Uniform Persistence and Global Attractivity of A Predator-Prey Model with Mutual Interference and Delays

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1 Uniform Persisence and Global Araciviy of A Predaor-Prey Model wih Muual Inerference and Delays KAI WANG Anhui Universiy of Finance and Economics School of Saisics and Alied Mahemaics 96 Caoshan Road, Bengbu, Anhui CHINA wangkai050318@163.com YANLING ZHU Universiy of Inernaional Business and Economics School of Inernaional Trade and Economics 10 Huixin Eas Sree, Chaoyang Disric, Beijing CHINA zhuyanling99@16.com Absrac: In his aer, a redaor-rey model wih muual inerference and delays is sudied by uilizing he comarison heorem and Lyaunov second mehod. Some sufficien condiions for uniform ersisence and global araciviy of osiive eriodic soluion of he model are obained. I is very ineresing ha he resuls obained are relaed o delays, esecially based on he muual inerference consan. Furhermore, i is he firs ime ha such a model is considered. An examle is illusraed o verify he feasibiliy of he resuls in he las ar. Key Words: Muual inerference, Delays, Uniform ersisence, Lyaunov second mehod, Global araciviy 1 Inroducion Recenly, he following redaor-rey(pp for shor) model wih muual inerference, ẋ = x(a b x) φ 1 (, x)y m, ẏ = y ( a () b ()y) + φ (, x)y m, where m (0, 1 is he muual inerference consan, which was inroduced by Hassell in 1971 (see 1 3 for more deails), has been sudied by some auhors, such as Wang ( ), Lin (009), Wang (010), Chen (009) and Wu (010), see 4 10 for more deails. They invesigaed he exisence and sabiliy of eriodic or almos eriodic soluions of some secial cases of he above model, and he models hey considered are all ordinary differenial sysems or difference sysems. I was oined by Kuang (1993) 11 ha any model of secies dynamics wihou delays is an aroximaion a bes, more deailed argumens on he imorance and usefulness of ime-delays in realisic models may also be found in he classical books of Macdonald (1989) 1 and Goalsamy (199) 13. There are many aers on sudy of dynamics of delayed oulaion models, see18,19,-6 and he references cied herein for more deails. Bu here are few lieraures considering ime-delays in PP model wih muual inerference. Moivaed by he above reasons, in his aer we invesigae a PP model wih muual inerference and Corresonding auhor. delays in he form: ẋ() = x()(a b x( τ ) c y m ( τ ())), ẏ() = y()( a () b ()y( τ 3 ()) + c ()x()y m ()), (1) where x(), y() denoe he size of rey and redaor a ime, resecively; m (0, 1 is muual inerference consan; a i (), b i () and c i ()(i = 1, ) are coninuous and bounded above and below by osiive consans on 0, + ), τ i ()(i = 1,, 3) are nonnegaive, bounded and coninuous funcions on 0, + ). If se τ = suτ i () : 0, + ), i = 1,, 3, hen we ge τ 0, + ). Considering he alicaion of model (1) o oulaion dynamics, we assume ha all osiive soluions of model (1) saisfy he following iniial condiions, x(θ) = φ(θ), θ τ, 0, φ(0) = φ0 > 0, y(θ) = ψ(θ), ψ τ, 0, ψ(0) = ψ 0 > 0, () where φ and ψ are given nonnegaive and bounded coninuous funcions on τ, 0. Remark 1 If m = 1, i.e., here is no muual inerference beween reys and redaors, hen model (1) ransforms o he classical PP model: ẋ() = x()(a b x( τ ) c y( τ ())), ẏ() = y()( a () b ()y( τ 3 ()) + c ()x()), E-ISSN: Volume 15, 016

