Dong et al. Advances in Difference Equations (2015) 2015:266 DOI 10.1186/s13662-015-0608-2

0 Advances in Difference Equations

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RESEARCH

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Partial permanence and extinction on stochastic Lotka-Volterra competitive systems

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Chunwei Dong1, Lei Liu1* and Yonghui Sun2

Correspondence: liulei_hust@hhu.edu.cn 'College of Science, Hohai University, Nanjing, Jiangsu 210098, China

Fulllist of author information is available at the end of the article

Abstract

This paper discusses an autonomous competitive Lotka-Volterra model in random environments. The contributions of this paper are as follows. (a) Some sufficient conditions for partial permanence and extinction on this system are established; (b) By using some novel techniques, the conditions imposed on permanence and extinction of one-species are weakened. Finally, a numerical experiment is conducted to validate the theoretical findings.

Keywords: Lotka-Volterra model; random environments; Brownian motions; Ito formula; persistence in mean; extinction

ft Spri

1 Introduction

It is a usual phenomenon for two or more species who live in proximity share the same basic requirements and compete for resources, habitat, food, or territory. It is therefore very important to study the competitive models for multi-species. As we know, the well-known Lotka-Volterra model concerning ecological population modeling has received great attention and has been studied extensively owing to its theoretical and practical significance. A deterministic, competitive Lotka-Volterra system with n interacting species is described by the n-dimensional differential equation

dxi(t)

~dT -xi(t)

bi - ^ aw (t) j=i

i = 1,2,..., n, (1.1)

ringer

where xi(t) represents the population size of species i at time t, bi is the growth rate of species i, and aj represents the effect of interspecific (if i = j) or intraspecific (if i = j) interaction.

On the other hand, from the biological point of view, population systems in the real world are inevitably affected by environmental noise. In practice, the growth rates are often subject to environmental noise. To obtain a more accurate description of such systems, we usually consider the stochastic perturbation of the growth rate bi by an average value plus an error term. Then the intrinsic growth rate depending on time becomes

bi ^ bi + OiBi(t),

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where Bi(t) is a white noise. As a result, system (1.1) becomes a stochastic Lotka-Volterra competition system with n interacting components as follows:

where bi, atj, ai are non-negative for i, j = 1,2,..., n, and of will be called the noise intensity matrix. Throughout this paper we always assume that the following hypothesis holds:

It is therefore necessary to reveal how the noise affects the population systems. As a matter of fact, stochastic Lotka-Volterra competitive systems have recently been studied by many authors, for example, [1-6].

In the study of population systems, permanence and extinction are two important and interesting topics, respectively meaning that the population system will survive or die out in the future, which have received much attention (see [7-18]). Luo and Mao [15] revealed that a large white noise will force the stochastic Lotka-Volterra systems to become extinct while the population may be persistent under a relatively small white noise. Li and Mao [9] investigated a non-autonomous stochastic Lotka-Volterra competitive system, and some sufficient conditions on stochastic permanence and extinction were obtained. Li etal. [11] showed that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. Tran and Yin [16] investigated stochastic permanence and extinction for stochastic competitive Lotka-Volterra systems using feedback controls.

However, most of the existing criteria are established for stochastic general Lotka-Volterra system. Hence, one natural question arises: How to derive some criteria with less conservatism for stochastic Lotka-Volterra competitive systems? This issue constitutes the first motivation of this paper.

Moreover, most of the existing criteria are established for total permanence and total extinction. To the best of our knowledge, partial permanence and partial extinction have scarcely been investigated, which are very important properties. Is it feasible to obtain some partial permanence and extinction conditions for stochastic Lotka-Volterra competitive systems? Thus, the second purpose of this paper is to solve this interesting problem.

The rest of the paper is arranged as follows. The main results of this paper are stated in Sections 3 and 4. In Section 2, some preliminaries, definitions and lemmas are given. Sufficient conditions on persistence in mean and extinction for one-species are obtained in Section 3. Based on these sufficient conditions on one-species, sufficient criteria on partial permanence and extinction on system (1.2) are established in Section 4. Section 5 provides some numerical examples to check the effectiveness of the derived results.

