Research Article Dynamics of a Predator-Prey System with Beddington-DeAngelis Functional Response and Delays

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1 Absrac and Applied Analysis Volume 214, Aricle ID 93762, 8 pages hp://dx.doi.org/1.1155/214/93762 Research Aricle Dynamics of a Predaor-Prey Sysem wih Beddingon-DeAngelis Funcional Response and Delays Nai-Wei Liu and Ting-Ting Kong School of Mahemaics and Informaion Science, Yanai Universiy, Yanai, Shandong 2645, China Correspondence should be addressed o Nai-Wei Liu; liunaiwei@aliyun.com Received 25 January 214; Acceped 14 May 214; Published 26 May 214 Academic Edior: Xinguang Zhang Copyrigh 214 N.-W. Liu and T.-T. Kong. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. We consider a predaor-prey sysem wih Beddingon-DeAngelis funcional response and delays, in which no only he sage srucure on prey bu also he delay due o digesion is considered. Firs, we give a sufficien and necessary condiion for he exisence of a unique posiive equilibrium by analyzing he corresponding locaions of a hyperbolic curve and a line. Then, by consrucing an appropriae Lyapunov funcion, we prove ha he posiive equilibrium is locally asympoically sable under a sufficien condiion. Finally, by using comparison heorem and he ω-limi se heory, we sudy he global asympoic sabiliy of he boundary equilibrium and he posiive equilibrium, respecively. Also, we obain a sufficien condiion o assure he global asympoic sabiliy. 1. Inroducion The purpose of his paper is o consider he following predaor-prey sysem wih Beddingon-DeAngelis funcional response and delays: x i () = ax () d ix i () ae d iτ 1 x( ), x () = ae d iτ 1 x( ) bx 2 () m 1 +m 2 x () +m 3 y (), y fx ( τ () = dy() + 2 )y( τ 2 ) m 1 +m 2 x( τ 2 )+m 3 y( τ 2 ), (1) where x i () denoes he immaure prey populaion densiy, x() denoes he maure prey populaion densiy, and y() denoes he maure predaor populaion densiy, and all he parameers τ 1, τ 2, a, d i, b, c, d, f, m 1, m 2,andm 3 are posiive. More precisely, he parameers τ 1 and τ 2 represen he ime ha he prey juveniles and he predaor juveniles ake o be maure, respecively. a is he inrinsic growh rae of he prey and d i is he deah rae of immaure preys. The erm ae d iτ 1 denoeshesurvivingraeofheimmaurepreyduring he delay period o be maure.the consan b denoes he inensiy of inraspecific acion on he prey and d is he deah rae of he predaor. The funcion cx()y()/(m 1 +m 2 x() + m 3 y()) represens he Beddingon-DeAngelis funcional response and he funcion fx( τ 2 )y( τ 2 )/(m 1 +m 2 x( τ 2 )+m 3 y( τ 2 )) represens he predaor s growh rae ha comes from he Beddingon-DeAngelis funcional response during ime period [ τ 2,]. Acually, he predaor-prey sysem wih Beddingon- DeAngelis funcional response has been exensively sudied in he lieraure. In he original work, Beddingon [1] and DeAngelis e al. [2] proposed he predaor-prey sysem as x () =ax( τ) bx 2 () y () = dy() + m 1 +m 2 x () +m 3 y (), fx () y () m 1 +m 2 x () +m 3 y (). Shaikhe [3] considered he sabiliy of a posiive poin of equilibrium of one nonlinear sysem wih afereffec and sochasic perurbaions. Indeed, he auhor in [3] no only obained sufficien condiions for asympoic mean square sabiliy of he considered sysems, bu also showed ha he rivial soluion is asympoically mean square sable and he (2)

