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Asymptotic behavior of a stochastic nonautonomous Lotka–Volterra competitive system with impulsive perturbations
Mathematical and Computer Modelling 57 (2013) 909–925
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Asymptotic behavior of a stochastic nonautonomous Lotka–Volterra competitive system with impulsive perturbations Meng Liu a,b,∗ , Ke Wang b a
School of Mathematical Science, Huaiyin Normal University, Huaian 223300, PR China
b
Department of Mathematics, Harbin Institute of Technology, Weihai 264209, PR China
article
info
Article history: Received 8 November 2011 Received in revised form 15 September 2012 Accepted 19 September 2012 Keywords: Competitive system Stochastic perturbations Impulsive effects Stochastic permanence Global attractivity
abstract This paper is concerned with an n-species stochastic nonautonomous Lotka–Volterra competitive system with impulsive effects. Some dynamical properties are investigated and the sufficient conditions for stochastic permanence, extinction and global stability are established. Moreover, the lower-growth rate and the upper-growth rate of the positive solution are studied. In addition, the limit of the average in time of the sample paths of solutions is estimated. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction In the real world, owing to many natural or man-made factors, such as earthquake, drought, flooding, fire, crop-dusting, planting, hunting and harvesting, the intrinsic discipline of species or environment often undergoes some discrete changes of relatively short time interval at some fixed times. From the viewpoint of mathematics, such sudden changes could be described by impulses. On the other hand, in the real world, it is a usual phenomenon that several species compete for limited territories or resources. Thus several authors (see e.g. [1–12]) have investigated the following nonautonomous Lotka–Volterra competitive system with impulses
n dxi (t ) = x (t ) r (t ) − aij (t )xj (t ) , t ̸= tk , k ∈ N i i dt j=1 + xi (tk ) − xi (tk ) = bik xi (tk ), i = 1, . . . , n, k ∈ N
(1)
where xi (t ) is the size of the ith population, ri (t ) and aij (t ) are continuous and bounded functions on R+ := [0, +∞), N denotes the set of positive integers, 0 < t1 < t2 < · · · , limk→+∞ tk = +∞. Recently, system (1) has been received great attention and many interesting dynamical properties, such as permanence, extinction, global stability and dynamical complexity, have been studied by [6–12] and the references cited therein. On the other hand, populations in nature are often subject to random fluctuations [13]. Therefore many investigations have been carried out to reveal the effect of environmental noise on the population dynamics (see e.g. [14–28]). Particularly,
∗
Corresponding author at: School of Mathematical Science, Huaiyin Normal University, Huaian 223300, PR China. Tel.: +86 051784183732. E-mail address: [email protected] (M. Liu).
0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2012.09.019
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M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Liu and Wang [28] considered the following stochastic single-species model with impulses
dx(t ) = x(t ) r (t ) − a(t )x(t ) dt + σ (t )x(t )dB(t ),
x(tk+ ) − x(tk ) = bk x(tk ),
t ̸= tk , k ∈ N
k∈N
where B(t ) is a standard Brownian motion, σ (t ) is a continuous and bounded function on t ≥ 0. The authors [28] studied some important properties of this model. However, in the real world, population does not exist alone, and it is a usual phenomenon that several species compete for limited resources, then it is of great importance to investigate the stochastic competitive system with impulses. To the best of our knowledge, no results related to the stochastic competitive system with impulses have been reported. Suppose that the environmental fluctuations will mainly manifest the growth rate ri (t ) (see e.g. [18–28]), with ri (t ) → ri (t ) + σi (t )B˙ i (t ), where B˙ i (t ) is the white noise, σi (t ) is a continuous and bounded function on t ≥ 0 and σi2 (t ) represents the intensity of the white noise. Then we obtain the following stochastic system:
dxi (t ) = xi (t ) ri (t ) −
n j =1
xi (tk ) − xi (tk ) = bik xi (tk ), +
aij (t )xj (t ) dt + σi (t )xi (t )dBi (t ),
t ̸= tk , k ∈ N
(2)
i = 1 , . . . , n, k ∈ N .
Because of biological meanings, we impose the additional restrictions −1 < bik , i = 1, . . . , n, k ∈ N, on (2). When bik > 0, the impulsive effects represent planting, while bik < 0 denote harvesting [28]. The remainder of the paper is organized as follows. In Section 2, we will show that when the noise is small enough the population system is stochastically permanent but a sufficiently large noise will force every species become extinct. In Section 3, we will study the lower-growth rate and the upper-growth rate of the positive solution. Moreover, we will prove that the limit of the average in time of the sample paths of the solutions is bounded with probability one and give an estimation for it. In Section 4, we will study the global stability. In the last section, we will give the conclusions. 2. Stochastic permanence and extinction Throughout this paper, let (Ω , F , {Ft }t ≥0 , P ) stand for a complete probability space. Let B(t ) = (B1 (t ), . . . , Bn (t )) be an n-dimensional Brownian motion. Set Rn+ = {x ∈ Rn : xi > 0 for all 1 ≤ i ≤ n}. If x ∈ Rn , its norm is |x| = inf aii (t ) > 0
t ≥0
n
i=1
x2i . Moreover, we assume that
for all i = 1, . . . , n
and
aij (t ) ≥ 0
for all i, j = 1, . . . , n with j ̸= i.
If f (t ) is a continuous and bounded function on [0, +∞), define f l = inf f (t ), t ≥0
f u = sup f (t ). t ≥0
For any constant sequences {cij } and {di }, 1 ≤ i, j ≤ n, define cˆ = max cij , 1≤i,j≤n
cˇ = min cij ; 1≤i,j≤n
dˆ = max di , 1≤i≤n
dˇ = min di . 1≤i≤n
To begin with, we introduce a concept proposed by Liu and Wang [28]. Definition 1 (Liu and Wang [28]). For ISDE: dX (t ) = F (t , X (t ))dt + G(t , X (t ))dB(t ), X (tk+ ) − X (tk ) = Bk X (tk ), k ∈ N
t ̸= tk , k ∈ N
(3)
with initial condition X (0), if a stochastic process X (t ) = (X1 (t ), . . . , Xn (t ))T , t ∈ [0, +∞), satisfies the following. (i) X (t ) is Ft -adapted and is continuous on (0, t1 ) and (tk , tk+1 ), k ∈ N; F (t , X (t )) ∈ L1 ([0, +∞); Rn ), G(t , X (t )) ∈ L2 ([0, +∞); Rn ), where Lk ([0, +∞); Rn ) is all Rn valued measurable {Ft }-adapted processes f (t ) satisfying T |f (t )|k dt < ∞ a.s. for every T > 0. 0
(ii) For each tk , k ∈ N , X (tk+ ) = limt →t + X (t ) and X (tk− ) = limt →t − X (t ) exist and X (tk ) = X (tk− ) with probability one. k k
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
911
(iii) X (t ) obeys the equivalent integral equation of (3) for almost every t ∈ [0, +∞) \ {tk } and satisfies the impulsive conditions at each t = tk , k ∈ N with probability one. Then X (t ) is said to be a solution of ISDE (3). Theorem 1. For any given initial value x(0) ∈ Rn+ , there is a solution x(t ) to Eq. (2) for all t ≥ 0 and x(t ) will remain in Rn+ with probability 1. Proof. The proof is motivated by Liu and Wang [28]. Consider the following SDE without impulses:
dyi (t ) = yi (t ) ri (t ) −
n
aij (t )
(1 + bjk )yj (t ) dt + σi (t )yi (t )dBi (t ),
i = 1, . . . , n
(4)
0 < tk < t
j =1
with initial value yi (0) = xi (0). By the theory of SDE (see e.g. [29]), Eq. (4) has a unique continuous maximal local solution y(t ) = (y1 (t ), . . . , yn (t ))T on [0, τe ), where τe is the explosion time. Now let us prove τe = +∞. The proof is standard and hence is omitted (see e.g. [17,22]). Let
xi (t ) =
(1 + bik )yi (t ).
(5)
0
We need only to show that x(t ) = (x1 (t ), . . . , xn (t ))T is the solution Eq. (2). In fact, xi (t ) is continuous on (tk , tk+1 ) ⊂ [0, +∞), k ∈ N, and for every t ̸= tk ,
dxi (t ) = d
(1 + bik )yi (t ) =
0
=
(1 + bik )dyi (t )
0
(1 + bik )yi (t ) ri (t ) −
0
n
aij (t )
= xi (t ) ri (t ) −
n
(1 + bjk )yj (t ) dt + σi (t )
0 < tk < t
j=1
(1 + bik )yi (t )dBi (t )
0 < tk < t
aij (t )xj (t ) dt + σi (t )x(t )dBi (t ).
j =1
At the same time, for every k ∈ N and tk ∈ [0, +∞), xi (tk+ ) = lim
+ t →tk 0
(1 + bij )yi (t ) =
(1 + bij )yi (tk+ )
0 < tj ≤ tk
= (1 + bik )
(1 + bij )yi (tk ) = (1 + bik )xi (tk ).
0
Moreover, xi (tk− ) = lim
−
(1 + bij )yi (t ) =
t → tk 0 < t < t ij
This completes the proof.
0 < tj < tk
(1 + bij )yi (tk− ) =
(1 + bij )yi (tk ) = xi (tk ).
0
Theorem 1 shows that Eq. (2) has a positive solution for any positive initial value. Now, let us study when Eq. (2) is stochastically permanent. Definition 2. Eq. (2) is said to be stochastically permanent if for every ε ∈ (0, 1), there is a pair of constants β > 0 and δ > 0 such that for any initial data x(0) ∈ Rn+ , the solution x(t ) has the property that lim inf P {|x(t )| ≥ β} ≥ 1 − ε, t →+∞
lim inf P {|x(t )| ≤ δ} ≥ 1 − ε. t →+∞
Assumption 1. There exist two constants m > 0 and M > 0 such that for all t > 0 and i = 1, . . . , n, m ≤ 0
ˇ l h = min inf hi (t ) > 0, 1≤i≤n
t ≥0
then Eq. (2) is stochastically permanent, where hi (t ) = ri (t ) − 0.5σi2 (t ),
i = 1, . . . , n.
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M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Proof. For y ∈ Rn+ , define
2 n
V1 (y) = 1/U (y) = 1 2
yi
.
i=1
By Itô’s formula, we can see that dV1 (y) = −
+
n
2
U 3 (y) i=1
y i ri ( t ) −
n
3
U 4 (y) i=1
n
aij (t )
(1 + bjk )yj dt
0 < tk < t
j =1
σi2 (t )y2i dt −
n
2
U 3 (y) i=1
σi (t )yi dBi (t ).
