stochastic lotka–volterra model with infinite delay

Statistics and Probability Letters 79 (2009) 698–706

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Statistics and Probability Letters

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Stochastic Lotka–Volterra model with infinite delayLi Wan a,∗, Qinghua Zhou ba Department of Mathematics and Physics, Wuhan University of Science and Engineering, Wuhan 430073, Chinab Department of Mathematics, Zhaoqing University, Zhaoqing 526061, China

a r t i c l e i n f o

Article history:Received 22 January 2008Received in revised form 18 October 2008Accepted 20 October 2008Available online 26 October 2008

a b s t r a c t

The stochastic Lotka–Volterra system with infinite delay is studied. We show that thesolution of such a system is a positive solution without explosion and give the conditionsto guarantee stochastic ultimate boundedness of the solution.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

The delay Lotka–Volterra model for n interaction species plays an important role in modelling the population growth ofcertain species. It can be written as the following n-dimensional delay differential equation

dx(t)dt= diagx1(t), . . . , xn(t)[b+ Ax(t − τ)],

where

x = (x1, . . . , xn)T, b = (b1, · · · bn)T, A = (aij)n×n.

For the study concernedwith the dynamics of such amodel; see, for example, Ahmad and Rao (1994), Bereketoglu and Gyori(1997), Freedman and Ruan (1995), Gopalsamy (1984), He and Gopalsamy (1997), Kuang (1993), Kolmanovskii andMyshkis(1992), Kuang and Smith (1993) and Teng and Yu (2000).However, environmental noise does exist in population systems. For the study on the stochastic Lotka–Volterra system

with discrete time delays, see, for example, Bahar andMao (2004) andMao et al. (2005). To the best of our knowledge, thereare few results about the stochastic Lotka–Volterra system with infinite delays in the literature today. Here we shall studythe following system

dxi(t) = xi(t)

(bi +

n∑j=1

aijxj(t)+n∑j=1

bijxj(t − τij)+n∑j=1

cij

∫ t

−∞

kij(t − s)xj(s)ds

)dt

+ xi(t)n∑j=1

σijxj(t)dwj, i = 1, 2, . . . , n, (1)

with the initial condition

xi(θ) = ϕi(θ) > 0 on −∞ < θ ≤ 0; sup−∞<θ≤0

|ϕ(θ)| <∞, (2)

∗ Corresponding author.E-mail addresses:[email protected] (L. Wan), [email protected] (Q. Zhou).

0167-7152/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2008.10.016

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http://dx.doi.org/10.1016/j.spl.2008.10.016

L. Wan, Q. Zhou / Statistics and Probability Letters 79 (2009) 698–706 699

where ϕi (i = 1, 2, . . . , n) is continuous on (−∞, 0], | · | denotes the Euclidean norm in Rn, kij (i, j = 1, 2, . . . , n) arereal valued non-negative continuous functions defined on [0,∞) and satisfy

∫∞

0 kij(t)dt = 1, w(t) = (w1(t), . . . , wn(t))T

is an n-dimensional Brownian motion defined on a complete probability space (Ω,F , P) with a natural filtration Ftt≥0generated by w(s) : 0 ≤ s ≤ t, where we associateΩ with the canonical space generated by w(t), and denote by F theassociated σ -algebra generated by w(t)with the probability measure P.As the ith state xi(t) of Eq. (1) is the size of the ith species in the system, it should be non-negative. In Section 2, we shall

show that for any b ∈ Rn and A, B, C ∈ Rn×n, the solution of Eq. (1) will remain in the positive cone

Rn+= x ∈ Rn, xi > 0, i = 1, . . . , n.

with probability one and it will not explode to infinity in a finite time.In addition, in a population dynamical system, the property of ultimate boundedness of the solution is more desired than

the non-explosion property. Moreover, the conditions for the ultimate boundedness are much more complicated than theconditions for the non-explosion; see, for example, Freedman and Ruan (1995), He and Gopalsamy (1997), Kuang (1993)and Teng and Yu (2000). In Section 3, the conditions to guarantee the stochastic ultimate boundedness of the solution arederived. In Section 4, the other properties of the solution are discussed.

2. Positive and global solutions

For the convenience, we introduce some notations. If A is a vector or matrix, its transpose is denoted by AT. If A is amatrix, its trace norm is denoted by |A| =

√trace(ATA). Denote by C((−∞, 0];Rn

+) the family of continuous functions from

(−∞, 0] to Rn+.

