COnsider the mutualism model

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1 IAENG Inernaional Journal of Applied Mahemaics, 46:2, IJAM_46_2_9 The Properies of Sochasic Muualism Model wih Time-lagged Delays Huizhen Qu, Xiaorong Gan and Tianwei Zhang Absrac Sufficien condiions are gained for almos sure permanence, global asympoic sabiliy and mean square period of he sochasic muualism model wih ime-lagged delays dn () = r ()N () ()+α ()N 2 (τ 2 ) +N 2 (τ 2 N ) () d +σ N ()db, dn 2() = r 2()N 2() 2 ()+α 2 ()N (τ ) +N (τ N ) 2() d +σ 2N 2()dB 2, where r i(), i(), α i() C(R, R + ) and α i() > i(), i =, 2. This paper implies ha under he condiion 2 σ2 i < r i k i, i =, 2, he inensiy of whie noise has a negaive impac on almos sure permanence, bu in any case, i makes no difference on global asympoic sabiliy. And he sysem is mean square periodic if ω is he period of r i(), i(), α i(), i =, 2. Index Terms Sochasic muualism model; Almos sure permanence; Iô formula; Global asympoic sabiliy; Mean square period. I. INTRODUCTION COnsider he muualism model dn () d = r N () +α N 2() +N 2() N (), dn 2() d = r 2 N 2 () 2+α 2N () +N () N 2 (), (.) where α i, i, r i R + are consans and α i > i, i =, 2. Couning on he naure of i (i =, 2), we classify sysem (.) as faculaive, obligae or a combinaion of boh. We refer o Dean, Boucher2, Vandermeer and Boucher3, Wolin and Lawlor4, and Boucher e al.5 for more deails of muualisic ineracions. A modificaion of sysem (.) leads o he ime-lagged model dn () d dn 2() d = r N () +α N 2(τ 2) +N 2(τ 2) N () = r 2 N 2 () 2+α 2N (τ ) +N (τ ) N 2 (),, (.2) where τ, τ 2, ) are consans. The cooperaive or muualisic effecs in sysem (.2) are no immediaely realized, bu happened wih ime goes on. However, due Manuscrip received June, 25; revised November 8, 25. Huizhen Qu is wih Deparmen of Applied Mahemaics of unming Universiy of Science and Technology, unming 6593, China. ( @qq.com). Xiaorong Gan is wih Deparmen of Applied Mahemaics of unming Universiy of Science and Technology, unming 6593, China. ( @qq.com). Tianwei Zhang is wih Ciy college of unming Universiy of Science and Technology, unming 655, China. ( @qq.com). Correspondence auhor: Tianwei Zhang. (zhang@kmus.edu.cn). o environmenal noise, may poin ou ha he random flucuaion6 should be showed by he rae of growh in he muualism model. Assume ha environmenal noise disurbs he growh rae r wih r r + σḃ, where σ 2 is he inensiy of whie noise and B is a sandard Brownian moion. Then we obain he sochasic model: dn () = r N () +α N 2(τ 2) +N 2(τ 2) N () d +σ N ()db, dn 2 () = r 2 N 2 () 2+α 2N (τ ) +N (τ ) N 2 () d +σ 2 N 2 ()db 2. Acually, he naural growh rae of many populaions varies wih, such as, due o he emperaure. Therefore i is significan and reasonable o consider he sochasic nonauonomous logisic model ()+α ()N 2(τ 2) +N 2(τ 2) dn () = r ()N () N () d + σ N ()db, dn 2 () = r 2 ()N 2 () N 2 () d + σ 2 N 2 ()db 2, 2()+α 2()N (τ ) +N (τ ) (.3) where α i (), i (), r i () C(R, R + ) and α i () > i (), i =, 2. In recen years, Eq.