2 Notation

Throughout this paper, unless otherwise specified, let F, {Ft}t>0, P) be a complete probability space with a filtration {Ft}t>o satisfying the usual conditions (i.e., it is increasing and right continuous, while Fo contains all P-nullsets). Let B(t) = (B^,...,-^) be an m-dimensional Brownian motion defined on the probability space. Let R+ = {x e Rn : xi > 0 for all 1 < i < n}.

bi > 0, au > 0, aij > 0 (i = j).

Lemma 2.1 ([19]) Assume that condition (1.3) holds. Then, for any given initial value x(0) e R+, there is a unique solution x(t) to system (1.2) and the solution will remain in R+ with probability 1, namely

P{x(t) e R+, Vt > 0} = 1.

Lemma 2.2 ([19]) Assume that condition (1.3) holds. Then, for any given initial value x(0) e R+, the solution xi(t) to system (1.2) obeys

sup Exp(t) < Kp, i = 1,...,n.

0<t<+œ . . — i=1

Definition 2.1 System (1.2) is said to be persistent in mean if there exist positive constants ai, ¡3i such that the solution to system (1.2) has the following property:

limsup- I xi(s)ds < fii a.s. i = 1,...,n,

t—œ t Jo

t—t 1 /,t

liminf- I Xi(s)ds > at a.s. i = 1,...,n.

t—TO t J0

To proceed with our study, we consider two auxiliary stochastic differential equations

dyi(t) = yi(t)[(bi - auyi(t)) dt + Oi dBi(t)], yi (0) = xi (0), i = 1,..., n,

dzi(t) = Zi(t)[(bi aiiyi (t) - auzi(t)) dt + Oi dBi(t)], Zi(0)= xi(0), i = 1,..., n,

(2.1) (2.2)

y(t) = (yi(t),...,yn (t))T, z(t) = (z1(t),...,Zn(t))T.

Lemma 2.3 Assume that condition (1.3) holds. Letx(t) be a solution to system (1.2) with x(0) e R+, then we have

z(t) < x(t) < y(t),

zi(t) <xi(t) <yi(t), i = 1,...,n. Proof By the Itô formula, we derive that

xi(t) xi (0)

02 - b^t+

J2ai> f

j=i j0

x,(s) ds - oiBi (t)

+ aii exp /0

- 0i(Bi(t)-Bi(s))

- bi (t - s) +

ds, i = 1,..., n,

I2ai>fs

x,(i) di

yi(t) xt(0)

- bi t - aiBi{t)

+ aii / exp

- bi (t - 5)

- a^Bi(t)-Bi(s))

ds, i = 1,..., n,

which means

xi(t) <yi(t), i = 1,...,n. Applying the Ito formula to equation (2.2) yields 11"

zi(t) xi (0)

+ aii I exp

Y^aij j yj(s)ds

T,a»[

- CTiBi(t)

- bi (t - s) +

yiW di

- ai(Bi(t)-Bi(s))

ds, i = 1,..., n.

From the representations of yi(t) and zi(t), and by (2.6) we have Zi(t) <Xi(t), i = 1,...,n.

3 Persistence in mean and extinction of one-species 3.1 Persistence of one-species

In this section, we investigate persistence in mean and extinction of one-species for system (1.2). Now, let us present some lemmas which are essential to the proof of Theorem 3.1.

Lemma 3.1 ([20]) Let condition (1.3) hold. The solution yi(t) to equation (2.1) has the following property:

r lQgyi(t) U \ A n lim-= bi —- A 0 a.s.

t^™ t V i 2 J

With the help of Lemma 3.1, we slightly improve Lemma 3.1 of [8] by weakening hypotheses posed on the coefficients of equation (2.2) as follows.

Lemma 3.2 Let condition (1.3) hold, and assume that bi - -y > 0, bj - -y > 0 (i = j) and bi - ^— Xj=i ij- ((bj —A 0) > 0. Then the solution to equation (2.2) has the prop-erty

!og zi(t) _ lim-= 0 a.s.

t^-ro t

Theorem 3.1 Let condition (1.3) and assumptions in Lemma 3.2 hold. Then the solution to system (1.2) has the following property:

1 ^ ^ , 1 limsup- I xi(s) ds <— I bi--I a.s.,

t^rn t J0 aii\ 2 /

- f* 1 liminf- I Xi(s)ds >—

liminf - I

t—œ t J0

b - '+)-E - °h - o

which means the species i of system (1.2) is persistent in mean.

