Pathwise estimation of stochastic functional Kolmogorov-type systems with infinite delay

Transcription

1 Zhu and Xu Journal of Inequalities and Applications 1, 1:171 R E S E A R C H Open Access Pathwise estimation of stochastic functional Kolmogorov-type systems with infinite delay Enwen Zhu 1 andyongxu * * Correspondence: xuyonghust@16com School of Mathematics, Central South University, Changsha, Hunan 4175, China Full list of author information is available at the end of the article Abstract In this paper, we study pathwise estimation of the global positive solutions for the stochastic functional Kolmogorov-type systems with infinite delay Under some conditions, the growth rate of the solutions for such systems with general noise structures is less than a polynomial rate in the almost sure sense To illustrate the applications of our theory more clearly, this paper also discusses various stochastic Lotka-Volterra-type systems as special cases MSC: 34K5; 6H1; 9D5; 93E3 Keywords: stochastic functional Kolmogorov systems; Lotka-Volterra systems; pathwise estimation; infinite delay 1 Introduction The stochastic functional Kolomogorov-type systems for n interacting species is described by the following stochastic functional differential equation: dx(t)=diag(x 1,,x n ) [ f (x t )dt + g(x t )dw(t) ], (11) where x =(x 1,,x n ) T, diag(x 1,,x n )representsthen n matrix with all elements zero except those on the diagonal which are x 1,,x n, w(t) is a scalar Brownian motion, and f =(f 1,,f n ) T : C ( [ τ,];r n) R n, g =(g 1,,g n ) T : C ( [ τ,];r n) R n There is an extensive literature concerned with the dynamics of this system and we here only mention [7, 9, 1] References [7, 9, 1] study existence and uniqueness of the global positive solution of Eq (11), and its asymptotic bound properties and moment average in time The nice positive property provides us with a great opportunity to discuss further how the solutions vary in R n + in more detail Our interest is to discuss pathwise estimation of the global positive solutions for stochastic functional Kolomogorov-type systems with infinite delay There is an extensive literature concerned with the dynamics of the Kolomogorov-type systems (11) without the stochastic perturbation and we here only mention [3 5, 1] As a special case, multispecies Lotka-Volterra-type systems with bounded delay and unbounded delay are studied For example, He and Gopalsamy [16] consider the global positive solution for a two-dimensional Lotka-Volterra system Kuang [19] examines global 1 Zhu and Xu; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

2 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page of 15 stability for infinite delay Lotka-Volterra-type systems For more details about the Lotka- Volterra-type systems, we refer the reader to see [6, 11 13, 15, ] and references therein In fact,population systems are often subject to environmental noise It is therefore useful to reveal how the noise affects on the Kolmogorov-type systems Recently, the stochastic Lotka-Volterra systems have received increasing attention References [1, 17] reveal that the noise plays an important role to suppress the growth of the solution References [, 18] show the stochastic system behaves similarly to the corresponding deterministic system under different stochastic perturbations, respectively These indicate clearly that different structures of environmental noise may have different effects on Lotka-Volterra systems However, little is yet known about the pathwise property of the stochastic functional Kolmogorov-type systems with infinite delay although they may be seen as a generalized stochastic functional Lotka-Volterra system This paper will examine the pathwise estimation of the stochastic functional Kolmogorov-type systems under the general noise structures Consider the n-dimensional unbounded delay stochastic functional Kolmogorovtype systems dx(t)=diag ( x 1 (t),,x n (t) )[ f (x t )dt + g(x t )dw(t) ] (1) on t, where w(t)isanm-dimensional Brownian motion, and f : BC ( (,];R n) R n, g : BC ( (,];R n) R n m, where the initial data space BC((,];R n ) denotes the family of bounded continuous R n -value functions ϕ defined on (,]withthenorm ϕ = sup θ ϕ(θ) < Here, f and g are local Lipschitz continuous Clearly, Eq (1) includes the following forms: dx(t)=diag ( x 1 (t),,x n (t) )[ f (x t )dt + g ( x(t) ) dw(t) ], (13) dx(t)=diag ( x 1 (t),,x n (t) )[ f ( x(t) ) dt + g(x t )dw(t) ] (14) Since this paper mainly examines the pathwise estimation of the solution for stochastic functional Kolmogorov-type systems with infinite delay, we assume that there exists a unique global positive solution for all discussed equations (see [8, 14]) In the next section, we give some necessary notations and lemmas To show our idea clearly, Section also studies the pathwise estimation for general stochastic functional differential equations with infinite delay Applying the result of Section, we give various conditions under which stochastic functional Kolmogorov systems with infinite delay show the nice pathwise properties in Section 3 As in the applications of Section and Section 3,Section4 discusses several special equations, including various stochastic Lotka-Volterra systems with infinite delay A general result Throughout this paper, unless otherwise specified, we use the following notations Let denote the Euclidean norm in R n IfA is a vector or matrix, its transpose is denoted by A T IfA is matrix, its trace norm is denoted by A = trace(a T A) Let R n = {x Rn : x i for all 1 i n}, R n + = {x Rn : x i for all 1 i n}, R n ++ = {x Rn : x i >forall1 i n} For any c =(c 1,,c n ) T R n ++,letĉ = min{c 1,,c n } and č = max{c 1,,c n }Let