(.3) has been researched inensively, see e.g.7, 8, 9,,. I is well-know ha, in mahemaical ecology, permanence is a very imporan and ineresing subjec, which means ha a populaion sysem will survive forever. Generally speaking, a definiive populaion sysem is permanen, if a sysem has he following propery < N x i() lim sup x i () M <, while i =, 2,..., n. On he oher hand, sudies on muualism model no only involve permanence bu also involve oher dynamic behaviors such as sabiliy and periodiciy. In recen years, on he basis of permanence resul, many scholars sudied he global asympoic sabiliy and he posiive periodic soluions of some kinds of nonlinear ecosysems by using periodic heory. For more deails we refer o 2, 3, 4, 5, 6 and he references herein. However, according o nowaday s lieraure, here are few people obain he permanence of sysem (.3). Therefore, he main purpose of his paper is o esablish some new sufficien condiions for he global asympoic sabiliy and posiive periodic soluions of sysem(.3). (Advance online publicaion: 4 May 26)

2 IAENG Inernaional Journal of Applied Mahemaics, 46:2, IJAM_46_2_9 Since all species suffer from he flucuaion of he environmen such as food supplies, harves and seasonal effecs of weaher ec. So i is usual o assume he periodic parameers in he sysem. However, in applicaion, if he various consiuen componens of he emporally nonuniform environmen are in commensurable periods, hen one has o consider he emporally environmen o be mean square periodic. Hence, if we consider he effecs of he environmen facors, mean square periodiciy is someimes more realisic and more general han periodiciy. Recenly, here are many papers dealing wih periodic soluions 7, 8, 9, 2, 2, 22, 23 and he references herein. I deserves o be menioned ha here have no resuls on muualism model wih mean square periodic soluions. We se where i =, 2. r + i = sup r i (), r i = inf r i(), (, ) (, ) k + i = sup k i (), k i = inf k i(), (, ) (, ) α + i = sup α i (), α i = inf α i(), (, ) (, ) Definiion. A sochasic populaion sysem is said o be almos surely sochasically permanen if for any iniial value x R n +, he soluion x() = (x (), x 2 (),..., x n ()) T has he propery < N i while i =, 2,..., n. x i() lim sup x i () M i <, Lemma. 24 Brownian moion saisfies he law of ieraed B() logarihm, ha is lim =, α > α 2. Remark. Seing α =, we ge lim =. Therefore, for ɛ >, i exiss posiive consan T such ha B() < ɛ for all > T. B() Remark 2. By Remark and he coninuiy of Brownian moion, for ɛ >, here exiss l R +, such ha B ɛ + l, R +. In Secion 2, hrough he prove of Lemma 2 and Lemma 3, we yield he Theorem, i.e., assuming 2 σ2 i < r i k i, i =, 2, almos sure permanence of he sochasic muualism model is considered, in Secion 3, we sudy he global asympoic sabiliy of sysem (.3), in Secion 4, we discuss he sysem s mean square period, in Secion 5, we give an example o illusrae he main resuls in he secion 2 and 3. Finally, we close he paper wih conclusions. II. PERMANENCE Lemma 2. If 2 σ2 i < r + i α+ i, i =, 2, hen he soluion o Eq.