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Volume 2011, Article ID 914270, 13 pagesdoi:10.1155/2011/914270 Research Article Stochastic Delay Lotka-Volterra Model 1 College of Science, Wuhan University of Science and Technology, W

Volume 2011, Article ID 914270, 13 pages

doi:10.1155/2011/914270

Research Article

Stochastic Delay Lotka-Volterra Model

1 College of Science, Wuhan University of Science and Technology, Wuhan, Hubei 430065, China

2 Department of Mathematics, Huazhong University of Science and Technology, Wuhan,

Hubei 430074, China

Correspondence should be addressed to Lian Baosheng,lianbs@163.com

Received 15 October 2010; Accepted 20 January 2011

Academic Editor: Alexander I Domoshnitsky

Copyrightq 2011 Lian Baosheng et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

This paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process, namely the delay Lotka-Volterra model The stochastic version

of this process appears to have some intriguing properties such as pathwise estimation and asymptotic moment estimation Indeed, their solutions will be stochastically ultimately bounded

1 Introduction

As is well known, Lotka-Volterra Model is nonlinear and tractable models of predator-prey system The predator-prey system is also studied in many papers In the last few years, Mao

et al change the deterministic model in this field into the stochastic delay model and give it more important properties1 8

Fluctuations play an important role for the self-organization of nonlinear systems;

we will study their influence on a simple nonlinear model of interacting populations, that

is, the Lotka-Volterra model A simple analysis shows the result that the system allows extreme behaviour, leading to the extinction of both of their species or to the extinction of the predator and explosion of the prey For example, in Mao et al.1 8, we can see that once the population dynamics are corporate into the deterministic subclasses of the delay Lotka-Voterra model, the stochastic model will bear more attractive properties: the solutions will be

be stochastically ultimately bounded, and their pathwise estimation and asymptotic moment estimation will be well done

The most simple stochastic model is given in the form of a stochastic delay differential equation also called a diffusion process; we call it a delay Lotka-Volterra model with diffusion The model will be

dx t  diagxtb  Axtdt  Bytdt  Gdwt, 1.1

where yt  xt−τ, xt  x1t, , x d t T where x1t, , x d t Tdenotes the transpose

of a vector or matrixx1t, , x d t, b  b1, b2, , b dT

, A  a ij  ∈ R d×m , B  b ij  ∈ R d×m,

G  γ ij  ∈ R d×m and wt is the m-dimensional Brownian motion, diag xt is the diag

matrix

This model of the stochastic delay Lotka-Volterra is different from Mao et al 3 10, which paid more attention to the mathematical properties of the model than the real background of the model However, our model has the following three characteristics First,

it is another stochastic delay subclass of the Lotka-Volterra model which is different from Mao et al Then we can obtain more comprehensive properties inTheorem 2.1 Second, in this field no paper gives more attention to it so far, especially for the stochastic delay model which is the focus in our model Third, this model has many real applications, for example,

in economic growth model it is different from the old delay Lotka-Volterra model which only palys a role in predator-prey system, for example, the stochastic R&D model9,10 is the best application of this model We hope our model can have new applications of the Lotka-Volterra model Throughout this paper, we impose the condition

−a ii > A i

j /  i

aij , 1 ≤ i ≤ d, 1.2

where aij  a ij if a ij > 0.

Of course, it is important for us to point that the condition1.2 may be not real in predator-prey interactions, but in the stochastic R&D model in economic growth model, it has a special meaning

−K θ max

i



a ii  A i− θ θ A1θi

1  θ1θ|a ii|θ





i

θ θ A1θi

1  θ1θ|a ii|θ < 0. 1.3

If θ  1/2 or 1, the inequality 1.3 can be deduced to

−K 1/2 max

i



a ii  A i− 2A

3/2 i

 27|aii|





i

2A 3/2 i

 27|aii|< 0, 1.4

−K1 max

i



a ii  A i− A2i

4|aii|





i

A2

i

If condition1.2 is satisfied, then

lim

θ → ∞

θ θ A1θi

1  θ1θ|a ii|θ  0. 1.6

Therefore, if θ is big enough, condition 1.2 implies condition 1.3.

It is obvious the conditions 1.3–1.5 are dependent on the matrix A, independent

on G.