3 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 3 of 15 (, F, P) be a complete probability space with a filtration {F t } t satisfying the usual conditions, that is, it is right continuous and increasing while F contains all P-null sets Let w(t) beanm-dimensional Brownian motion defined on the complete probability space If x(t) isanr n -valued stochastic process on t R, weletx t = {x(t + θ):θ (,]} for t Let M be the family of the probability measures μ on (,] Let ε > For any ε [, ε ], we define a set of the probability measure { } M ε := μ M, μ ε := e εθ dμ(θ)< Clearly, M ε M ε M and μ ε is continuously dependent on ε on [, ε ], and μ ε 1 as ε For the arbitrary V(x) C (R n ; R), define ( V(x) V x (x)=,, V(x) ), V xx (x)= x 1 x n ( ) V(x) x i x j n n The following lemma shows boundedness of polynomial functions Lemma 1 For any positive constants b, α, and the function h(x) C(R n + ; R), ifh(x) = o( x α ),as x Then [ sup b x α + h(x) ] < x R n + Proof We may choose c >suchthat h(x) < x α for any x α > c, which implies b x α + h(x) < Therefore, we have [ sup b x α + h(x) ] [ = sup b x α + h(x) ] <, x R n + x R n +, x <c as required To show our idea about the pathwise estimation of solutions clearly, we first consider the following general n-dimensional stochastic functional differential equation: dx(t)=f(t, x t )dt + G(t, x t )dw(t) (1) on t Here, F : R + BC ( R ; R n) R n, G : R + BC ( R ; R n) R n m are local Lipschitz continuous and w(t) isanm-dimensional Brownian motion Assume that Eq (1) almost surely admits a unique global positive solution Then we have the following pathwise estimation Theorem 1 Let p >and x(t)=x(t, ξ) be the global positive solution of Eq (1) If there exists a vector c R n ++ such that for any ε (, ε ] and an integer J >, there are constants