2 which has been exensively sudied wih or wihou delays, such as exisence of eriodic soluions, ermanence, sabiliy of equilibria and so on, see 14 4 and he references cied herein. Remark If we neglec he influence of delays in model (1), i.e., τ i () 0, i = 1,, 3, hen model (1) reduces o he following ordinary differenial oulaion sysem: ẋ() = x()(a b x() c y m ()), ẏ() = y()( a () b ()y() + c ()x()y m ). (3) In aer 5, we sudied he exisence and global asymoic sabiliy of osiive eriodic soluion of model (3) wih c () = kc, and in aer 6, we invesigaed he ermanence, and global asymoic sabiliy of he osiive soluions of model (3). The aim of his aer is o esablish sufficien condiions for he uniform ersisence of model (1) wih m (0, 1), and resen sufficien condiions for he global araciviy of model (1) wih m (0, 1). The srucure of he aer is: In secion, some useful lemmas and definiions are given. In secion 3, some main resuls on he uniform ersisence and he exisence of osiive eriodic soluions for model (1) are esablished. In secion 4, an examle is given o verify he feasibiliy of our resuls by simulaion. The significance is ha we verify ha he muual inerference consan m has inrinsic effec on he u- niform ersisence, he exisence of osiive eriodic soluions and he global araciviy of model (1). Lemmas and definiions In his secion, we give some imoran lemmas and definiions which will be used in nex secions. For he sake of convenience, we le f L = inf E f(), f U = inf E f(), where f is a coninuously bounded funcion defined on inerval E, and denoe by φ i () he inverse funcion of φ i () = τ i (), i = 1,, 3, resecively. Lemma 3 ( See 3 ) If a > 0, b > 0 and ż() ( ) z()(b a z()) for 0, z(0) > 0, hen he following inequaliy hold: z() ( ) b a 1 + ( b a z(0) 1 ) ex b. Lemma 4 If a > 0, b > 0, ż() z m ()(b az 1 m ()) wih 0 < m < 1 and z(0) > 0, hen for any small consan ε > 0 we have z() (b/a) 1/(1 m) ε for T, where T is a large enough osiive consan. Proof: I follows from ha ż() z m ()(b az 1 m ()) d(z 1 m )()/d (1 m)(b az 1 m ()). From Lemma 3 we ge z 1 m () b/a + ( z 1 m (0) b/a ) e a(1 m) for 0 i.e., z() b/a + ( z 1 m (0) b/a ) e a(1 m) 1/(1 m) for 0. Then for any small osiive consan ε here exiss a osiive consan T such ha z() (b/a) 1/(1 m) ε for T. Definiion 5 Model (1) is said o be uniformly ersisen if here exiss a comac region D In R such ha every soluion (x(), y()) of model (1) wih iniial condiion () evenually eners and remains in region D. Definiion 6 Model (1) is called global araciviy if for any osiive soluions (x(), y()), (x 0 (), y 0 ()), lim + ( x() x 0() + y() y 0 () ) = 0. 