(.3) saisfies he following inequaliies lim sup N () r α 2 σ2 r lim sup N 2 () r 2 α 2 2 σ2 2 r2 := M, := M 2. Proof: Denoe N () = x (), by Iô formula o he firs equaion of Eq.(.3), we obain dx () = N 2()dN () + N 3()(dN ()) 2 = N () r () + α ()N 2 ( τ 2 ) () + N 2 ( τ 2 ) N () d σ N () db + σ 2 N () d Thus Seing = r ()x () () + α ()N 2 ( τ 2 ) d + N 2 ( τ 2 ) +r ()d σ x ()db + σx 2 ()d r ()α ()x ()d + r ()d σ x ()db +σ 2 x ()d. dx () + (r ()α () σ 2 )x ()d + σ x ()db r ()d. (2.) dx () + r ()α () σ 2 x ()d + σ x ()db =. We rewrie he above equaion as x () dx () = (r ()α () + σ 2 )d σ db. (2.2) By Iô formula, i follows d ln x () = x () dx () 2 x 2 ()(dx ()) 2. (2.3) From (2.2) and (2.3), i leads Then x () dx () = d ln x () + 2 σ2 d = (r ()α () + σ 2 )d σ db. d ln x () = (r ()α () + 2 σ2 )d σ db. Inegraing boh sides from o ges x () = x () exp r (u)α (u)du + 2 σ2 σ B }. (2.4) From (2.) and (2.4), i yields dx () exp r (u)α (u)du } 2 σ2 + σ B r () exp r (u)α (u)du 2 σ2 + σ B }d.(2.5) Since B is Brown moion, lim B =. So here exiss ɛ > small enough and T = T (ɛ) > such ha B ɛ, T. Leing T X (T ) = x (T ) exp r (u)α (u)du 2 σ2 T +σ B T }. (Advance online publicaion: 4 May 26)

3 IAENG Inernaional Journal of Applied Mahemaics, 46:2, IJAM_46_2_9 Hence, inegraing (2.5) from T o leads o x () exp r (u)α (u)du } 2 σ2 + ɛσ x () exp r (u)α (u)du } 2 σ2 + σ B x () exp r (u)α (u)du } 2 σ2 + σ B X (T ) + r (s) exp r (u)α (u)du T 2 σ2 s + σ B s }ds X (T ) + r (s) exp r (u)α (u)du T } 2 σ2 s σ B s ds X (T ) + r (s) exp r (u)α (u)du T } 2 σ2 s ɛσ s ds, which yields x () X (T ) exp r (u)α (u)du + 2 σ2 } ɛσ + r (s) exp r (u)α (u)du T } 2 σ2 (s ) ɛσ ( + s) ds X (T ) exp ( } 2 σ2 r + α+ ) ɛσ +r exp r α (s ) T 2 σ2 (s ) } ɛσ (s + ) ds Leing follows x () lim X (T ) exp ( 2 σ2 r + α+ } ɛσ ) + r expr α (s ) T } 2 σ2 (s ) ɛσ (s + )}ds r α (s ) Consequenly = lim r = lim sup N () = exp T } 2 σ2 (s ) ds r r α. 2 σ2 x () r α 2 σ2 r := M. By he same way, we ge lim sup N 2 () r 2 α 2 2 σ2 2 r2 This complees he proof. := M 2. In he following, we give a crucial assumpion for he permanence of sysem (.3): (H ) r i i > 2 σ2 i, i =, 2. From (H ), here exiss ɛ > small enough, such ha r i i 2 σ2 i ɛ σ i >, i =, 2. By Remarks -2, here mus exis T > and l i > such ha B i ɛ for all T, B i ɛ + l i for all, where l i := sup B is, i =, 2. s,t Lemma 3. If (H ) holds, hen he soluion o Eq.(.3) saisfies he following inequaliies N () r+ + 2 σ2 ɛ σ r + eσl := N, N 2() r σ2 2 ɛ σ 2 r + 2 eσ2l2 := N 2. Proof: Denoe N () = x (), by Iô formula o he firs equaion of Eq.