Note that the coefficients of Eq. (1) donot satisfy the linear growth condition, though they are locally Lipschitz continuous.Therefore the solution of Eq. (1) may explode at a finite time; see Mao (1991, 1997). The following theorem shows that thesolution of Eq. (1) is positive solution without explosion.

Theorem 2.1. For any b ∈ Rn, A, B, C ∈ Rn×n and any given initial data x(s) : −∞ < s ≤ 0 satisfying (2), there is a uniquesolution x(t) to Eq. (1) and the solution will remain in Rn

+with probability one.

Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given initial data x(s) : −∞ < s ≤ 0satisfying (2), there is a unique maximal local solution x(t) on t ∈ (−∞, τ ), where τ is explosion time; see Mao (1994).To show this solution is global, we need to show that τ = ∞ a.s. From (2), we know the initial data is both bounded andbounded away from zero. Therefore, we can choose a sufficiently large constantm0 > 0 such that

1m0≤ min

t≤0|x(t)| ≤ max

t≤0|x(t)| ≤ m0.

For each integerm ≥ m0, we define the stopping time

τm = inft ∈ [0, τ ) : xi(t) is not belong to

(1m,m), for some i = 1, 2, . . . , n

,

where we set inf∅ = ∞ (as usual ∅ denotes the empty set). Clearly, τm is increasing as m → ∞. Set τ∞ = limm→∞ τm,then τ∞ ≤ τ a.s. If we can show that τ∞ = ∞ a.s., then τ = ∞ a.s. and x(t) ∈ Rn

+a.s. for all t ≥ 0. To this end, we define a

C2-function V : Rn+→ R+ by

V (x) =n∑i=1

[√xi − 1−

12log xi

]and set

V1(x(t)) =12n

n∑i=1

n∑j=1

∫ t

t−τijx2j (s)ds,

V2(x(t)) =12n

n∑i=1

n∑j=1

∫∞

0kij(s)

∫ t

t−sx2j (u)duds.

It is clear that these function are non-negative. Let m ≥ m0 and T > 0 be arbitrary. For 0 ≤ t ≤ τm ∧ T , we apply the Itôformula to V (x(t))+ V1(x(t))+ V2(x(t)) and obtain

d[V (x(t))+ V1(x(t))+ V2(x(t))]

= d[V1(x(t))+ V2(x(t))] +n∑i=1

0.5[x−0.5i (t)− x−1i (t)]xi(t)

(bi +

n∑j=1

aijxj(t)+n∑j=1

bijxj(t − τij)

700 L. Wan, Q. Zhou / Statistics and Probability Letters 79 (2009) 698–706

+

n∑j=1

cij

∫ t

−∞

kij(t − s)xj(s)ds

)dt + 0.5[x−0.5i (t)− x−1i (t)]xi(t)

n∑j=1

σijxj(t)dwj

+ 0.5[−0.25x−1.5i (t)+ 0.5x−2i (t)]x2i (t)

n∑j=1

σ 2ij x2j (t)dt

≤ d[V1(x(t))+ V2(x(t))] +n∑i=1

0.5[x0.5i (t)− 1]

(bi +

n∑j=1

aijxj(t)+n∑j=1

bijxj(t − τij)

+

n∑j=1

cij

∫ t

−∞

kij(t − s)xj(s)ds

)dt + 0.5[−0.25x0.5i (t)+ 0.5]

n∑j=1

σ 2ij x2j (t)dt

+M(t),

where

M(t) =n∑i=1

0.5[x0.5i (t)− 1]n∑j=1

σijxj(t)dwj.

Note that

dV1(x(t)) =n∑i=1

n∑j=1

12n[x2j (t)− x

2j (t − τij)]

= 0.5|x(t)|2 −n∑i=1

n∑j=1

12nx2j (t − τij),

dV2(x(t)) =12n

n∑i=1

n∑j=1

∫∞

0kij(s)[x2j (t)− x

2j (t − s)]ds

= 0.5|x(t)|2 −12n

n∑i=1

n∑j=1

∫∞

0kij(s)x2j (t − s)ds,

n∑i=1

[x0.5i (t)− 1]

(n∑j=1

aijxj(t)+n∑j=1

bijxj(t − τij)+n∑j=1

cij

∫ t

−∞

kij(t − s)xj(s)ds

)

≤

n∑i=1

n∑j=1

(n4[x0.5i (t)− 1]

2a2ij +1nx2j (t)+

n4[x0.5i (t)− 1]