Proof Applying the Itô formula to equation (2.1) yields

logy.(t) = logy. (0) + ( b. - ) * - a.. j^t y.(s) ds + a.Bi(t).

Then we have

y. (s) ds = —-^ t + —B.(t) —— (log y. (t)-log y. (0)).

aii aii a;:;:

Dividing both sides of (3.5) by t yields

y.(s) ds = —

Jo aii

log y. (0) logy.(t)

2 ) + 1 i

+ bi - + + - / a.dBi(s)

Note that

logyi(0) lim-= 0,

t—œ t

aidBi(s) = 0 a.s.

This implies

lim \f ty.(s)ds = -1 lim + 1 f b. - Î ).

t—œ t

a..t—œ t an

By Lemma 3.1, letting t — m on both sides of (3.6) yields

1t lim -

t—œ t j0

t1 yj(s) ds = —

J0 ajj

= — bi - j v 0 a.s.

Combining Lemma 3.2 and (3.7), we can claim that

1 ft 1 ft 1 ( a2

limsup- I xi(s) ds < lim - I yi(s) ds = — I bi —- | a.s.

t—M t Jo t—M t Jo aii \

Now we process to show assertion (3.3). Applying the Ito formula to logzi(t) yields

logZi(t) = logz. (0) - ^ ^^ - b^j ds - a.. f^t z.(s) ds

+ > I a.jZj(s) ds +/ o.dBi(s).

j=i j0 j0

Dividing both sides of (3.9) by t yields

log zi(t) log Zi (0) 1 ft / a

11X i-b¥- tL tzi(s) ds

T^f aijyj(s)ds +1/" OidBi(s). (3.10)

t j=i Jo t Jo

Using Lemma 3.2 and the law of strong large number for martingale, we have

lim 1 i ffidBi(s) = 0, lim l0gZi(t) = 0 a.s. (3.11)

t^-ro t Jo t^-ro t

Combining Lemma 3.2 and (3.11), letting t ^ +to on both sides of (3.9) yields

1 pt 1 r / tj2\ _ a■■ ft

lim - I zi(s) ds = — I bi —- I + > lim — / y,(s) ds t Jo i() au\i 2) ¿-at^TO t Jo y() _

aii Abi

aii Abi

bi- j-f) *o

> o. (3.12)

Since Xi(t) > Zi(t), assertion (3.3) is true. Therefore this theorem is proved. □

Remark 3.1 Compared with the existing literature [8], the conditions imposed on the permanence of one-species are weaker.

Applying Lemma 3.1 to system (1.2), we have the following corollary, which coincides with Theorem 3.1 in [8].

Corollary 3.1 Let condition (1.3) hold and assume that bi - > 0, bi - -J— a-- (bj -

a-2 11

j) > 0for all i = 1,..., n. System (1.2) is persistent in mean. 3.2 Extinction of one-species

Theorem 3.2 Let condition (1.3) holdandxi(t) be the solution to system (1.2) with positive initial value xi (0). Then we have the following assertions:

(i) If of > 2bi, the solution xi(t) to system (1.2) has the property that

log Xi(t) of

limsup-< bi —- a.s. (3.13)

t^ro t 2

That is, the species i of system (1.2) will become extinct.

(ii) If of = 2bi, the solution xi(t) to system (1.2) has the property that

lim Xi(t) = 0 a.s. (3.14)

That is, the species i of system (1.2) still become extinct with probability one.

Proof The proof is rather technical, so we will divide it into two steps. The first step is to show the exponential extinction of species i when of > 2bi. The second step is to show the extinction in the case of of = 2bi.

Step 1. Applying the Ito formula to logXi(t) yields

log Xi (t) = log Xi (0) + (bids - Xa xi(s) ds + oidBi(s). (3.15)

Dividing both sides of (3.15) by t yields

:' (». -al

t t h\ 2

logXi(t) logXi(0) + 1 /"/». - al Us

1 n ft 1 ft

— y"^aj I Xj(s) ds +- I oidBi(s). (3.16)

t j=1 Jo t Jo

Using the law of strong large number for martingales, we can claim that 1 it

lim - I oidBi(s) = 0 a.s.

t—TO t Jo

Letting t — ^ yields

log Xi(t) of

lim sup-< bi--a.s.

t—t 2

Step 2. Now, let us finally show assertion (3.14). Decompose the sample space into three mutually exclusive events as follows:

= \ w : lim sup xi(t) > liminf xi(t) = yi > 0 f;

I t—m t—m )

Qi2 = |« : limsupxi(t) > liminfxi(t) = 0 f;

I t—m t—m )

Qi3 = « : lim xi(t) = 0.

