Unifor a Predator-Prey Model with Beddington-DeAngelis Functional Response*

XU Changjin,YAO Lingyun

(1.Guizhou Key Laboratory of Economics System Simulation,Guizhou University of Finance and Economics,Guiyang 550004,China；2.Library,Guizhou University of Finance and Economics,Guiyang 550004,China)

UniforaPredator-PreyModelwithBeddington-DeAngelisFunctionalResponse*

XU Changjin1,YAO Lingyun2

(1.Guizhou Key Laboratory of Economics System Simulation,Guizhou University of Finance and Economics,Guiyang 550004,China；2.Library,Guizhou University of Finance and Economics,Guiyang 550004,China)

An asymptotically periodic predator-prey model with Beddington-DeAngelis functional response is investigated.Some sufficient conditions for the uniformly strong persistence of the system are obtained.

predator-prey model;uniform persistence;asymptotically periodic;Beddington-DeAngelis functional responseCLCnumberO175.13DocumentcodeA

10.3969/j.issn.1007-2985.2014.01.003

1 Introduction

The qualitative properties such as boundedness,stability,permanence and existence of periodic solutions have attracted a lot of attention and many good results have already been reported.For example,GYLLENBERG M et al[1]studied limit cycles of a competitor-competitor-mutualist Lotka-Volterra model.SONG X Y et al[2]made a discussion on the linear stability of trivial periodic solution and semi-trivial periodic solutions and the permanence of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect.AGGELIS G et al[3]considered the coexistence of both prey and predator populations of a prey-predator model.AGIZA H N et al[4]investigated the chaotic phenomena of a discrete prey-predator model with Holling type II.SEN M et al[5]analyzed the bifurcation behavior of a ratio-dependent prey-predator model with the Allee effect.ZHANG Z Q et al[6]gave a theoretical study on the existence of multiple positive periodic solutions for a delayed predator-prey system with stage structure for the predator.ZHANG Z Q et al[7]focused on the existence of at least four positive periodic solutions for a ratio-dependent predator-prey system with multiple exploited (or harvesting) terms.KO W et al[8]discussed the coexistence states of a nonlinear Lotka-Volterra type predator-prey model with cross-diffusion.FAZLY M et al[9]dealt with periodic solutions of a predator-prey system with monotone functional responses.One can see ref.[10-19] etc.for more related studies.However,the research work on asymptotically periodic predator-prey model is very few at present.

In2011,HAQUEM[20]investigatedthestability(localandglobal)andbifurcation(saddle-node,transcritical,Hopf-Andronov,Bogdanov-Takens)ofthefollowingBeddington-DeAngelispredator-preymodel

(1)

wherex(t)andy(t)denotethedensitiesofpreyandpredator,respectively,attimet;r,k,m,a,b,c,e,d,harepositiveconstantsthatstandforpreyintrinsicgrowthrate,carryingcapacityoftheenvironment,consumptionrate,preysaturationconstant,predatorinterference,anothersaturationconstant,conversionrate,predatordeathrate,predatorinterspeciescompetition,respectively.Indetails,onecanseeref. [20].

(2)

withinitialconditionsx(0)=φ1(0)≥0,y(0)=φ2(0)≥0.

Theprincipleobjectofthisarticleistoexploretheuniformlystrongpersistenceofsystem(2).Thereareveryfewpaperswhichdealwiththistopic,seeref. [10,21].

Inordertoobtainourresults,weassumethatsystem(2)alwayssatisfies:

2 Uniformly Strong Persistence

fl-ε≤f(t)≤fu+εfort≥T.

(3)

Lemma1 Both the positive and nonnegative cones ofR2are invariant with respect to system (2).

It follows from lemma 1 that any solution of system (2) with a nonnegative initial condition remains nonnegative.

In what follows,we will establish our result.

Theorem2 LetA1,A2andB1be defined by (5),(7) and (9),respectively.Assume that conditions (H) andblrl>mu,elmlB1>du(auA1+buA2+cuhold，then system (2) is uniformly strong persistence.

ProofIt follows from (3) that for anyε>0,there existsT1>0 such that

(4)

Substitute (4) into the first equation of system (2),then we have

By lemma 2,we get

(5)

Then for anyε>0,there existsT2>T1>0 such that

x(t)≤A1+εt≥T2.

(6)

Similarly,from (3) and the second equation of system (2),we obtain that for anyε>0,there existsT3>T2>0 such that

In view of lemma 2,we derive

(7)

Then for anyε>0,there existsT4>T3>0 such that

y(t)≤A2+εt≥T4.

(8)

According to (6),(8) and the first equation of system (2),we obtain that for anyε>0,there existsT5>T4>0 such that

Using lemma 2 again,we have

(9)

Thus for anyε>0,there existsT6>T5>0 such thatx(t)≥B1-ε.

According (6),(8) and the second equation of system (2),we obtain that for anyε>0,there existsT7>T6>0 such that

Using lemma 2 again,we have

Thus the proof of theorem 1 is complete.

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具有Beddington-DeAngelis功能反应的捕食模型

徐昌进1，姚凌云2

(1.贵州财经大学贵州省经济系统仿真重点实验室,贵州 贵阳 550004;2.贵州财经大学图书馆,贵州 贵阳 550004)

研究了一类具有Beddington-DeAngelis 功能反应的渐近周期捕食模型，得到了该系统一致强持久的充分条件.

O175.13

A

1007-2985(2014)01-0008-04

date：2013-05-04

National Natural Science Foundation of China (11261010,11201138)；Soft Science and Technology Program of Guizhou Province (2011LKC2030)；Natural Science and Technology Foundation of Guizhou Province (J[2012]2100)；Governor Foundation of Guizhou Province ([2012]53)

Biography： XU Changjin (1970-),male,was born in Huaihua City,Hunan Provinve,professor,Ph.D.;research areas are theory and its applications of delay differential equations.

键词:捕食模型;一致持久;渐近周期;Beddington-DeAngelis 功能反应