具有線性脈沖的周期捕食系統的持久性

陳丹,許宗文,張樹文

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

具有線性脈沖的周期捕食系統的持久性

陳丹1,許宗文1,張樹文2

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

研究具有Holling IV功能性反應和脈沖的周期捕食食餌系統.找到了影響該系統動力學行為的閾值R0.證明了當R0＜1時,該系統的食餌滅絕周期解是局部漸近穩(wěn)定的;當R0＞1時,該系統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;脈沖;Holling IV功能性反應;持續(xù)生存;局部漸近穩(wěn)定

1 引言

脈沖微分方程是20世紀末發(fā)展非常迅速的一個數學分支,這是因為它比普通微分方程具有更加能貼合實際.許多學者對此進行了深入研究,得到許多結論[13].但現有成果多見于具有Holling I,Holling II,Holling III功能性反應的脈沖捕食-食餌系統[45],具有Holling IV功能性反應的脈沖捕食-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和Holling IV功能性反應的周期捕食食餌系統:

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[6]中的方法,容易得到y?(t)是全局穩(wěn)定的.

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

證明定義函數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(2.1)的結論,有當t充分大時,y(t)＞m2.下面要找到一個m1＞0使得當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：生物數學.

2010 MSC:34D05