4 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 4 of 15 K, K j >, α j and probability measures μ j M ε (1 j J) such that c T F(t, ϕ) c T ϕ() K + + p 1 j=1 [ c T ] G(t, ϕ) + ε log ( c T ϕ() ) c T ϕ() J ( K j ϕ(θ) αj dμ j (θ) μ jε ϕ() ) α j () for any t and ϕ BC(R ; R n ++ ) Then for any given initial data ξ BC(R ; R n ++ ) L q (R ; R n ),whereq= min 1 j J (α j ),thesolutionx(t) of Eq (1) has the following property: log( x(t) ) log t 1 p as (3) Proof Define V(x)=c T x, onx R n + For any ε (, ε ], applying the Itô formula to e εt log V(x(t)) yields where e εt log V ( x(t) ) = log V ( x() ) + d [ e εs log V ( x(s) )] = log V ( ξ() ) + M(t)= e εs [ c T F(s, x s ) e εs ct G(s, x s ) 1 dw(s) ( c T ) G(s, x s ) + ε log V ( x(s) )] ds + M(t), (4) is a continuous local martingale with the quadratic variation t ( c M(t), M(t) = e εs T ) G(s, x s ) ds For any given k N and δ > 1, the exponential martingale inequality yields { [ P sup M(t) p t k e εk ( c e εs T ) G(s, x s ) } ds] δeεk log n 1 p k δ Since k=1 n δ <, the well-known Borel-Cantelli lemma yields that there exists an with P( ) = 1 such that for any ω there exists an integer k (ω), when k k (ω) and k 1 t k, M(t) p ( c e εs T ) G(s, x s ) ds + δeε(t+1) log(t +1) (5) p

5 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 5 of 15 Substituting the inequalities (5) into(4), and letting t sufficiently large, it is obtained almost surely that e εt log V ( x(t) ) log V ( ξ() ) + p 1 δe ε(t+1) log(t +1)+I, where I = e εs [ c T F(s, x s ) + p 1 By the condition (), we obtain that ( c T ) G(s, x s ) + ε log V ( x(s) )] ds I e εs {K + J [ K j x(s + θ) αj dμ j (θ) μ jε x(s) j] } α ds j=1 ε 1 Ke εt + J ( ) K j e εs x(s + θ) α j dμ j (θ) μ jε x(s) α j ds j=1 We may compute that e εs = = x(s + θ) αj dμ j (θ)ds dμ j (θ) dμ j (θ) +θ [ e εθ dμ j (θ) θ e εs x(s + θ) α j ds e ε(s θ)x(s) αj ds e εs x(s) α j ds + μ jε C αj + μ jε e εsx(s) αj ds, e εs ] x(s) α j ds where C αj := ξ(s) α j ds <, μ jε := e εθ dμ j (θ)<, for all 1 j J Therefore, e εt log V ( x(t) ) log V ( ξ() ) + ε 1 Ke εt + J K j μ jε C αj + p 1 δe ε(t+1) log(t +1) j=1 This implies log(v(x(t))) log t δeε p, as Noting the inequality <ĉ x V(x) c x, forx R n ++,

6 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 6 of 15 and letting δ 1, ε, we obtain that log( x(t) ) log t 1 p, as as required The result (3) showsthatforanyε >, there is a positive random time T ε such that, with probability one, for any t > T ε, x(t) p t 1+ε (6) In other words, with probability one, the solution will not grow faster than t (1+ε)/p In this theorem, it is a key to compute the condition () In the following section, we apply Theorem 1 to examine stochastic functional Kolmogorov-type systems 3 Main result This section mainly applies the result of Theorem 1 to Eq (1) For any x R n ++ and ϕ BC(R ; R n ++ ), we firstly list the following conditions for both f and g that we will need (H1) These exist α >, κ, κ and the probability measure μ M ε,whereε (, ε ] and α ε,suchthat f (ϕ) κϕ() α + κ ϕ(θ) α ( dμ(θ)+o ϕ() α) (H) There exist β >, λ, λ and the probability measure ν M ε,whereε (, ε ] and β ε,suchthat g(ϕ) λ ϕ() β + λ ϕ(θ) β dν(θ)+o (ϕ() β ) (H3) There exist c =(c 1,,c n ) T R n ++, β, b >, σ and the probability measure ν M ε,whereε (, ε ] and β ε,suchthat ϕ() [ ϕ T ()Cg(ϕ) ] bϕ() β σ ϕ(θ) β ( dν(θ)+o ϕ() β), where C = diag(c 1,,c n ) (H4) There exist c R n ++, α, b >, σ and the probability measure μ M ε,whereε (, ε ] and α ε,suchthat ϕ() 1 ϕ T ()Cf (ϕ) b ϕ() α + σ ϕ(θ) α dμ(θ)+o (ϕ() α ), where C = diag(c 1,,c n ) When f (x) and g(x) replace f (ϕ) and g(ϕ), the above conditions may be rewritten as (H1 ) Theseexistα >, κ such that f (x) κ x α +o ( x α)