2b2ij

+1nx2j (t − τij)+

∫ t

−∞

kij(t − s)[n4[x0.5i (t)− 1]

2c2ijds+1nx2j (s)

]ds)

=

n∑i=1

n∑j=1

(n4[x0.5i (t)− 1]

2a2ij +1nx2j (t)+

n4[x0.5i (t)− 1]

2b2ij


n4[x0.5i (t)− 1]

2c2ij +∫∞

0kij(s)

1nx2j (t − s)ds

),

n∑i=1

x0.5i (t)n∑j=1

σ 2ij x2j (t) ≥

n∑i=1

σ 2ii x2.5i (t),

and

n∑i=1

n∑j=1

σ 2ij x2j (t) ≤

n∑i=1

[n∑j=1

σ 2ij

n∑j=1

x2j (t)

]≤ |σ |2|x(t)|2.

Thus, we obtain

d[V (x(t))+ V1(x(t))+ V2(x(t))] ≤ F(x(t))dt +M(t), (3)


where

F(x) = (2+ 0.25|σ |2)|x(t)|2 +n∑i=1

0.5[x0.5i (t)− 1]bi +n∑i=1

n∑j=1

[x0.5i (t)− 1]2 n8[a2ij + b

2ij + c

2ij ]

− 0.125n∑i=1

σ 2ii x2.5i (t). (4)

Note that for x ∈ Rn+, lim|x|→∞ F(x) < 0. Then, there existsM > 0 such that F(x) < 0 for x ∈ Rn

+and |x| > M. Therefore,

there exists C > 0 such that maxx∈Rn+F(x) = maxx∈Rn

+,|x|≤M F(x) ≤ C .

Integrating both side of (3) from 0 to τm ∧ T and taking expectations, we obtain

EV (x(τm ∧ T )) ≤ E[V (x(τm ∧ T ))+ V1(x(τm ∧ T ))+ V2(x(τm ∧ T ))]

≤ V (x(0))+ V1(x(0))+ V2(x(0))+ E∫ τm∧T

0F(x(s))ds

≤ V (x(0))+ V1(x(0))+ V2(x(0))+ CE(τm ∧ T )≤ V (x(0))+ V1(x(0))+ V2(x(0))+ CT . (5)

Note that for every ω ∈ τm ≤ T , there is some i such that xi(τm, ω) equals either m or 1/m. Hence V (x(τm, ω)) is no lessthan either

√m− 1− 1

2 logm or√1/m− 1−

12log(1/m) =

√1/m− 1+

12logm,

that is,

V (x(τm, ω)) ≥[√m− 1−

12logm

]∧

[√1/m− 1+

12logm

].

From (5), we obtain

P[τm ≤ T ][√m− 1−

12logm

]∧

[√1/m− 1+

12logm

]≤ E[1τm≤T (ω)V (x(τm, ω))]

≤ V (x(0))+ V1(x(0))+ V2(x(0))+ CT ,

where 1τm≤T is the indicator function of τm ≤ T . Let m → ∞, we obtain limm→∞ P[τm ≤ T ] = 0 and henceP[τ∞ ≤ T ] = 0. Since T > 0 is arbitrary, we must have

P[τ∞ <∞] = 0 and P[τ∞ = ∞] = 1.

The proof is complete.

3. Stochastically ultimate boundedness

In this section, we shall show that the solution of Eq. (1) is stochastically ultimately bounded. Now, we give the definitionof stochastically ultimate boundedness.

Definition 3.1. Eq. (1) is said to be stochastically ultimately bounded if for any ε ∈ (0, 1), there is a positive constantC = C(ε) such that for any initial data x(s) : −∞ < s ≤ 0 satisfying (2), the solution x(t) of Eq. (1) satisfies

lim supt→∞

P|x(t)| ≤ C ≥ 1− ε.

To our aim, we first prove the following theorem.

Theorem 3.2. Assume that µ ∈ (0, 1) and there exists some constant λ > 0 such that∫∞

0kij(s)eλsds = kij <∞. (6)

Then there is a positive constant C = C(µ), which is independent of the initial data x(s) : −∞ < s ≤ 0 satisfying (2), suchthat the solution x(t) of Eq. (1) satisfies

lim supt→∞

E|x(t)|µ ≤ C .