Condition 1.4 will be used in a further topic in the paper; the condition 1.4 is

complicated, we can find many matrixes A that have a property like this For example,

A  diag a11, a22, a dd  a ii < 0 for 1 ≤ i ≤ d 1.7

satisfy the condition1.4 Furthermore, if i / j, a ij ≤ 0, or a ijare proper small enough positive numbers, condition1.4 holds too Particularly, if d  2, the condition can be induced into

a11 a

12 < −2a21 3/2



27a22

, a22 a

12< −2a12 3/2



27a11

1.8

It is clear that the upper inequalities are the key conditions in the stochastic R&D model in economic growth model

Let

I θ x 

ij

a ij x θ i x j 1.9

The homogeneous function I θ x of degree 1  θ has the following key property.

Lemma 1.1 Suppose the matrix A satisfies condition 1.2 Let

then

sup

x∈S

I θ x ≤ −K θ , θ > 0, 1.11

where K θ is given in condition1.3

Proof Fix x ∈ S, so 0 < x j ≤ x∞ 1 We will show I θ x ≤ −K θ We have

I θ x ≤

i

a ii x1θi 

i



j /  i

aij x θ i x j

≤

i

a ii x1θi  A i x i θ



i

ϕ i x i

1.12

with A isatisfying condition1.2, where ϕi x i   a ii x1θi  A i x i θ Since, from condition1.2,

ϕ i 0  0, ϕ i 1  a ii  A i < 0, and

ϕi t  0 ⇒ t  t0 − θA i

1  θa ii  θA i

1  θ|a ii| ∈ 0, 1, 1.13 then

max

0≤t≤1ϕ i t  ϕ i t o  θ θ A1θi

1  θ1θ|a ii|θ  M i 1 ≤ i ≤ d. 1.14

Since x ∈ S, we have 0 < x i ≤ 1, 1 ≤ i ≤ d, x ∈ S, and there exits at least x i  1, such that

ϕ i x i  ≤ M i , 1 ≤ i ≤ d and at least ϕ i x i   ϕ i 1  a ii  A i for some i Thus



ϕ i x i ≤ max

i

⎛

⎝a ii  A i

j /  i

M j

⎞

⎠

 max

i a ii  A i − M i 

i

M i

1.15

Now, from condition1.3, the right hand of the upper equation is just −Kθ , so I θ x ≤ −K θ; Lemma 1.1is proved

We use the ordinary result of the polynomial functions

Lemma 1.2 Let f i 1 ≤ i ≤ n be a homogeneous function of degree θ i , θ > θ i ≥ 0, and a > 0; then

the function as follows has an upper bound for some constant K.

F x n

i1

f i − ad

i1

x i θ ≤ K. 1.16

2 Positive and Global Solutions

LetΩ, F, {F t}t≥0 , P  be a complete probability space with filtration {F t}t≥0satisfying the usual

conditions, that is, it is increasing and right continuous while F0 contains all P -null sets 8 Moreover, let wt be an m-dimensional Brownian motion defined on the filtered space and

R d

  {x ∈ R d

 : x i > 0 for all 1 ≤ i ≤ d} Finally, denote the trace norm of a matrix A

by|A|  traceAT A where A T denotes the transpose of a vector or matrix A and its

operator norm byA  sup{|Ax| : |x|  1} Moreover, let τ > 0 and denote by C−τ, 0; R d

 the family of continuous functions from−τ, 0 to R d

 The coefficients of 1.1 do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of1.1 may explode at a finite time

let us emphasize the important feature of this theorem It is well known that a deterministic equation may explode to infinity at a finite time for some system parameters

b ∈ R d and A ∈ R d×m However, the explosion will no longer happen as long as conditions

1.2 and 1.3 hold In other words, this result reveals the important property that conditions

1.2 and 1.3 suppress the explosion for the equation The following theorem shows that this solution is positive and global

Theorem 2.1 Let us assume that K 1/2 satisfy

3K 1/2 > d

β i  2β

i

, β i

j

bij , βj

i

bij 2.1

Then for any given initial data {x t : −τ ≤ t ≤ 0} ∈ C−τ, 0, R d

, there exists a unique global

solution x  xt to 1.1 on t ≥ −τ Moreover, this solution remains in Rd

 with probability 1, namely, x t ∈ R d

for all t ≥ −τ almost surely.