Note that hˇl > 0, then we can choose a constant θ > 0 such that
2 )u . hˇl > θ max (σi2 )u = θ (σ
(6)
1≤i≤n
Define V2 (y) = (1 + V1 (y))θ . In view of Itô’s formula, one can derive that dV2 (y) = θ (1 + V1 (y))θ−1 dV1 (y) + 0.5θ (θ − 1)(1 + V1 (y))θ−2 (dV1 (y))2
= θ (1 + V1 (y))
θ−2
+ (1 + V1 (y))
−(1 + V1 (y)) n
3
− θ (1 + V1 (y))θ−1
(y)
= θ (1 + V1 (y))θ−2 − +
(y)
+
n n
2 U3
U 4 (y)
+
U 6 (y)
+
≤ θ (1 + V1 (y))
−
− θ (1 + V1 (y))θ−1 = θ (1 + V1 (y))
y i ri ( t ) −
n
U 6 (y) i=1 n
U 3 (y) i=1
−
U 6 (y)
2
2
n
4
U 6 (y) i=1
n
n
2
U 5 (y) i=1
(1 + bjk )yi yj +
2(θ − 1)
2
θ−2
θ −1 2
aij (t )
σi2 (t )y2i dt
yi ri (t )
2
n n
(y)
i =1 j =1
U5
aij (t )
σi2 (t )y2i dt − θ (1 + V1 (y))θ−1
i =1
y2i ri
+
(t ) +
M au U (y)
M au
+
U 3 (y)
+
2 )u 3(σ U 2 (y)
+
≤ θ (1 + V1 (y))
−
− θ (1 + V1 (y))θ−1
n
2
U 3 (y) i=1
σi (t )yi dBi (t )
n 2θ + 1
U 6 (y)
σ (t ) 2 i
y2i
n
U 6 (y) i=1
y2i hi (t ) − θ σi2 (t )
2
n
U 3 (y) i=1
σi (t )yi dBi (t )
V2 2 1
1.5 0 . 5 ˇ l 2 u u 2 u u (y)[h − θ (σ ) ] + V1 (y)M a + V1 (y)3(σ ) + V1 (y)M a dt
2
n
2 n
U3
(y)
i =1
dt
i=1
σi (t )yi dBi (t )
2 )u + V 0.5 (y)M a u dt − θ (1 + V1 (y))θ−1 (y)M au + V1 (y)3(σ 1 θ−2
(1 + bjk )yi yj
0
V11.5
(1 + bjk )yj
0 < tk < t
j=1
0
3
θ−2
yi ri (t ) −
n
σi (t )yi dBi (t )
U 3 (y) i=1
aij (t )
i =1 n
2
i =1 j =1
3
n
2 U3
U 3 (y) i=1
σi2 (t )y2i +
U 4 (y) i=1
n
2
σi (t )yi dBi (t ).
(7)
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
913
Here, in the proof of the last inequality, we have used the fact that: if yi ≥ 0, i = 1, 2, . . . , n, then
n
2
2
=n
≤ n max yi
yi
2
max 1≤i≤n
1≤i≤n
i =1
y2i
≤ n2
n
y2i .
i=1
Now, let κ be sufficiently small such that
2 )u . 0 < 0.5n2 κ/θ < hˇl − 0.5θ (σ Define V3 (y(t )) = eκ t V2 (y(t )). An application of Itô’s formula results in dV3 (y) = κ eκ t V2 (y)dt + eκ t dV2 (y)
κt
2 )u /n2 κ(1 + V1 (y))2 /θ − 2V12 (y) hˇl /n2 − θ (σ
θ −2
≤ θ e (1 + V1 (y))
+
V11.5
2 )u + V 0.5 (y)M a u dt (y)M au + V1 (y)3(σ 1
− θ eκ t (1 + V1 (y))θ−1
n
2
U 3 (y) i=1
σi (t )yi dBi (t )
2 2 ˇ l 2 u = θ e (1 + V1 (y)) − (y) h /n − θ (σ ) /n − 0.5κ/θ 2 )u + 2κ/θ + V 0.5 (y)M a u + κ/θ dt + V11.5 (y)M au + V1 (y) 3(σ 1 κt
θ −2
2V12
− θ eκ t (1 + V1 (y))θ−1
n
2
U 3 (y) i=1
= eκ t J (y)dt − θ eκ t (1 + V1 (y))θ−1
σi (t )yi dBi (t ) 2
U3
(y)
n
σi (t )yi dBi (t ),
i =1
where
2 )u /n2 − 0.5κ/θ J (y) = θ (1 + V1 (y))θ−2 −2V12 (y) hˇl /n2 − θ (σ
2 )u + 2κ/θ + V 0.5 (y)M a u + κ/θ . + V11.5 (y)M au + V1 (y) 3(σ 1 By the definition of κ, J (y) is upper bounded in Rn+ , namely J1 := supy∈Rn J (y) < +∞. Then + dV3 (y(t )) ≤ J1 eκ t dt − θ eκ t (1 + V1 (y))θ −1
2 U3
(y)
n
σi (t )yi dBi (t ).
i=1
In other words, we have shown that
κt
θ
E V3 (y(t )) = E e (1 + V1 (y(t )))
≤ 1 + V1 (y(0))
θ
+ J1 eκ t /κ.
Consequently
lim sup E t →+∞
n
−2θ yi (t )
i =1
≤ lim sup E (1 + V1 (y(t )))θ ≤ J1 /κ =: J2 . t →+∞
Therefore lim sup E t →+∞
n i =1
−2 θ xi (t )
= lim sup E t →+∞
n i=1 0
−2 θ (1 + bik )yi
≤ m−θ J2 := J3 .
(8)
914
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Then we obtain lim supt →+∞ E |x(t )|−2θ
0.5/θ
≤ n2θ J3 =: J4 . For any ε > 0, set β = ε 0.5/θ /J4
P {|x(t )| < β} = P {|x(t )|−2θ > β −2θ } ≤
E [|x(t )−2θ |]
β −2θ
, by the Chebyshev inequality,
= β 2θ E [|x(t )|−2θ ].
Consequently, lim supt →+∞ P {|x(t )| < β} ≤ β 2θ J4 = ε . That is to say lim inf P {|x(t )| ≥ β} ≥ 1 − ε. t →+∞
Now let us prove that for ε > 0, there exists a constant δ > 0 such that lim inft →+∞ P {|x(t )| ≤ δ} ≥ 1 − ε . arbitrary q n For y ∈ Rn+ , define V (y) = i=1 yi , where p > 0. Using Itô’s formula yields
n
dV (y) =
q qyi
ri ( t ) −
i =1 q qyi
riu
−
alii myi
aij (t )
(1 + bjk )yj + 0.5(q − 1)σi2 (t ) dt + qσi (t )yqi dBi (t )
0
j =1
n
≤
n
+ 0.5q(σ ) dt + qσi (t )yqi dBi (t ). 2 u i
i =1
An application of Itô’s formula again leads to d[et V (y)] = et V (y)dt + et dV (y)
≤ qet
n q
q
yi 1/q + riu − alii myi + 0.5q(σi2 )u dt + qσi (t )yi dBi (t )
i=1
≤ et K (q)dt + qσi (t )yqi dBi (t ), where K (q) is a positive number. Consequently, lim sup E
n q yi
t →+∞
≤ K1 (q).
(9)
i=1
In other words, we have shown that
lim sup E
n
t →+∞
q xi
= lim sup E
i=1
t →+∞
n i =1
q (1 + bik ) yqi
≤ M q K1 (q).
0
Then the desired assertion follows from the Chebyshev inequality.
k+1 2 Remark 1.It is useful to point out that Assumption 1 is easy to be satisfied. An example is bik = e(−1) /k − 1, i = 1, . . . , n, then 1 ≤ 0
In the above we have demonstrated that if the noise is sufficiently small, the original system (1) and the associated stochastic model (2) behave similarly in the sense that both will be permanent. In other words, we show that if the noise is sufficiently small the noise will not spoil this nice property. However, in the following theorem we will show that if the noise is sufficiently large, the solution to the associated stochastic system (2) will become extinct with probability one, although the solution to the original model (1) may be persistent. Theorem 3. If x(t ) = (x1 (t ), . . . , xn (t ))T is a solution of Eq. (2), then for every 1 ≤ i ≤ n,
lim sup t
−1
t →+∞
ln xi (t ) ≤ lim sup t t →+∞
−1
ln(1 + bik ) +
0 < tk < t
t
[ri (s) − 0.5σ (s)]ds =: λi , 2 i
0
In particular, if λi < 0, then limt →+∞ xi (t ) = 0 a.s. Proof. Applying Itô’s formula to Eq. (4) gives
d ln yi (t ) =
hi (t ) −
n j=1
aij (t )
0 < tk < t
(1 + bjk )yj (t ) dt + σi (t )dBi (t ).
a.s.
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
915
In other words, we have shown that
ln yi (t ) − ln yi (0) =
t
hi (s)ds −
n
0
where Mi (t ) =
t
aij (s)xj (s)ds + Mi (t ),
(10)
0
j=1
t
σi (s)dBi (s). Clearly, Mi (t ) is a local martingale with quadratic variation t σi2 (s)ds ≤ (σi2 )u t . ⟨Mi (t ), Mi (t )⟩ = 0
0
It then follows from the strong law of large numbers for local martingales (see e.g. [29, Theorem 3.4]) that lim Mi (t )/t = 0
a.s.
t →+∞
(11)
At the same time, by (10),
ln(1 + bik ) + ln yi (t ) − ln yi (0) =
0 < tk < t
t
ln(1 + bik ) +
0
hi (s)ds − 0
n
t
aij (s)xj (s)ds + Mi (t ). 0
j=1
That is to say
ln xi (t ) − ln xi (0) =
ln(1 + bik ) +
0 < tk < t
≤
t
hi (s)ds − 0
ln(1 + bik ) +
j =1
t
aij (s)xj (s)ds + Mi (t ) 0
t
0 < tk < t
n
hi (s)ds + Mi (t ). 0
By (11), we obtain the required assertion.
3. Asymptotic properties Theorem 4. If Assumption 1 holds, then the solution of Eq. (2) has the property that lim sup
ln |x(t )|
t →+∞
≤ 1 a.s.
ln t
(12)
If Assumption 1 holds and moreover, hˇl > 0, then lim inf
ln |x(t )|
≥−
ln t
t →+∞
2 )u (σ 2hˇl
a.s.
