Persistence and non-persistence of a stochastic food chain model with finite delay

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1 Available online a J. Nonlinear Sci. Appl., (7), 7 87 Research Aricle Journal Homepage: Persisence non-persisence of a sochasic food chain model wih finie delay Haihong Li a,, Haixia Li b, Fuzhong Cong c a Deparmen of Basic Science, Jilin Consrucion Universiy, Changchun 4, China. b School of Business, Changchun Guanghua Universiy, Changchun 4, China. c Deparmen of Basic Courses, Air Force Aviaion Universiy, Changchun, Jilin, China. Communicaed by A. Aangana Absrac In his paper, we sudy a hree species predaor-prey ime-delay chain model wih sochasic perurbaion. Firs, we analyze ha his sysem has a unique posiive soluion. Then, we deduce he condiions ha he sysem is persisen in ime average. Afer ha, condiions for he sysem going o be exincion in probabiliy are esablished. A las, numerical simulaions are carried ou o suppor our resuls. c 7 All righs reserved. Keywords: Sochasic differenial equaion, persisen, non-persisen, exincion. MSC: 9E.. Inroducion The mos exciing modern applicaion of mahemaics is used in biology. As we know, wo species sysems such as predaor-prey, plan-pes sysems eceera has been one of he dominan heme in he imporance of ecology mahemaics because of is widespread. Afer ha, he predaor-prey chain model is he ypical represenaive. As we know, unil he lae 7s, some ineres mahemaics rirophic food chain model appeared [5, 6]. One of he mos famous populaion dynamics model on ecological sysem has received a lo of aenion exensive research which named Loka-Volerra predaor-prey sysem, refer o [,, ]. Especially he persisence exincion of his model is very ineresing opic. The hree species predaor-prey chain model is described as follows: x () = x () (a b x () b x ()), x () = x () ( a + b x () b x () b x ()), (.) x () = x () ( a + b x () b x ()), Corresponding auhor address: lihaihong888999@6.com (Haihong Li) doi:.46/jnsa..6.8 Received 7--4

2 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), where x i ()(i, =,, ) represen he densiies of prey, mid-level predaor, op predaor species a ime, respecively. The parameers a, a, a, b ii (i, =,, ) are posiive consans ha s for inrinsic growh rae, predaor deah rae of he second species, predaor deah rae of he hird species, coefficien of inernal compeiion, respecively. b, b represen sauraed rae of he second he hird predaor, b, b represen he decremen rae of predaor o prey. Sysem (.) describes a hree species predaor-prey chain model in which he laer preys on he former. From a biological viewpoin, we no only require he posiive soluion of he sysem bu also require is unexploded propery in any finie ime sabiliy. We know ha he global asympoic sabiliy of a posiive equilibrium x = (x, x, x ) holds is global sable if he following condiion holds: a b b a b b + b b b b a >, which could refer o [9]. In a recen period of ime, i is easy o unders in he process of boh naural man-made involve he ime delay, such as biology, medicine so on. Kuang [9] menioned ha animals mus ake ime o diges heir food before furher aciviies responses ake place. So, he dynamic model of any species lack of delay is an approximae model. Sard for hree classes dynamic model of a single species wih independen discree delay o have global asympoic sabiliy equilibrium poin was se up by friedman Gopalsamy [4]. By consrucing proper lyapunov funcionals, he global sabiliy of he ime-delay sysems is sudied by Xiaoqing e al. [8]. Hence, we inroduce ime-delays in sysem (.) suppose ha he middle carnivore specie needs ime τ o have he abiliy o hun afer birh i jus capures prey τ maure adul. Similarly, suppose ha he op carnivore specie needs ime τ o have he abiliy o hun afer birh i jus capures mid-level τ ([8, 4, ]) maure adul predaor. Then we ge x () = x () (a b x () b x ( τ)), x () = x () ( a + b x ( τ) b x () b x ( τ)), (.) x () = x () ( a + b x ( τ) b x ()). However, populaion dynamic sysem in he real environmen is unavoidable influenced by environmenal noise (see, e.g., [7, 8]). The parameers in he sysem are no absolue consans, hey are always in some near average. So we can no ignore he influence of noise on he sysem. Recenly many auhors have discussed populaion sysems subjec o whie noise (see, e.g., [,, 5 7,, 9]). May (see, e.g., [5]) poined ou ha due o coninuous flucuaion in he environmen, he equilibrium disribuion does no reach a sable value, bu flucuaes romly near he average value. Therefore, Loka-Volerra predaor-prey chain models in rom environmens are becoming popular. Ji e al. [5, 6] invesigaed he asympoic behavior of he sochasic predaor-prey sysem wih perurbaion. Polansky [6] Barra e al. [] have given some special sysems of heir invarian disribuion. Afer ha, Gard [9] analysed ha under some condiions he sochasic food chain model exiss an invarian disribuion. Mao e al. [4] have discussed nonexplosion, persisence, asympoic sabiliy of he sochasic delay equaions, hey poined ou ha he noise will no only suppress a poenial populaion explosion in he delay model bu will also make he populaion o be sochasically uaely bounded. However, seldom people invesigae he persisen non-persisen of he food chain ime-delay model wih sochasic perurbaion. Li e al. have sudied he food chain model wih sochasic perurbaion in [], his paper is a coninuaion of he previous aricle. In his paper, we inroduce he whie noise ino he inrinsic growh rae of sysem (.), suppose a i a i + σ i B i () (i =,, ), hen we obain he following sochasic sysem dx () = x () (a b x () b x ( τ)) d + σ x ()db (), dx () = x () ( a + b x ( τ) b x () b x ( τ)) d σ x ()db (), (.) dx () = x () ( a + b x ( τ) b x ()) d σ x ()db (),