2 2 Absrac and Applied Analysis posiive equilibrium is sable in probabiliy by consrucing appropriae Lyapunov funcions. In paricular, Shaikhe expounded a general mehod of consrucion of Lyapunov funcions for sochasic funcional differenial equaions and obained he sabiliy condiions for funcional differenial equaions in [4]. See also [5 1] for he corresponding resuls abou he predaor s funcional response. Paricularly, Huo [8] sudied he global araciveness of delay diffusive prey-predaor sysems wih he Michaelis-Menen funcional response. In he real world, many species usually have a life hisory ha akes hem hrough wo sages, immaure sage and maure sage. Recenly, he sysem wih he sage srucure on prey has been exensively sudied. In paricular, She and Li [11] analyzed he following Beddingon-DeAngelis model wih he sage srucure on prey: x i () = ax () d ix i () ae d iτ x ( τ), x () = ae d iτ x ( τ) bx 2 () m 1 +m 2 x () +m 3 y (), y fx () y () () = dy() + m 1 +m 2 x () +m 3 y (), (3) where he sage srucure was inroduced by a consan ime delay. In fac, i is he case of τ 2 =in sysem (1). See also [12 15] for predaor-prey models wih Beddingon-DeAngelis funcional response and sage srucure. More recenly, Xiang e al. [16] esablisheda virusdynamics model wih Beddingon-DeAngelis funcional response and delays. In [16], hey no only incorporaed he delay τ 1, which described he ime beween viral enry ino a arge cell and he producion of new virus paricles, bu also incorporaed he delay τ 2, which described he mauraion ime of he newly produced viruses. Addiionally, i is well known ha he growh of he predaor is affeced by he ime delay due o digesion. This moivaes us o resolve he issue. Moivaed by [16], in he presen paper, we sudy he predaor-prey sysem (1) wih Beddingon-DeAngelis funcional response and delays, in which no only he sage srucureonpreybualsohedelaydueodigesionis considered. More precisely, le a= ae d iτ 1 and assume ha x i () = a e dis x (s) ds (4) τ 1 holds. Then, based on similar conclusions for Lemma 3.1 in [1], we obain x i () = a e dis x (+s) ds, (5) τ 1 which means ha x i () depends on x() compleely. Thus, we can obain he following sysem, which is separaed from (1): x () =ax( ) bx 2 () y () = dy() + m 1 +m 2 x () +m 3 y (), fx ( τ 2 )y( τ 2 ) m 1 +m 2 x( τ 2 )+m 3 y( τ 2 ). I is obvious ha, for sudying he dynamics of he predaor-prey sysem (1) wih Beddingon-DeAngelis funcional response and delays, i is sufficien o sudy he dynamics of (6). The res of he organizaion of his paper is as follows. In Secion 2, we firs esablish he local sabiliy of he boundary equilibria of (6) by analyzing he characerisic equaions, and hen, we sudy he local asympoic sabiliy of he posiive equilibria of (6) byconsrucinganappropriae Lyapunov funcion. In Secion 3, we consider he global asympoic sabiliy of he boundary equilibrium and he posiive equilibrium, respecively. 2. Local Sabiliy of Equilibria In his secion, we sudy he local sabiliy of he equilibria of (6). By direc calculaions, we have ha he sysem (6)has he origin equilibrium E (, ) and he boundary equilibrium E 1 ((a/b), ). In order o invesigae he exisence of he posiive equilibrium, we consider he equaions (a bx) (m 1 +m 2 x+m 3 y) cy = d (m 1 +m 2 x+m 3 y) + fx = and obain he following resul. Theorem 1. The sysem (6) has a unique posiive equilibrium E (x,y ) if and only if he condiion (6) (7) (f dm 2 ) a b >dm 1 (8) holds, which is equivalen o he condiion (f dm 2 ) ae d iτ 1 b >dm 1. (9) Proof. We prove he exisence of he posiive equilibrium by analyzing (7). For he firs equaion of (7), i is easy o see ha is corresponding curves go hrough he poins (a/b, ), ( m 1 /m 2,),and(, am 1 /(c am 3 )) when c am 3 =. Furhermore, in case of ((c am 3 )/bm 3 ) =m 1 /m 2,hefirs equaionisahyperbolicequaionandhaswoasympoic lines x+((c am 3 )/bm 3 )=and y+(m 2 /m 3 )x + ((bm 1 m 3 cm 2 )/bm 2 3 ) =.If((c am 3)/bm 3 ) = m 1 /m 2,hefirs equaion can be simplified as (m 1 +m 2 x)(am 2 bm 2 x bm 3 y) =,whicharewolines.