3 Main resuls In his secion, we will resen some sufficien condiions on uniform ersisence, exisence of osiive eriodic soluions and global araciviy of model (1), resecively. 3.1 Uniform ersisence and osiive eriodic soluions Theorem 7 If he following condiions hold: (1) K 1 := (a 1 c 1 M m)l > 0, () K := ( c M 3 M m ) L a > 0 or K 3 := ( c M 3 b M m ) L > 0. Then model (1) is uniformly ersisen. Proof: I follows from model (1)-() ha x() x(0) = e 0 (a 1(s) b 1 (s)x(s τ 1 (s)) c 1 (s)y m (s τ (s)))ds, y() y(0) = e 0 ( a (s) b (s)y(s τ 3 (s))+c (s)x(s)y m (s))ds, E-ISSN: Volume 15, 016

3 which ogeher wih he osiiviy of (x(0), y(0)) yields he exisence of osiive soluion (x(), y()) of model (1)-(). Now we esimae he evenually uer bounds of all osiive soluion (x(), y()) of model (1). From he firs equaion of model (1), we ge ẋ() x() ( a U 1 b L 1 x( τ ) ), i follows from 6, Lemma.3 ha here exiss consan T 1 > 0 such ha x() au 1 b L 1 exb L 1 τ U 1 := M 1 for > T 1. (4) Similarly, from he second equaion of model (1) we ge d(y 1 m ()) d (1 m) ( c U M 1 a L y 1 m () ). This inequaliy and Lemma 3 yields ( ) c U y() M 1 c U 1 + M 1 a L y1 m (0) 1 a L ex (m 1) c U M m Noe ha ex (m 1) c U M 1 0 for +, which yields ha, for any small consan ε 0 > 0 here exiss a consan T > 0 such ha, for > T ( c U y() M 1 a L + ε 0 ) 1 1 m := M. (5) We now esimae he evenually lower bounds of all osiive soluion (x(), y()) of sysem (1). From (5) and he firs equaion of model (1) we ge ẋ() x() (a 1 c 1 M m ) L b U 1 x( τ ). Condiion (1) and 6, Lemma.4 imly ha, for any small consan ε 1 > 0 here is a T 3 > 0 such ha x() M 3 for > T 3, (6) ε 1. where M 3 = min K1 e (K 1 b U b U 1 M 1)τ1 U, K 1 1 b U 1 The second equaion of model (1) imlies ẏ() y() (c M 3 M m a ) L b U y( τ 3 ()) So from 6, Lemma.4 and condiion () we know ha for any small consan ε > 0 here mus exis consan T 4 > 0 such ha y() M 4 for > T 4, (7).. where M 4 = min K b U e (K b U M )τ U 3, K b U ε. On he oher hand, from he second equaion of sysem (1) one can obain ẏ() y() (c M 3 b M m ) L a U y () 1 m. Thus Lemma 4 and assumion (3) yield ha, for any small consan ε 3 > 0 here is a T 5 > 0 such ha ( ) 1 K3 1 m y() ε3 := M 5 for > T 5. (8) a U Se T = max 1 i 5 T i, M 6 = maxm 4, M 5 and D = (u, v) M 3 u M 1, M 6 v M, hen (4)-(8) yields (x(), y()) D for > T. Thus (x(), y()) is uniformly ersisen. Remark 8 If all coefficiens in model (1) are coninuously eriodic funcions, i.e., i is a eriodic sysem, hen Theorem 7 and Brouwer fixed oin heorem yields he following resul. Theorem 9 If model (1) is a ω eriodic sysem and condiion (1)-() in Theorem 7 holds, and all deviaing argumens τ i ()(i = 1,, 3) are coninuous ω eriodic funcions, hen model (1) has a leas one osiive ω eriodic soluion. 3. Global araciviy In his secion we will resen some sufficien condiions for he global asymoic sabiliy of model (1). Theorem 10 Assume ha all condiions in Theorem 7 hold, and furher assume ha he following condiions hold: A 1 τ i, i = 1,, 3 are coninuously differeniable on 0, + ) and inf 0 (1 τ i ()) > 0; A There exis osiive consans α and β such ha lim inf + C i(), i = 1,, 3 > 0, where C = αb β c () M m α(a + M 1 b + M m c ) βm M m 6 c () αm 1 b 1 (φ ()) 1 τ (φ ()) b 1 (l)dl 3 () b (l)dl 1 (φ ()) φ () b 1 (l)dl, E-ISSN: Volume 15, 016