3 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), where B i () (i =,, ) are independen whie noises wih B i () =, σ i > (i =,, ) represening he inensiies of he noise. The aim of his paper is o discuss he long ime behavior of sysem (.) by sochasic comparison heorem which is differen from Mao e al. [4]. We have menioned ha x = (x, x, x ) is also he posiive equilibrium of sysem (.). Bu, when i is suffered sochasic perurbaions, here is no posiive equilibrium. Hence, i is impossible ha he soluion of sysem (.) will end o a fixed poin. In his paper, we show ha sysem (.) is persisen in ime average. Furhermore, under cerain condiions, we prove he populaion of sysem (.) will die ou in probabiliy which will no happen in deerminisic sysem could reveal ha large whie noise may lead o exincion. The res of his paper is organized as follows. In Secion, we show ha here is a unique non-negaive soluion of sysem (.). In Secion, we show ha sysem (.) is persisen in ime average. While in Secion 4, we consider hree siuaions when he populaion of he sysem will be exincion. In Secion 5, numerical simulaions are carried ou o suppor our resuls. Throughou his paper, unless oherwise specified, le (Ω, {F }, P) be a complee probabiliy space wih a filraion {F } saisfying he usual condiions (i.e., i is righ coninuous F conains all P-null ses). Le R + denoe he posiive cone of R, namely R + = {x R : x i >, i }, R + = {x R : x i, i }.. Exisence uniqueness of he nonnegaive soluion To invesigae he dynamical behavior, he firs concern hing is wheher he soluion is global exisence. Moreover, for a populaion model, wheher he soluion is nonnegaive is also considered. Hence, in his secion we show ha he soluion of sysem (.) is global nonnegaive. As we have known, in order for a sochasic differenial equaion o have a unique global (i.e., no explosion a a finie ime) soluion wih any given iniial value, he coefficiens of he equaion are generally required o saisfy he linear growh condiion local Lipschiz condiion (see, e.g., []). I is easy o see ha he coefficiens of sysem (.) are locally Lipschiz coninuous, so sysem (.) has a local soluion. By Theorem., we show he global exisence of his soluion. Le N() be he soluion of he non-auonomous logisic equaion wih rom perurbaion dn() = N()[(a() b()n())d + α()db()], (.) where B() is one-dimensional sard Brownian moion, N() = N > N is independen of B(). Lemma. ([8]). There exiss a unique coninuous soluion N() of (.) for any iniial value N() = N >, which is global represened by N() = exp{ [a(s) σ (s) ]ds + σ(s)db(s)} /N + b(s) exp{ s [a(τ),. (.) σ (τ) ]dτ + σ(τ)db(τ)}ds In order o ge he conclusion, we should inroduce wo sysems firs, dφ () = Φ () (a b Φ ()) d + σ Φ ()db (), dφ () = Φ () ( a + b Φ ( τ) b Φ ()) d σ Φ ()db (), dφ () = Φ () ( a + b Φ ( τ) b Φ ()) d σ Φ ()db (), Φ i () = ξ i () C([, ]; R + ), i =,,, (.) di () = I () (a b I () b Φ ( τ)) d + σ I ()db (), di () = I () ( a + b I ( τ) b I () b Φ ( τ)) d σ I ()db (), di () = I () ( a + b I ( τ) b I ()) d σ I ()db (), I i () = ξ i () C([, ]; R + ) i =,,, (.4)

