田德生,朱長(zhǎng)青,朱永松

(湖北工業(yè)大學(xué)理學(xué)院,湖北武漢 430068)

H olling IV捕食-食餌時(shí)滯系統(tǒng)的多個(gè)周期解

田德生,朱長(zhǎng)青,朱永松

(湖北工業(yè)大學(xué)理學(xué)院,湖北武漢 430068)

應(yīng)用重合度定理研究了一類具有Holling IV類功能性反應(yīng)時(shí)滯捕食-食餌系統(tǒng)的周期解的存在性問題,建立了該系統(tǒng)具有至少兩個(gè)正周期解的充分條件.

捕食-食餌模型；Holling IV；多個(gè)周期解；重合度

1 引言

近年來,一種強(qiáng)有力的方法—重合度理論廣泛應(yīng)用于研究生態(tài)方程的周期解問題[13],本文考慮如下的具有Holling IV類功能性反應(yīng)時(shí)滯擴(kuò)散捕食系統(tǒng)

其中g(shù)為連續(xù)的ω-周期函數(shù).

1 主要結(jié)果

首先,我們引入重合度理論中的延拓定理[8].

設(shè)X,Z是賦范向量空間,L:Dom L?X→Z為線性映射,N:X→Z為連續(xù)映射. 稱L為指標(biāo)為零Fredholm算子,如果dim Ker L=codim Im L＜∞且Im L為Z中的閉集.如果L為指標(biāo)為零Fredholm算子,又存在連續(xù)投影P:X→X和Q:Z→Z滿足Im P=Ker L和Im L=Ker Q=Im(I?Q),那么L|DomL∩KerP:(I?P)X→Im L是可逆的,記其逆為KP.設(shè)?是X的有界開集,若(QN)()有界且KP(I?Q)N:→X是緊的,則稱N在是L-緊的.由于Im Q與Ker L同構(gòu),因此存在同構(gòu)映射J:Im Q→Ker L.

引理設(shè)L是指標(biāo)為零Fredholm算子,N在ˉ?是L緊的.假設(shè)

(i)對(duì)任意的λ∈(0,1),x∈??∩dom L,都有Lx/=λN x;

(ii)對(duì)任意的x∈??∩Ker L,都有QN x/=0;

(iii)deg{JQN,?∩Ker L,0}/=0.

[1]Zhang Zhengqiu,Wang Zhichen.Periodic solution of a two-species ratio-dependent predator-prey system with time delay in a two-patch environm ent[J].Anziam J.,2003,45:233-244.

[2]Tian Desheng,Zeng Xianwu.Existence of at least two periodic solutions of a ratio-dependent p redator-p rey model with exp loited term[J].Acta M ath.App l.Sinica,English series,2005,23(3):489-494.

[3]T ian Desheng,Zeng X ianwu.Periodic solution of a class of predator-prey model exp loited with functional response[J].數(shù)學(xué)雜志,2005,25(5):480-484.

[4]Sm ith F E.Popu lation dynam ics in Daphnia m agna and a new model for popu lation grow th[J].Ecology, 1963,44:651-663.

[5]G ilpin M E,Ayala F J.G lobalmodels of grow th and com petition[J].Proc.Nat.Acad.Scis.Usa.,1973, 70:3590-3593.

[6]Jicai Huang.Bifurcation and chaos in a discrete predator-prey system with Holling type-IV functional response[J].Acta Math.App l.Sinica,English series,2005,21(1):157-176.

[7]馬知恩.種群生態(tài)系統(tǒng)的數(shù)學(xué)建模與研究[M].合肥:安徽教育出版社,1996.

[8]Gaines R E,Mawhin J L.Coincidence Degree and Non-linear Differential Equations[M].Berlin:Sp ringer, 1977.

Multip leperiod icsolutions of adelayed predator-prey system with Holling IV

TIAN De-sheng,ZHU Chang-qing,ZHU Yong-song

(College of Sciences,Hubei University of Technology,Wuhan 430068,China)

By m eans of the coincidence degree theory,we study the existence of positive periodic solutions for a delayed predator-prey system with Holling IV functional response.A set of suficient conditions for this system to have at least two positive periodic solutions is estab lished.

predator-prey model,Holling IV,multiple periodic solutions,coincidence degree

O175.14

A

1008-5513(2009)02-0339-07

2007-09-25.

田德生(1966-),副教授,研究方向：常微分方程定性分析,生物數(shù)學(xué).

2000M SC:34K 13,92D 25