4 C () = b () M 3 M6 m c () (a () + M b () + M 1 M m 6 c ()) M b (φ 3 ()) 1 τ 3 (φ 3 ()) 3 () b (l)dl 3 (φ 3 ()) C 3 () = βm 3c () αc 1(φ ()) M 1 τ (φ ( ()) 1 + M 1 b 1 (φ ()) φ 3 () b (l)dl, 1 (φ ()) Then sysem (1)-() is globally aracive. φ () b 1 (l)dl Proof: Suose (x 0 (), y 0 ()) and (x(), y()) are any wo osiive soluions of model (1)-(), by Theorem 7 one know ha here exiss T > 0 such ha, for > T, M 3 x 0 (), x() M 1 ; M 6 y 0 (), y() M. Define V = α ln x() ln x 0 () +β ln y() ln y 0 (), where α, β are osiive consans. Along sysem (1) we ge is Dini derivaive as follows, D + V (1) = α sgn(x() x 0 ()) b x() x 0 () c y m (φ ()) y0 m (φ ()) + b (ẋ(s) ẋ 0 (s))ds + β sgn(y() y 0 ()) φ b ()y() y 0 () + c ()x()y m () x 0 ()y0 m () + b () (ẏ(s) ẏ 0 (s))ds φ 3 () αb x() x 0 () βb () y() y 0 () + αc y m (φ ()) y m 0 (φ ()) + βc () sgn(y() y 0 ())x()y m () x 0 ()y0 m () + αb sgn(x() x 0 ()) + βb () sgn(y() y 0 ()) φ φ 3 () ) (ẋ(s) ẋ 0 (s))ds (ẏ(s) ẏ 0 (s))ds. (9). Meanwhile, we have sgn(y() y 0 ())x()y m () x 0 ()y m 0 () =sgn(y() y 0 ())x()(y m () y m 0 ()) + y m 0 ()(x() x 0 ()) = x() y m () y m 0 () + y m 0 ()sgn(y() y 0 ())(x() x 0 ()) x() y m () y m 0 () + y m 0 () x() x 0 (), (10) = = φ φ (ẋ(s) ẋ 0 (s))ds x(s)a 1 (s) b 1 (s)x(φ 1 (s)) c 1 (s)y m (φ (s)) x 0 (s)a 1 (s) b 1 (s)x 0 (φ 1 (s)) c 1 (s)y m 0 (φ (s)))ds φ (x(s) x 0 (s))a 1 (s) b 1 (s)x 0 (φ 1 (s)) c 1 (s)y m 0 (φ (s))ds + φ x(s)b 1 (s)x 0 (φ 1 (s)) x(φ 1 (s)) + c 1 (s)y m 0 (φ (s)) y m (φ (s))ds (11) and = = = φ 3 () φ 3 (s) (ẏ(s) ẏ 0 (s))ds y(s) a (s) b (s)y(φ 3 (s)) + c (s)x(s)y m (s) y 0 (s) a (s) b (s)y 0 (φ 3 (s)) + c (s)x 0 (s)y0 m (s)ds φ 3 () (y(s) y 0 (s)) a (s) b (s)y 0 (φ 3 (s)) + c (s)x 0 (s)y m 0 (s)ds + y(φ 3 (s)) y 0 (φ 3 (s)) φ 3 () y(s) b (s) + c (s)x(s)y m (s) x 0 (s)y0 m (s)ds φ 3 () (y(s) y 0 (s)) a (s) b (s)y 0 (φ 3 (s)) + c (s)x 0 ()y0 m (s)ds b (s)y(s)y(φ 3 (s)) φ 3 () y 0 (φ 3 (s))ds + c (s)y(s)x(s)(y m (s) φ 3 () y0 m (s)) + y0 m (s)(x(s) x 0 (s))ds. (1) E-ISSN: Volume 15, 016

5 Subsiuion of (10)-(1) ino (9) yields Le D + V (1) αb x() x 0 () βb () y() y 0 () + αc y m (φ ()) y m 0 (φ ()) + βc ()( x() y m () y0 m () + y0 m () x() x 0 () ) + αb x(s) x 0 (s) a 1 (s) b 1 (s)x 0 (φ 1 (s)) φ c 1 (s)y m 0 (φ (s))ds + φ x(s)b 1 (s) x 0 (φ 1 (s)) x(φ 1 (s)) + c 1 (s) y0 m (φ (s)) y m (φ (s)) ds + βb () y(s) y 0 (s) a (s) φ 3 () b (s)y 0 (φ 3 (s)) + c (s)x 0 (s)y0 m (s)ds + + φ 3 () φ 3 () b (s)y(s) y(φ 3 (s)) y 0 (φ 3 (s)) ds c (s)y(s) x(s) y m (s) y0 m (s) + y m 0 (s) x(s) x 0 (s)) ds β c () M m