4 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), where Φ() = (Φ (), Φ (), Φ ()), I() = (I (), I (), I ()), are he soluions of he above sochasic differenial equaions wih ime delay. Theorem.. For any iniial daa x() = {(ξ (), ξ (), ξ ()) : } C([, ]; R +), he posiive soluion of sysem (.) has he propery ha i.e., where I() x() Φ(), I i () x i () Φ i (), i =,,, Φ() = (Φ (), Φ (), Φ ()), I() = (I (), I (), I ()), are soluions of sysems (.) (.4) sochasic differenial equaions wih ime delay. Proof. Le z () = x (). Then, by Iô s formula, we have Tha is, Then dz () = d( x () ) = [( a x () b x ( τ) x () z () =e ( σ =e ( σ e ( σ b )d + σ x () db ()] + σ x () d =[(σ a )z () + b + b x ( τ) ]d σ z ()db () x () =[(σ a )d σ db ()]z () + (b + b x ( τ) )d. x () dz () = [(σ a )d σ db ()]z () + (b + b x ( τ) )d. x () a )ds σ db (s) [ a ) σ B () [ a ) σ B () [ =Φ (). x () + x () + x () + (b + b x ( τ) x () (b + b x ( τ) x () b e (a σ )s+σ B (s) ds] s )e (a σ )dτ+σ db (τ) ds] )e (a σ )s+σ B (s) ds] By Lemma., we obain ha Φ () is he soluion of he following equaion dφ () = Φ () (a b Φ ()) d + σ Φ ()db (). Hence, we have x () Φ (), a.s..

5 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), On he oher h, le z () = x (). Then, by Iô s formula, we could derive ha hen dz () = d( x () ) = [( a x () + b x ( τ) x () Therefore b b x ( τ) )d σ x () x () db ()] + σ x () d =[(σ + a )z () + b b x ( τ)z () b x ( τ)z ()]d + σ z ()db () =[(σ + a b x ( τ) b x ( τ))d + σ db ()]z () + b d, z () = σ x () e( +a )+σ B (s) b x (s)ds+b x (s)ds + b e (a + σ )( s)+σ (B () B (s)) b s x (µ)dµ+b s x (µ)dµ ds σ x () e( +a )+σ B (s) b x (s)ds + b =Φ (). e (a + σ )( s)+σ (B () B (s)) b s Φ (µ)dµ ds x () Φ (), a.s.. By Lemma., we obain ha Φ () is he soluion of he following equaion dφ () = Φ () ( a + b Φ ( τ) b Φ ()) d σ Φ ()db (). A las, le z () = x (). Then, by Iô s formula, we could derive ha hen dz () = d( x () ) = [( a x () + b x ( τ) x () b )d σ x () db ()] + σ x () d =[(σ + a )z () + b b x ( τ)z ()]d + σ z ()db () =[(σ + a b x ( τ))d + σ db ()]z () + b d, z () = σ x () e( +a )+σ B (s) b x (s)ds + b e (a + σ )( s)+σ (B () B (s)) b s x (µ)dµ ds σ x () e( +a )+σ B (s) b Φ (s)ds + b =Φ (), e (a + σ )( s)+σ (B () B (s)) b s Φ (µ)dµ ds hen, i is easy o see ha Φ () is he soluion of he following equaion dφ () = Φ () ( a + b Φ ( τ) b Φ ()) d σ Φ ()db (),