I t—rn I

When of = 2»i, equation (3.16) has the following form:

log xi(t) log xi (0) 1

\y\aij f xi(s)ds + 1 i t J0 t J0

xi(s) ds + - OidBi(s). (3.17)

Furthermore, we decompose the sample space into the following two mutually exclusive events according to the convergence of xi(s) ds:

Ei1 = 1« :j xi(s) ds < , Ei2 = j« : j xi(s) ds = . (3.18)

The proof of limt—TOxi(t) = 0 a.s. is equivalent to showing Ei1 c ^i3, Ei2 c ^i3 a.s. The strategy of the proof is as follows.

* First, by using the techniques proposed in [21], we show that Ei1 c ^i3. It is sufficient to show P(En n = 0 and P(Efl n Q.a) = 0.

* Second, using some novel techniques, we prove that P(Ei2 n = 0 and P(Ei2 n = 0, which means Ei2 c ^i3 a.s.

Now we realize this strategy as follows.

Case 1. Let us now show Ei1 c Œi3. Clearly, xi(t) e C(R+,R) a.s. It is straightforward to see from Ei1 that liminft^TO xi(t) = 0 a.s. Therefore, we have obtained that P(Ei1 n = 0. Now we only need to prove that P(Ei1 n Œi2) = 0. We prove it by contradiction. If P(Ei1 n ni2) > 0, there exists a number e >0 such that

where / = {limsupt^TO xi(t) > 2ej. Let us now define a sequence of stopping times

Ti = inf{t > 0 : xi(t) > 2e}, T2k = inf{t > r2k-i: xi(t) < e}, Tik+i = inf{ t > T2k: xi (t) > 2e}, k = 1,2,....

From Eii, we also have EE /0to xi(s) <is) < to, then we compute it

where is the indicator function for all sets A. Since T2k < to whenever T2k-1 < to, by the above formula, so we have

P(J1 n Efl) > 2e,

> e IZE[/{T2k-1<TO)nE,1(T2k - T2k-0],

e y^E[/{t2k-1<TO)nE,i(T2k-1 - ?2k)] < TO.

On the other hand, integrating equation (1.2) from 0 to t yields

(3.21)

A simple computation shows that

< 2E(xf(s)) + 2E^bt + 6b2 ¿a;2x;2(s) + ¿aj))

E(o? ■ x2(s)) = of ■ E(x2(s)) < of ■ K2 =: Nf,

where K2 and K4 are defined in Lemma 2.2. Using the Holder inequality and Burkholder-Davis-Gundy inequality (see [19]), we compute

I{r2k-1<TO}nEii SUp |xi(t2k-1 + t)-xi(r2k-1)|

/{t2k-1<~}n£i1 sup 0<t<T

/{t2k-1<~}n£i1 sup 0<t<T

£ |xi(s) ■ ^»i - X]aijxj(s)J j ds

1 /• t2k-1+t 2" / (oi ■ xi(s)) dBi(s) •'t2k-1

^ ^ ^x2 (s) ■ ^»i - aijxj(s) ^ j (s) ds

f t2k-1+t

I^k-K^E! ■ x2(s^ d

J T2k-1

< 2T(Mf + 4N2).

Furthermore, we choose T = T(e) > 0 sufficiently small for

2T(m2 + 4N2) < e3.

It then follows from (3.22) that

P({r2k-1 < TO}n{Hk n Ei1}) < 2(T + 4)T M2+ N2) < e,

Hk = sup |xi(T2k-1 + t) - xi (l*2k-1) 1 > e , k = 1,2, ...,n.

(3.23)

Recalling the fact that Tk < to, for k = 1,2,..., whenever w e J1, we further compute

P({T2k-1< {Hk n Ei1}) = P({T2k-1 < to} n En) - P({T2k-1 < to} n {Hk n Ei1}) > 2e - e = e.