Proof. Define W (y) = ln U (y) = ln dW (y) =
=
dU (y)
−
n
U (y) i=1
−
n
(dU (y)) 2U 2 (y)
i =1
yi . By Itô’s formula, we can observe that
2
U (y) 1
(13)
yi ri (t ) −
n
aij (t )
n
2U 2 (y) i=1
(1 + bjk )yj dt
0
j =1
1
σi2 (t )y2i dt +
n
1
U (y) i=1
σi (t )yi dBi (t ).
(14)
Using Itô’s formula again leads to d(et W (y)) = et W (y)dt + et dW (y)
t
= e ln
n
yi +
i =1
−
et 2U 2
(y)
n i =1
1
n
U (y) i=1
yi ri (t ) −
σi2 (t )y2i dt +
n j =1
et
n
U (y) i=1
aij (t )
0
σi (t )yi dBi (t ).
(1 + bjk )yj
dt
916
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Consequently, et ln
n
n
yi (t ) − ln
i =1
yi (0)
i =1
t
=
e
s
ln
0
n
yi (s) +
U (y(s)) i=1
i=1 t
n
es
− 0
2U 2 (y(s)) i=1
t
n
1
yi (s) ri (s) −
n
aij (s)
n
(1 + bjk )yj (s)
ds
0
j =1
σi2 (s)y2i (s)ds +
Ni (t ),
(15)
i =1
where Ni (t ) =
es
σi (s)yi (s)dBi (s).
U (y(s))
0
The quadratic form of Ni (t ) is
⟨Ni (t ), Ni (t )⟩ =
t
e2s U 2 (y(s))
0
σi2 (s)y2i (s)ds.
By virtue of the exponential martingale inequality (see e.g [29, p. 44, Theorem 7.4]), for any positive constants T , δ , and α , we have
Ni (t ) −
sup
P
δ 2
0 ≤t ≤T
⟨Ni (t ), Ni (t )⟩ > β
Let T = γ k, δ = nε e−γ k and β =
nε e−γ k
Ni (t ) −
sup
P
2
0≤t ≤γ k
ρ eγ k ln k , nε
≤ e−δβ .
where ρ > 1, γ > 0 and 0 < ε < 1, we obtain
ρ eγ k ln k ⟨Ni (t ), Ni (t )⟩ > nε
≤ k−ρ .
In view of the Borel–Cantelli lemma (see e.g [29, p. 7, Lemma 2.4]), for almost all ω ∈ Ω , there exists k0 (ω) such that for every k ≥ k0 (ω), nε e−γ k
Ni (t ) ≤
2
⟨Ni (t ), Ni (t )⟩ +
ρ eγ k ln k , nε
0 ≤ t ≤ γ k, 1 ≤ i ≤ n.
That is to say nε e−γ k
Ni (t ) ≤
t
2
0
e2s U 2 (y(s))
σi2 (s)y2i (s)ds +
ρ eγ k ln k nε
for 0 ≤ t ≤ γ k and 1 ≤ i, j ≤ n. When these inequalities are used in Eq. (15), we get et ln
n
yi (t ) − ln
i =1
≤
e
s
ln
0 t
−
es
t s
=
e
ln
0 t
n
yi (s) +
es
t
e
n
2U 2 (y(s)) i=1
0
0
n
i=1
−
≤
yi (s) +
2U 2 (y(s)) i=1
0
n i =1
yi (0)
i =1
t
n
s
ln
n i =1
n
1
U (y(s)) i=1
yi (s) ri (s) −
y2i
n nε e−γ k i =1
1
n
U (y(s)) i=1
aij (s)
yi (s) ri (s) −
t
e2s
0
U 2 (y(s))
n
2
j =1
yi (s) + ru − m min
1≤i≤n
n i=1
yi (s) ds
ds
σi2 (s)y2i (s)ds + ρ eγ k ln k/ε
aij (s)
0
σi2 (s)y2i (s)[1 − nεes−γ k ]ds + ρ eγ k ln k/ε alii
(1 + bjk )yj (s)
0
j =1
σ (s) (s)ds + 2 i
n
(1 + bjk )yj (s)
ds
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925 t
+
2U 2
0
t
s
≤
e
ln
0
es
n
(y(s))
i=1
n
917
σi2 (s)y2i (s)[1 + nε es−γ k ]ds + ρ eγ k ln k/ε
yi (s) + ru − m min alii 1≤i≤n
i=1
n
2 )u yi (s) ds + 0.5(σ
t
es [1 + nε]ds + ρ eγ k ln k/ε,
0
i=1
where
u u r = max ri = max sup ri (t ) . 1≤i≤n
1≤i≤n
t ≥0
Here in the proof of the last inequality, we have used s ≤ γ k. Since min1≤i≤n alii > 0, then there is a constant C > 0 independent of k such that ln
n
yi + ru − m min alii
n
1≤i≤n
i=1
2 )u [1 + nε] ≤ C . yi + 0.5(σ
i=1
Thus for any 0 ≤ t ≤ γ k, we obtain et ln
n
yi (t ) − ln
n
i=1
yi (0) ≤ C [et − 1] + ρ eγ k ln k/ε.
i=1
In other words, ln
n
yi (t ) ≤ e−t ln
i=1
n
yi (0) + C [1 − e−t ] + ρ e−t eγ k ln k/ε.
i=1
If γ (k − 1) ≤ t ≤ γ k and k ≥ k0 (ω), we get ln
n
yi (t )/ ln t ≤ e−t ln
n
i=1
yi (0)/ ln t + C [1 − e−t ]/ ln t + ε −1 ρ e−γ (k−1) eγ k ln k/ ln t .
i =1
Consequently
lim sup t →+∞
ln
ln |y(t )|
yi (t )
i=1
≤ lim sup
ln t
n
ln t
t →+∞
≤ ρ eγ /ε.
Letting ρ → 1, γ → 0 and ε → 1 gives lim sup t →+∞
ln |y(t )| ln t
≤ 1.
(16)
It then follows that lim sup t →+∞
ln |x(t )| ln t
≤ lim sup
ln(M |y(t )|) ln t
t →+∞
≤ 1.
Now we turn to (15). By (8), there is a positive constant M2 such that E [V2 (y)] = E
1 + V1 (y(t ))
θ
≤ M2 ,
t ≥ 0.
(17)
On the other hand, by (6)
dV2 (x) ≤ θ (1 + V1 (y))
θ−2
2
2 )u ] + V 2 (y) − V12 (y)[hˇl − θ (σ 1 n
−
V12
(y) +
V11.5
2 )u + V 0.5 (y)M a u dt (y)M au + V1 (y)3(σ 1
− θ (1 + V1 (y))θ−1
2 U3
(y)
n i =1
σi (t )yi dBi (t )
918
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
≤ θ (1 + V1 (y))
θ−2
−
− θ (1 + V1 (y))θ−1
2
V12
n
2
2 ˇ l 2 u (y)[h − θ (σ ) ] + V1 (y) + M3 dt n
U 3 (y) i=1
σi (t )yi dBi (t )
≤ θ M4 (1 + V1 (y))θ dt − θ (1 + V1 (y))θ−1
2
n
U 3 (y) i=1
σi (t )yi dBi (t ),
(18)
where M3 and M4 are positive numbers. Let µ > 0 be sufficiently small such that
2 )u < 0.5. θ M3 µ + 8nθ µ0.5 (σ
(19)
Let k = 1, 2 . . . , applying (18) yields
E
sup
(1 + V1 (y(t )))θ
(k−1)µ≤t ≤kµ
θ ≤ E 1 + V1 (y((k − 1)µ))
θ t θ M4 1 + V1 (y(s)) ds +E sup (k−1)µ≤t ≤kµ (k−1)µ θ−1 n t 2 θ 1 + V1 (y(s)) +E sup σi (s)yi (s)dBi (s) . 3 U (y(s)) i=1 (k−1)µ≤t ≤kµ (k−1)µ
(20)
We compute
θ θ t kµ θ M4 1 + V1 (y(s)) ds ≤ E E sup θ M4 1 + V1 (y(s)) ds (k−1)µ≤t ≤kµ (k−1)µ (k−1)µ θ
≤ θ M4 µE
sup
(k−1)µ≤t ≤kµ
1 + V1 (y(t ))
.
(21)
On the other hand, by the famous Burkholder–Davis–Gundy inequality (see e.g. [29, p. 40, Theorem 7.3]), we have
θ−1 n t 2 θ 1 + V1 (y(s)) E sup σ ( s ) y ( s ) dB ( s ) i i i U 3 (y(s)) i=1 (k−1)µ≤t ≤kµ (k−1)µ θ−1 n t 2 θ 1 + V1 (y(s)) ≤ E sup σi (s)yi (s)dBi (s) 3 (y(s)) U ( k − 1 )µ≤ t ≤ k µ ( k − 1 )µ i=1 0 . 5 2θ−2 n kµ ≤4 E 4θ 2 1 + V1 (y(s)) V13 (y(s))σi2 (s)y2i (s)ds
(k−1)µ
i =1
0.5
=4
2θ n kµ E 4θ 2 1 + V1 (y(s)) (k−1)µ i=1
V12 (y(s))
σi2 (s)y2i (s) 2 ds 2 n yi (s) 1 + V1 (y(s)) i =1
n 2 )u ≤ 8θ (σ E i=1
2 )u E ≤ 8nθ µ0.5 (σ
kµ
(k−1)µ
1 + V1 (y(s))
2θ 0.5 ds
sup
(k−1)µ≤t ≤kµ
1 + V1 (y(t ))
θ .
When (22) and (21) are used in (20), we obtain
E
sup
(1 + V1 (y(t )))
(k−1)µ≤t ≤kµ
θ
≤ E 1 + V1
θ y((k − 1)µ)
(22)
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
0.5 2 u + θ M3 µ + 8nθ µ (σ ) E
919
sup
(k−1)µ≤t ≤kµ
1 + V1 (y(t ))
θ
.
By an application of (17) and (19), one can see that
E
sup
(k−1)µ≤t ≤kµ
(1 + V1 (y(t )))θ
≤ 2M2 .
Let ε > 0 be arbitrary. It then follows from the famous Chebyshev inequality that
P ω:
θ
(1 + V1 (y(t ))) > (kµ)
sup
1+ε
2M2
≤
(kµ)1+ε
(k−1)µ≤t ≤kµ
,
k = 1, 2, . . . .
By the Borel–Cantelli lemma, for almost all ω ∈ Ω , there exists a integer k0 = k0 (ω) such that ln(1 + V1 (y(t )))θ ln t
≤
(1 + ε) ln(kµ) ln((k − 1)µ)
for k ≥ k0 and (k − 1)µ ≤ t ≤ kµ. That is to say ln(1 + V1 (y(t )))θ
lim sup
≤ 1 + ε.
ln t
t →+∞
Letting ε → 0 gives ln(|y(t )|−2θ )
lim sup
ln t
t →+∞
≤ 1.