3 Absrac and Applied Analysis 3 For he second equaion of (7), i implies a line which goes hrough he poins (, m 1 /m 3 ) and (dm 1 /(f dm 2 ), ) when dm 1 /(f dm 2 ) =. Their corresponding figures are as shown in Figures 1, 2,and3. Consequenly, by analyzing Figures 1, 2, and 3 and discussions, we obain ha (6)hasauniqueposiiveequilibrium. Condiion (9) indicaes ha he posiive equilibrium E exiss for all τ 1 in he inerval I=(,τ ),where τ = 1 d i ln a(f dm 2) bdm 1. (1) Furhermore, E will coincide wih E 1 when τ 1 = τ and here exiss no posiive equilibrium for τ 1 >τ. Now, we sudy he sabiliy of equilibria E (, ), E 1 ((a/b), ) and E (x,y ),respecively. A he beginning, we rewrie sysem (6) asz () = F(Z(), Z( τ 1 ), Z( τ 2 )),wherez() = (x(), y()), Z( τ 1 ) = (x( τ 1 ), y( τ 1 )), Z( τ 2 ) = (x( τ 2 ), y( τ 2 )). Then, for an arbirary fixed poin Z =(x,y),byleing P = ( F/ Z()) Z, Q = ( F/ Z( τ 1 )) Z,andR = ( F/ Z( τ 2 )) Z,wehave where P=( 2bx cq x cq y d ), Q = ( a ), X X R=( fq, x fq y )X q(x,y)= q x = q y = xy m 1 +m 2 x+m 3 y, y(m 1 +m 3 y) (m 1 +m 2 x+m 3 y) 2, x(m 1 +m 2 x) (m 1 +m 2 x+m 3 y) 2. (11) (12) Therefore, he characerisic equaion of he sysem (6) a he poin Z is as follows: P+Qe λτ 1 +Re λτ 2 λi = 2bx cq x +ae λτ 1 λ cq y fq x e λτ 2 d + fq y e λτ 2 λ = cy (m 2bx 1 +m 3 y) (m 1 +m 2 x+m 3 y) 2 +ae λτ 1 λ cx (m 1 +m 2 x) (m 1 +m 2 x+m 3 y) 2 fy (m 1 +m 3 y) (m 1 +m 2 x+m 3 y) 2 e λτ 2 d + fx (m 1 +m 2 x) (m 1 +m 2 x+m 3 y) 2 e λτ 2 λ =. (13) Thus, according o he characerisic equaion a he poin E, we obain he following resul. Theorem 2. The equilibrium E (, ) is unsable. Proof. The characerisic equaion of he sysem (6) a he poin E is P+Qe λτ 1 +Re λτ 2 λi = ae λτ 1 λ (,) d λ = (λ+d) (λ ae λτ 1 )=. (14) Obviously, λ= disanegaiveeigenvalueandheoher eigenvalue depends on he soluion of λ ae λτ 1 =. Noicing ha he line y=λand he curve y=ae λτ 1 mus inersec a a unique poin (λ, y),whereλ is a posiive value, we can obain ha E is a saddle. Thus, E is unsable. According o he characerisic equaion of he poin E 1, we have he following heorem. Theorem 3. The equilibrium E 1 ((a/b), ) is unsable if (f dm 2 )(a/b) > dm 1 and is locally asympoically sable if (f dm 2 )(a/b) < dm 1. Proof. The characerisic equaion of he sysem (6) a he poin E 1 is P+Qe λτ 1 +Re λτ 2 λi ((a/b),) = 2a + ae λτ 1 ac λ bm 1 +am 2 af d+ e λτ 2 λ bm 1 +am 2 =(λ+2a ae λτ 1 )(λ+d =. af bm 1 +am 2 e λτ 2 ) (15) For he equaion λ+2a ae λτ 1 =, i is clear ha he line y=λ+2aand he curve y=ae λτ 1 mus inersec a a unique poin (λ, y),whereλis a negaive value. And for he equaion