If w e {r2k-1 < to} n {Hk n Ei1}, note that

T2k (w) - T2k-1(w) > T.

We derive from (3.20) and (3.24) that

TO > f ^ £[/{T2*_I (T2k - T2k-l)]

- f iz£[7{t2k-i<TO}n{^kn£,i}(r2k-l - T2k)]

- {T2k-i < TO} n {Hck n £;i})

- e = TO, (.25)

which is a contraction. So that P(Eil n = 0 holds. Therefore, we obtain that Eil c ^i3.

Case 2. Now, we turn to prove that Ei2 c ^i3 a.s. It is sufficient to show P(Ei2 n = 0 and P(Ei2 n &i2) = 0. We prove it by contradiction.

If P(Ei2 n > 0 holds, for any w e Ei2 n e0 e (0, Yi), there exists T = (e0, w) such that

Xi(t) >Yi - e0 > "2" Vt > T a.s. It then follows from (3.l7) that

l ft l fT l ft l fT t - T yi

-J X"(s)ds =^J x"(s)ds x"(s)ds J x"(s)ds +—-— y a.s.

Letting t — to, we obtain that

liminf - i X"(s)ds > — >0 a.s.

t^TO t J0 2

This implies

log x"(t) -A Y" limsup-< - > aij— < 0 a.s.,

t—>to t ^—f 2

which contradicts the definition of Ei2 and nil. So P(Ei2 n = 0 must hold.

Now we process to show P(Ei2 n > 0 is false. For this purpose, we need more notations as follows:

nf(i) := {0 < s < t: xi(s) - e}, Sf(i) := ^^, Se(i) := liminf Sf i, Af(i) := {w e Ei2 n ^i2 : Se(i) > 0},

t—TO 1 '

where m(nf (i)) indicates the length of nf (i). It is easy to see that A0(i) = Ei2 n For any el < e2, simple computations show that

nfl(i) d nf2(i), m(nfl(i)) - m(nf2(i)),

.... m(nfl)(i^ m(nf2)(i) V(i) =-;-- V(i) =-;-,

which implies

Sl2 (i) < 5ei (i), Al2 (i) c A11 (i) Vei < e2-

It is easy to observe from the continuity of probability that

P(Ae (i)) — P(A0(i)) = P(E2 n ^2) as e — 0.

If P(Ei2 n &i2) > 0, there exists e >0 such that P(De) > 0. For any w e Ae (i), simple computations show that

1 ft 1 f 1 f 1 f

- I xi(s) ds = - I xi(s) ds + - / xi(s) ds > - / xi(s) ds a.s.

t Jo t Jul(i) t ,/[0,t]\ne t Jul(i)

By letting t — to, we have

1 ft 1 f liming I xi(s) ds > liming I xi(s) ds > Se(i)e a.s. (3.26)

t—TO t Jo t—TO t Ju"

Substituting (3.26) into (3.17), we obtain that

log xi(t) ^ limsup-< - > aijS (i)e < 0 a.s.

t—t . j=1

This contradicts the definition of Ei2 and ^i2. It yields the desired assertion P(Ei2 n = 0 immediately. Combining the fact Ei1 c P(Ei2 n = 0 and P(Ei2 n = 0, we can claim that

lim xi(t) = 0 a.s.

The proof is completed. □

Remark 3.2 In comparison with [8] and [11], we point out that species i is still extinct when of = 2bi by using some novel stochastic analysis techniques.

Corollary 3.2 Let condition (1.3) hold andx(t) be a solution to system (1.2) with positive initial value x(0). Assume that there exists an integer m, 1 < m < n, such that

of > 2bi, i = 1,..., m, (.27)

of = 2bi, i = m + 1,..., n. (3.8)

Then we have the following assertions:

(i) For all i = 1,..., n, the solution xi(t) to system (1.2) has the property that

log xi(t) of

lim-= bi —- a.s. i = 1,...,m. (3.29)

t—TO t 2

(ii) For all i = m +1,..., n, the solution Xi(t) to system (1.2) has the property that log Xi(t) .

lim-= 0 a.s. i = m + 1,...,n. (.0)

t^-ro t

Proof By virtue of Theorem 3.2, for all of > 2bi, i = 1,..., m,we obtain that

log xi(t) °f limsup-< bi--a.s. i = 1,...,m.

t^ro t 2

This shows that for any e e (0, -L - bi) there is a positive random variable T(ei) such that

Xi(t) < expjybi )t + e^} Vt > T(fi),a.s. i = 1,...,m, which means

I xi(s)ds < ro a.s. i = 1,...,m. Jo

Then letting t ^ro on both sides of (3.16) yields

log Xi(t) af lim-= bi--a.s. i = 1,...,m,

t^ro t 2

which is the required assertion (3.29).