αb x() x 0 () βb () y() y 0 () βm 3 c () y m () y0 m () + αc y m (φ ()) y0 m (φ ()) + αb x(s) x 0 (s) a 1 (s) + M 1 b 1 (s) φ + M m c 1 (s)ds + M 1 b 1 (s) x 0 (φ 1 (s)) φ x(φ 1 (s)) + c 1 (s) y0 m (φ (s)) y m (φ (s)) ds + βb () y(s) y 0 (s) a (s) + M b (s) φ 3 () + M 1c (s) ds M6 1 m + M b (s) y(φ 3 (s)) φ 3 () y 0 (φ 3 (s)) ds M 6 M 3 c (s) y m (s) φ 3 () y0 m (s) ds + M M6 1 m c (s) x(s) x 0 (s) ds. φ 3 () (13) =α V () φ () + βm M m 4 c 1 (φ (s)) 1 τ (φ (s) y m (s)) ym 0 (s) ds x 0 (s) dsdl + α 3 () φ 3 (l) b (l)c (s) x(s) φ 1 (l) b 1 (l)a 1 (s) + M 1 b 1 (s) + M m c 1 (s) x(s) x 0 (s) dsdl + αm 1 φ 1 (l) b 1 (l)b 1 (s) x(φ (s)) x 0 (φ (s)) + c 1 (s) y m (φ (s)) y m 0 (φ (s)) dsdl + β 3 () φ 3 (l) b (l) y(s) y 0 (s) a (s) + M b (s) + M 1 M6 m c (s)dsdl + βm 3 () φ 3 (l) hen is derivaive is as follows, V () b (l)b (s) y(φ 3 (s)) y 0 (φ 3 (s)) dsdl, = αc 1(φ ()) 1 τ (φ (s) y m ()) ym 0 () αc y m (φ ()) y0 m (φ ()) + αa + M 1 b + M m c b 1 (l)dl x() x 0 () αb a 1 (s) + M 1 b 1 (s) + M m c 1 (s) φ x(s) x 0 (s) ds + αm 1 b b 1 (l)dl x(φ ()) x 0 (φ ()) + c y m (φ (s)) y0 m (φ ()) αm 1 b b 1 (s) x(φ (s)) φ x 0 (φ (s)) + c 1 (s) y m (φ (s)) y0 m (φ (s)) ds + β a () + M b () + M 1 M6 m c () 3 () b (l)dl y() y 0 () βb () E-ISSN: Volume 15, 016

6 Subsiuing (13)-(15) ino (16), we obain y(s) y 0 (s) a (s) + M b (s) φ 3 () + M 1 M m 6 c (s)ds + βm b () y(φ 3 ()) y 0 (φ 3 ()) βm b () 3 () φ 3 () b (l)dl b (s) y(φ 3 (s)) y 0 (φ 3 (s)) ds + βm M6 m c () 3 () βm M m 6 b () Define V 3 () b (l)dl x() x 0 () φ 3 () 1 (φ (s)) c (s) x(s) x 0 (s) ds. (14) b 1 (φ = αm 1 b 1 (l) (s)) φ () φ (s) 1 τ (φ (s)) x(s) x 0 (s) + c 1 (φ (s)) ym (s) y0 m (s) dlds 3 (φ 3 (s)) + βm φ 3 () φ 3 (s) b (l) y(s) y 0 (s) dlds. b (φ 3 (s)) 1 τ 3 (φ 3 (s)) By simly calculaing we ge is derivaive in he following form, V 3() 1 (φ ()) b 1 (φ = αm 1 b 1 (l)dl ()) φ () 1 τ (φ ()) x() x 0 () + c 1 (φ ()) ym () y0 m () αm 1 b b 1 (l)dl x(φ ()) x 0 (φ ()) +c y m (φ ()) y m 0 (φ ()) + βm y() y 0 () 3 (φ 3 ()) φ 3 () b (l)dl βm b () 3 () b (φ 3 ()) 1 τ 3 (φ 3 ()) Define he Lyaunov funcional by hen b (l)dl y(φ 3 ()) y 0 (φ 3 ()). V () = V + V () + V 3 (), (15) D + V () = D + V + V () + V 3(). (16) D + V () (1) β c () M m αb x() x 0 () βb () y() y 0 () βm 3 c () y m () y0 m () βm 6 M 3 b () φ 3 () c (s) y m (s) y m 0 (s) ds + αc 1(φ ()) 1 τ (φ () y m ()) ym 0 () + αa +M 1 b + M m c b 1 (l)dl x() x 0 () + β a () + M b () + M 1 M6 m c () 3 () 3 () 1 (φ ()) b (l)dl y() y 0 () + βm M m 6 c () b (l)dl x() x 0 () + αm 1b 1 (φ ()) 1 τ (φ ()) b 1 (l)dl x() x 0 () + c 1 (φ φ () ()) y m () y m 0 () + βm 3 (φ 3 ()) φ 3 () b (l)dl b (φ 3 ()) 1 τ 3 (φ 3 ()) y() y 0() ( αb β c () M m α a + M 1 b ) + M m c ( 3 () 1 (φ ()) b (l)dl αm 1 φ () b 1 (l)dl b 1 (l)dl βm M m 6 c () b 1 (φ ()) 1 τ (φ ()) x() x 0 () β a () + M b () + M ) 1c () φ M b (φ 3 ()) 1 τ 3 (φ 3 ()) M6 1 m 3 (φ 3 ()) φ 3 () b (l)dl 3 () b () b (l)dl y() y 0 () βm 3 c () y m () y0 m () + αc 1(φ ()) 1 τ (φ ()) φ 1 + M 1 b 1 (φ ()) 1 (φ ()) b 1 (l)dl φ () y m () y m 0 (). (17) E-ISSN: Volume 15, 016