Proof Since the coefficients of the equation are locally Lipschitz continuous, for any given initial data{x t : −τ ≤ t ≤ 0} ∈ C−τ, 0, R d

, there is a unique maximal local solution xt

on t ∈ 0, ρ, where ρ is the explosion time 3 10 To show this solution is global, we need to

show that ρ  ∞ a.s Let k0be sufficiently large for

1

k0 < min

−τ≤t≤0 |xt| ≤ max

For each integer k ≥ k0, define the stopping time

τ k inf t ∈

0, ρ

: x i t /∈k−1, k , for some i  1, , d

, 2.3

where throughout this paper we set inf φ  ∞ as usual φ denotes the empty set Clearly, τ k

is increasing as k → ∞ Set τ∞ limk → ∞ τ k , whence τ∞≤ ρ a.s If we can show that τ∞ ∞

a.s., then ρ  ∞ a.s and xt ∈ R d

a.s for all t ≥ −τ In other words, to complete the proof all

we need to show is that τ∞ ∞ a.s Or for all t > 0, we have Pτ k ≤ T → 0, k → ∞ To show this statement, let us define a C2-functions V : R d− Rby

u t  t − lnt, V t u√x i x ∈ R d

 . 2.4

The nonnegativity of this function can be seen from

u t  t − lnt > 0 on t > 0. 2.5

Let k ≥ k0and T > 0 be arbitrary For 0 ≤ t ≤ T ∧ τ k, we apply the Ito formula to V x to obtain that

LV x  1

2



i

√x i− 1

⎡

⎣b i

j

a ij x j  b ij y j

⎤

⎦  1 8



ij

2 −√x i r2

ij

 1 2



i

b i√x i− 1  1

8



ij



−4a ij x j  r2

ij2 −√x i

1 2



ij

b ij√x i− 1 1

2



ij

a ij√

x i x j

 φx 12

ij

b ij

√

x i x j−1 2



ij

a ij y j1

2I x,

2.6

where φx  1/2

i b i√x i −11/8ij −4a ij x j r2

ij2−√x i is a homogeneous function

of a degree not above 1, G  γ ij  ∈ R d×m, and by1.9, Ix  I1/2 x, and let z  x/x∞, for

all x ∈ R d

; thenz∞ 1 ByLemma 1.1, we obtain

I x  Izx∞  Izx 3/2

∞

≤ −k 1/2 x 3/2

∞ ≤ −d −3/2 k 1/2 |x| 3/2

,

2.7

where we use the fact V 3/2 x d

i1 x 3/2 i and V 3/2 x ≤ dx 3/2

∞ , K 1/2 > 0, and



ij

b ij√

x i y j≤

ij

b ij

⎛

⎝x 3/2 i

3 2y

3/2 j

3

⎞

⎠

 1 3



i



j

bij x 3/2 i 2

3



j



i

bij y j 3/2

 1 3



i

β i x 3/2 i 2

3



j

βj y 3/2 j

−

ij

b ij y j ≤ −

ij

b−ij y j −

j

ρ j y j ,

2.8

where b ij−  −b ij , if b ij < 0, and ρ ji b ij−

Thus

LV x ≤ φx  1

6



i

β i x 3/2 i − 1

2d K 1/2 V 3/2 x 

i

 1

3βi y 3/2 i  1

2ρ i y i



. 2.9

W t, xt  V x 

t

t−τ



i

 1

3βi x 3/2 i s 1

2ρ i x i s

ds. 2.10

Then, if t ≤ τ k, byLemma 1.2, we obtain

LW t, xt  LV x 

i

 1

3βi x 3/2 i t − y 3/2

i t  1

2ρ i



x i t − y i t 

≤ φx 12

i

ρ i x i t − 1

6d



i



3k 1/2 − dβ i  2β

i



x 3/2 i t

≤ K

2.11

with a constant K.

Consequently,

EW xτ k ∧ T ≤ EWτ k ∧ T, xτ k ∧ T

 W0 · x0  E

τ k ∧T

0

LW t, xtdt

≤ W0 · x0  KT.

2.12

On the other hand, if τ k ≤ T, then x i τ k  / ∈ k−1, k for some i; therefore,

V xτ k  ≥ u

 1

√

k



∧ u

k −→ ∞,

EV xτ k ∧ T ≥ Pτ k ≤ T



u

 1

√

k



∧ u

k

2.13

so limk → ∞ P τ k ≤ T  0;Theorem 2.1is proved

3 Stochastically Ultimate Boundedness

Theorem 2.1 shows that under simple hypothesis conditions 1.2, 1.3, and 2.1, the solutions of 1.1 will remain in the positive cone Rd

 This nice positive property provides

us with a great opportunity to construct other types of Lyapunov functions to discuss how

the solutions vary in R din more detail

As mentioned in Section 2, the nonexplosion property in a population dynamical system is often not good enough but the property of ultimate boundedness is more desired Let us now give the definition of stochastically ultimate boundedness