6 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), In he same mehod, we could derive ha x () Φ (), a.s.. x i () I i () a.s., i =,,, where I() = (I (), I (), I ()) is he soluion of sysem (.4). Remark.. From Lemma., we know Φ () = σ x () e( a ) σ B () + b e ( σ a )( s) σ (B () B (s)) ds, Φ () = σ x () e( +a )+σ B () b Φ(s)ds + b e ( σ +a )( s)+σ (B () B (s)) b s Φ(µ)dµ ds, Φ () = σ x () e( +a )+σ B () b Φ(s)ds + b e ( σ +a )( s)+σ (B () B (s)) b s Φ(µ)dµ ds; I () = σ x () e( a ) σ B ()+b Φ(s)ds + b e ( σ a )( s) σ (B () B (s))+b Φ(µ)dµ ds, I () = σ x () e( +a )+σ B () b I (s)ds+b Φ (s)ds + b e ( σ +a )( s)+σ (B () B (s)) b s I (µ)dµ+b Φ(µ)dµ ds, I () = σ x () e( +a )+σ B () b I(s)ds + b e ( σ +a )( s)+σ (B () B (s)) b s I(µ)dµ ds. From he represenaions of Φ i () I i (), (i=,,), Theorem. ells us he species will no reach zero in finie ime. From now on, we denoe he unique global posiive soluion of sysem (.) wih he given iniial daa ξ = {ξ() = (ξ (), ξ (), ξ ()) : } C([, ]; R + ) by x(, ξ). In he same way, we define he soluions of sysems (.) (.4) by Φ(, ξ), I(, ξ).. Persisen in ime average There is no equilibrium of sysem (.). Hence we can no show he permanence of he sysem by proving he sabiliy of he posiive equilibrium as he deerminisic sysem. In his secion we firs show ha his sysem is persisen in mean. Before we give he resul, we should do some preparaion work. Chen e al. in [] proposed he definiion of persisence in mean for he deerminisic sysem. Here, we also use his definiion for he sochasic sysem. Definiion.. Sysem (.) is said o be persisen in mean, if inf x (s)ds >, a.s.. Before giving he resul, we do some preparaion work.

7 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), Lemma. ([7]). Le f C ([, + ) Ω, (, + )), F C ([, + ) Ω, R). If here exis posiive consans λ, λ, such ha F() = a.s., hen log f() λ λ f(s)ds + F(), a.s., inf f(s)ds λ λ, From Lemma., i is easy o see ha we could ge Lemmas..4 wih he same mehod. Lemma.. Le f C ([, + ) Ω, (, + )), F C ([, + ) Ω, R). If here exis posiive consans λ, λ, such ha a.s.. log f() λ λ f(s)ds + F(),, a.s., F() =, a.s., hen sup f(s)ds λ λ, a.s.. Lemma.4. Le f C ([, + ) Ω, (, + )), F C ([, + ) Ω, R). If here exis posiive consans λ, λ, such ha log f() = λ λ f(s)ds + F(),, a.s., F() =, a.s., hen Assumpion.5. f(s)ds = λ λ, a.s.. r b r b b + b b r >, r = a σ b b b >, r i = a i + σ i i =,. Lemma.6. If Assumpion.5 is saisfied, hen he soluion Φ(, ξ) of sysem (.) has he following propery: log Φ i () =, Φ i (s)ds = M i, a.s., where M = r b, M = r b r b b, M = r b b r b b r b b b b b. Proof. From he resuls in [6] if Assumpion.5 is saisfied, we know log Φ () =, Φ (s)ds = a σ = r = M, a.s., (.) b b besides, according o Iô s formula, he second populaion of sysem (.) is changed ino I hen follows d log Φ () = ( r + b Φ ( τ) b Φ ())d σ db (). log Φ () = log Φ () r + b Φ (s τ)ds b Φ (s)ds σ B (). (.)

8 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), Noice ha Φ (s τ)ds = Φ (s)ds = from he second equaion of (.), we ge sides by, aking, yields so Φ (s τ)ds = ξ (s)ds + Φ (s)ds Φ (s)ds, (.) Φ (s)ds =, dividing he equaion (.) boh Φ (s)ds = M, log Φ () r + b Φ (s τ)ds σ B () r = r + b. b Wih Lemma.4 Assumpion.5 we could ge Φ (s)ds = r + b r b b = r b r b b b = M >. Le (.) divide,, ogeher wih (.) (.), consequenly log Φ () =. Similarly, according o Io s formula, he hird populaion of sysem (.) is changed ino i hen follows d log Φ () = ( r + b Φ ( τ) b Φ ())d σ db (), log Φ () = log Φ () r + b Φ (s τ)ds b Φ (s)ds σ B (), Φ (s)ds = r r + b b r b b b log Φ () = M >, =. b From his, ogeher wih Theorem. Lemma.6, he following resul is obviously rue. Theorem.7. If Assumpion.5 is saisfied, hen he soluion x(, ξ) of sysem (.) has he following propery: sup log x i () By above all, we could ge he following resul., i =,,. Theorem.8. If Assumpion.5 is saisfied, hen he soluion x(, ξ) of sysem (.) has he following propery inf x (s)ds x, where x = ( x, x, x ) is he only nonnegaive soluion of he following equaion, r b x b x =, r + b x b x b x =, r + b x b x =. a.s.,