Consequently lim inf
ln |y(t )|
≥ −0.5/θ .
ln t
t →+∞
But this holds for any θ that satisfies (6), we therefore have lim inf
ln |y(t )|
≥−
ln t
t →+∞
2 )u (σ . 2hˇl
It then follows that lim inft →+∞
(23)
ln |x(t )| ln t
≥ lim inft →+∞
ln(m|x(t )|) ln t
≥ − (σ ˇ)l . 2 u
2h
2 u ˇl Theorem 4 shows that for any ε > 0, there exists a random variable Tε > 0 such that t −(0.5(σ ) /h +ε) ≤ |x(t )| ≤ t 1+ε for 2 u ˇl
t ≥ Tε almost surely. That is to say, the solution will not decay faster than t −(0.5(σ ) /h +ε) and will not grow faster than t 1+ε with probability one. We are now in the position to estimate the limit of the average in time of the sample paths of solutions. Theorem 5. Suppose aˇl > 0, then the solution x(t ) of (2) obeys
lim sup t
−1
t →+∞
hu + ru + lim sup t −1
t
t →+∞
|x(s)|ds ≤
n i=1 0
ln(1 + bik ) a.s.
aˇl
0
(24)
If moreover, hˇl > 0, then
lim inf t t →+∞
−1
hˇl /n2 + lim inf t −1
t
t →+∞
|x(s)|ds ≥
n i=1 0
ln(1 + bik ) a.s.
au n
0
(25)
Proof. By (14), we have ln
n i=1
yi (t ) − ln
n i=1
yi (0) =
t
0
1
n
U (y(s)) i=1
− 0
t
yi (s) ri (s) −
1 2U 2
(y(s))
n
aij (s)
j =1 n i =1
σi2 (s)y2i (s)ds +
(1 + bjk )yj (s) ds
0
t 0
1 U (y(s))
σi (s)yi (s)dBi (s)
920
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925 n t
=
y2i (s)[ri (s) − 0.5σi2 (s)] +
i=1
n n i=1 j=1,j̸=i
ri (s)yi (s)yj (s) ds
U 2 (y(s))
0 n n t
aij (s)yi (s)xj (s)
i=1 j=1
−
ds +
U (y(s))
0
n i=1
t 0
1 U (y(s))
σi (s)yi (s)dBi (s)
t n n ˇ u u l ≤ h +r t −a xj (s)ds + Qi (t )
0
where Qi (t ) = t −1 ln
t
1 0 U (y(s))
n
≤t
i=1
σi (s)yi (s)dBi (s). That is to say
xi (t ) − t −1 ln
i=1
−1
i=1
n
xi (0)
i=1
n
ln(1 + bik ) + hu + ru − t −1 aˇl
t n
i=1 0
0
i=1
xi (s)ds + t −1
n
Qi (t ).
(26)
i=1
Note that the quadratic variation of Qi (t ) is t
⟨Qi (t ), Qi (t )⟩ =
1 U2
0
(y(s))
2 )u t . σi2 (s)y2i (s)ds ≤ (σ
By the strong law of large numbers for martingales, we have lim Qi (t )/t = 0,
1 ≤ i ≤ n.
t →+∞
Thus for given ε > 0, there exists T such that for t ≥ T , t −1 ln
n
xi (0) ≤ ε/3,
t −1
i=1
t −1
n
Qi (t ) ≤ ε/3;
i =1
n
ln(1 + bik ) ≤ lim sup t −1 t →+∞
i=1 0
n
ln(1 + bik ) + ε/3 =: λ + ε/3.
i=1 0
In view of (26), we obtain ln
n
xi (t ) ≤ Ψ t − Γ
t n 0
i=1
xi (s)ds,
i =1
where
Ψ = λ + hu + ru + ε, Γ = aˇl . t n Let g (t ) = 0 i=1 xi (s)ds, we get ln(dg (t )/dt ) ≤ Ψ t − Γ g (t ),
t ≥ T.
In other words eΓ g (t ) (dg /dt ) ≤ eΨ t ,
t ≥ T.
Rewriting this inequality gives eΓ g (t ) ≤ eΓ g (T ) + Γ Ψ −1 eΨ t − Γ Ψ −1 eΨ T . Taking the logarithm of both sides results in g (t ) ≤ Γ
−1
ln Γ Ψ
−1 Ψ t
e
Γ g (T )
+e
−ΓΨ
−1 Ψ T
e
.
That is to say lim sup t −1 t →+∞
t n 0
i=1
xi (s)ds ≤ lim sup Γ −1 t −1 ln Γ Ψ −1 eΨ t + eΓ g (T ) − Γ Ψ −1 eΨ T t →+∞
.
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
921
By l’Hospital’s rule, we get lim sup t −1
t n
t →+∞
0
xi (s)ds ≤ Ψ /Γ .
i =1
Making use of the arbitrariness of ε results in lim sup t −1
t →+∞
t
|x(s)|ds ≤ lim sup t −1 t →+∞
0
t n 0
λ + hu + ru . aˇl
xi (s)ds ≤
i =1
Now let us turn to proving (25). Note that n
ln
n
yi (t ) − ln
i=1
n
yi (0) =
t
y2i (s)[ri (s) − 0.5σi2 (s)] +
i =1
i=1 j=1,j̸=i
ri (s)yi (s)yj (s) ds
U 2 (y(s))
0
i=1
n n
n n t
−
aij (s)
i =1 j =1
(1 + bjk )yi (s)yj (s)
0 < tk < s
ds +
U (y(s))
0
≥ hˇl /n2 − au
t n 0
xi (s)ds +
i=1
n
Qi (t )
i =1 n
Qi (t ).
i =1
In other words, we have shown that t
−1
ln
n
xi (t ) − t
−1
ln
i =1
n
xi (0) ≥ t
−1
n
ln(1 + bik ) + hˇl /n2 − au
i=1 0
i =1
t n 0
i =1
xi (s)ds + t −1
n
Qi (t ).
i=1
Then similar to the proof of (24), we can shown that
lim inf t −1 t →+∞
t n 0
hˇl /n2 + lim inf t −1 xi (s)ds ≥
t →+∞
n i=1 0
au
i=1
Then the required assertion follows from n|x| >
n
i=1
xi .
ln(1 + bik )
.
4. Global stability In this section we will study the global stability of Eq. (2). Definition 3. Let x(t ) = (x1 (t ), . . . , xn (t ))T , z (t ) = (z1 (t ), . . . , zn (t ))T be two arbitrary solutions of Eq. (2) with initial values x(0), z (0) ∈ Rn+ respectively. If for every 1 ≤ i ≤ n, limt →+∞ |xi (t ) − zi (t )| = 0 a.s., then we say Eq. (2) is globally stable. To begin with, we prepare some useful lemmas. Lemma 6 (See e.g. [30]). Let X (t ) be an n-dimensional stochastic process on t ≥ 0. Suppose that there exist positive constants α, β, c such that E |X (t ) − X (s)|α ≤ c |t − s|1+β ,
0 ≤ s, t < ∞.
Then X (t ) has a continuous version, X˜ (t ). Moreover, almost every sample path of X˜ (t ) is locally but uniformly Hölder continuous with exponent ϑ < α/β . Lemma 7. Let Assumption 1 hold. If y(t ) = (y1 (t ), . . . , yn (t ))T is a solution of (4) with initial values y(0) ∈ Rn+ , then, almost every sample path of yi (t ) (1 ≤ i ≤ n) is uniformly continuous for t ≥ 0. q
Proof. By (9), there exists T > 0, such that E ( i=1 yi (t )) ≤ 1.5K1 (q) for all t ≥ T . Moreover, it follows from the continuity n q n q of E ( i=1 yi (t )) that there is a K2 (q) > 0 such that E ( i=1 yi (t )) ≤ K2 (q) for t ≤ T . Let K (q) = max{1.5K1 (q), K2 (q)}, then for all t ≥ 0,
n
E
n i=1
q yi
(t ) ≤ K (q).
922
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Clearly, Eq. (4) is equivalent to the following equation yi (t ) = yi (0) +
t
n
yi (s) ri (s) − 0
aij (s)
0 < tk < s
j=1
t
(1 + bjk )yj (s) ds +
σi (s)yi (s)dBi (s). 0
Therefore
q n E yi (t ) ri (t ) − aij (t ) (1 + bjk )yj (t ) j=1 0 < tk < t q n = E |yi (t )|q ri (t ) − aij (t ) (1 + bjk )yj (t ) j =1 0
≤ 0.5K (2q) + (n + 1)2q−2
i =1
max |ri |u
2q
1≤i≤n
+ n(au M )2q K (2q) =: G1 (q).
By the famous moment inequality for stochastic integrals (see e.g. [29, p. 39, Theorem 7.1]), we obtain that for 0 ≤ t1 ≤ t2 and q > 2,
E
t2
q/2 q q q(q − 1) 2 u (t2 − t1 )(q−2)/2 σi (s)yi (s)dBi (s) ≤ ((σ ) ) 2
t1
q(q − 1) 2 )u )q ≤ ((σ 2
t2
q
E (yi (s))ds
t1
q/2 (t2 − t1 )q/2 K (q).
Thus for 0 < t1 < t2 < ∞, t2 − t1 ≤ 1, 1/q + 1/p = 1, we can see that E (|yi (t2 ) − yi (t1 )|q )
q t2 n t2 yi (s) ri (s) − aij (s) (1 + bjk )yj (s) ds + σi (s)yi (s)dBi (s) =E t1 t 1 j =1 0
=2
q−1
q/p+1
(t2 − t1 )
G1 (q) + 2
q −1
q −1
≤2
q/ 2
(t2 − t1 )
(t2 − t1 )
q −1
≤2
q/ 2
(t2 − t1 )
q(q − 1) 2 )u )q ((σ 2
1+
q/2
+
q(q − 1) 2
q(q − 1)
q/2 (t2 − t1 )q/2 K (q)
q/2
2
G2 (q)
q/2 G2 (q),
2 )u )q K (q)}. Then Lemma 6 means that almost every sample path of y (t ) is uniformly Hölderwhere G2 (q) = max{G1 (q), ((σ i q −2 continuous with exponent ϑ ∈ (0, 2q ).