4 4 Absrac and Applied Analysis Figure 1: m 1 /m 2 < (c am 3 )/bm 3 <. Figure 3: (c am 3 )/bm 3 > where A=a, B=2bx + C= cx (m 1 +m 2 x ) (m 1 +m 2 x +m 3 y ) 2, E= cy (m 1 +m 3 y ) (m 1 +m 2 x +m 3 y ) 2, fx (m 1 +m 2 x ) (m 1 +m 2 x +m 3 y ) 2, D = d, (18) Figure 2: (c am 3 )/bm 3 < m 1 /m 2 <. λ+d (af/(bm 1 +am 2 ))e λτ 2 =, noicing ha he line y= λ+dand he curve y = (af/(bm 1 +am 2 ))e λτ 2 mus inersec a a unique poin (λ, y),wehavehaλ is posiive when (f dm 2 )(a/b) > dm 1 and is negaive when (f dm 2 )(a/b) < dm 1.Consequenly,E 1 is unsable if (f dm 2 )(a/b) > dm 1 and locally asympoically sable if (f dm 2 )(a/b) < dm 1. In he res of his secion, by consrucing an appropriae Lyapunov funcion, we sudy he sabiliy of he posiive equilibrium E (x,y ) under condiion (8). Le x () =x +u(), y () =y + V () ; hen, he linearizaion of he sysem (6) is u () =Au( ) Bu() CV (), V () = DV () +EV ( τ 2 )+Fu( τ 2 ), (16) (17) F= fy (m 1 +m 3 y ) (m 1 +m 2 x +m 3 y ) 2. Obviously, all he consans defined above are posiive, and he sysem (17)canberecasas u () = (A B) u () CV () A u (s) ds = (A B) u () CV () A [Au (s τ 1 ) Bu(s) CV (s)]ds V () = (E D) V () E V (s) ds + Fu () τ 2 F u (s) ds τ 2 = (E D) V () +Fu() F [Au (s τ 1 ) Bu(s) CV (s)]ds τ 2 E [ DV (s) +EV (s τ 2 )+Fu(s τ 2 )] ds. τ 2 (19)

5 Absrac and Applied Analysis 5 By consrucing an appropriae Lyapunov funcion, we have he following heorem. Theorem 4. Le τ 1 assume ha (,τ ) wih τ defined in (1), and τ 1,τ 2 (,τ ), F C < min {2 (D E),2(B A)}, (2) also holds, where 2 (D E) F C τ = min { AC + F (A+B+E) +2(CF+DE+E 2 ), 2 (B A) F C A (2A + 2B + C) +F(A+B+E) }. (21) Then, he equilibrium (, ) of he sysem (17) is locally asympoically sable, which implies ha he posiive equilibrium E (x,y ) of he sysem (6) is locally asympoically sable. Proof. To illusrae he equilibrium (, ) of he sysem (17) is locally asympoically sable, i is sufficien o sudy he exisence of a sric Lyapunov funcion. By leing W 11 () = u 2 () and