Now we process to show assertion (3.30). By utilizing Theorem 3.1 and conditions (3.28), we derive

lim xi(t) = 0 a.s. i = m + 1,...,n.

This implies 1 ft

lim - I xi(s)ds = 0 a.s. i = m + 1,...,n. (.31)

t^ro t J0

By the law of strong large numbers for martingales and (3.31), letting t ^ro on both sides of (3.17) yields

log xi(t) _ . . lim-=0 a.s. i = m + 1,...,n.

t ^ro t

The proof is completed. □

4 Partial permanence and extinction

Now in this section we present conditions for system (1.2) to be partially permanent and extinct. To proceed with our study, we consider the following auxiliary stochastic equation:

' d$i (t) = ®i(t)(bi - aij)) dt + -i^i(t) dBi(t), (4 1)

(0)= xi(0), i = 1, ...,m. .

Theorem 4.1 Let condition (1.3) hold. Assume that there exists an integer m, 1 < m < n, such that

: - X) aji > 0,

2 m / 2 \

bimHbk-t)>0, i = 1,...,m,

2 l=lak^ 2'

bi < y, i = m + 1,..., n.

Then we have the following assertions:

(i) For all i = 1,..., m, the solution x(t) to system (1.2) has the property that

1 ft ,, , 1 /, .

limsup- I xi(s)ds <— I bi--) a.s. i = 1,...,m.

t—ro t Jo aiA 2

liming I xi(s)ds

t—TO t Jo

a.s. i = 1,...,m.

That is, for each i = 1,..., m, the species i of system (1.2) is persistent in mean; (ii) For all i = m +1,..., n, the solution x(t) to system (1.2) has the property that

limsup

log xi(t) oi

< bi - 0l-y a " i 2 j=r ajj

E- bk-akk

bi -■

a.s. i = m + 1,..., n.

That is, for each i = m + 1,..., n, the species i will become extinct.

Proof We will divide the proof into two steps. The first step is to show the permanence of the top m species of system (1.2). The second step is to show the extinction for the bottom n - m species of system (1.2). Step 1. Applying the Ito formula to (4.1) yields

dlog (t) = ^bi - -2- - ^ aij&j(t)j dt + Oi dBi(t), i = 1,..., m.

Simple computations show that

d(log xi(t) - log ^i(t)) = - J2 aijj) - (t)) dt

- y^ auxi(t)dt, i = 1,...,m.

Applying the Ito formula to V(t) = J2mi I logxi(t) - log ^i(t)| yields

D+V(t) = £sgn(xi(t) - ^(t)) • [dlogxi(t)-dlog ^(t)]

^fl,j(x;(t) - $j(t)) + ^ (t)

_ j=1 /=m+1

mm m n

< - ^ A;; |xi(t) - ^i(t) I dt + ai> \xi(t) - j) I dt + m aiiXi(t) dt

1=1 /=1 i=1 i=m+1

— ^ ^ aii\X i(t)-^(t)\dt 5Zayi\xi(t)-^i(t)\dt aiixi(t)dt

i=1 /=1 i=j i=1 l=m+1

m mm m n

= aii\xi (t)-^i(t)\dt + ^ ^a;ï\xi(t) - $i(t)\ dt + ^ ^ aiixi (t) dt

i=1 j=i

i=1 i=m+1

< - ^I au - ^a/i I \xi(t) - ^i(t) \ dt + aiixi(t) dt.

i=1 i=m+1

Hence we get

D+ V (t) < -¡^ |xi(t)-^i (t)| dt + ^ dixi (t) dt, i = 1,..., m,

i=1 l=m+1

where ¡x = min1<i<m(flii - Xm j > 0, = Xm1 ^l > 0. We therefore have /. t m

V (t)+x E|xi(s)-*i(s)| ds

n /. t

< V(0) + V" 0l I xl(s)ds, i = 1,...,m.

l=m+1 ^

Letting t —^ ro on both sides of (4.10) yields

/> ro /> ro m

/ |xi(s)-^i(s)| ds < / VVs)-^(s)| ds

j0 j0 i=1

V(0)+V 0i xi (s) ds

i=m+1 ^

By Theorem 3.2 and condition (4.3), we have

I xi(s)ds <+œ a.s. i = m + 1,...,n.