(.3), i leads Thus dx () = N 2()dN () + N 3()(dN ()) 2 = N () r () + α ()N 2 ( τ 2 ) () + N 2 ( τ 2 ) N () d σ N () db + N () σ2 d N () r () () N ()d σ N () db + N () σ2 d = r () ()x ()d + r ()d + σ 2 x ()d σ x ()db. dx () + r () () σ 2 x ()d + σ x ()db r ()d. Tha is dx () exp r (u) (u)du } 2 σ2 + σ B r () exp r (u) (u)du 2 σ2 + σ B }d. Inegraing boh sides from T o obains x () exp r (u) (u)du } 2 σ2 + σ B X (T ) r (s) exp r (u) (u)du T 2 σ2 + σ B s }ds, (Advance online publicaion: 4 May 26)

4 IAENG Inernaional Journal of Applied Mahemaics, 46:2, IJAM_46_2_9 where T X (T ) = x (T ) exp r (u) (u)du 2 σ2 T +σ B T }. By Remark 2, i follows ha x () X (T ) exp r (u) (u)du + 2 σ2 } σ B + r (s) exp r (u) (u)du T } 2 σ2 (s ) + σ (B s B ) ds X (T ) exp ( } 2 σ2 r ) + ɛ σ +r + exp(r eσl + + T 2 σ2 ɛ σ )(s )ds } = X (T ) exp 2 σ2 r + ɛ σ r + + eσl r σ2 ɛ σ exp (r + + } 2 σ2 ɛ σ )( T ). Leing, we ge Consequenly r + lim sup x () eσl r σ2 ɛ. σ N () = lim sup By he same way, we ge x () r+ + 2 σ2 ɛ σ r + eσl := N. N 2() r σ2 2 ɛ σ 2 r + 2 eσ2l2 := N 2. This complees he proof. According o Lemma 2 and Lemma 3, we obain he following heorem. Theorem. If (H ) holds, hen he soluion o Eq. (.3) is almos surely sochasically permanen, ha is, N =: r+ + 2 σ2 ɛ σ r + eσl lim sup N () r α 2 σ2 r N 2 =: r σ2 2 ɛ σ 2 r + 2 eσ2l2 lim sup N 2 () r 2 α 2 2 σ2 2 r2 N () := M, (2.6) N 2() III. GLOBAL ASYMPTOTIC STABILITY := M 2. (2.7) Theorem 2. Assume ha (H 2 ) here exiss wo posiive consan λ and λ 2 such ha Γ = λ r λ 2r + 2 (+ 2 + α+ 2 ) >, Γ 2 = λ 2 r 2 λ r + (+ + α+ ) >. Then sysem (.3) is globally asympoically sable. Proof: Assuming (N (), N 2 ()) T and ( N (), N 2 ()) T are any wo soluions of Eq.(.3). Le (y, y 2 ) T = (ln N (), ln N 2 ()) T and (ȳ, ȳ 2 ) T = (ln N (), ln N 2 ()) T. Denoe y () = ln N (), by Iô formula o he firs equaion of sysem (.3), we obain dy () = N () dn () 2 N 2()(dN ()) 2 = N () dn () 2 σ2 d () + α ()N 2 ( τ 2 ) = r () N () d + N 2 ( τ 2 ) +σ db 2 σ2 d, by he same way, i ransforms sysem (.3) ino he following sysem, dy () = r () ()+α ()N 2(τ 2) +N 2(τ 2) N () d +σ db 2 σ2 d, dy 2 () = r 2 () 2()+α 2()N (τ ) +N (τ ) N 2 () d +σ 2 db 2 2 σ2 2d, dȳ () = r () ()+α () N 2(τ 2) + N 2(τ 2) N (3.) () d +σ db 2 σ2 d, dȳ 2 () = r 2 () 2()+α 2() N (τ ) + N (τ ) N 2 () d +σ 2 db 2 2 σ2 2d. Define V () = V () + V () + V 2 (), (3.2) where V () = λ y () ȳ () + λ 2 y 2 () ȳ 2 (), V () = λ 2 r 2 + (+ 2 + α+ 2 ) N (s) N (s) ds, τ V 2 () = λ r + (+ + α+ ) N 2(s) N 2 (s) ds. τ 2 Calculaing he upper righ derivaive of V () along sysem (3.), D + V () = λ sgny () ȳ ()y () ȳ () +λ 2 sgny 2 () ȳ 2 ()y 2() ȳ 2() λ r () N () N () +λ r () () + α () N 2 ( τ 2 ) N 2 ( τ 2 ) λ 2 r 2 () N 2 () N 2 () +λ 2 r 2 () 2 () + α 2 () N ( τ ) N ( τ ) λ r N () N () +λ r + (+ + α+ ) N 2( τ 2 ) N 2 ( τ 2 ) λ 2 r2 N 2() N 2 () +λ 2 r 2 + (+ 2 + α+ 2 ) N ( τ ) N ( τ ). (3.3) (Advance online publicaion: 4 May 26)