differeniaing W 11 () wih respec o,wehave W 11 () =2(A B) u2 () 2Cu() V () 2Au() [Au (s τ 1 ) Bu(s) CV (s)]ds 2(A B) u 2 () 2Cu() V () +A(A+B+C) τ 1 u 2 () +A [Au 2 (s τ 1 )+Bu 2 (s) +CV 2 (s)]ds. (22) Leing W 12 () = A z [Au2 (s τ 1 )+Bu 2 (s)+cv 2 (s)]ds dz and differeniaing W 12 () wih respec o,wehave W 11 () +W 12 () 2(A B) u2 () 2Cu() V () +A(A+B+C) τ 1 u 2 () +Aτ 1 [Au 2 ( τ 1 )+Bu 2 () +CV 2 ()]. (23) Furhermore, by leing W 13 () = A 2 τ 1 u 2 (s)ds and W 1 () = W 11 () + W 12 () + W 13 (),wehave W 1 () [2(A B) +A(2A + 2B + C) τ 1]u 2 () +ACτ 1 V 2 () 2Cu() V (). (24) Nex, by leing W 21 () = V 2 () and differeniaing W 21 () wih respec o,weobain W 21 () =2(E D) V 2 () +2Fu() V () 2V () {F [Au (s τ 1 ) Bu(s) CV (s)] τ 2 + E [ DV (s) +EV (s τ 2 )+Fu(s τ 2 )]} ds 2(E D) V 2 () +2Fu() V () + [F (A+B+C) +E(D+E+F)] τ 2 V 2 () + {F [Au 2 (s τ 1 )+Bu 2 (s) +CV 2 (s)] τ 2 +E[DV 2 (s) +EV 2 (s τ 2 )+Fu 2 (s τ 2 )]} ds. (25) Leing W 22 () = τ 2 z {F[Au2 (s τ 1 )+Bu 2 (s) + CV 2 (s)] + E[DV 2 (s)+ev 2 (s τ 2 )+Fu 2 (s τ 2 )]}ds dz and differeniaing W 22 () wih respec o,wehave W 21 () +W 22 () 2(E D) V 2 () +2Fu() V () + [F (A+B+C) +E(D+E+F)] τ 2 V 2 () +{F[Au 2 ( τ 1 )+Bu 2 () +CV 2 ()] +E[DV 2 () +EV 2 ( τ 2 )+Fu 2 ( τ 2 )]} τ 2. (26) By leing W 23 () = AFτ 2 u 2 (s)ds, W 24 () = E 2 τ 2 V 2 (s)ds, and W 25 () = EFτ 2 u 2 (s)ds, we τ 2 have W 23 () =AFτ 2 [u 2 () u 2 ( τ 1 )], W 24 () =E2 τ 2 [V 2 () V 2 ( τ 2 )], W 25 () =EFτ 2 [u 2 () u 2 ( τ 2 )]. (27) By leing W 2 () = W 21 () + W 22 () + W 23 () + W 24 () + W 25 (), we have W 2 () {2(E D) +τ 2 [F (A+B+E) +2(CF+DE+E 2 )]} V 2 () +F(A+B+E) τ 2 u 2 () +2Fu() V (). (28)

6 6 Absrac and Applied Analysis Finally, we le W() = W 1 () + W 2 (). Obviously,W() and W() = if and only if u() = V() =. Besides, when u() = V() =, W () =.Furhermore, W () {2(E D) +ACτ 1 +τ 2 [F (A+B+E) +2(CF+DE+E 2 )]} V 2 () +2(F C) u () V () +[2(A B) +A(2A + 2B + C) τ 1 +F(A+B+E) τ 2 ] u 2 () {2(E D) +ACτ 1 +τ 2 [F (A+B+E) +2(CF+DE+E 2 )]} V 2 () +2 F C u () V () +[2(A B) +A(2A + 2B + C) τ 1 +F(A+B+E) τ 2 ] u 2 () {2(E D) + F C +ACτ 1 +τ 2 [F (A+B+E) +2(CF+DE+E 2 )]} V 2 () +[2(A B) + F C +A(2A + 2B + C) τ 1 +F(A+B+E) τ 2 ]u 2 (). (29) Based on he condiions in Theorem 4, wehavew () < if u 2 () + V 2 () >.Thus,W() is a sric Lyapunov funcion. According o he Lyapunov sabiliy heorem, he equilibrium (, ) of he sysem (17) is locally asympoically sable, which implies ha he posiive equilibrium E (x,y ) of he sysem (6) is locally asympoically sable. 