7 In view of a a m b m a m b m b m b a for a > 0, b > 0, which yields βm 3 c () y m () y m 0 () βm 3 M m 6 c () y() y 0 () βm 3c () y m () y0 m (). M Thus, combinaion of (18) and (17) yields D + V () C x() x 0 () βc () y() y 0 () C 3 () y m () y m 0 (), (18) which ogeher wih A 1 and A yield here mus exis hree osiive consans λ 1, λ and λ 3 such ha D + V () λ 1 x() x 0 () λ y() y 0 () λ 3 y m () y m 0 () for > T. Thus, V () is non-increasing on 0, + ). Inegraing he above inequaliy from T o we obain λ 1 x() x 0 () d + λ y() y 0 () d T + λ 3 y m () y0 m () d < + for > T. T By Barbala s Lemma 7, we ge he resul. Theorem 11 Assume ha all condiions in Theorem 7 and A 1 hold, and furher assume ha here exis muual rime osiive inegers, q wih > q such ha A 3 : m = q and A 4: There exis osiive consans α and β such ha lim inf + C, B() > 0, where B() :=βb T u i v i B () wih M 6 u, v M, q u i v q i, B = b () + M 3 M m c () (a () + M b () + M 1 M m 6 c ()) M b (φ 3 ()) 1 τ 3 (φ 3 ()) 3 () b (l)dl 3 (φ 3 ()) B () = βm 3c () + αc 1(φ ()) M 6 1 τ (φ ( ()) 1 + M 1 b 1 (φ ()) φ 3 () b (l)dl, 1 (φ ()) φ () b 1 (l)dl ), where C is defined in Theorem 10, M, M 6 is defined in he roof of Theorem 7 Then model (1) is globally aracive. Proof: One see a a m b m b m b a a m b m for a > 0, b > 0, hus βm 3 c () y m () y m 0 () βm 3c () y m () y0 m () M 6 βm 3 c ()M m () y() y 0 (). Subsiuing (19) ino (17), one has D + V () C x() x 0 () βb y() y 0 () + B () y m () y m 0 (). Assumion A 3 yields y() y 0 () = y 1 1 () y0 () y m () y0 m () = y 1 1 q () y0 () which ogeher wih (0) give y i (19) (0) i ()y0 (), y i q i ()y0 (), D + V () C x() x 0 () w() y 1 1 () y0 (), where w() =βb B () q y i y i i ()y0 () q i ()y0 (), which ogeher wih assumion A 4 imlies lim x() x 0() = lim + + y 1 1 () y0 () = 0. Remark 1 Comare Theorem 10 wih Theorem 11, we find ha he assumions in he former is more simle han hose in he laer. Bu we will show ha he assumions in Theorem 11 is more weak han Theorem 10. In fac, if q = 1, hen B() = βb B () βm6 B B (), which is more simler han ha of q 1, and hus can be checked easily. u i v i E-ISSN: Volume 15, 016

8 If τ i () = τ i > 0, i = 1,, 3, ha is, sysem (1) reduces o he following sysem: ẋ() = x()(a b x( τ 1 ) c y m ( τ )), (1) ẏ() = y()( a () b ()y( τ 3 ) + c ()x()y m ()), hen corresonding o Theorems we ge he following resuls. Theorem 13 Assume all condiions in Theorem 7 hold, and here exis osiive consans α, β such ha lim inf C i(), i = 4, 5, 6 > 0, where + C 4 () = αb β c () M m α(a + M 1 b +M m c ) +τ3 +τ1 b (l)dl αm 1 b 1 ( + τ ) b 1 (l)dl βm M6 m c () +τ1 +τ +τ b 1 (l)dl, C 5 () = b () M 3 M6 m c () M b ( + τ 3 ) +τ3 +τ 3 + M 1 M m 6 c ()) b (l)dl (a () + M b () +τ3 C 6 () = βm 3c () αc 1 ( + τ ) M ( 1 + M 1 b 1 ( + τ ) b (l)dl, +τ1 +τ +τ Then model (1) is globally aracive. ) b 