譚德君

(集美大學(xué)教師教育學(xué)院,福建廈門 361021)

具脈沖效應(yīng)的非自治隨機(jī)干擾的捕食-食餌系統(tǒng)的研究

譚德君

(集美大學(xué)教師教育學(xué)院,福建廈門 361021)

建立一個(gè)具有脈沖效應(yīng)的非自治隨機(jī)的比例依賴的捕食-食餌模型,通過研究具有脈沖效應(yīng)的非自治隨機(jī)系統(tǒng)與無脈沖效應(yīng)的非自治隨機(jī)系統(tǒng)的等價(jià)性,證明該模型的有界性,均值一致有界和滅絕性等動(dòng)力學(xué)性質(zhì).

脈沖效應(yīng);隨機(jī)擾動(dòng);捕食-食餌系統(tǒng);有界性

1 引言

捕食-食餌系統(tǒng)是一個(gè)重要的生態(tài)系統(tǒng),有許多學(xué)者對(duì)此進(jìn)行深入研究,得到許多結(jié)論[1-2],具有比例依賴功能反應(yīng)的捕食-食餌系統(tǒng)也得到廣泛的研究[3-4].生物種群生活在不斷變化的環(huán)境中,如:季節(jié)的變化,食物供給等,人類的活動(dòng),環(huán)境的突然變化,如：洪水、地震、海嘯時(shí)種群產(chǎn)生瞬間的影響,環(huán)境的噪音對(duì)種群的發(fā)展產(chǎn)生不斷的影響,所以研究脈沖效應(yīng)[5-6]和隨機(jī)環(huán)境[7-8]的非自治種群動(dòng)力學(xué)行為,成為現(xiàn)代生物數(shù)學(xué)的一個(gè)主要課題,但是考慮脈沖與環(huán)境噪音同時(shí)對(duì)生態(tài)種群作用的成果不多.本文借鑒文獻(xiàn)[2,9]的方法,利用具有脈沖效應(yīng)的非自治隨機(jī)系統(tǒng)與無脈沖效應(yīng)的非自治隨機(jī)系統(tǒng)的等價(jià)性,研究具有脈沖效應(yīng)和隨機(jī)擾動(dòng)的非自治捕食-食餌系統(tǒng)的動(dòng)力學(xué)性質(zhì).

本文研究具有脈沖效應(yīng)的非自治隨機(jī)的與比例依賴功能反應(yīng)的捕食-食餌系統(tǒng).

2 主要結(jié)果

3 結(jié)論

本文給出了具有脈沖擾動(dòng)和比例依賴功能反應(yīng)的非自治隨機(jī)捕食-食餌系統(tǒng),首先建立一個(gè)脈沖擾動(dòng)和無脈沖擾動(dòng)的比例依賴功能性反應(yīng)的非自治隨機(jī)捕食-食餌系統(tǒng)的平衡關(guān)系;其次,利用這種關(guān)系通過對(duì)無脈沖非自治的隨機(jī)捕食-食餌系統(tǒng)的研究,得到有脈沖非自治的隨機(jī)捕食-食餌系統(tǒng)的動(dòng)力學(xué)性質(zhì).這種對(duì)具有脈沖擾動(dòng)和隨機(jī)擾動(dòng)的生態(tài)種群研究的方法是一種特殊的方法,更一般的方法有待進(jìn)一步研究.

[1]Aziz-A laoui MMA,Daher Okiye M.Boundedness and global stability for a predator-prey model with m odified Leslied Gower and Holling type 2 schem es[J].App l.Math.Lett.,2003,16:1069-1075.

[2]張樹文,張?jiān)偶?譚德君.具脈沖效應(yīng)和Beddington-DeAnglis功能反應(yīng)時(shí)滯周期捕食系統(tǒng)[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2010,4:534-540.

[3]Hsa B S,Hwang T W,Kuang Y.Global analysis of the Michaelis-Menten ratio-dependent p redator-p rey system[J].Math.Biol.,2001,42:489-506.

[4]Xu R R,Chen L S.Persistence and global stability for n-species ratio-dependent p redator-p rey system with tim e delay[J].Math.Anal.App l.,2002,275:27-43.

[5]Liu B,Teng Z D,Chen L S.Analysis of predator-prey m odelwith Holling 2 functional response concerning im pu lsive control strategy[J].Com put.App l.Math.,2006,193:347-362.

[6]Nie L L,Teng Z D,Hu L.Existence and stability of periodic solution of a predator-prey m odel state dependent im pulsive affects[J].Com put.App l.Math.,2009,224:544-555.

[7]Jiang D D,Shi N,Li X.Global stability and stochastic perm anence of non-autonom ous logistic equation with random perturbation[J].Math.Anal.App l.,2008,340:588-597.

[8]Ji C J,Jiang D J,Li X Y.Qualitative analysis of a stochastic ratio-dependent predator-prey system[J]. Journal of Com putational and App lied Mathematics.,2011,235:1326-1341.

[9]Li C X.Si J P,Jiang S.Stability of im pu lsive stochastic differential delay system s and its app lication to im pu lsive stochastic neural networks[J].Nonlinear Analysis,2011:74:3099-3111.

[10]K lebaner F F.Introdution to Stochastic Calculus with App lication[M].London:Im perial College Press, 1998.

Study of non-autonom ous predator-prey system with impulsive effects and random perturbation

Tan Dejun
(College of Education of Teacher,Jimei University,Xiamen 361021,China)

A m odel of a non-autonom ous ratio-dependent p redator-p rey system with im pulsive effects and random perturbation is builded.The equivalent relation between the solution of non-autonom ous stochastic differential system with im pu lsive effects and that of a corresponding non autonomous stochastic differential system with im pulsive effect is researched.Moreover,we prove som e dynam ic behavior of this system for the boundedness,uniform ly bounded in them ean and extinction of this system.

im pu lsive effect,random perturbation,predator-prey system,boundedness

0231

A

1008-5513(2012)03-0285-09

2012-02-02.

福建省自然科學(xué)基金(2008J0199).

譚德君(1965-),碩士,教授,研究方向：生物數(shù)學(xué).

2010 MSC:34D 05,34D 20