5 IAENG Inernaional Journal of Applied Mahemaics, 46:2, IJAM_46_2_9 Furher, calculaing he upper righ derivaive of V (), V 2 () along sysem (3.), i follows ha D + V () = λ 2 r 2 + (+ 2 + α+ 2 ) N () N () λ 2 r 2 + (+ 2 + α+ 2 ) N ( τ ) N ( τ ), (3.4) D + V 2 () = λ r + (+ + α+ ) N 2() N 2 () λ r + (+ + α+ ) N 2( τ 2 ) N 2 ( τ 2 ). (3.5) Togeher wih (3.2) (3.5), for R, we ge D + V () λ r + λ 2r + 2 (+ 2 + α+ 2 ) N () N () +λ 2 r 2 + λ r + (+ + α+ ) N 2() N 2 () Γ N () N () Γ 2 N 2 () N 2 (), Hence, for R, V () is nonincreasing, inegraing he above formula from o yields V () + Γ N (s) N (s) ds +Γ 2 N 2 (s) N 2 (s) ds V () < +,, implies ha, ha is, N (s) N (s) ds < +, N 2 (s) N 2 (s) ds < +, lim N (s) N (s) = lim N 2(s) N 2 (s) =. s + s + This complees he proof. Remark 3. The heorem illusraes ha he inensiy of whie noise has a negaive impac on almos sure permanence, bu i makes no difference on global asympoic sabiliy. IV. PERIODIC SOLUTION In his secion, we assume ha (H 3 ) here exiss a posiive consan ω such ha (H 4 ) max A+B r α A = 2 r i ( + ω) = r i (), α i ( + ω) = α i (), i =, 2. r α +σ 2 r 2 α 2, A2+B2 r 2 α 2, i ( + ω) = i (), } <, where 4r + (+ α )2 + 4r + (M + ɛ) 2 B = 8 r r α + (M + ɛ)( + α )2, A 2 = 2 4r 2 + (+ 2 α 2 )2 + 4r 2 + (M 2 + ɛ) 2 +σ 2 2, B 2 = 8 r2 r α 2 + (M 2 + ɛ)( 2 + α 2 )2. 2 Definiion 2. A funcion f is called mean square periodic if here exiss a posiive consan ω such ha E N ( + ω) N () 2 =, E N 2 ( + ω) N 2 () 2 =, R. Theorem 3. Assume ha (H 3 ) and (H 4 ) hold, and for any ɛ >, here exiss > T, such ha N i () < M i +ɛ, i =, 2, hen sysem (.3) is mean square periodic, ha is E N ( + ω) N () 2 =, E N 2 ( + ω) N 2 () 2 =. Proof: From he firs equaion of sysem (.3), we ge () + α ()N 2 ( τ 2 ) dn () = r ()N () + N 2 ( τ 2 ) N () d + σ N ()db where Therefore So = r ()α ()N ()d + σ N ()db +r () () α () + N 2 ( τ 2 ) N ()d r ()N 2 ()d = r ()α ()N ()d + f()d + σ N ()db, f() = r () () α () + N 2 ( τ 2 ) N () r ()N 2 (). dn ( + ω) = r ( + ω)α ( + ω)n ( + ω)d dn ( + ω) dn () +f( + ω)d + σ N ( + ω)db (+ω) = r ()α ()N ( + ω)d + f( + ω)d +σ N ( + ω)db (+ω). = r ()α ()N ( + ω) N ()d + f( + ω) f()d +σ N ( + ω)db (+ω) N ()db. (4.) Seing Y i () = N i ( + ω) N i (), i =, 2, from (4.) we ge dy () = r ()α ()Y ()d + f( + ω) f()d +σ N ( + ω)d B N ()db, where B = B +ω B = B B d = B. Then, i follows dy ()e r(s)α(s)ds = e r(s)α(s)ds f( + ω) f()d +σ e r(s)α(s)ds N ( + ω)d B N ()db. Inegraing boh sides from o T ges = Y (T )e T r (s)α (s)ds Y ()e r(s)α(s)ds T +σ T e s r(u)α(u)du f(s + ω) f(s)ds N (s)db s, e s r(u)α(u)du N (s + ω)d B s (Advance online publicaion: 4 May 26)