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
923
Lemma 8 (See e.g. [31]). Let f be a non-negative function defined on t ≥ 0 such that f is integrable on t ≥ 0 and is uniformly continuous on t ≥ 0. Then limt →+∞ f (t ) = 0. Theorem 9. If Assumption 1 holds and there exist positive constants α1 , . . . , αn and δ > 0 such that for 1 ≤ i ≤ n and t ≥ 0, n
αi aii (t ) −
αj aji (t ) ≥ δ,
(27)
j=1, j̸=i
then Eq. (2) is globally stable. Proof. Let x(t ) = (x1 (t ), . . . , xn (t ))T and z (t ) = (z1 (t ), . . . , zn (t ))T be two arbitrary solutions of Eq. (2) with initial values x(0), z (0) ∈ Rn+ respectively. Let yi (t ) be the solution of the following equation
dyi (t ) = yi (t ) ri (t ) −
n
aij (t )
(1 + bjk )yj (t ) dt + σi (t )yi (t )dBi (t ),
i = 1 , . . . , n;
y(0) = x(0)
i = 1 , . . . , n;
y(0) = z (0).
0 < tk < t
j=1
and let y˜ i (t ) be the solution of the following equation
dyi (t ) = yi (t ) ri (t ) −
n
aij (t )
(1 + bjk )yj (t ) dt + σi (t )yi (t )dBi (t ),
0 < tk < t
j=1
Then for every 1 ≤ i ≤ n, one can see that
xi (t ) =
(1 + bik )yi (t ),
zi (t ) =
0
Define V¯ (t ) =
(1 + bik )˜yi (t ).
0
n
i =1
d+ V¯ (t ) =
αi | ln yi (t ) − ln y˜ i (t )|. By Itô’s formula,
n
sgn(yi (t ) − y˜ i (t ))d(ln yi (t ) − ln y˜ i (t ))
i=1
=
n
sgn(yi (t ) − y˜ i (t )) −
i=1
≤−
aij (t )
n
αi aii (t )
0
n
αi aii (t ) −
n n
n
n
αj aji (t ) |xi (t ) − zi (t )|dt
αi aii (t ) −
n
αj aji (t )
j=1, j̸=i
n
0
|yi (t ) − y˜ i (t )|dt .
i=1
By integrating and then taking the expectation, we have V¯ (t ) ≤ V¯ (0) − mδ
n i=1
t
|yi (s) − y˜ i (s)|ds.
0
That is to say, n i =1
αj aij (t )|xj (t ) − zj (t )|dt
j=1, j̸=i
i =1
V¯ (t ) + mδ
αj aij (t )
i=1 j=1, j̸=i
n
≤ −δ m
n n i=1 j=1, j̸=i
αi aii (t )|xi (t ) − zi (t )|dt +
i=1
=−
(1 + bjk )(yj (t ) − y˜ j (t )) dt
(1 + bik )|yi (t ) − y˜ i (t )|dt +
i=1
=−
0 < tk < t
j=1
i=1
≤−
n
t
|yi (s) − y˜ i (s)|ds ≤ V¯ (0) < ∞. 0
(1 + bik )|yi (t ) − y˜ i (t )|dt
0 < tk < t
(1 + bjk )|yj (t ) − y˜ j (t )|dt
924
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Noting that V¯ (t ) ≥ 0, then |yi (t ) − y˜ i (t )| ∈ L1 [0, ∞). Consequently, by Lemmas 7 and 8, we get limt →+∞ |yi (t ) − y˜ i (t )| = 0 a.s. Consequently lim |xi (t ) − zi (t )| = lim
t →+∞
This completes the proof.
t →+∞
0 < tk < t
(1 + bik )|yi (t ) − y˜ i (t )| ≤ M lim |yi (t ) − y˜ i (t )| = 0, t →+∞
a.s.
5. Concluding remarks This paper is concerned with an n-species stochastic non-autonomous Lotka–Volterra competitive system with impulsive effects. We first establish the sufficient conditions for stochastic permanence and extinction. Then we estimate the limit of the average in time of the sample paths of solutions. Finally, we study the global stability. Our key contributions are the following. (A) This article deals with the stochastic n-species population system with impulsive effects, while most of the existing results (see, e.g. [3–12]) are concerned with the deterministic case although [28] considers the stochastic single-species case. (B) Most of the existing papers (see, e.g. [5,8,9]) suppose that the coefficients are periodic, while we here do not have this restriction. That is to say, our model covers more practical situations. (C) Comparing with deterministic results [3–12], our work reveals that if the noise is sufficiently small, then the stochastic model keeps many desired properties that the associated deterministic system owns, for example, permanence. However, if the noise is sufficiently large, the stochastic noises may change the properties of the system greatly. For example, if the noise is sufficiently large, the solution to the associated stochastic system is extinct with probability one, although the solution to the original deterministic model may be persistent. Some interesting topics deserve further investigation. One may investigate some realistic but complex systems, for example, the stochastic model with Markovian switching (see e.g. [21,23]). It is also interesting to study what happens if aij is stochastic. Acknowledgments The authors thank the editor and reviewers for their important and valuable comments. The authors also thank the NSFC of PR China (Nos 11126219, 11171081 and 11171056). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific Press, Singapore, 1989. D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993. G. Ballinger, X. Liu, Permanence of population growth models with impulsive effects, Math. Comput. Modelling 26 (1997) 59–72. X. Liu, G. Ballinger, Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear Anal. 53 (2003) 1041–1062. X. Liu, L. Chen, Global dynamics of the periodic logistic system with periodic impulsive perturbations, J. Math. Anal. Appl. 289 (2004) 279–291. S. Ahmad, I.M. Stamova, Asymptotic stability of competitive systems with delays and impulsive perturbations, J. Math. Anal. Appl. 334 (2007) 686–700. S. Ahmad, I.M. Stamova, Asymptotic stability of an n-dimensional impulsive competitive system, Nonlinear Anal. RWA 8 (2007) 654–663. S. Ahmad, G. Stamov, Almost periodic solutions of n-dimensional impulsive competitive systems, Nonlinear Anal. RWA 10 (2009) 1846–1853. M. He, Z. Li, F. Chen, Permanence, extinction and global attractivity of the periodic Gilpin–Ayala competition system with impulses, Nonlinear Anal. RWA 11 (2010) 1537–1551. J. Hou, Z. Teng, S. Gao, Permanence and global stability for nonautonomous N-species Lotka–Volterra competitive system with impulses, Nonlinear Anal. RWA 11 (2010) 1882–1896. X. Yang, W. Wang, J. Shen, Permanence of a Logistic type impulsive equation with infinite delay, Appl. Math. Lett. 24 (2011) 420–427. H. Hu, K. Wang, D. Wu, Permanence and global stability for nonautonomous n-species Lotka–Voalterra competitive system with impulses and infinite delays, J. Math. Anal. Appl. 377 (2011) 145–160. E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge, 1991. L. Arnold, W. Horsthemke, J.W. Stucki, The influence of external real and white noise on the Lotka–Volterra model, Biom. J. 21 (1979) 451–471. H.I. Freedman, P. Waltman, Periodic solutions of perturbed Lotka–Volterra systems, in: H.A. Antosiewicz (Ed.), Proc. Int. Conf. on Differ. Equations, Academic Press, New York, 1975, pp. 312–316. T.C. Gard, Persistence in stochastic food web models, Bull. Math. Biol. 46 (1984) 357–370. X. Mao, G. Marion, E. Renshaw, Environmental Brownian noise suppresses explosions in populations dynamics, Stochastic Process. Appl. 97 (2002) 95–110. J.R. Beddington, R.M. May, Harvesting natural populations in a randomly fluctuating environment, Science 197 (1977) 463–465. R. Rudnicki, K. Pichor, Influence of stochastic perturbation on prey–predator systems, Math. Biosci. 206 (2007) 108–119. S. Pang, F. Deng, X. Mao, Asymptotic properties of stochastic population dynamics, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 15 (2008) 603–620. C. Zhu, G. Yin, On hybrid competitive Lotka–Volterra ecosystems, Nonlinear Anal. 71 (2009) e1370–e1379. X. Li, X. Mao, Population dynamical behavior of non-autonomous Lotka–Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst. 24 (2009) 523–545. M. Liu, K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment II, J. Theoret. Biol. 267 (2010) 283–291.
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[24] M. Liu, K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Modelling 54 (2011) 2139–2154. [25] M. Liu, K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching II, Math. Comput. Modelling 55 (2012) 405–418. [26] M. Liu, K. Wang, Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol. 73 (2011) 1969–2012. [27] M. Liu, K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl. 375 (2011) 443–457. [28] M. Liu, K. Wang, On a stochastic logistic equation with impulsive perturbations, Comput. Math. Appl. 63 (2012) 871–886. [29] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997. [30] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin, 1991. [31] I. Barbalat, Systems dequations differentielles d’osci d’oscillations nonlineaires, Rev. Roumaine Math. Pures Appl. 4 (1959) 267–270.
Contents lists available at SciVerse ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Asymptotic behavior of a stochastic nonautonomous Lotka–Volterra competitive system with impulsive perturbations Meng Liu a,b,∗ , Ke Wang b a
School of Mathematical Science, Huaiyin Normal University, Huaian 223300, PR China
b
Department of Mathematics, Harbin Institute of Technology, Weihai 264209, PR China
article
info
Article history: Received 8 November 2011 Received in revised form 15 September 2012 Accepted 19 September 2012 Keywords: Competitive system Stochastic perturbations Impulsive effects Stochastic permanence Global attractivity
abstract This paper is concerned with an n-species stochastic nonautonomous Lotka–Volterra competitive system with impulsive effects. Some dynamical properties are investigated and the sufficient conditions for stochastic permanence, extinction and global stability are established. Moreover, the lower-growth rate and the upper-growth rate of the positive solution are studied. In addition, the limit of the average in time of the sample paths of solutions is estimated. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction In the real world, owing to many natural or man-made factors, such as earthquake, drought, flooding, fire, crop-dusting, planting, hunting and harvesting, the intrinsic discipline of species or environment often undergoes some discrete changes of relatively short time interval at some fixed times. From the viewpoint of mathematics, such sudden changes could be described by impulses. On the other hand, in the real world, it is a usual phenomenon that several species compete for limited territories or resources. Thus several authors (see e.g. [1–12]) have investigated the following nonautonomous Lotka–Volterra competitive system with impulses
n dxi (t ) = x (t ) r (t ) − aij (t )xj (t ) , t ̸= tk , k ∈ N i i dt j=1 + xi (tk ) − xi (tk ) = bik xi (tk ), i = 1, . . . , n, k ∈ N
(1)
where xi (t ) is the size of the ith population, ri (t ) and aij (t ) are continuous and bounded functions on R+ := [0, +∞), N denotes the set of positive integers, 0 < t1 < t2 < · · · , limk→+∞ tk = +∞. Recently, system (1) has been received great attention and many interesting dynamical properties, such as permanence, extinction, global stability and dynamical complexity, have been studied by [6–12] and the references cited therein. On the other hand, populations in nature are often subject to random fluctuations [13]. Therefore many investigations have been carried out to reveal the effect of environmental noise on the population dynamics (see e.g. [14–28]). Particularly,
∗
Corresponding author at: School of Mathematical Science, Huaiyin Normal University, Huaian 223300, PR China. Tel.: +86 051784183732. E-mail address: [email protected] (M. Liu).