7 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 7 of 15 (H ) Thereexistβ >, λ such that g(x) λ x β +o ( x β) (H3 ) Thereexistc R n ++, β, b >such that [ x T Cg(x) ] b x β+ +o ( x β+), where C = diag(c 1,,c n ) (H4 ) Thereexistc R n ++, α, b >such that x T Cf (x) b x α+1 +o ( x α+1), where C = diag(c 1,,c n ) Here, we emphasize that the same letter represents the same parameter when we use the several conditions simultaneously In the following sections, we always assume that the initial data x BC(R n ; Rn ++ ) L(α β) (R n ; Rn ) Applying the conditions (H1) and (H3) to stochastic functional Kolmogorov systems (1)gives Theorem 31 Under the conditions (H1) and (H3), for the global positive solution x(t) of Eq (1), if one of the following conditions p =1, α <β, b > σ ; (31a) <p <1, α =β, b > σ + c (κ + κ) 1 p (31b) holds, then log( x(t) p ) log(t) 1, as (3) Proof To apply Theorem 1, we need to compute the condition () Substituting F(t, ϕ) =diag(ϕ 1 (),,ϕ n ())f (ϕ) andg(t, ϕ) =diag(ϕ 1 (),,ϕ n ())g(ϕ) into the condition ()yields ε := ϕt ()Cf (ϕ) c T ϕ() + p 1 [ ϕ T ] ()Cg(t, ϕ) + ε log ( c T ϕ() ) c T ϕ() =: I 1 + I + ε log ( c T ϕ() ), (33) where C = diag(c 1,,c n ) By the condition (H1), { I 1 max fi (ϕ) } 1 i n f (ϕ) κ ϕ() α + κ ϕ(θ) α dμ(θ)+o (ϕ() α)

8 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 8 of 15 Noting p 1, by the condition (H3), p 1 [ I ϕ T Cg(ϕ) ] c ϕ() p 1 [b ϕ() β σ ϕ(θ) β ( dν(θ)+o ϕ() β) ] c Substituting I 1 and I into (33) and noting that ε log(c T ϕ()) o( ϕ() α ), we get where ε κ ϕ(θ) α σ (1 p) dμ(θ)+ ϕ(θ) β dν(θ) c + κ ϕ() α b(1 p) ϕ() β ( +o ϕ() β) c [ = κ ϕ(θ) α dμ(θ) μ ε ϕ() ] α [ + σ (1 p) c ] ϕ(θ) β dν(θ) νεϕ() β I = 1 p c (b σ ν ε) ϕ() β +(κ + κμ ε ) ϕ() α +o (ϕ() β ) (34) If the condition (31a)holds,choosingδ (, 1) to replace p in (34), then we have I = 1 δ c (b σ ν ε) ϕ() β +o (ϕ() β) =: b1 ϕ() β +o (ϕ() β) Noting b > σ and ν ε 1asε, choose ε sufficiently small such that b 1 > Lemma 1 gives I const, which implies the condition () is satisfied Letting δ 1, Theorem 1 gives the desired result under p =1 If the condition (31b)holds, I = 1 p ( ) b σ ν c ε c (κ + κμ ε ) ϕ() β ( +o ϕ() β) 1 p =: b ϕ() β +o (ϕ() β ) By the condition (31b), choose ε sufficiently small such that b >,sowehavei const, which implies the condition () and furthergives the desired result by Theorem 1 Applying Theorem 31 to Eq (13), the condition (H3) should be replaced by (H3 ), which implies σ = This result may be described as follows Corollary 3 Under the conditions (H1) and (H3 ), for the global positive solution x(t) of Eq (13), if one of the following conditions p =1, α <β, b > σ ; (35a) <p <1, α =β, b(1 p)> c (κ + κ) (35b) is satisfied, then (3)holds + I,