1 (l)dl. Theorem 14 Assume ha all condiions in Theorem 7 and A 1 hold, and here exis muual rime osiive inegers, q wih > q such ha A 3 hold, and here exis osiive consans α and β such ha lim inf + C 4 (), B 0 () > 0, where B 0 () =βb 3 () M 6 u, v M, u i v i B 4 () q u i v q i, B 3 () = b () + M 3 M m c () M b ( + τ 3 ) +τ3 +τ 3 + M 1 M m 6 c ()) b (l)dl (() + M b () +τ3 B 4 () = βm 3c () + αc 1 ( + τ ) M ( M 1 b 1 ( + τ ) b (l)dl, +τ1 +τ +τ ) b 1 (l)dl. Then model (1) is globally aracive. Corollary 15 Assume ha all condiions in Theorem 7 and A 1 hold, and furher assume ha here exiss a osiive ineger wih > 1 such ha m = 1, and here exis osiive consans α and β such ha lim inf C 4(), B 0 () > 0, + where B 0 () := βm6 B 3 () B 4 (). Then model (1) is globally aracive. 4 Alicaion and simulaion Examle: Le a = sin, a () = sin, b = 6, b () = 3.4, c = sin, c () = sin, τ 1 = 0.005, τ = 0.03, τ 3 = 0, m = 1/3. Choose ε 0 = ε 1 = , ε = ε 3 = 10 0, α = 1., β = By calculaing we obain M 1 =.0667, M = , K 1 = , M 3 = , K =.07577, K 3 = , M 4 = , M 5 = , M 6 = , lim inf C 4() = , lim inf C 5() = , + + lim inf C 6() = , lim inf B 0() = According o Theorems 7,10, his sysem is uniformly ersisen and has π-eriodic osiive soluion. Meanwhile, according o Corollary 15 we asser ha he sysem is globally aracive, see Figs.1- for more deails. Remark 16 Examle shows ha lim inf B 0() > 0 is + weak han lim inf C 5(), C 6 () > 0, which verifies + Remark 1. Remark 17 Obviously, if m = 1, hen here is no muual inference beween reys and redaors, hen from Figs.3-4. one can easily see ha he rey is uniformly ersisen bu he redaor is exinc evenually. So he muual inerference can effec he oulaion of he rey and redaor. In he real world we mus consider some PP models under he influence of muual inerference. E-ISSN: Volume 15, 016

9 3.8 x1()=, in 0.03,0 x()=1.6, in 0,03,0 x3()=3, in 0.03, y1()=0.05, in 0.03,0 y()=1, in 0.03,0 y3()=0.5, in 0.03,0 Poulaions x1(),x(),x3().6.4. Poulaion y1(),y(),y3() Time Fig.1. The inegral curves of examle wih m = 1/ Time Fig.4. The inegral curves of examle wih m = 1. Poulaions y1(),y(),y3() y1()=0.05, in 0.03,0 y()=1, in 0.03,0 y3()=0.5, in 0.03, Time Fig.. The inegral curves of examle wih m = 1/3. Poulaions x1(),x(),x3() x1()=, in 0.03,0 x()=1.6, in 0.03,0 x3()=3, in 0.03, Time Fig.3. The inegral curves of examle wih m = 1. Acknowledgemens: The research was suored by he NSF of China ( , , ), he NSF of Anhui Province ( QA13), he Key NSF of Educaion Bureau of Anhui Province (KJ013A003, KJ014A003) and he Suor Plan of Excellen Youh Talens in Colleges and Universiies in Anhui Province. References: 1 L. Chen, Mahemaical Models and Mehods in Ecology, Science Press Beijing, M. Hassell, Muual Inerference beween Searching Insec Parasies, J. Anim. Ecol. 40, 1971, M. Hassell, Densiy deendence in singlesecies oulaion, J. Anim. Ecol. 44, 1975, K. Wang, Y. Zhu, Global araciviy of osiive eriodic soluion for