3. Global Sabiliy of Equilibria In his secion, we sudy he global asympoic sabiliy of he boundary equilibrium E 1 ((a/b), ) and he posiive equilibrium E (x,y ), respecively. We firs give wo lemmas which will be used laer. Lemma 5 (see [1, Lemma 3.2]). For a given sysem x () =b 1 x ( τ) a 1 x () a 2 x 2 (), (3) where b 1 >, a 1, a 2 >, τ,andx() > for all τ,wehave (i) lim + x() = (b 1 a 1 )/a 2 if and only if b 1 >a 1 ; (ii) lim + x() = ifandonlyifb 1 a 1. Lemma 6 (see [11, Lemma 3.2]). Le S={(x,y):x>,y> } and S={(x,y):x,y }. Then, he ses S and S are boh invarian ses. BasedonLemmas5 and 6, we prove ha he boundary equilibrium E 1 is globally aracive. Theorem 7. Le (x(), y()) be he soluion of (6) wih posiive iniial condiion in S. (i) If (f dm 2 )(a/b) < dm 1,henE 1 ((a/b), ) is globally aracive. (ii) If (f dm 2 )(a/b) > dm 1,henE 1 ((a/b), ) is unsable. Proof. If (f dm 2 )(a/b) < dm 1, we firs prove ha lim + y() =. Due o he firs equaion (6), here is x () ax( ) bx 2 (). (31) In view of Lemma 6 and by considering he following comparison equaion p () =ap( ) bp 2 (), p() =x() >, (32) we have x() p() for all. In addiion, by Lemma 5, we obain ha lim + p() = (a/b). Thus, here exiss a sufficienly small posiive consan ε wih (f dm 2 )((a/b) + ε) < dm 1 such ha, for his ε, here exiss a T ε >such ha x() < (a/b) + ε for all >T ε.subsiuingiinohesecond equaion of (6), we have y () dy() + f ((a/b) +ε) m 1 +m 2 ((a/b) +ε) y( τ 2), (33) for all > T ε. Considering he following comparison equaion: q f ((a/b) +ε) () = dq() + m 1 +m 2 ((a/b) +ε) q( τ 2), q(t ε +τ 2 )=y(t ε +τ 2 )>, and leing q() = e λ,wehave e λ [λ+ d (34) f ((a/b) +ε) m 1 +m 2 ((a/b) +ε) e λτ 2 ]=. (35) Then he line y=λ+dand he curve y = (f((a/b)+ε)/(m 1 + m 2 ((a/b) + ε)))e λτ 2 mus inersec a a unique poin (λ, y). Since (f dm 2 )((a/b) + ε) < dm 1,wecanobainhad> f((a/b)+ε)/(m 1 +m 2 ((a/b)+ε)).thus,wededucehaλ<. Therefore, lim + q() = lim + e λ =. Obviously, based on he comparison heorem, we ge ha y() q() for all T+τ 2.Thus,lim + y() =. Nex, we prove ha x() (a/b) as +.

