曾曉云

(海軍航空工程學(xué)院系統(tǒng)科學(xué)與數(shù)學(xué)研究所,山東 煙臺(tái) 264001)

一類離散Lotka-Volterra系統(tǒng)持久性態(tài)的研究

曾曉云

(海軍航空工程學(xué)院系統(tǒng)科學(xué)與數(shù)學(xué)研究所,山東 煙臺(tái) 264001)

單時(shí)滯的Lotka-Volterra競(jìng)爭(zhēng)系統(tǒng)的持久性,Saito等已經(jīng)做了比較詳盡的討論,也得到了比較好的結(jié)論．但是Saito等只討論了單時(shí)滯系統(tǒng)的情形,對(duì)于多時(shí)滯的較為復(fù)雜的系統(tǒng)并沒(méi)有進(jìn)一步討論．本文將討論多時(shí)滯的Lotka-Volterra競(jìng)爭(zhēng)系統(tǒng)的持久性．持久性對(duì)于一個(gè)生態(tài)系統(tǒng)而言是一個(gè)非常重要的性質(zhì)．在對(duì)系統(tǒng)做了一些合理的限定后,得到了關(guān)于該系統(tǒng)持久的一些充分性的結(jié)論．

周期性;離散的Lotka-Volterra系統(tǒng);持久性

Saito等[1]考慮下列帶有時(shí)滯的離散的Lotka-Volterra競(jìng)爭(zhēng)系統(tǒng)

(1)

其中:

(2)

r1,r2,μ1,μ2是常數(shù),滿足r1>0,r2>0,μ1≥0,μ2≥0,k1,k2,l1,l2是非負(fù)整數(shù)．

最近,Zeng等[2]討論了下列系統(tǒng)

(3)

其中:b(n),r(n),ai(n),cj(n),di(n),ej(n)都是正的周期為T的序列．

(4)

持久性對(duì)于一個(gè)生態(tài)系統(tǒng)而言是一個(gè)非常重要的性質(zhì)．很多學(xué)者在這方面做了大量的研究,見(jiàn)文獻(xiàn)[3-14].本文將討論下列系統(tǒng)

(5)

其中:i=1,2,…,n;j=1,2,…,m.且

(6)

這里,x(k)表示被捕食種族在第k代的密度,y(k)表示捕食種族在第k代的密度,ai(k),ej(k)分別測(cè)度的是捕食與被捕食種族競(jìng)爭(zhēng)行為的強(qiáng)度,b(k)表示的是被捕食種族內(nèi)在生長(zhǎng)率,r(k)表示的是捕食種族的死亡率．進(jìn)一步假定:b(k),r(k)是有界正序列,ai(k),cj(k),di(k),ej(k)是有界非負(fù)序列,用{f(k)}表示任意有界序列,且

1 主要結(jié)果

為了證明本文的主要結(jié)果,引入以下引理．

(7)

其中:

分2種情況討論.

(8)

并且

(9)

(10)

成立．進(jìn)一步,由式(5)和(10)可以得到:

這與式(9)矛盾．因而,有

(11)

因此,由式(5)和(11),得到

(12)

下面將證明

也分2種情況討論:

(13)

且

(14)

另一方面,由式(13)和(14),可以證得,當(dāng)0≤k≤m時(shí),有

(15)

進(jìn)而,由式(5)和(15)可得

這與式(14)相矛盾．由此可以得到:

(16)

進(jìn)而,由式(5)和(16)可以得到:

因而,有

其中：

(17)

證明 由定理1知,對(duì)于充分大的K可以得到

(18)

由式(5)和(18)可以得到

即

或

(19)

由式(19)與(5)可以得到，當(dāng)k充分大時(shí)，有

由引理1及上述不等式可以得到:

類似于上述證明過(guò)程,可以得到:

定理3 假定(H1),(H2)成立,則滿足條件(6)的系統(tǒng)(5)是持久的．

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(責(zé)任編輯 李春梅)

Permanence for a Discrete Periodic Lotka-Volterra System with Delays

ZENG Xiao-yun

(Institute of Applied Mathematics, Naval Aeronautical Engineering Academy, Yantai 264001, China)

Saito et al investigate the permanence of a discrete periodic Lotka-Volterra competition predator-prey system with delay and get some good results, by focusing on the case of single delay. In this paper, we investigate a discrete periodic Lotka-Volterra competition predator-prey system with delays. The sufficient and realistic results are obtained for the permanence for the above system.

periodicity; discrete Lotka-Volterra system; permanence

1004-8820(2015)04-0239-07

10.13951/j.cnki.37-1213/n.2015.04.002

2015-02-10

曾曉云(1969- ),女,陜西禮泉人,副教授,碩士,研究方向:時(shí)滯微分與差分方程的穩(wěn)定性理論.

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