LI F,L H W.Hopf bifurcation of predator-prey model with time delay and stage structure for the prey[J].Math Comput Model 2012,55:672-679.
[6]Hsu S B.On global stability of a predator-prey system[J].Math Biosci,1978,39(2):1-10.
[7]Cheng Z,Wang Y,Cao J.Stability and Hopf bifurcation of a neural network model with distributed delays and strong kernel[J].Nonlin Dyn,2016(86):323-335.
[8]Dong T,Liao X,Wang A.Stability and Hopf bifurcation of a complex-valued neural network with two time delays [J].Nonlin Dyn, 2015(82):173-184.
[9]Culshaw R,Ruan S.A delay differential equation model of HIV infection of CD4 T-cells[J].Mathematical Biosciences,2000, 165:27-39.
[10]Ruan Shigui,Wei Junjie.On the zeros of transcendental functions with applications to stability of delay differential equations with two delays[J].Dyn Contin Disctete Impuls Syst Ser A:Math Anal,2003,10:863-874.
Hopf bifurcation of a predator-prey system with two delays and Holling- II functional response function
YU Liqi1,HE Shuli1,WANG Qiang2
(1. Department of Basic Course,2. School of Information Engineering,East University of Heilongjiang,Harbin 150066,China)
Two time-delay parameters are introduced into a predator-prey system with Holling-Ⅱfunctional response function,they are used to describe the growth delay of predators and prey.The local stability of the equilibrium of the system was analyzed,the results exhibited that the equilibrium point of the system is disturbed,and then a periodic solution appears with the change of parameters.The explicit algorithms for Hopf bifurcation are derived,the conclusion is verified by numerical experiments.
Holling-II type functional response function;stability;time delay;Hopf bifurcation;predator-prey system
1007-9831(2023)10-0016-06
O175∶Q-332
A
10.3969/j.issn.1007-9831.2023.10.004
2022-12-10
黑龍江省自然科學(xué)基金項(xiàng)目(LH2022A022);黑龍江省教育科學(xué)“十四五”規(guī)劃2022年度重點(diǎn)課題(GJB1422487);高等教育2023年度黑龍江省教育科學(xué)規(guī)劃重點(diǎn)課題(GJB14230003)
于莉琦(1983-),女,黑龍江哈爾濱人,副教授,碩士,從事微分方程穩(wěn)定性研究.E-mail:85972693@qq.com