程永玲
(山西大學商務學院基礎教學部,山西太原030031)
具有時滯和反饋控制的捕食
——被捕食系統(tǒng)的全局吸引性
程永玲
(山西大學商務學院基礎教學部,山西太原030031)
研究一類具有時滯和反饋控制的非自治捕食-被捕食Lotka-Volterra系統(tǒng).通過構(gòu)造合適的Lyapunov泛函,得到了系統(tǒng)全局吸引的新的準則.
Lyapunov泛函;反饋控制;時滯;全局吸引
經(jīng)典的Lotka-Volterra競爭系統(tǒng)已經(jīng)得到了廣泛的研究.在文獻[1-8]中,已經(jīng)得到了很好的結(jié)果.最近,具有反饋控制的生態(tài)系統(tǒng)得到了許多科學家的關(guān)注,如[6-8].文獻[7],通過構(gòu)造合適的Lyapunov泛函,建立了系統(tǒng)持久與穩(wěn)定的新的充分條件.文獻[8],討論了具有時滯與反饋控制的多種群非自治Lotka-Volterra競爭系統(tǒng).
受到這些文獻所研究內(nèi)容的啟發(fā),我們研究了如下非自治捕食-被捕食系統(tǒng):
文中,對系統(tǒng)(1)我們始終假定:i,j=1,2,…n.
(H1)bi(t),aij(t),ci(t),di(t),ei(t),f1(t)為[0,∞)上的有界、連續(xù)函數(shù),且aij(t)≥0,b2(t)≥0,ci(t)≥0,di(t)≥0,ei(t)≥0,f1(t)≥0.
定義1 若對系統(tǒng)(1)的所有解(x1(t),x2(t),u1(t),u2(t)),存在正常數(shù)M,T使得,當t>T時,有xi(t)≤M,|ui(t)|≤M,則稱系統(tǒng)(1)最終有界.
[9]中,定理3.1的證明類似,可得如下引理.
引理1 假定(H1)-(H4)成立,則系統(tǒng)(1)最終有界.
定理1假定(H1)-(H4)成立,且存在正常數(shù)ki>0(i=1,2,3,4)使得
其中
且ψ-1(t)和分別φ-1(t)為t-τ(t)與t-δ(t)的反函數(shù).
證明 設(x1(t),x2(t),u1(t),u2(t))為系統(tǒng)(1)的任意正解.由引理1,存在正常數(shù)M,T使得,當t>T時,有
構(gòu)造Lyapunov函數(shù)V(t)
計算V(t)的上導數(shù)且化簡,有
定理證畢.
參考文獻:
[1]AHMAD S,LAZER A C.Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system[J].Nonlinear Anal,2000,40:37-49
[2]CHEN FENGDE.The permanence and global attractivity of Lotka-Volterra competition system with feedbackcontrols[J].Nonlinear Anal.:Real World Appl,2006,7:133-143
[3]師向云,郭 振.一類具有時滯和階段結(jié)構(gòu)的捕食系統(tǒng)的全局分析[J].信陽師范學院學報(自然科學版),2008,21(1):32-35
[4]GOPALSAMY K.Stability and oscillations in Delay Different Equations of Population Dynamics[M].Kluwer Academic,Dordrecht/Norwell,MA,1992.
[5]王 豪,鄭麗麗.一類具有階段結(jié)構(gòu)和時滯的捕食與被捕食系統(tǒng)[J].信陽師范學院學報(自然科學版),2004,17(2):140-145
[6]CHEN FENGDE.Permanence in nonautonomous multi-species predator-prey system with feedback controls[J].Appl Math Comput,2006,173:694-709
[7]NIE LINFEI,TENG ZHIDONG.Lin Hu,Jigen Peng,Permanence and stability in non-autonomous predator-preyLotka-Volterra systems with feedback controls[J].Comput Math Appl,2009,58:436-448
[8]NIE LINFEI,Peng Jigen,Teng Zhidong.Permanence and stability in multi-species nonautonomous Lotka-Volterra competitive systems with delays and feedback controls[J].Math Comput Modelling,2009,49:295-306
Global Stability in Nonautonomous Predator-prey Systems with Delays and Feedback Controls
CHENG Yongling
(Department of Mathematics,Business College of Shanxi University,Taiyuan 030031,China)
A nonautonomous predator-prey Lotka-Volterra systems with delays and feedback controls is considered.By constructing suitable Lyapunov functional,some new criteria for global stability is established.
Lyapunov functional;Feedback control;Delay;Global stability
1672-2027(2016)04-0052-03
O175.14
A
2016-08-27
程永玲(1980-),女,山西晉城人,碩士,山西大學商務學院講師,主要從事動力系統(tǒng)研究.