6 IAENG Inernaional Journal of Applied Mahemaics, 46:2, IJAM_46_2_9 ha is Y () = Y (T )e T r (s)α (s)ds T T N (s)db s. Leing T +, we obain Y () = N (s)db s. e s r(u)α(u)du f(s + ω) f(s)ds σ e s r(u)α(u)du N (s + ω)d B s e s r(u)α(u)du f(s + ω) f(s)ds σ e s r(u)α(u)du N (s + ω)d B s Using Hölder inequaliy and isomeric ransformaion, i follows E Y () 2 = E e s r(u)α(u)du f(s + ω) f(s)ds +σ e s r(u)α(u)du N (s + ω)d B s 2 N (s)db s 2E +2E σ N (s)db s 2E e s r(u)α(u)du f(s + ω) f(s)ds e s r(u)α(u)du N (s + ω)d B s 2 e s r(u)α(u)du ds f(s + ω) f(s) 2 ds + 2σ 2 E 2 r α where + 2σ2 r α 2 e s r(u)α(u)du e s r(u)α(u)du ds e s r(u)α(u)du N (s + ω) N (s) 2 ds e r α (s) Ef(s + ω) f(s) 2 ds f( + ω) f() 2 = r ()( () α ()) e r α (s) EY 2 (s)ds, (4.2) N ( + ω) + N 2 ( + ω τ 2 ) r ()N 2 ( + ω) N 2 () N () + N 2 ( τ 2 ) 2r ()( () α ()) 2 N ( + ω) + N 2 ( + ω τ 2 ) 2 + 2r()N 2 2 ( + ω) N 2 () 2 N () + N 2 ( τ 2 ) 2r + (+ N ( + ω) α )2 + N 2 ( + ω τ 2 ) } 2 2 N () + 2(r + + N 2 ( τ 2 ) )2 N 2 ( + ω) N 2 () 2. (4.3) Since 2 N ( + ω) + N 2 ( + ω τ 2 ) N () + N 2 ( τ 2 ) = + N 2 ( + ω τ 2 ) (N ( + ω) N ()) N () ( + ξ) 2 N 2( + ω τ 2 ) N 2 ( τ 2 ) N ( + ω) N () N ()N 2 ( + ω τ 2 ) 2 N 2 ( τ 2 ) 2N ( + ω) N () 2 +2N 2 ()N 2 ( + ω τ 2 ) N 2 ( τ 2 ) 2 = 2Y 2 () + 2(M + ɛ) 2 Y 2 2 ( τ 2 ), (4.4) where ξ is beween N 2 ( + ω τ 2 ) and N 2 ( τ 2 ), and N 2 ( + ω) N 2 () = N ( + ω) + N ()N ( + ω) N () 2(M + ɛ)n ( + ω) N () = 2(M + ɛ)y (). (4.5) From (4.3)-(4.5) we ge f( + ω) f() 2 2r + (+ α )2 2Y 2 () + 2(M + ɛ) 2 Y2 2 ( τ 2 ) +2(r + )2 2(M + ɛ)y () 2 = 4r + (+ α )2 + 4r + (M + ɛ) 2 Y 2 () +4r + (M + ɛ)( + α )2 Y 2 2 ( τ 2 ). (4.6) From (4.2) and (4.6) i leads E Y () 2 2 r α e r α (s) E 4r + (+ α )2 + 4r + (M + ɛ) 2 Y 2 () } +4r + (M + ɛ)( + α )2 Y2 2 ( τ 2 ) ds + 2σ2 r α = A +B By he same way, we obain Seing E Y 2 () 2 A 2 +B 2 X = e r α (s) EY 2 (s)ds e r α (s) EY 2 (s)ds 2 e r α (s) EY 2 2 (s τ 2 )ds. (4.7) e r 2 α 2 (s) EY 2 2 (s)ds e r 2 α 2 (s) EY 2 (s τ )ds. (4.8) max EY 2 (s), EY2 2 (s)}, T s + (Advance online publicaion: 4 May 26)

7 IAENG Inernaional Journal of Applied Mahemaics, 46:2, IJAM_46_2_9 from (4.7) and (4.8), we yield E Y () 2 A e r α (s) dsx +B e r α (s) dsx = A + B r X, (4.9) α E Y 2 () 2 A 2 e r 2 α 2 (s) dsx From (4.9) and (4.), we have ha is By (H 4 ) we ge hus ha is +B 2 e r 2 α 2 (s) dsx = A 2 + B 2 r2 X, (4.) α 2 max T + E Y () 2, E Y 2 () 2 } max A + B r α X max A + B r α This complees he proof. X =,, A 2 + B 2 r2 }X, α 2, A 2 + B 2 r2 }X. α 2 E Y () 2 =, E Y 2 () 2 =, E N ( + ω) N () 2 =, E N 2 ( + ω) N 2 () 2 =. V. AN EXAMPLE In his secion we use an example o illusrae he main resuls. Consider he sysem dn () = ( cos2 )N () ( cos2 )+( cos2 )N 2(e ) +N 2(e ) N () d + 8 N ()db, dn 2 () = ( 2 + (5.) 