Proof. For x ∈ Rn+, we define V (x) =

∑ni=1 x

µ

i ,

V1(x(t)) =1n

n∑i=1

n∑j=1

∫ t

t−τijeλ(s+τij)x2j (s)ds,

and

V2(x(t)) =1n

n∑i=1

n∑j=1

∫∞

0kij(s)

∫ t

t−seλ(u+s)x2j (u)duds.

By the Itô formula, we obtain

d[eλtV (x(t))+ V1(x(t))+ V2(x(t))]

= d[V1(x(t))+ V2(x(t))] + λeλtV (x(t))dt +M(t)+ eλt

n∑i=1

µxµi

(bi +

n∑j=1

aijxj(t)+n∑j=1

bijxj(t − τij)

+

n∑j=1

cij

∫ t

−∞

kij(t − s)xj(s)ds

)dt −

µ(1− µ)2

n∑i=1

xµi

[n∑j=1

σ 2ij x2j (t)

]dt

≤ λeλtV (x(t))dt +M(t)+1n

n∑i=1

n∑j=1

[eλ(t+τij)x2j (t)− eλtx2j (t − τij)]dt

+1n

n∑i=1

n∑j=1

∫∞

0kij(s)[eλ(t+s)x2j (t)− e

λtx2j (t − s)]dsdt

+ eλt

n∑i=1

µxµi bidt +n∑i=1

n∑j=1

[n4a2ijµ

2x2µi +1nx2j (t)+

n4b2ijµ

2x2µi


n4c2ijµ

2x2µi +1n

∫∞

0kij(s)x2j (t − s)ds

]dt −

µ(1− µ)2

n∑i=1

σ 2ii x2+µi dt

≤ λeλtV (x(t))dt + eλt [eλτ + k]|x(t)|2dt +M(t)+ eλt

n∑i=1

µxµi bi + |x(t)|2

+

n∑i=1

n∑j=1

n4µ2x2µi

[a2ij + b

2ij + c

2ij

]−µ(1− µ)2

n∑i=1

σ 2ii x2+µi

dt

= M(t)+ eλtF(x)dt, (7)

where τ = max1≤i,j≤n τij, k = max1≤i,j≤n kij,

M(t) = eλtn∑i=1

µxµin∑j=1

σijxj(t)dwj,

F(x) =n∑i=1

(λ+ µbi)xµ

i +

n∑i=1

n∑j=1

n4µ2x2µi [a

2ij + b

2ij + c

2ij ]

−µ(1− µ)2

n∑i=1

σ 2ii x2+µi + [eλτ + k+ 1]|x|2.

Similar to the previous argument, there exists C1 > 0 such that supx∈Rn+F(x) ≤ C1. Integrating both side of (7) from 0 to t

and taking expectations, we obtain

eλtEV (x(t)) ≤ V (x(0))+ V1(x(0))+ V2(x(0))+ E∫ t

0eλsF(x(s))ds

≤ V (x(0))+ V1(x(0))+ V2(x(0))+ λ−1C1eλt

which implies that

lim supt→∞

EV (x(t)) ≤ λ−1C1.


Since

|x|µ ≤ nµ/2 max1≤i≤n

xµi ≤ nµ/2V (x),

we obtain

lim supt→∞

E|x|µ ≤ C,

where C = λ−1nµ/2C1. The proof is complete.

From the above theorem, it is easy to obtain

Theorem 3.3. Suppose that the condition (6) holds. Then the solution for Eq. (1) is stochastically ultimately bounded.

Proof. From Theorem 3.2, we know that there is C∗ > 0 such that

lim supt→∞

E|x|1/2 ≤ C∗.

For any ε > 0, set C = C2∗/ε2. By Chebyshev’s inequality, we obtain

P|x(t)| > C ≤ E|x|1/2/√C .

Thus we have

lim supt→∞

P|x(t)| > C ≤ C∗/√C = ε,

which implies

lim supt→∞

P|x(t)| ≤ C ≥ 1− ε.

4. Other properties of the solution

Theorem 3.3 shows that the solution of Eq. (1) will be ultimate bounded with large probability. The following resultshows that the average in time of pth (0 < p ≤ 2) moment of solution will be bounded.

Theorem 4.1. Let 0 < p ≤ 2. Then there is a positive constant C, which is independent of the initial data x(s) : −∞ < s ≤ 0satisfying (2), such that the solution x(t) of Eq. (1) satisfies

lim supT→∞

1T

∫ T

0E|x(t)|pdt ≤ C .