9 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), Proof. From sysem (.), log x () log x () = r b = r b b x (s)ds b x (s)ds b ( x (s)ds + σ B (), x (s τ)ds + σ B () ξ (s)ds x (s)ds) similarly, log x () log x () = r + b b x (s)ds + b ( x (s)ds b ( ξ (s)ds ξ (s)ds x (s)ds) b x (s)ds) σ B (), x (s)ds Hence log x () log x () = r + b b x (s)ds + b ( x (s)ds σ B (). ξ (s)ds x (s)ds) c (log x () log x ()) + c (log x () log x ()) + c (log x () log x ()) = (r c r c r c ) + ( b c + b c ) c b ( From Theorem., we ge + ( b c b c + b c ) ξ (s)ds x (s)ds x (s)ds (b c + b c ) x (s)ds) + c b ( c b ( ξ (s)ds x (s)ds) + c b ( + c σ B () c σ B () c σ B (). ξ (s)ds x (s)ds ξ (s)ds x (s)ds) x (s)ds) (.4) x i () Φ i (), (i =,, ), hen x i (s)ds =. (.5) Le c = b, c = b, c = b b + b b, ogeher wih Assumpion.5, we know b r c r c r c >.

10 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), B According o Theorem.7 equaion (.5), ogeher wih i () =, (i =,, ), we could ge Such ha, sup c (log x () log x ()) + c (log x () log x ()) + c (log x () log x ()) = (r c r c r c ) (c b + c b ) inf x (s)ds. inf x (s)ds r c r c r c c b + c b = x, where x = ( x, x, x ) is he only nonnegaive soluion of he following equaion when Assumpion.5 is saisfied, r b x b x =, r + b x b x b x =, r + b x b x =. 4. Non-persisence In he previous secion, we showed he soluion x(, ξ) of sysem (.) is sable in ime average, in his secion, we discuss he dynamics of sysem (.) when he whie noise is geing larger. We show he siuaion when he populaion of sysem (.) will be non-persisen of he whie noise is large, which does no happen in he deerminisic sysem in hree cases. Definiion 4.. Sysem (.) is said o be non-persisen, if here are posiive consans c i, (i =,, ) such ha i= x c i i () =, a.s.. Now we presen condiions for all species or some species of (.) o be exinc. Consider he following cases. Case (i): r <. According o Iô s formula, he firs populaion of sysem (.) is changed ino If r <, we could ge d log Φ () (r b Φ ())d σ db (). sup log Φ () = r < a.s., from he sochasic comparison heorem, we have sup log x () < a.s., hence x () =, a.s..

11 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), From he second populaion of sysem (.) equaion (.), we have hence Similarly, log Φ () log Φ () sup log Φ () r + b r + b sup = r + b sup sup log Φ () Φ (s τ)ds σ db () Φ (s τ)ds Φ (s)ds = r, a.s.. = r, a.s., a.s., (4.) x i() =, a.s. i =,. Case (ii): r >, r b b r <. I is clear ha from he equaions (4.) (.), we ge Similarly hus, sup sup log Φ () log Φ () r + b sup r + b r b <, a.s.. Φ (s)ds = r <, a.s., x i() =, a.s., i =,. By above all, from he conclusion in [4], we could easily know ha he disribuion of x () converges weekly o he probabiliy measure wih densiy where C = (b /σ )r /σ /Γ(r /σ ), f (ζ) = C ζ r /σ e b ζ/σ, Case (iii): r b b r b b + b b b b r <. we ge x (s)ds = r b, a.s.. I is clear ha from (.4), leing c = b, c = b, c = b b + b b sup b, wih B i () =, i =,,, c (log x () log x ()) + c (log x () log x ()) + c (log x () log x ()) = (r c r c r c ) (c b + c b ) inf r c r c r c, x (s)ds

12 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), moreover, hen sup log x c ()xc ()xc () r c r c r c <, xc ()xc ()xc () =, a.s.. Therefore, by he above argumens, we ge he following conclusion. Theorem 4.. Le x(, ξ) be he soluion of sysem (.), he following conclusions are founded. () If r <, hen () If r >, r b b r <, hen x i() =, a.s., i =,,. x i() =, a.s., i =,, he disribuion of x () converges weekly o he probabiliy measure wih densiy where C = (b /σ )r /σ /Γ(r /σ ), f (ζ) = C ζ r /σ e b ζ/σ, () If r b r b b + b b r <, hen b b b where c = b, c = b, c = b b + b b b. x (s)ds = r b, a.s.. xc ()xc ()xc () =, a.s., Tha is o say, he large whie noise will lead o he populaion sysem non-persisen. 5. Numerical simulaion In his secion, we give ou he numerical experimen o suppor our resuls. Consider he equaion dx () = x () (a b x () b x ( τ)) d + σ x ()db (), dx () = x () ( a + b x ( τ) b x () b x ( τ)) d σ x ()db (), dx () = x () ( a + b x ( τ) b x ()) d σ x ()db (). By he mehod in [], we have he difference equaion σ x,k+ = x,k + x,k [(a b x,k b x,k m ) + σ ɛ,k + (ɛ,k )], σ x,k+ = x,k + x,k [( a + b x,k m b x,k b x,k m ) σ ɛ,k + (ɛ,k )], σ x,k+ = x,k + x,k [( a + b x,k m b x,k ) σ ɛ,k + (ɛ,k )],