0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2012.09.019
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M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Liu and Wang [28] considered the following stochastic single-species model with impulses
dx(t ) = x(t ) r (t ) − a(t )x(t ) dt + σ (t )x(t )dB(t ),
x(tk+ ) − x(tk ) = bk x(tk ),
t ̸= tk , k ∈ N
k∈N
where B(t ) is a standard Brownian motion, σ (t ) is a continuous and bounded function on t ≥ 0. The authors [28] studied some important properties of this model. However, in the real world, population does not exist alone, and it is a usual phenomenon that several species compete for limited resources, then it is of great importance to investigate the stochastic competitive system with impulses. To the best of our knowledge, no results related to the stochastic competitive system with impulses have been reported. Suppose that the environmental fluctuations will mainly manifest the growth rate ri (t ) (see e.g. [18–28]), with ri (t ) → ri (t ) + σi (t )B˙ i (t ), where B˙ i (t ) is the white noise, σi (t ) is a continuous and bounded function on t ≥ 0 and σi2 (t ) represents the intensity of the white noise. Then we obtain the following stochastic system:
dxi (t ) = xi (t ) ri (t ) −
n j =1
xi (tk ) − xi (tk ) = bik xi (tk ), +
aij (t )xj (t ) dt + σi (t )xi (t )dBi (t ),
t ̸= tk , k ∈ N
(2)
i = 1 , . . . , n, k ∈ N .
Because of biological meanings, we impose the additional restrictions −1 < bik , i = 1, . . . , n, k ∈ N, on (2). When bik > 0, the impulsive effects represent planting, while bik < 0 denote harvesting [28]. The remainder of the paper is organized as follows. In Section 2, we will show that when the noise is small enough the population system is stochastically permanent but a sufficiently large noise will force every species become extinct. In Section 3, we will study the lower-growth rate and the upper-growth rate of the positive solution. Moreover, we will prove that the limit of the average in time of the sample paths of the solutions is bounded with probability one and give an estimation for it. In Section 4, we will study the global stability. In the last section, we will give the conclusions. 2. Stochastic permanence and extinction Throughout this paper, let (Ω , F , {Ft }t ≥0 , P ) stand for a complete probability space. Let B(t ) = (B1 (t ), . . . , Bn (t )) be an n-dimensional Brownian motion. Set Rn+ = {x ∈ Rn : xi > 0 for all 1 ≤ i ≤ n}. If x ∈ Rn , its norm is |x| = inf aii (t ) > 0
t ≥0
n
i=1
x2i . Moreover, we assume that
for all i = 1, . . . , n
and
aij (t ) ≥ 0
for all i, j = 1, . . . , n with j ̸= i.
If f (t ) is a continuous and bounded function on [0, +∞), define f l = inf f (t ), t ≥0
f u = sup f (t ). t ≥0
For any constant sequences {cij } and {di }, 1 ≤ i, j ≤ n, define cˆ = max cij , 1≤i,j≤n
cˇ = min cij ; 1≤i,j≤n
dˆ = max di , 1≤i≤n
dˇ = min di . 1≤i≤n
To begin with, we introduce a concept proposed by Liu and Wang [28]. Definition 1 (Liu and Wang [28]). For ISDE: dX (t ) = F (t , X (t ))dt + G(t , X (t ))dB(t ), X (tk+ ) − X (tk ) = Bk X (tk ), k ∈ N
t ̸= tk , k ∈ N
(3)
with initial condition X (0), if a stochastic process X (t ) = (X1 (t ), . . . , Xn (t ))T , t ∈ [0, +∞), satisfies the following. (i) X (t ) is Ft -adapted and is continuous on (0, t1 ) and (tk , tk+1 ), k ∈ N; F (t , X (t )) ∈ L1 ([0, +∞); Rn ), G(t , X (t )) ∈ L2 ([0, +∞); Rn ), where Lk ([0, +∞); Rn ) is all Rn valued measurable {Ft }-adapted processes f (t ) satisfying T |f (t )|k dt < ∞ a.s. for every T > 0. 0
(ii) For each tk , k ∈ N , X (tk+ ) = limt →t + X (t ) and X (tk− ) = limt →t − X (t ) exist and X (tk ) = X (tk− ) with probability one. k k
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
911
(iii) X (t ) obeys the equivalent integral equation of (3) for almost every t ∈ [0, +∞) \ {tk } and satisfies the impulsive conditions at each t = tk , k ∈ N with probability one. Then X (t ) is said to be a solution of ISDE (3). Theorem 1. For any given initial value x(0) ∈ Rn+ , there is a solution x(t ) to Eq. (2) for all t ≥ 0 and x(t ) will remain in Rn+ with probability 1. Proof. The proof is motivated by Liu and Wang [28]. Consider the following SDE without impulses:
dyi (t ) = yi (t ) ri (t ) −
n
aij (t )
(1 + bjk )yj (t ) dt + σi (t )yi (t )dBi (t ),
i = 1, . . . , n
(4)
0 < tk < t
j =1
with initial value yi (0) = xi (0). By the theory of SDE (see e.g. [29]), Eq. (4) has a unique continuous maximal local solution y(t ) = (y1 (t ), . . . , yn (t ))T on [0, τe ), where τe is the explosion time. Now let us prove τe = +∞. The proof is standard and hence is omitted (see e.g. [17,22]). Let
xi (t ) =
(1 + bik )yi (t ).
(5)
0
We need only to show that x(t ) = (x1 (t ), . . . , xn (t ))T is the solution Eq. (2). In fact, xi (t ) is continuous on (tk , tk+1 ) ⊂ [0, +∞), k ∈ N, and for every t ̸= tk ,
dxi (t ) = d
(1 + bik )yi (t ) =
0
=
(1 + bik )dyi (t )
0
(1 + bik )yi (t ) ri (t ) −
0
n
aij (t )
= xi (t ) ri (t ) −
n
(1 + bjk )yj (t ) dt + σi (t )
0 < tk < t
j=1
(1 + bik )yi (t )dBi (t )
0 < tk < t
aij (t )xj (t ) dt + σi (t )x(t )dBi (t ).
j =1
At the same time, for every k ∈ N and tk ∈ [0, +∞), xi (tk+ ) = lim
+ t →tk 0
(1 + bij )yi (t ) =
(1 + bij )yi (tk+ )
0 < tj ≤ tk
= (1 + bik )
(1 + bij )yi (tk ) = (1 + bik )xi (tk ).
0
Moreover, xi (tk− ) = lim
−
(1 + bij )yi (t ) =
t → tk 0 < t < t ij
This completes the proof.
0 < tj < tk
(1 + bij )yi (tk− ) =
(1 + bij )yi (tk ) = xi (tk ).
0
Theorem 1 shows that Eq. (2) has a positive solution for any positive initial value. Now, let us study when Eq. (2) is stochastically permanent. Definition 2. Eq. (2) is said to be stochastically permanent if for every ε ∈ (0, 1), there is a pair of constants β > 0 and δ > 0 such that for any initial data x(0) ∈ Rn+ , the solution x(t ) has the property that lim inf P {|x(t )| ≥ β} ≥ 1 − ε, t →+∞
lim inf P {|x(t )| ≤ δ} ≥ 1 − ε. t →+∞
Assumption 1. There exist two constants m > 0 and M > 0 such that for all t > 0 and i = 1, . . . , n, m ≤ 0
ˇ l h = min inf hi (t ) > 0, 1≤i≤n
t ≥0
then Eq. (2) is stochastically permanent, where hi (t ) = ri (t ) − 0.5σi2 (t ),
i = 1, . . . , n.
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M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Proof. For y ∈ Rn+ , define
2 n
V1 (y) = 1/U (y) = 1 2
yi
.
i=1
By Itô’s formula, we can see that dV1 (y) = −
+
n
2
U 3 (y) i=1
y i ri ( t ) −
n
3
U 4 (y) i=1
n
aij (t )
(1 + bjk )yj dt
0 < tk < t
j =1
σi2 (t )y2i dt −
n
2
U 3 (y) i=1
σi (t )yi dBi (t ).
Note that hˇl > 0, then we can choose a constant θ > 0 such that
2 )u . hˇl > θ max (σi2 )u = θ (σ
(6)
1≤i≤n
Define V2 (y) = (1 + V1 (y))θ . In view of Itô’s formula, one can derive that dV2 (y) = θ (1 + V1 (y))θ−1 dV1 (y) + 0.5θ (θ − 1)(1 + V1 (y))θ−2 (dV1 (y))2
= θ (1 + V1 (y))
θ−2
+ (1 + V1 (y))
−(1 + V1 (y)) n
3
− θ (1 + V1 (y))θ−1
(y)
= θ (1 + V1 (y))θ−2 − +
(y)
+
n n
2 U3
U 4 (y)
+
U 6 (y)
+
≤ θ (1 + V1 (y))
−
− θ (1 + V1 (y))θ−1 = θ (1 + V1 (y))
y i ri ( t ) −
n
U 6 (y) i=1 n
U 3 (y) i=1
−
U 6 (y)
2
2
n
4
U 6 (y) i=1
n
n
2
U 5 (y) i=1
(1 + bjk )yi yj +
2(θ − 1)
2
θ−2
θ −1 2
aij (t )
σi2 (t )y2i dt
yi ri (t )
2
n n
(y)
i =1 j =1
U5
aij (t )
σi2 (t )y2i dt − θ (1 + V1 (y))θ−1
i =1
y2i ri
+
(t ) +
M au U (y)
M au
+
U 3 (y)
+
2 )u 3(σ U 2 (y)
+
≤ θ (1 + V1 (y))
−
− θ (1 + V1 (y))θ−1
n
2
U 3 (y) i=1
σi (t )yi dBi (t )
n 2θ + 1
U 6 (y)
σ (t ) 2 i
y2i
n
U 6 (y) i=1
y2i hi (t ) − θ σi2 (t )
2
n
U 3 (y) i=1
σi (t )yi dBi (t )
V2 2 1
1.5 0 . 5 ˇ l 2 u u 2 u u (y)[h − θ (σ ) ] + V1 (y)M a + V1 (y)3(σ ) + V1 (y)M a dt
2
n
2 n
U3
(y)
i =1
dt
i=1
σi (t )yi dBi (t )
2 )u + V 0.5 (y)M a u dt − θ (1 + V1 (y))θ−1 (y)M au + V1 (y)3(σ 1 θ−2
(1 + bjk )yi yj
0
V11.5
(1 + bjk )yj
0 < tk < t
j=1
0
3
θ−2
yi ri (t ) −
n
σi (t )yi dBi (t )
U 3 (y) i=1
aij (t )
i =1 n
2
i =1 j =1
3
n
2 U3
U 3 (y) i=1
σi2 (t )y2i +
U 4 (y) i=1
n
2
σi (t )yi dBi (t ).