Proof. Let F1(x) = F(x)+ |x|p, where F(x) is defined by (4). Similar to the previous argument, there exists C > 0 such thatsupx∈Rn

+F1(x) ≤ C . From (5) with F(x) = F1(x)− |x|p, we obtain∫ τm∧T

0E|x(s)|pds ≤ V (x(0))+ V1(x(0))+ V2(x(0))+ CE(τm ∧ T ).

Letm→∞, we obtain

E∫ T

0|x(s)|pds ≤ V (x(0))+ V1(x(0))+ V2(x(0))+ CT ,

which implies the result.

Wehave discussed how the solution of Eq. (1) inRn+varies in probability or inmoment. In order to investigate the solution

pathwise, we need the exponential martingale inequality; see Mao (1994).

Lemma 4.2. Let M = (Mt) be a continuous local martingale. Then for any positive constants τ , γ , δ, we have

P(sup0≤t≤τ

[Mt −

γ

2< M,M >t

]> δ

)≤ e−γ δ,

where< M,M > is the quadratic variation of M.

Now we state the following result on the solution pathwise.


Theorem 4.3. Suppose that the following condition

σii > 0, i = 1, . . . , n; σij ≥ 0, i 6= j

and (6) hold. Then, for any initial data x(s) : −∞ < s ≤ 0 satisfying (2), the solution x(t) of Eq. (1) satisfies

lim supt→∞

log |x(t)|log t

≤ 1 a.s. (8)

Proof. Let λ > 0 be arbitrary. For x ∈ Rn+, we define V (x) =

∑ni=1 xi,

V1(x(t)) =n∑i=1

n∑j=1

|bij|∫ t

t−τijeλ(s+τij)xj(s)ds,

and

V2(x(t)) =n∑i=1

n∑j=1

|cij|∫∞

0kij(s)

∫ t

t−seλ(u+s)xj(u)duds.

By the Itô formula, we obtain

d[eλt log(V (x(t)))+ V1(x(t))+ V2(x(t))]

= d[V1(x(t))+ V2(x(t))] +M(t)+ λeλt log(V (x(t)))dt + eλtn∑i=1

xi

(bi +

n∑j=1

aijxj(t)+n∑j=1

bijxj(t − τij)

+

n∑j=1

cij

∫ t

−∞

kij(t − s)xj(s)ds

)V−1(x(t))dt − 0.5eλt

n∑j=1

x2j (t)

[n∑i=1

xi(t)σij

]2V−2(x(t))dt

≤ d[V1(x(t))+ V2(x(t))] +M(t)+ λeλt log(V (x(t)))dt + eλtn∑i=1

xi

(|bi| +

n∑j=1

|aij|xj(t)+n∑j=1

|bij|xj(t − τij)

+

n∑j=1

|cij|∫ t

−∞

kij(t − s)xj(s)ds

)V−1(x(t))dt − 0.5eλt

n∑j=1

x2j (t)

[n∑i=1

xi(t)σij

]2V−2(x(t))dt

≤ d[V1(x(t))+ V2(x(t))] +M(t)+ λeλt log(V (x(t)))dt + eλtn∑i=1

(|bi| +

n∑j=1

|aij|xj(t)+n∑j=1

|bij|xj(t − τij)

+

n∑j=1

|cij|∫∞

0kij(s)xj(t − s)ds

)dt − 0.5eλt

n∑j=1

x2j (t)

[n∑i=1

xi(t)σij

]2V−2(x(t))dt

≤

n∑i=1

n∑j=1

|bij|eλ(t+τij)xj(t)dt +n∑i=1

n∑j=1

|cij|∫∞

0kij(s)eλ(t+s)xj(t)dsdt +M(t)+ λeλt log(V (x(t)))dt

+ eλtn∑i=1

(|bi| +

n∑j=1

|aij|xj(t)

)dt − 0.5eλt

n∑j=1

x2j (t)

[n∑i=1

xi(t)σij

]2V−2(x(t))dt

≤ (b+ c + a)eλtn∑j=1

xj(t)dt +M(t)+ λeλt log(V (x(t)))dt + eλtn∑i=1

|bi|dt

− 0.5eλtn∑j=1

x2j (t)

[n∑i=1

xi(t)σij

]2V−2(x(t))dt,

where

b = max1≤j≤n

n∑i=1

[|bij|eλτij ], c = max1≤j≤n

n∑i=1

[|cij|kij], a = max1≤j≤n

n∑i=1

|aij|

M(t) = eλtV−1(x(t))n∑i=1

xin∑j=1

σijxj(t)dwj(t).