9 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 9 of 15 Applying Theorem 31 to Eq (14), the condition (H1) should be replaced by (H1 ), which is equivalent to choose κ = in (H1) Then we may obtain the following corollary Corollary 33 Under the conditions (H1 ) and (H3), for the global positive solution x(t) of Eq (14), if one of the following conditions p =1, α <β, b > σ ; (36a) <p <1, α =β, b > σ + c κ 1 p (36b) holds, then log( x(t) p ) log(t) 1, as (37) In these results above, the condition (H3) or (H3 ) plays an important role to suppress growth of the solution To use the condition (H3) or (H3 ), it is necessary to require p 1 To avoid the condition (H3) or (H3 ), we may impose another condition on the function f to suppress growth of the solution This idea leads to the following theorem Theorem 34 Under the conditions (H) and (H4), for the global positive solution x(t) of Eq (1), if α >β, b > c σ /ĉ (38) hold, then If p 1 and log( x(t) ) log(t), as (39) α =β, b > c σ ĉ + c (λ + λ) (p 1) (31) hold, then log( x(t) p ) log(t) 1, as (311) Proof Repeating the proof process of Theorem 31 obtains Eq (33) Then by the conditions (H) and (H4), we may estimate I 1 and I By the condition (H4), [ I 1 ϕ() 1 b ϕ() α + σ ϕ(θ) α dμ(θ)+o (ϕ() α ) ] c T ϕ() b ϕ() α σ + c ĉ ϕ(θ) α dμ(θ)+o (ϕ() α)

10 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 1 of 15 Note the elementary inequality: for any x, y andu (, 1) (x + y) x u + y 1 u Then for any u, v (, 1), by the condition (H) and the Hölder inequality, I [ n (p 1)+ i=1 (p 1)+ (p 1)+ (p 1)+ u (p 1)+ u c i ϕ i () gi (ϕ) c T ϕ() ( ) max gi ϕ() 1 i n [λ ϕ() β + λ [λ ϕ() β + λ [ λ ϕ() β λ + v 1 v Substituting I 1 and I into (33)yields where ε σ ĉ [ + (p 1)+ λ u(1 v) ] ϕ(θ) β dν(θ)+o (ϕ() β) ] ϕ(θ) β dν(θ) ] +o ( ϕ() β) ϕ(θ) α dμ(θ) μ ε ϕ() ] α [ ϕ(θ) β dν(θ) ] +o ( ϕ() β) ] ϕ(θ) β dν(θ) νεϕ() β + I, ( b I = c σ μ ) ε ϕ() α (p 1) + ( λ + ĉ u v + λ ) νε ϕ() β ( +o ϕ() α) 1 v If the condition (38) holds, we can choose sufficiently small ε such that b > c σ μ ε /ĉ > Then, by Lemma 1,wehave ( b I = c σ μ ) ε ϕ() α ( +o ϕ() α) const ĉ Then, for any p 1, Theorem 1 gives log( x(t) ) log t 1 p, as Letting p, we get the desired assertion (39) If the condition (31) holds, noting α =β and p 1, we have [ b I = c σ μ ε ĉ ( (p 1) λ u v + λ )] νε ϕ() α ( +o ϕ() α) 1 v