7 Absrac and Applied Analysis 7 Since lim + y() =, foranygivenε 1 (, (a/b)), here exiss a T 1 >such ha <y()<(bm 1 /2c)ε 1 for all T 1.Thus,wehave ax ( τ 1 ) bx 2 () x () for all T 1.Thais, =ax( ) bx 2 () ax ( τ 1 ) bx 2 () x () m 1 +m 2 x () +m 3 y () ax( ) bx 2 () m 1 ax( ) bx 2 () bε 1 2 x (), (36) ax( ) bx 2 () bε 1 2 x (). (37) Then, we consider he following comparison equaion: p () =a p( ) b p 2 () bε 1 2 p(t 1 )=x(t 1 )>. p (), (38) Since a > bε 1, based on Lemma 5, we have ha lim + p() = (a/b) (ε 1 /2). Addiionally, we have ha p() x() p() for all T 1. Since lim + p() = (a/b),forheaboveε 1, here exiss a T 2 >such ha p() (a/b) + ε 1 for all >T 2.Similarly, since lim + p() = (a/b) (ε 1 /2) for he above ε 1,wealso ge ha here exiss a T 3 >such ha p() (a/b) + (ε 1 /2) > (ε 1 /2) for all >T 3. Therefore, if we le T = max{t 1,T 2,T 3 }, here is ε 1 < x() (a/b) < ε 1 for all >T.Thus,wehave lim + x() = (a/b). A las, we prove he second par of his heorem. We can assure ha (f dm 2 )(a/b) > dm 1 and ry o derive a conradicion. If (f dm 2 )(a/b) > dm 1,hen(6) hasa unique posiive equilibrium (x,y ) by Theorem 1, whichis also a soluion of (6); ha is, (x,y )=(x (), y ()) saisfies (6) and saisfies lim + (x (), y ()) = ((a/b), ) by he above argumens, which conradics he fac ha (x,y ) is a posiive equilibrium. Thus, conclusion (ii) holds. This complees he proofs. Remark 8. In he above heorem, we derive he global araciveness of he boundary equilibrium E 1 ((a/b), ). Unforunaely, we jus obain a sufficien condiion ha (f dm 2 )(a/b) < dm 1 o ensure ha E 1 is globally asympoically sable. If (f dm 2 )(a/b) = dm 1,sincehesecondequaion of (6) is a delay diffusion equaion, we canno derive ha lim + y() = by using comparison heorem. We leave iforafurhersudy. Furhermore, for he firs equaion of (1), we can solve explicily as lim x + i () = a2 (e diτ1 1). (39) bd i Thus, by Theorem 7, we can obain he following corollary direcly. Corollary 9. Le (x i (), x(), y()) be he soluion of (1) wih heposiiveiniialcondiion.iff dm 2,hen lim (x + i (),x(),y()) =( a2 (e d iτ 1 1), a,). (4) bd i b In he res of his secion, we will sudy he global araciveness of he posiive equilibrium E and obain he following resul. Theorem 1. Assume ha (8) and he following condiions am 3 >c, f(am 3 c) >d, (41) bm 1 m 3 +m 2 (am 3 c) hold, and le (x(), y()) be he soluion of (6) wih he posiive iniial condiion in S such ha lim + (x(), y()) = E. Then,heposiiveequilibriumE of (6) is globally aracive. Proof. Provided ha condiions (8) and(41) hold,basedon comparison heorem and by similar argumens in [11], we can obain ha, for any soluion (x(), y()) of (6) wihhe posiive iniial condiion in S, here exiss a poin E (x, y ) such ha lim + (x(), y()) = (x, y ). Indeed, jus by modifying he coefficiens of (6),hediscussionin[11] also holds. We omi he proof here and refer o [11] for deails. Then, in view of he propery of he ω-limi se in [17] and by he uniqueness resul of he posiive equilibrium in Theorem 1, wehavee = E. Thus, we derive ha E is globally aracive. 4. Conclusion In he presen paper, we considered he sabiliy of he equilibria of a predaor-prey sysem wih Beddingon-DeAngelis funcional response and delays. More precisely, we derived a sufficien and necessary condiion for he exisence of a unique posiive equilibrium and proved ha he posiive equilibrium is locally asympoically sable under a sufficien condiion. Also, we esablished he local sabiliy of he boundary equilibria by analyzing he characerisic equaions. Finally, by using comparison heorem and he ω-limi se heory, we sudied he global asympoic sabiliy for boh he boundary equilibrium and he posiive equilibrium. In paricular, in Theorem 7, we derived he global araciveness of he boundary equilibrium E 1 ((a/b), ). Bu unforunaely, we jus obained a sufficien condiion ha (f dm 2 )(a/b) < dm 1 o ensure ha E 1 was globally asympoically sable. If (f dm 2 )(a/b) = dm 1,sincehe second equaion of (6) was a delay diffusion equaion,we