13 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), where ɛ,k, ɛ,k ɛ,k, i =,, are he Gaussian rom variables N(, ), r = a σ >, r i = a i + σ i, (i =, ), m represens he ineger par τ/. Choose (x (), x (), x ()) R +, [, ], suiable parameers, by Malab, we ge Figs.,, x ()..5.5 x ().5 x () Figure : The soluions of sysem (.) sysem (.) wih (x (), x (), x ()) = (.9,.,.), [, ], a =.7, a =., a =., b =., b =., b =., b =.5, b =., b =.4, b =.8. The ligh lines represen he soluion of sysem (.), while he dark lines represen he soluion of sysem (.) wih σ =., σ =., σ = x (). x ().5 x () Figure : Two of he species will die ou in probabiliy. The soluions of sysem (.) sysem (.) wih (x (), x (), x ()) = (.9,.,.), [, ], a =.5, a =., a =., b =.6, b =., b =., b =.5, b =., b =.4, b =.8. The ligh lines represen he soluion of sysem (.), while he dark lines represen he soluion of sysem (.) wih σ =., σ =., σ =..

14 H. Li, H. Li, F. Cong, J. Nonlinear Sci. Appl., (7), x ().5 x (). x () Figure : One of he species or boh species will die ou in probabiliy. The soluions of sysem (.) sysem (.) wih (x (), x (), x ()) = (.9,.,.), [, ], a =.7 a =., a =., b =., b =., b =., b =.5, b =., b =.4, b =.8. The ligh lines represen he soluion of sysem (.), while he dark lines represen he soluion of sysem (.) wih σ =., σ =., σ =.. In Fig., when he noise is small, choose parameers saisfying he condiion of Theorem.7, hen he soluion of sysem (.) will persis in ime average. In Fig., we observe Case (iii) in Theorem 4. choose parameers r >, r b b r <. As Theorem 4. indicaed ha wo predaors will die ou in probabiliy, hen he prey soluion of sysem (.) will persis in ime average. In Fig., we observe Case (i) in Theorem 4. choose parameers r <. As Theorem 4. indicaed ha no only predaors bu also prey will die ou in probabiliy when he noise of he prey is large, i does no happen in he deerminisic sysem, hese simulaed resuls are consisen wih our heorems. Acknowledgmen The work was suppored by Naural Science Foundaion of Jilin Province of China (No. 5). References [] M. Barra, G. Del Grosso, A. Gerardi, G. Koch, F. Marchei, Some basic properies of sochasic populaion models, Sysems heory in immunology, Proc. Working Conf., Rome, (978), Lecure Noes in Biomah., Springer, Berlin- New York, (979), [] L.-S. Chen, J. Chen, Nonlinear biological dynamical sysem, Science Press, Beijing, (99). [] H. I. Freedman, Deerminisic mahemaical models in populaion ecology, Monographs Texbooks in Pure Applied Mahemaics, Marcel Dekker, Inc., New York, (98). [4] H. I. Freedman, K. Gopalsamy, Global sabiliy in ime-delayed single-species dynamics, Bull. Mah. Biol., 48 (986), [5] H. I. Freedman, J. W.-H. So, Global sabiliy persisence of simple food chains, Mah. Biosci., 76 (985), [6] H. I. Freedman, P. Walman, Mahemaical analysis of some hree-species food-chain models, Mah. Biosci., (977), [7] T. C. Gard, Persisence in sochasic food web models, Bull. Mah. Biol., 46 (984), [8] T. C. Gard, Sabiliy for mulispecies populaion models in rom environmens, Nolinear Anal., (986), 4 49., [9] T. C. Gard, Inroducion o sochasic differenial equaions, Monographs Texbooks in Pure Applied Mahemaics, Marcel Dekker, Inc., New York, (988).,

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