具有線性脈沖的周期捕食系統(tǒng)的持久性

陳丹,許宗文,張樹文

(1.閩南理工學(xué)院信管系,福建石獅 362700;2.集美大學(xué)理學(xué)院,福建廈門 361021)

具有線性脈沖的周期捕食系統(tǒng)的持久性

陳丹1,許宗文1,張樹文2

(1.閩南理工學(xué)院信管系,福建石獅 362700;2.集美大學(xué)理學(xué)院,福建廈門 361021)

研究具有Holling IV功能性反應(yīng)和脈沖的周期捕食食餌系統(tǒng).找到了影響該系統(tǒng)動力學(xué)行為的閾值R0.證明了當(dāng)R0＜1時,該系統(tǒng)的食餌滅絕周期解是局部漸近穩(wěn)定的;當(dāng)R0＞1時,該系統(tǒng)的食餌滅絕周期解變得不穩(wěn)定且食餌將一致持久.

捕食食餌系統(tǒng);脈沖;Holling IV功能性反應(yīng);持續(xù)生存;局部漸近穩(wěn)定

1 引言

脈沖微分方程是20世紀(jì)末發(fā)展非常迅速的一個數(shù)學(xué)分支,這是因為它比普通微分方程具有更加能貼合實際.許多學(xué)者對此進(jìn)行了深入研究,得到許多結(jié)論[13].但現(xiàn)有成果多見于具有Holling I,Holling II,Holling III功能性反應(yīng)的脈沖捕食-食餌系統(tǒng)[45],具有Holling IV功能性反應(yīng)的脈沖捕食-捕食模型至今研究較少.因此,本文建立了在固定時刻具有脈沖效應(yīng)和Holling IV功能性反應(yīng)的周期捕食食餌系統(tǒng):

這里r(t),a(t),c1(t),c2(t),d(t)都是以T為周期的,并且存在整數(shù)q使得τk+q=τk+T.x(t) 與y(t)分別表示食餌與捕食者的種群密度,r(t)代表內(nèi)稟增長率,a(t)表示密度制約率,d(t)是捕食者的死亡率.

2 引理

利用文獻(xiàn)[6]中的方法,容易得到y(tǒng)?(t)是全局穩(wěn)定的.

引理2.1當(dāng)t充分大時,存在一個常數(shù)M＞0,使得系統(tǒng)(1.1)的解X(t)=(x(t),y(t))滿足x(t)≤M,y(t)≤M.

證明定義函數(shù)V(t,X(t)),使得

3 持續(xù)生存與滅絕

令m2=y?(t)-ε1,ε1＞0,由比較定理和系統(tǒng)(2.1)的結(jié)論,有當(dāng)t充分大時,y(t)＞m2.下面要找到一個m1＞0使得當(dāng)t充分大時,x(t)≥m1.將分為兩步來做.

1.因為R0＞1,可以選擇足夠小的m3＞0,ε2＞0,ε3＞0,ε=ε2+ε3,使得

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Permanence in a periodic predator-prey system with linear impulsive perturbations

Chen Dan1,Xu Zongwen1,Zhang Shuwen2
(1.Information Management Department,Minnan University of Science and Technology, Shishi362700,China; 2.College of Science,Jimei University,Xiamen361021,China)

In this paper,a non-autonomous periodic predator-prey system with Holling IV functional response and impulsive perturbation is considered.The threshold value R0which determines the dynamical behavior of the model is provided.Furthermore,we prove that the prey-eradication periodic solution is locally asymptotically stable provided R0＜1,the prey-eradication periodic solution is unstable and the pest will be uniform persistent when R0＞1.

predator-prey system,impulsive perturbation,Holling IV functional response,permanence, locally asymptotically stable

O175.12

A

1008-5513(2013)02-0208-06

10.3969/j.issn.1008-5513.2013.02.015

2012-09-12.

福建省教育廳科技項目(JB12252).

陳丹(1986-),碩士生,研究方向：生物數(shù)學(xué).

2010 MSC:34D05