(7)
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
913
Here, in the proof of the last inequality, we have used the fact that: if yi ≥ 0, i = 1, 2, . . . , n, then
n
2
2
=n
≤ n max yi
yi
2
max 1≤i≤n
1≤i≤n
i =1
y2i
≤ n2
n
y2i .
i=1
Now, let κ be sufficiently small such that
2 )u . 0 < 0.5n2 κ/θ < hˇl − 0.5θ (σ Define V3 (y(t )) = eκ t V2 (y(t )). An application of Itô’s formula results in dV3 (y) = κ eκ t V2 (y)dt + eκ t dV2 (y)
κt
2 )u /n2 κ(1 + V1 (y))2 /θ − 2V12 (y) hˇl /n2 − θ (σ
θ −2
≤ θ e (1 + V1 (y))
+
V11.5
2 )u + V 0.5 (y)M a u dt (y)M au + V1 (y)3(σ 1
− θ eκ t (1 + V1 (y))θ−1
n
2
U 3 (y) i=1
σi (t )yi dBi (t )
2 2 ˇ l 2 u = θ e (1 + V1 (y)) − (y) h /n − θ (σ ) /n − 0.5κ/θ 2 )u + 2κ/θ + V 0.5 (y)M a u + κ/θ dt + V11.5 (y)M au + V1 (y) 3(σ 1 κt
θ −2
2V12
− θ eκ t (1 + V1 (y))θ−1
n
2
U 3 (y) i=1
= eκ t J (y)dt − θ eκ t (1 + V1 (y))θ−1
σi (t )yi dBi (t ) 2
U3
(y)
n
σi (t )yi dBi (t ),
i =1
where
2 )u /n2 − 0.5κ/θ J (y) = θ (1 + V1 (y))θ−2 −2V12 (y) hˇl /n2 − θ (σ
2 )u + 2κ/θ + V 0.5 (y)M a u + κ/θ . + V11.5 (y)M au + V1 (y) 3(σ 1 By the definition of κ, J (y) is upper bounded in Rn+ , namely J1 := supy∈Rn J (y) < +∞. Then + dV3 (y(t )) ≤ J1 eκ t dt − θ eκ t (1 + V1 (y))θ −1
2 U3
(y)
n
σi (t )yi dBi (t ).
i=1
In other words, we have shown that
κt
θ
E V3 (y(t )) = E e (1 + V1 (y(t )))
≤ 1 + V1 (y(0))
θ
+ J1 eκ t /κ.
Consequently
lim sup E t →+∞
n
−2θ yi (t )
i =1
≤ lim sup E (1 + V1 (y(t )))θ ≤ J1 /κ =: J2 . t →+∞
Therefore lim sup E t →+∞
n i =1
−2 θ xi (t )
= lim sup E t →+∞
n i=1 0
−2 θ (1 + bik )yi
≤ m−θ J2 := J3 .
(8)
914
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Then we obtain lim supt →+∞ E |x(t )|−2θ
0.5/θ
≤ n2θ J3 =: J4 . For any ε > 0, set β = ε 0.5/θ /J4
P {|x(t )| < β} = P {|x(t )|−2θ > β −2θ } ≤
E [|x(t )−2θ |]
β −2θ
, by the Chebyshev inequality,
= β 2θ E [|x(t )|−2θ ].
Consequently, lim supt →+∞ P {|x(t )| < β} ≤ β 2θ J4 = ε . That is to say lim inf P {|x(t )| ≥ β} ≥ 1 − ε. t →+∞
Now let us prove that for ε > 0, there exists a constant δ > 0 such that lim inft →+∞ P {|x(t )| ≤ δ} ≥ 1 − ε . arbitrary q n For y ∈ Rn+ , define V (y) = i=1 yi , where p > 0. Using Itô’s formula yields
n
dV (y) =
q qyi
ri ( t ) −
i =1 q qyi
riu
−
alii myi
aij (t )
(1 + bjk )yj + 0.5(q − 1)σi2 (t ) dt + qσi (t )yqi dBi (t )
0
j =1
n
≤
n
+ 0.5q(σ ) dt + qσi (t )yqi dBi (t ). 2 u i
i =1
An application of Itô’s formula again leads to d[et V (y)] = et V (y)dt + et dV (y)
≤ qet
n q
q
yi 1/q + riu − alii myi + 0.5q(σi2 )u dt + qσi (t )yi dBi (t )
i=1
≤ et K (q)dt + qσi (t )yqi dBi (t ), where K (q) is a positive number. Consequently, lim sup E
n q yi
t →+∞
≤ K1 (q).
(9)
i=1
In other words, we have shown that
lim sup E
n
t →+∞
q xi
= lim sup E
i=1
t →+∞
n i =1
q (1 + bik ) yqi
≤ M q K1 (q).
0
Then the desired assertion follows from the Chebyshev inequality.
k+1 2 Remark 1.It is useful to point out that Assumption 1 is easy to be satisfied. An example is bik = e(−1) /k − 1, i = 1, . . . , n, then 1 ≤ 0
In the above we have demonstrated that if the noise is sufficiently small, the original system (1) and the associated stochastic model (2) behave similarly in the sense that both will be permanent. In other words, we show that if the noise is sufficiently small the noise will not spoil this nice property. However, in the following theorem we will show that if the noise is sufficiently large, the solution to the associated stochastic system (2) will become extinct with probability one, although the solution to the original model (1) may be persistent. Theorem 3. If x(t ) = (x1 (t ), . . . , xn (t ))T is a solution of Eq. (2), then for every 1 ≤ i ≤ n,
lim sup t
−1
t →+∞
ln xi (t ) ≤ lim sup t t →+∞
−1
ln(1 + bik ) +
0 < tk < t
t
[ri (s) − 0.5σ (s)]ds =: λi , 2 i
0
In particular, if λi < 0, then limt →+∞ xi (t ) = 0 a.s. Proof. Applying Itô’s formula to Eq. (4) gives
d ln yi (t ) =
hi (t ) −
n j=1
aij (t )
0 < tk < t
(1 + bjk )yj (t ) dt + σi (t )dBi (t ).
a.s.
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
915
In other words, we have shown that
ln yi (t ) − ln yi (0) =
t
hi (s)ds −
n
0
where Mi (t ) =
t
aij (s)xj (s)ds + Mi (t ),
(10)
0
j=1
t
σi (s)dBi (s). Clearly, Mi (t ) is a local martingale with quadratic variation t σi2 (s)ds ≤ (σi2 )u t . ⟨Mi (t ), Mi (t )⟩ = 0
0
It then follows from the strong law of large numbers for local martingales (see e.g. [29, Theorem 3.4]) that lim Mi (t )/t = 0
a.s.
t →+∞
(11)
At the same time, by (10),
ln(1 + bik ) + ln yi (t ) − ln yi (0) =
0 < tk < t
t
ln(1 + bik ) +
0
hi (s)ds − 0
n
t
aij (s)xj (s)ds + Mi (t ). 0
j=1
That is to say
ln xi (t ) − ln xi (0) =
ln(1 + bik ) +
0 < tk < t
≤
t
hi (s)ds − 0
ln(1 + bik ) +
j =1
t
aij (s)xj (s)ds + Mi (t ) 0
t
0 < tk < t
n
hi (s)ds + Mi (t ). 0
By (11), we obtain the required assertion.
3. Asymptotic properties Theorem 4. If Assumption 1 holds, then the solution of Eq. (2) has the property that lim sup
ln |x(t )|
t →+∞
≤ 1 a.s.
ln t
(12)
If Assumption 1 holds and moreover, hˇl > 0, then lim inf
ln |x(t )|
≥−
ln t
t →+∞
2 )u (σ 2hˇl
a.s.
Proof. Define W (y) = ln U (y) = ln dW (y) =
=
dU (y)
−
n
U (y) i=1
−
n
(dU (y)) 2U 2 (y)
i =1
yi . By Itô’s formula, we can observe that
2
U (y) 1
(13)
yi ri (t ) −
n
aij (t )
n
2U 2 (y) i=1
(1 + bjk )yj dt
0
j =1
1
σi2 (t )y2i dt +
n
1
U (y) i=1
σi (t )yi dBi (t ).
(14)
Using Itô’s formula again leads to d(et W (y)) = et W (y)dt + et dW (y)
t
= e ln
n
yi +
i =1
−
et 2U 2
(y)
n i =1
1
n
U (y) i=1
yi ri (t ) −
σi2 (t )y2i dt +
n j =1
et
n
U (y) i=1
aij (t )
0
σi (t )yi dBi (t ).
(1 + bjk )yj
dt
916
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Consequently, et ln
n
n
yi (t ) − ln
i =1
yi (0)
i =1
t
=
e
s
ln
0
n
yi (s) +
U (y(s)) i=1
i=1 t
n
es
− 0
2U 2 (y(s)) i=1
t
n
1
yi (s) ri (s) −
n
aij (s)
n
(1 + bjk )yj (s)
ds
0
j =1
σi2 (s)y2i (s)ds +
Ni (t ),
(15)
i =1
where Ni (t ) =
es
σi (s)yi (s)dBi (s).
U (y(s))
0
The quadratic form of Ni (t ) is
⟨Ni (t ), Ni (t )⟩ =
t
e2s U 2 (y(s))
0
σi2 (s)y2i (s)ds.
By virtue of the exponential martingale inequality (see e.g [29, p. 44, Theorem 7.4]), for any positive constants T , δ , and α , we have
Ni (t ) −
sup
P
δ 2
0 ≤t ≤T
⟨Ni (t ), Ni (t )⟩ > β
Let T = γ k, δ = nε e−γ k and β =
nε e−γ k
Ni (t ) −
sup
P
2
0≤t ≤γ k
ρ eγ k ln k , nε
≤ e−δβ .
where ρ > 1, γ > 0 and 0 < ε < 1, we obtain
ρ eγ k ln k ⟨Ni (t ), Ni (t )⟩ > nε
≤ k−ρ .
