于莉琦,賀樹立,王強
具有Holling-II型功能反應函數(shù)的雙時滯捕食者-食餌系統(tǒng)的Hopf分支
于莉琦1,賀樹立1,王強2
(黑龍江東方學院 1. 基礎部,2. 信息工程學院,黑龍江 哈爾濱 150066)
在具有Holling-Ⅱ型功能反應函數(shù)的捕食者-食餌系統(tǒng)中引入2個時滯參數(shù),用來刻畫捕食者和食餌的生長時滯,研究了系統(tǒng)平衡點的局部穩(wěn)定性.結果表明,隨著參數(shù)的變化,系統(tǒng)平衡點發(fā)生了擾動,進而出現(xiàn)了周期解.給出了Hopf分支存在條件的顯示表達式,并通過數(shù)值實驗驗證了結論.
Holling-Ⅱ型功能反應函數(shù);穩(wěn)定性;時滯;Hopf分支;捕食者-食餌系統(tǒng)
研究物種間的捕食關系,有助于預測和估計捕食者與食餌的種群數(shù)量,對于種群發(fā)展和保護有著重要意義.捕食者-食餌系統(tǒng)是生物數(shù)學中的重要研究對象,長時間以來受到研究者們的廣泛關注,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[8,10]在研究的系統(tǒng)中引入了2個時滯量,分析了時滯變化對系統(tǒng)穩(wěn)定性的影響.
文獻[6]研究了一個具有HollingII型功能反應函數(shù)的捕食者-食餌系統(tǒng)
鑒于時滯量在生物系統(tǒng)的廣泛使用,本文在系統(tǒng)(1)中引入2個時滯參數(shù)用來描述捕食者和食餌成長的滯后量,得到系統(tǒng)
式(3)的特征方程為
式(7)等價于
當條件H2成立時,方程(12)沒有正根;當條件H3成立時,方程(12)至少有一個正根.
(14)
綜合上述分析可得到定理4.
圖1 當時,平衡點是漸近穩(wěn)定的
圖2 當時,平衡點不穩(wěn)定
圖4 當時,平衡點不穩(wěn)定
在具有Holling-Ⅱ型功能反應函數(shù)的捕食者-食餌系統(tǒng)中引入2個時滯參數(shù),用來刻畫捕食者和食餌的生長時滯,以2個時滯為參數(shù),證明了隨著參數(shù)的變化,系統(tǒng)平衡點的穩(wěn)定性會發(fā)生改變,出現(xiàn)Hopf分支,經計算給出了分支存在條件的顯示表達式,數(shù)值實驗驗證了所給結果.
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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
黑龍江省自然科學基金項目(LH2022A022);黑龍江省教育科學“十四五”規(guī)劃2022年度重點課題(GJB1422487);高等教育2023年度黑龍江省教育科學規(guī)劃重點課題(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