H olling IV捕食-食饵时滞系统的多个周期解

田德生,朱长青,朱永松

(湖北工业大学理学院,湖北武汉 430068)

H olling IV捕食-食饵时滞系统的多个周期解

田德生,朱长青,朱永松

(湖北工业大学理学院,湖北武汉 430068)

应用重合度定理研究了一类具有Holling IV类功能性反应时滞捕食-食饵系统的周期解的存在性问题,建立了该系统具有至少两个正周期解的充分条件.

捕食-食饵模型；Holling IV；多个周期解；重合度

1 引言

近年来,一种强有力的方法—重合度理论广泛应用于研究生态方程的周期解问题[13],本文考虑如下的具有Holling IV类功能性反应时滞扩散捕食系统

其中g为连续的ω-周期函数.

1 主要结果

首先,我们引入重合度理论中的延拓定理[8].

设X,Z是赋范向量空间,L:Dom L⊂X→Z为线性映射,N:X→Z为连续映射. 称L为指标为零Fredholm算子,如果dim Ker L=codim Im L＜∞且Im L为Z中的闭集.如果L为指标为零Fredholm算子,又存在连续投影P:X→X和Q:Z→Z满足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,记其逆为KP.设Ω是X的有界开集,若(QN)()有界且KP(I−Q)N:→X是紧的,则称N在是L-紧的.由于Im Q与Ker L同构,因此存在同构映射J:Im Q→Ker L.

引理设L是指标为零Fredholm算子,N在¯Ω是L紧的.假设

(i)对任意的λ∈(0,1),x∈∂Ω∩dom L,都有Lx/=λN x;

(ii)对任意的x∈∂Ω∩Ker L,都有QN x/=0;

(iii)deg{JQN,Ω∩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].数学杂志,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]马知恩.种群生态系统的数学建模与研究[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,生物数学.

2000M SC:34K 13,92D 25