11 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 11 of 15 Assume that λ, λ (whenλ, λ =, the computation is obvious) Choose v = λ/(λ + λ) (, 1) such that b := b c σ μ ε ĉ ( (p 1) λ u νε v + λ 1 v ) = b c σ μ ε ĉ (p 1) u (λ + λ)(λ + λν ε) Noting that the condition (31)holds,chooseε sufficiently small and u sufficiently near 1 such that b >SoI const Applying Theorem 1 therefore gives the desired assertion (311) The result (39) is a very strong result It shows that growth of the solution x(t)issoslow that it is near to be bound By this result, we know that for any ε >, there is a positive random time T ε such that, with probability one, for any t T ε, x(t) t ε (31) In other words, with probability one, the solution will not grow fast than t ε Applying Theorem 34 to Eq (13), the condition (H) should be replaced by (H ), which implies that λ = in (H) We mayobtain thefollowingcorollary Corollary 35 Under the conditions (H ) and (H4), for the global positive solution x(t) of Eq (13), if α >β, b > c σ /ĉ, (313) then (39)holdsIfp 1 and α =β, then (311)holds b > c σ ĉ + c λ (p 1), (314) Applying Theorem 34 to Eq (14), the condition (H4) should be replaced by (H4 ), which implies that σ = in (H4) The corollaryfollows Corollary 36 Under the conditions (H) and (H4 ), for the global positive solution x(t) of Eq (14), if α >β, (315) then (39)holdsIfp 1 and α =β, b > c (λ + λ) (p 1), (316) then (311)holds Comparing Theorem 31 with Theorem 34, we conclude that in Theorem 31,thefunction g plays an important role to suppress growth of the solution in condition (H3) While in Theorem 34, the function f is the main factor to suppress growth of the solution in condition (H4)

12 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 1 of Somespecialcase In this section, some special Kolmogorov-type equations are considered by applying the results in the previous two sections to obtain some pathwise estimation, where some special noises are chosen We firstly consider the n-dimensional stochastic functional equations dx(t)=diag ( x 1 (t),,x n (t) )[ f (x t )dt + q dw(t) ], (41) which is equivalent to choosing g(x) q R n in Eq (1) Clearly, the condition (H )is satisfied Applying Corollary 35 to Eq (41) gives the following theorem Theorem 41 Under the condition (H4), if b > c σ /ĉ, for the global positive solution x(t) of Eq (41), then the result (39)holds Then we consider the functional Lotka-Volterra-type system of Eq (41) dx(t)=diag ( x 1 (t),,x n (t) )[( ) ] a + Ax(t)+B x(t + θ)dμ(θ) dt + q dw(t), (4) where a R n, A, B R n n and μ M ε for some suitable ε (, ε ] For any c = (c 1,,c n ) T R n ++ LetC = diag(c 1,,c n ), Then we have [ ] ϕ T ()Cf (ϕ)=ϕ T ()C a + Aϕ() + B ϕ(θ)dμ(θ) ϕ T ()CAϕ() + ϕ T ()CB 1 λ+ max( CA + A T C )ϕ() + č B ϕ() ϕ(θ) dμ(θ)+o (ϕ() ) ϕ(θ) dμ(θ)+o (ϕ() ) =: [ ϕ() b ] ϕ() + σ ϕ(θ) dμ(θ) +o ( ϕ() ), (43) where b = λ + max (CA + AT C), σ = č B Hereweusethenotation λ + max (M)= sup x R +, x =1 x T Mx, for a symmetric matrix M R n n, which was introduced by Bahar and Mao [] Let us emphasize that this is different from the largest eigenvalue λ max (M) ofthematrixm Recall the nice property of the largest eigenvalue: λ max (M)= sup x R, x =1 x T Mx Clearly, λ + max (M) λ max(m), which implies that λ + max (M) <ifthematrixm is negative definite (43) showsthatf satisfies the condition (H4) when we choose α = 1 The condition b > c σ /ĉ is equivalent to λ + max( CA + A T C ) > c č B /ĉ, (44) then Theorem 41implies thefollowingcorollary