Theorem 3.1 Suppose 2.1 and the following condition:

min

i −a ii − A i  > max

hold Then for all θ > 0 and any initial data {x t: −τ ≤ t ≤ 0} ∈ C−τ, 0, R d

, there is a positive

constant K, which is independent of the initial data, such that the solution xt of 1.1 has the

property that

lim sup

t → ∞ E |xt| θ

Proof If condition1.2 is satisfied, then

lim

θ → ∞

θ θ A1θi

1  θ1θ|a ii|θ  0. 3.3

By Liapunov inequality,



E |x| r 1/r

≤E |x| θ 1/θ , if 0 < r < θ < ∞. 3.4

So in the proof, we suppose θ is big enough, and these hypotheses will not effect the

conclusion of the theorem

Define the Lyapunov functions

V xt  V θ xt d

i1

x θ i ,

It is sufficient to prove

lim sup

t → ∞

E |V xt| ≤ K0, 3.6

with a constant K0, independent of initial data{x t:−τ ≤ t ≤ 0} ∈ C−τ, 0, R d



We have

LV θ



x, y



i

θx θ i

⎡

⎣b i

j



a ij x j  b ij y j

⎤

⎦  θθ −1

2



ij

γ ij2x i θ



i

θx θ i

⎛

⎝b iθ − 1

2



j

γ ij2

⎞

⎠  θI θ x  θ

ij

b ij x θ i y j

≤ cV θ x  θI θ x  θ

ij

b ij x i θ y j ,

3.7

where c  max i θb i θ−1/2j γ2

ij  is constant and I θ x is given in 1.9 Let z  x/x∞,

for all x ∈ R d

; byLemma 1.1, we have

I θ x  I θ zx∞  I θ zx1θ∞ ≤ −K θ d −1−θ |x|1θ. 3.8 Then,

I θ x ≤ −K θ x1θ

∞ ≤ −K θ d−1V θ1 x,



ij b ij x i θ y j≤

ij bij 1

1 θ

θx1θi  y1θ

Thus we obtain

LV θ



x, y

≤ cV θ x − θ

d 1  θ



i



1  θK θ x i1θ− dβ i θx i1θ− dβ

i y i1θ

, 3.10

and from1.3

lim

θ → ∞ K θ min

i −a ii − A i , 3.11

if θ is big enough, then

1  θK θ > d

θβ i  e τ β i

ByLemma 1.2and inequality3.12,

e s EV θ xs| t

0

 E

t

0

e s V θ xs  LV θ xsds

≤

t

0

e s



c1V θ xs − θ

d 1  θ



i



1  θK θ − dβ i θ

x1θi s



ds

 E

t

0

e s

i

βi θ

1 θ x1θi s − τds c1 c  1

≤ E

t

0

e s



c1V θ xs − θ

d 1  θ



i



1  θK θ − dβ i θ − de τ βi

x1θi s



ds

 Ee τ

0

−τ e s

i

βi θ

1 θ x i1θsds

≤ E

t

0

e s c1V θ xs − c2V1θxsds  Ee τ

0

−τ e s

i

βi θ

1 θ x1θi sds

≤

t

0

K0e s ds  Ee τ

0

−τ e s

i

βi θ

1 θ x1θi sds

≤ K0e t − K0 Ee τ

0

−τ e s

i

βi θ

1 θ x1θi sds,

3.13

where c2 infi θ/d1 θ1 θK θ −dβ i θ − de τ βi  > 0 is a constant Then 3.2 follows from

the above inequality andTheorem 3.1is proved

4 Asymptotic Pathwise Estimation

In the previous sections, we have discussed how the solutions vary in R d

in probability or in moment In this section, we will discuss the solutions pathwisely

Theorem 4.1 Suppose 2.1 holds and the following condition:

is satisfied, where K1is given by1.5 and B  supx1 |Bx| Then for any initial data {x t:−τ ≤

t ≤ 0} ∈ C−τ, 0, R d

, the solution xt of 1.1 has the property that

lim sup

t → ∞

1

t

 ln|xt| K1d −3/2 − B t

0

|xs|ds



≤ |b| − λ

where λ  λminGG T

Proof Define the Lyapunov functions

V x  V1x, for x ∈ R d

By Ito’s formula, we have

LV

x, y

V x 

b T x  I1x  x T By

V x

≤ |b||x| − K1d−1|x|

2 |x|By

V x

≤ |b| − K1d −3/2 |x|  By.

4.4

Theorem 3.1 Suppose... longer happen as long as conditions