Substituting (4.12) into (4.11) yields

i \x;(s)- ^(s) ds <+œ a.s. i = 1,...,m.

(4.10)

(4.11)

(4.13)

By virtue of the similar techniques proposed in Step 2 of Theorem 3.2, we have

lim |xi(t)-#i(t)| =0 a.s. i = 1,...,m. (.14)

t—TO1 1

When condition (4.2) is satisfied, by applying Corollary 3.1 to system (4.1), we have

1 t 1 i2 limsup- / 0i(s)ds <— \bi--a.s. i = 1,...,m,

t Jo a 1

liminf 1 f 4(s) ds > - (b, - -

t Jo ~ an \l -J ¿-f akk \ k -J

- ' akk\ -

a.s. i = l,...,m.

A simple computation shows that

1 ft 1 ft 1 ft

limsup / xi(s) ds < limsup -I (xi(s)-0i(s)) ds + limsup- / ^i(s) ds

t—TO t Jo t—TO t Jo t—TO t Jo

1 /, < , < _ bi - tt I a.s. i = l,...,m,

liming I xi(s) ds

t^œ t Jo

l ft l ft

liming I (xi(s)- 4i(s)) ds + liming / 4i(s) ds t Jo t Jo

bi--)-

akk\ -

a.s. i = l,...,m.

(4.15)

(4.16)

Therefore, we obtain thatxi(t) is persistent in mean, for all i = 1,...,m. Step 2. For all i = m + 1,...,n, applying Ito to logxi(t) yields

log Xi(t) log Xi (0) 1 -=-+ -

jo> 4) "s- m? joj"s

It follows from (4.15) that

V — f xi(s)ds +- / oidBi(s), i = m + 1, ...,n.

_.„.1 t J0 t Jo

liminf

t^-ro t

m Ç t

J2ai> /

j=1 •/o

Xj (s) ds

> y^ an

b-j) -E Ëh-T

- k=j a -

a.s. j = 1, ...,m.

(4.17)

By letting t —to on both sides of (4.16) yields. We can conclude that

limsupiOgX« < bi - O--

t^TO t - . 1 Ojj

-Y °iL[bk - Ok

a.s. i = m + 1,...,n.

Finally, we can get xi(t) will become extinct for all i = m + 1,...,n. The proof is completed. □

5 Numerical simulations

In this paper, we have discussed the persistence in mean and extinction of system (1.2). Moreover, sufficient conditions have been established in Theorems 3.1, 3.2 and 4.1. Thus, in this section, we give out the numerical experiment for the case n = 2 as follows to support to our results.

dx:(t) = x:(t)[(0.9 - 0.3x1 (t) - 1.2x2(t)) dt + 01 dB^t)],

dx2(t) = x2(t)[(1.1 - 0.3x1 (t) - 0.4x2(t)) dt + 02 dB2(t)].

The existence and uniqueness of the solution follows from Lemma 2.1. We consider the solution with initial data xi(0) = 1.4, x2(0) = 2.1. By Matlab software, we simulate the solution to system (5.1) with different values of 01 and o2.

In Figure 1, 01 = 0.03, o2 = \/22. By Theorems 3.1, 3.2 and 4.1, species 1 is persistent in mean and species 2 is extinct with zero exponential extinction rate.

In Figure 2, 01 = 0.03, o2 = 0.04. By the conditions of Corollary 3.1, all of the species are persistent in mean.

In Figure 3,01 = \/L8, o2 = 1.7. By Corollary 3.2, species 1 is extinct with zero exponential extinction rate and species 2 is exponentially extinct.

Figure 1 The solution of system (5.1)with oT = 0.03, o2 = V2.2. The blue line represents species 1, while the green line represents species 2.

Figure 3 The solution of system (5.1)with

o1 = -/T8, o2 = 1.7. The blue line represents species 1, while the green line represents species 2.