a Volerra model, Al. Mah. Comu. 03, 008, K. Wang, Exisence and global asymoic sabiliy of osiive eriodic soluion for a Predaor- Prey sysem wih muual inerference, Nonlinear Anal. RWA 10, 009, K. Wang, Permanence and global asymoical sabiliy of a redaor-rey model wih muual inerference, Nonlinear Anal. RWA 1, 011, X. Lin, F. Chen, Almos eriodic soluion for a Volerra model wih muual inerference and Beddingon-DeAngelis funcional resonse, Al. Mah. Comu. 14, 009, X. Wang, Z. Du, J. Liang, Exisence and global araciviy of osiive eriodic soluion o E-ISSN: Volume 15, 016

10 a Loka-Volerra model, Nonlinear Anal. Real World Al. 11, 010, L. Chen, L. Chen, Permanence of a discree eriodic volerra model wih muual inerference, Disc. Dyna. Nau. Soci. 009, ar. no R. Wu, Permanence of a discree eriodic Volerra model wih muual inerference and Beddingon-Deangelis funcional resonse, Disc. Dyna. Nau. Soci. 010, ar. no Y. Kuang, Delay Differenial Equaions wih Alicaions in Poulaion Dynamics, Academic Press New York, N. Macdonald, Biological Delay Sysems: Linear Sabiliy Theory, Cambridge Universiy Press Cambridge, K. Goalsamy, Sabiliy and Oscillaions in Delay Differenial Equaions of Poulaion Dynamics, Kluwer Academic Publisher Boson, T. Buron, V. Huson, Permanence for nonauonomous redaor-rey sysems, Differ. Inegral Equa. 4, 1991, A. Tineo, Permanence of a large class of eriodic redaor-rey sysems, J. Mah. Anal. Al. 41, 000, Z. Teng, On he ersisence and osiive eriodic soluion for lanar comeing Loka-Volerra sysems, Ann. Diff. Eqs. 13(3), 1997, Z. Teng, L. Chen, Necessary and sufficien condiions for exisence of osiive eriodic soluions of eriodic redaor-rey sysems, Aca. Mah. Scienia 18, 1998, Z. Teng, L. Chen, The osiive eriodic soluions in eriodic Kolmogorov ye sysems wih delays, Aca Mah. Al. Sinica, 1999, M. Fan, Global exisence of osiive eriodic soluion of redaor-rey sysem wih deviaing argumens, Aca. Mah. Al. Sinica 3, 000, J. Zhao, W. Chen, Global asymoic sabiliy of a eriodic ecological model, Al. Mah. Comu. 147, 004, F. Chen, The ermanence and global araciviy of Loka-Volerra comeiion sysem wih feedback conrols, Nonlinear Anal. RWA 7, 006, Y. Xia, J. Cao, S. Cheng, Periodic soluions for a Loka-Volerra muualism sysem wih several delays, Al. Mah. Model. 31, 007, C. Chen, F. Chen, Condiions for global araciviy of mulisecies ecological comeiionredaor sysem wih Holling III ye funcional resonse, J. Biomah. 19, 004, X. He, Sabiliy and delays in a redaor-rey sysem, J. Mah. Anal. Al. 198, 1996, F. Chen, Permanence and global sabiliy of nonauonomous Loka-Volerra sysem wih redaor-rey and deviaing argumens, Al. Mah. Comu. 173, 006, K. Wang, Y. Zhu, Permanence and global asymoic sabiliy of a delayed redaor-rey model wih Hassell-Varley ye funcional resonce, Bulle. Iranian Mah. Soci. 37, 011, I. Barbala, Sysem d equaions differenial d oscillaions nonlineariies, Rev. Roumaine Mah. Pure Al. 4, 1959, E-ISSN: Volume 15, 016