13 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 13 of 15 Corollary 4 If there exists c =(c 1,,c n ) T R n ++ such that C = diag(c 1,,c n ) satisfies the condition (44), for the globalpositive solution x(t) of Eq (4) hastheproperty(39) If we choose μ is the Dirac measure at some point < τ <, theneq(4) maybe rewritten as dx(t)=diag ( x 1 (t),,x n (t) )[( a + Ax(t)+Bx(t τ) ) dt + q dw(t) ], (45) which is another Lotka-Volterra type delay equation that has been investigated by Bahar et al [] Applying Corollary 4 to Eq (45) gives the following corollary Corollary 43 If there exists c =(c 1,,c n ) T R n ++ such that C = diag(c 1,,c n ), λ + max( CA + A T C ) > B c č/ĉ, (46) then the global positive solution x(t) of Eq (45) hasthe property(39) Now we consider another Kolmogorov-type equation dx(t)=diag(x 1,,x n ) [ f (x t )dt + Qx(t)dw(t) ], (47) where Q =[q ij ] R n n Clearly, here g satisfies the condition (H ) when we choose that λ = Q and β = 1IfthematrixQ satisfies the condition q ii > if1 i n, while q ij ifi j, (48) then g satisfies the condition (H3 ) when we choose that b = max 1 i n {c i q ii } Then Corollary 3 gives the following theorem Theorem 44 Under the conditions (48) and (H1), for the global positive solutions of Eq (47), if one of the following two conditions p =1, α <; (49a) <p <1, α =, max c i i n{ i qii } c (κ + κ) >, 1 p (49b) is satisfied, where c =(c 1,,c n ) T R n ++,then(3)holds Roughly speaking, Theorem 44 shows that under the condition (48), if the growth order α of f is smaller than (or α = under some conditions), (3) is satisfied for all p 1 To satisfy (3)underp 1, the condition (48)playsacrucialroleButheref is only required to satisfy an unrestrictive condition, which includes the linear growth condition If f is the linear growth function, namely, f (ϕ)=a + Aϕ() + B ϕ(θ)dμ(θ)

14 Zhuand Xu Journal of Inequalities and Applications 1, 1:171 Page 14 of 15 then Eq (47)mayberewrittenas dx(t)= diag ( x 1 (t),,x n (t) ) [( ) ] a + Ax(t)+B x(t + θ)dμ(θ) dt + Qx(t)dw(t), (41) where Q R n n, which is another stochastic functional Lotka-Volterra systems Clearly, f satisfies the condition (H1) when we choose that α =1,κ =1,κ = B So,forp =1,we have the following theorem Theorem 45 Under the condition (48), the global positive solution of Eq (41) has the property log x(t) log t 1, as We further choose that μ is a Dirac measure at the point τ,theneq(41)iswrittenas the delay stochastic Lotka-Volterra systems dx(t)=diag ( x 1 (t),,x n (t) )[( a + Ax(t)+Bx(t τ) ) dt + Qx(t)dw(t) ], (411) which is discussed by [1] where Theorem 53 is obtained This means that our result is more general Competing interests The authors declare that they have no competing interests Authors contributions The authors contributed equally and significantly in writing this article The authors read and approved the final manuscript Author details 1 School of Mathematics and Computing Sciences, Changsha University of Science and Technology, Changsha, Hunan 4176, China School of Mathematics, Central South University, Changsha, Hunan 4175, China Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grant nos , , the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grant no 11FEFM11, the Scientific Research Funds of Hunan Provincial Science and Technology Department of China under Grant no 1SK396, and Hunan Provincial Natural Science Foundation of China under Grant no 1jj45 Received: 8 October 11 Accepted: July 1 Published: 1 August 1 References 1 Bahar, A, Mao, X: Stochastic delay Lotka-Volterra model J Math Anal Appl 9, (4) Bahar, A, Mao, X: Stochastic delay population dynamics Int J Pure Appl Math 4, (4) 3 Tineo, A: Existence of global attractors for a class of periodic Kolmogorov systems J Math Anal Appl 79, (3) 4 Granados, B: Permanence for a large class of planar periodic Kolmogorov systems J Math Anal Appl 68, () 5 Tang, B, Kuang, Y: Permanence in Kolmogorov-type systems of nonautonomous functional differential equations J Math Anal Appl 197, (1996) 6 Chen, F: Global asymptotic stability in n-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control Appl Math Comput 17, (5) 7 Wu, F, Hu, S: Stochastic functional Kolmogorov-type population dynamics J Math Anal Appl 347, (8) 8 Wu, F, Hu, S: On the pathwise growth rates of large fluctuations of stochastic population systems with innite delay Stoch Models 7, (11) 9 Wu, F, Hu, S, Liu, Y: Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system J Math Anal Appl 364, (1)

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