0.8 0.6

£. 0 2 i

f i!' -0.6 -0.8 -1,

6 Conclusions

This paper is devoted to partial permanence and extinction on a stochastic Lotka-Volterra competitive model. Firstly, by using some novel techniques, we established some weaker sufficient conditions on the persistence in mean and extinction for one-species. Secondly, based on these sufficient conditions for one-species and some stochastic analysis techniques, sufficient criteria for ensuring the partial permanence and extinction of the populations of the n different species in the ecosystem have been obtained. Finally, numerical experiment is provided to illustrate the effectiveness of our results.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

The authors have made the same contribution. All authors read and approved the final manuscript. Author details

1 College of Science, Hohai University, Nanjing, Jiangsu 210098, China. 2College of Energy and Electrical Engineering, Hohai University, Nanjing, Jiangsu 210098, China.

Acknowledgements

The authors would like to thank the editor and referees for their very important and helpful comments and suggestions. We also thank the National Natural Science Foundation of China (Grant Nos. 61304070,11271146,61104045,51190102), the National Key Basic Research Program of China (973 Program) (2013CB228204).

Received: 11 February 2015 Accepted: 12 August 2015 Published online: 28 August 2015

References

1. Jiang, D, Shi, N, Li, X: Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation. J. Math. Anal. Appl. 340, 588-597 (2008)

2. Jiang, D, Shi, N: A note on non-autonomous logistic with random perturbation. J. Math. Anal. Appl. 303,164-172 (2005)

3. Liu, M, Wang, K: Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system. Appl. Math. Lett. 25,1980-1985 (2012)

4. Liu, M, Wang, K: Persistence and extinction in stochastic nonautonomous logistic systems. J. Math. Anal. Appl. 375, 443-457(2011)

5. Mao, X, Marion, G, Renshaw, E: Environmental noise suppresses explosion in population dynamics. Stoch. Process. Appl. 97, 95-110(2002)

6. Mao, X: Stationary distribution of stochastic population systems. Syst. Control Lett. 60,398-405 (2011)

7. Bao, J, Mao, X, Yin, G, Yuan, C: Competitive Lotka-Volterra population dynamics with jumps. Nonlinear Anal. 74, 6601-6616(2011)

8. Jiang, D, Ji, C, Li, X, O'Regan, D: Analysis of autonomous Lotka-Volterra competition systems with random perturbation. J. Math. Anal. Appl. 390,582-595 (2012)

9. Li, X, Mao, X: Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete Contin. Dyn. Syst. 24,523-545 (2009)

10. Li, X, Jiang, D, Mao, X: Population dynamical behavior of Lotka-Volterra system under regime switching. Comput. Appl. Math. 232,427-448 (2009)

11. Li, X, Gray, A, Jiang, D, Mao, X: Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching. J. Math. Anal. Appl. 376,11-28 (2011)

12. Liu, M, Wang, K: Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations. Discrete Contin. Dyn. Syst. 33, 2495-2522 (2013)

13. Liu, M, Wang, K: Dynamics of a two-prey one-predator system in random environments. J. Nonlinear Sci. 23, 751-775 (2013)

14. Liu, M, Wang, K: Dynamics of a Leslie-Gower Holling-type II predator-prey system with Levy jumps. Nonlinear Anal. 85, 204-213 (2013)

15. Luo, Q, Mao, X: Stochastic population dynamics under regime switching II. J. Math. Anal. Appl. 355,577-593 (2009)

16. Tran, K, Yin, G: Stochastic competitive Lotka-Volterra ecosystems under partial observation: feedback controls for permanence and extinction. J. Franklin Inst. 351,4039-4064 (2014)

17. Zhu, C, Yin, G: On competitive Lotka-Volterra model in random environments. J. Math. Anal. Appl. 357,154-170 (2009)

18. Zhu, C, Yin, G: On hybrid competitive Lotka-Volterra ecosystems. Nonlinear Anal. 71, 1370-1379 (2009)

19. Mao, X: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)

20. Liu, L, Shen, Y: Sufficient and necessary conditions on the existence of stationary distribution and extinction for stochastic generalized logistic system. Adv. Differ. Equ. 2015 10 (2015)

21. Mao, X: A note on the LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268, 125-142 (2002)