8 8 Absrac and Applied Analysis could no derive ha lim + y() = by using comparison heorem. Moreover, as poined ou in [11], here are wo ineresing problems abou sysem (1), which are relaed o he phenomena of bifurcaions and he exisence of periodic orbis of (1). We leave hem for a furher sudy. Conflic of Ineress The auhors declare ha here is no conflic of ineress regarding he publicaion of his paper. Acknowledgmens The firs auhor was parially suppored by NSFC (112142, ) and NSF of Shandong Province of China (ZR21AQ6). The second auhor was parially suppored by Graduae Innovaion Foundaion of Yanai Universiy (GIFYTU). References [12] C.-Y. Huang, Y.-J. Li, and H.-F. Huo, The dynamics of a sage-srucured predaor-prey sysem wih impulsive effec and Holling mass defence, Applied Mahemaical Modelling, vol. 36, no. 1, pp , 212. [13] H. Li and Y. Takeuchi, Dynamics of he densiy dependen predaor-prey sysem wih Beddingon-DeAngelis funcional response, Mahemaical Analysis and Applicaions, vol. 374, no. 2, pp , 211. [14] H. Li, Persisence in non-auonomous Loka-Volerra sysem wih predaor-prey raio-dependence and densiy dependence, Applied Mahemaics, vol. 2, no. 9, pp , 211. [15] H. Li and R. Zhang, Sabiliy and opimal harvesing of a delayed raio-dependen predaor-prey sysem wih sage srucure, JournalofBiomahemaics, vol. 23, no. 1, pp. 4 52, 28. [16] H. Xiang, L.-X. Feng, and H.-F. Huo, Sabiliy of he virus dynamics model wih Beddingon-DeAngelis funcional response and delays, Applied Mahemaical Modelling, vol. 37, no.7,pp ,213. [17] W. Hahn, Sabiliy of Moion, Springer, New York, NY, USA, [1] J. R. Beddingon, Muual inerference beween parasies or predaors and is effec on searching efficiency, Animal Ecology,vol.44,pp ,1975. [2] D.L.DeAngelis,R.A.Goldsein,andR.V.O Neil, Amodelof rophic ineracion, Ecology, vol. 56, pp , [3] L. Shaikhe, Sabiliy of a posiive poin of equilibrium of one nonlinearsysemwihafereffecandsochasicperurbaions, Dynamic Sysems and Applicaions, vol.17,no.1,pp , 28. [4] L. Shaikhe, Lyapunov Funcionals and Sabiliy of Sochasic Funcional Differenial Equaions, Springer, New York, NY, USA, 213. [5] S.B.Hsu,S.P.Hubbell,andP.Walman, Compeingpredaors, SIAM Journal on Applied Mahemaics, vol.35,no.4,pp , [6] H.-F. Huo, W.-T. Li, and J. J. Nieo, Periodic soluions of delayed predaor-prey model wih he Beddingon-DeAngelis funcional response, Chaos, Solions & Fracals, vol. 33, no. 2, pp ,27. [7] H.-F. Huo, Periodic soluions for a semi-raio-dependen predaor-prey sysem wih funcional responses, Applied Mahemaics Leers, vol. 18, no. 3, pp , 25. [8] H.-F. Huo, Permanence and global araciviy of delay diffusive prey-predaor sysems wih he Michaelis-Menen funcional response, Compuers & Mahemaics wih Applicaions, vol. 49, no. 2-3, pp , 25. [9] T.-W. Hwang, Global analysis of he predaor-prey sysem wih Beddingon-DeAngelis funcional response, Mahemaical Analysis and Applicaions,vol.281,no.1,pp , 23. [1] S. Liu, L. Chen, and Z. Liu, Exincion and permanence in nonauonomous compeiive sysem wih sage srucure, Mahemaical Analysis and Applicaions,vol.274,no. 2, pp , 22. [11] Z. She and H. Li, Dynamics of a densiy-dependen sagesrucured predaor-prey sysem wih Beddingon-DeAngelis funcional response, Mahemaical Analysis and Applicaions,vol.46,no.1,pp ,213.

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