2 sin2 )N 2 () ( sin2 )+( sin2 )N (e ) +N (e ) N 2 () d + 8 N 2()dB 2. Since, 2 σ2 i < r i k i, i =, 2, saisfies he condiion of Theorem, we choose σ i = 8, l i, i =, 2. From sysem (5.) and (2.6), (2.7), we yield N =: r+ + 2 σ2 ɛ σ r + 64 e 8, eσl N 2 =: r σ2 2 ɛ σ 2 r e 8, eσ2l2 M =: r α 2 σ2 r = 7 64, So, M 2 =: r 2 α 2 2 σ2 2 r 2 64 e 8 64 e 8 = N () lim sup N () 7 64, N 2() lim sup N 2 () Therefore, sysem(5.) is almos surely sochasically permanen. On he oher hand, seing λ = λ 2 =, we yield Γ = λ r λ 2r 2 + (+ 2 + α+ 2 ) >, Γ 2 = λ 2 r2 λ r + (+ + α+ ) >, herefore, sysem (5.) is global asympoic sabiliy. VI. CONCLUSION This paper concerns he sochasic and ime-lagged muualism model. We know ha permanence is a very imporan and ineresing subjec in mahemaical ecology, which means ha a populaion sysem will survive forever. A definiion of almos sure permanence is presened here, which is similar o he definiion in definiive models. Under he condiion 2 σ2 i < r i k i, i =, 2, he sochasic model (.3) is almos surely sochasically permanen and he inensiy of whie noise has a negaive impac on i, bu makes no difference on global asympoic sabiliy. And in some cerain condiions, we deduce he sysem (.3) is mean square periodic. REFERENCES A.M. Dean, A simple model of muualism, Amer. Naural. 2, 49-47, D.H. Boucher, The Biology of Muualism: Ecology and Evoluion, Croom Helm, London, J.H. Vandermeer and D.H. Boucher, Varieies of muualisic ineracion models, J. Theor. Biol. 74, , C.L. Wolin and L.R. Lawlor, Models of faculaive muualism: densiy effecs, Amer. Naural., 44, , D.H. Boucher, S. James and.h. eeler, The ecology of muualism, Ann. Rev. Sys. 3, , R.M. May, Sabiliy and Complexiy in Model Ecosysems, Princeon Universiy Press, NJ, 2. 7 D. Jiang, N. Shi, X. Li, Global sabiliy and sochasic permanence of a non-auonomous logisic equnaion wih random perurbaion, J. Mah. Anal. Appl., 34, , M. Liu,. Wang, Persisence and exincion in sochasic nonauonomous logisic sysems, J. Mah. Anal. Appl. 375, , 2. 9 X. Li, X. Mao, Populaion dynamical behavior of nonauonomous Loka-Volerra compeiion sysem wih random perurbaion, Discree Conin. Dyn. Sys. 24, , 29. M. Liu,. Wang, Saionary disribuion, ergodiciy and exincion of a sochasic generaized logisic sysem, Appl. Mah. Le. 25, , 22. Y.H. Fan, L.L. Wang, Permanence for a discree model wih feedback conrol and delay, Discree Dyn. Na. Soc. 28, Aricle ID 9459, X. Gu, Y.H. Xia, Sabiliy analysis in a nonlinear ecological model, Appl. Mah. Compu. 39, 89-2, C.J. Xu, M.X. Liao, Sabiliy and bifurcaion analysis in a seir epidemic model wih nonlinear incidence raes, IAENG Inernaional Journal of Applied Mahemaics, 4, 9-98, 2. 4 J. Zhou, J.P. Shi, The exisence, bifurcaion and sabiliy of posiive soluions of a diffusive leslie-gower predaor-prey model wih hollingype II suncional responses, Mah. Anal. Appl,, 45, 68-63, R.Z. Yang, J.J. Wei, Sabiliy and bifurcaion analysis of a diffusive prey-predaor sysem in holling ype III wih a prey refuge Nonlinear Dynamics, 79, , azuyoshi MORI, General paramerizaion of sabilizing conrollers wih doubly coprime facorizaion over commuaive rings, IAENG Inernaional Journal of Applied Mahemaics, 44, 26-2, 24. (Advance online publicaion: 4 May 26)

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