Hence, we have

eλt log(V (x(t))) ≤ log(V (x(0)))+ V1(x(0))+ V2(x(0))+∫ t

0λeλs log(V (x(s)))ds

+

∫ t

0eλs[(b+ c + a)

n∑j=1

xj(s)+n∑i=1

|bi|]ds+M∗(t)−∫ t

00.5eλs

n∑j=1

x2j (s)

[n∑i=1

xi(s)σij

]2V−2(x(s))ds,

where

M∗(t) =∫ t

0eλsV−1(x(s))

n∑i=1

xi(s)n∑j=1

σijxj(s)dwj(s)

is a real-valued continuous local martingale vanishing at t = 0 and its quadratic form is given by

〈M∗(t),M∗(t)〉 =∫ t

0e2λs

n∑j=1

x2j (s)

[n∑i=1

xi(s)σij

]2V−2(x(s))ds.

Let θ > 1 and ε ∈ (0, 1) be arbitrary. By Lemma 4.2, we obtain that for each integerm ≥ 1

Psup0≤t≤m

[M∗(t)−

ε

2eλm〈M∗(t),M∗(t)〉

]>θeλm

εlogm

≤ m−θ .

Since the series∑∞

m=1m−θ converges, it follows from the Borel–Cantelli lemma that there is anΩ∗ ⊂ Ω with P(Ω∗) = 1

such that for any ω ∈ Ω∗ there exists an integerm0 = m0(ω) such that

M∗(t) ≤ε

2eλm< M∗(t),M∗(t) > +

θeλm

εlogm

for all 0 ≤ t ≤ m andm ≥ m0. Hence, we obtain

eλt log(V (x(t))) ≤ log(V (x(0)))+ V1(x(0))+ V2(x(0))+∫ t

0λeλs log(V (x(s)))ds+

∫ t

0eλs[(b+ c + a)

n∑j=1

xj(s)

+

n∑i=1

|bi|

]ds−

∫ t

00.5(1− ε)eλs

n∑j=1

x2j (s)

[n∑i=1

xi(s)σij

]2V−2(x(s))ds+

θeλm

εlogm

≤ log(V (x(0)))+ V1(x(0))+ V2(x(0))+∫ t

0λeλs log(V (x(s)))ds

+

∫ t

0eλs[(b+ c + a)

√n|x(s)| +

n∑i=1

|bi|

]ds− σ∗

∫ t

00.5(1− ε)eλs

|x(s)|2

n2ds+

θeλm

εlogm

= C1 +∫ t

0eλsF(x(s))ds+

θeλm

εlogm

for all 0 ≤ t ≤ m andm ≥ m0, since

V 2(x)n≤ |x|2 ≤ nV 2(x), (9)

n∑j=1

x2j

[n∑i=1

xiσij

]2V−2(x) ≥ V−2(x)

n∑j=1

σ 2jj x4j ≥ σ∗V

−2(x)n∑j=1

x4j

≥σ∗

nV−2(x)|x|4 ≥

σ∗

n2|x|2,

where

σ∗ = min1≤i≤n

σ 2ii > 0, C1 = log(V (x(0)))+ V1(x(0))+ V2(x(0)),

F(x) = λ log(V (x))+ (b+ c + a)√n|x| +

n∑i=1

|bi| − σ∗0.5(1− ε)|x|2

n2.


Clearly, there exists a positive constant C2 such that

F(x) ≤ C2,∀x ∈ Rn+.

Hence, we obtain

eλt log(V (x(t))) ≤ C1 +C2λeλt +

θeλm

εlogm

for all 0 ≤ t ≤ m andm ≥ m0. Thus, for any ω ∈ Ω0, ifm− 1 ≤ t ≤ m andm ≥ m0,we have

log V (x(t))log t

≤1

log(m− 1)

[C1e−λ(m−1) +

C2λ+θeλ

εlogm

],

which implies

lim supt→∞

log V (x(t))log t

≤θeλ

ε, a.s.

Let ε→ 1, θ → 1 and λ→ 0, we have

lim supt→∞

log V (x(t))log t

≤ 1, a.s.

From (9), we obtain (8). The proof is complete.

Acknowledgments

The authors would like to thank the referee and the editor for their helpful comments and suggestions. This research waspartly supported by the National Natural Science Foundation of China (10801109) and the Natural Science Foundation ofWuhan University of Science and Engineering (2008Z25).

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stochastic lotka–volterra model with infinite delay

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