In view of the Borel–Cantelli lemma (see e.g [29, p. 7, Lemma 2.4]), for almost all ω ∈ Ω , there exists k0 (ω) such that for every k ≥ k0 (ω), nε e−γ k
Ni (t ) ≤
2
⟨Ni (t ), Ni (t )⟩ +
ρ eγ k ln k , nε
0 ≤ t ≤ γ k, 1 ≤ i ≤ n.
That is to say nε e−γ k
Ni (t ) ≤
t
2
0
e2s U 2 (y(s))
σi2 (s)y2i (s)ds +
ρ eγ k ln k nε
for 0 ≤ t ≤ γ k and 1 ≤ i, j ≤ n. When these inequalities are used in Eq. (15), we get et ln
n
yi (t ) − ln
i =1
≤
e
s
ln
0 t
−
es
t s
=
e
ln
0 t
n
yi (s) +
es
t
e
n
2U 2 (y(s)) i=1
0
0
n
i=1
−
≤
yi (s) +
2U 2 (y(s)) i=1
0
n i =1
yi (0)
i =1
t
n
s
ln
n i =1
n
1
U (y(s)) i=1
yi (s) ri (s) −
y2i
n nε e−γ k i =1
1
n
U (y(s)) i=1
aij (s)
yi (s) ri (s) −
t
e2s
0
U 2 (y(s))
n
2
j =1
yi (s) + ru − m min
1≤i≤n
n i=1
yi (s) ds
ds
σi2 (s)y2i (s)ds + ρ eγ k ln k/ε
aij (s)
0
σi2 (s)y2i (s)[1 − nεes−γ k ]ds + ρ eγ k ln k/ε alii
(1 + bjk )yj (s)
0
j =1
σ (s) (s)ds + 2 i
n
(1 + bjk )yj (s)
ds
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925 t
+
2U 2
0
t
s
≤
e
ln
0
es
n
(y(s))
i=1
n
917
σi2 (s)y2i (s)[1 + nε es−γ k ]ds + ρ eγ k ln k/ε
yi (s) + ru − m min alii 1≤i≤n
i=1
n
2 )u yi (s) ds + 0.5(σ
t
es [1 + nε]ds + ρ eγ k ln k/ε,
0
i=1
where
u u r = max ri = max sup ri (t ) . 1≤i≤n
1≤i≤n
t ≥0
Here in the proof of the last inequality, we have used s ≤ γ k. Since min1≤i≤n alii > 0, then there is a constant C > 0 independent of k such that ln
n
yi + ru − m min alii
n
1≤i≤n
i=1
2 )u [1 + nε] ≤ C . yi + 0.5(σ
i=1
Thus for any 0 ≤ t ≤ γ k, we obtain et ln
n
yi (t ) − ln
n
i=1
yi (0) ≤ C [et − 1] + ρ eγ k ln k/ε.
i=1
In other words, ln
n
yi (t ) ≤ e−t ln
i=1
n
yi (0) + C [1 − e−t ] + ρ e−t eγ k ln k/ε.
i=1
If γ (k − 1) ≤ t ≤ γ k and k ≥ k0 (ω), we get ln
n
yi (t )/ ln t ≤ e−t ln
n
i=1
yi (0)/ ln t + C [1 − e−t ]/ ln t + ε −1 ρ e−γ (k−1) eγ k ln k/ ln t .
i =1
Consequently
lim sup t →+∞
ln
ln |y(t )|
yi (t )
i=1
≤ lim sup
ln t
n
ln t
t →+∞
≤ ρ eγ /ε.
Letting ρ → 1, γ → 0 and ε → 1 gives lim sup t →+∞
ln |y(t )| ln t
≤ 1.
(16)
It then follows that lim sup t →+∞
ln |x(t )| ln t
≤ lim sup
ln(M |y(t )|) ln t
t →+∞
≤ 1.
Now we turn to (15). By (8), there is a positive constant M2 such that E [V2 (y)] = E
1 + V1 (y(t ))
θ
≤ M2 ,
t ≥ 0.
(17)
On the other hand, by (6)
dV2 (x) ≤ θ (1 + V1 (y))
θ−2
2
2 )u ] + V 2 (y) − V12 (y)[hˇl − θ (σ 1 n
−
V12
(y) +
V11.5
2 )u + V 0.5 (y)M a u dt (y)M au + V1 (y)3(σ 1
− θ (1 + V1 (y))θ−1
2 U3
(y)
n i =1
σi (t )yi dBi (t )
918
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
≤ θ (1 + V1 (y))
θ−2
−
− θ (1 + V1 (y))θ−1
2
V12
n
2
2 ˇ l 2 u (y)[h − θ (σ ) ] + V1 (y) + M3 dt n
U 3 (y) i=1
σi (t )yi dBi (t )
≤ θ M4 (1 + V1 (y))θ dt − θ (1 + V1 (y))θ−1
2
n
U 3 (y) i=1
σi (t )yi dBi (t ),
(18)
where M3 and M4 are positive numbers. Let µ > 0 be sufficiently small such that
2 )u < 0.5. θ M3 µ + 8nθ µ0.5 (σ
(19)
Let k = 1, 2 . . . , applying (18) yields
E
sup
(1 + V1 (y(t )))θ
(k−1)µ≤t ≤kµ
θ ≤ E 1 + V1 (y((k − 1)µ))
θ t θ M4 1 + V1 (y(s)) ds +E sup (k−1)µ≤t ≤kµ (k−1)µ θ−1 n t 2 θ 1 + V1 (y(s)) +E sup σi (s)yi (s)dBi (s) . 3 U (y(s)) i=1 (k−1)µ≤t ≤kµ (k−1)µ
(20)
We compute
θ θ t kµ θ M4 1 + V1 (y(s)) ds ≤ E E sup θ M4 1 + V1 (y(s)) ds (k−1)µ≤t ≤kµ (k−1)µ (k−1)µ θ
≤ θ M4 µE
sup
(k−1)µ≤t ≤kµ
1 + V1 (y(t ))
.
(21)
On the other hand, by the famous Burkholder–Davis–Gundy inequality (see e.g. [29, p. 40, Theorem 7.3]), we have
θ−1 n t 2 θ 1 + V1 (y(s)) E sup σ ( s ) y ( s ) dB ( s ) i i i U 3 (y(s)) i=1 (k−1)µ≤t ≤kµ (k−1)µ θ−1 n t 2 θ 1 + V1 (y(s)) ≤ E sup σi (s)yi (s)dBi (s) 3 (y(s)) U ( k − 1 )µ≤ t ≤ k µ ( k − 1 )µ i=1 0 . 5 2θ−2 n kµ ≤4 E 4θ 2 1 + V1 (y(s)) V13 (y(s))σi2 (s)y2i (s)ds
(k−1)µ
i =1
0.5
=4
2θ n kµ E 4θ 2 1 + V1 (y(s)) (k−1)µ i=1
V12 (y(s))
σi2 (s)y2i (s) 2 ds 2 n yi (s) 1 + V1 (y(s)) i =1
n 2 )u ≤ 8θ (σ E i=1
2 )u E ≤ 8nθ µ0.5 (σ
kµ
(k−1)µ
1 + V1 (y(s))
2θ 0.5 ds
sup
(k−1)µ≤t ≤kµ
1 + V1 (y(t ))
θ .
When (22) and (21) are used in (20), we obtain
E
sup
(1 + V1 (y(t )))
(k−1)µ≤t ≤kµ
θ
≤ E 1 + V1
θ y((k − 1)µ)
(22)
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
0.5 2 u + θ M3 µ + 8nθ µ (σ ) E
919
sup
(k−1)µ≤t ≤kµ
1 + V1 (y(t ))
θ
.
By an application of (17) and (19), one can see that
E
sup
(k−1)µ≤t ≤kµ
(1 + V1 (y(t )))θ
≤ 2M2 .
Let ε > 0 be arbitrary. It then follows from the famous Chebyshev inequality that
P ω:
θ
(1 + V1 (y(t ))) > (kµ)
sup
1+ε
2M2
≤
(kµ)1+ε
(k−1)µ≤t ≤kµ
,
k = 1, 2, . . . .
By the Borel–Cantelli lemma, for almost all ω ∈ Ω , there exists a integer k0 = k0 (ω) such that ln(1 + V1 (y(t )))θ ln t
≤
(1 + ε) ln(kµ) ln((k − 1)µ)
for k ≥ k0 and (k − 1)µ ≤ t ≤ kµ. That is to say ln(1 + V1 (y(t )))θ
lim sup
≤ 1 + ε.
ln t
t →+∞
Letting ε → 0 gives ln(|y(t )|−2θ )
lim sup
ln t
t →+∞
≤ 1.
Consequently lim inf
ln |y(t )|
≥ −0.5/θ .
ln t
t →+∞
But this holds for any θ that satisfies (6), we therefore have lim inf
ln |y(t )|
≥−
ln t
t →+∞
2 )u (σ . 2hˇl
It then follows that lim inft →+∞
(23)
ln |x(t )| ln t
≥ lim inft →+∞
ln(m|x(t )|) ln t
≥ − (σ ˇ)l . 2 u
2h
2 u ˇl Theorem 4 shows that for any ε > 0, there exists a random variable Tε > 0 such that t −(0.5(σ ) /h +ε) ≤ |x(t )| ≤ t 1+ε for 2 u ˇl
t ≥ Tε almost surely. That is to say, the solution will not decay faster than t −(0.5(σ ) /h +ε) and will not grow faster than t 1+ε with probability one. We are now in the position to estimate the limit of the average in time of the sample paths of solutions. Theorem 5. Suppose aˇl > 0, then the solution x(t ) of (2) obeys
lim sup t
−1
t →+∞
hu + ru + lim sup t −1
t
t →+∞
|x(s)|ds ≤
n i=1 0
ln(1 + bik ) a.s.
aˇl
0
(24)
If moreover, hˇl > 0, then
lim inf t t →+∞
−1
hˇl /n2 + lim inf t −1
t
t →+∞
|x(s)|ds ≥
n i=1 0
ln(1 + bik ) a.s.
au n
0
(25)
Proof. By (14), we have ln
n i=1
yi (t ) − ln
n i=1
yi (0) =
t
0
1
n
U (y(s)) i=1
− 0
t
yi (s) ri (s) −
1 2U 2
(y(s))
n
aij (s)
j =1 n i =1
σi2 (s)y2i (s)ds +
(1 + bjk )yj (s) ds
0
t 0
1 U (y(s))
σi (s)yi (s)dBi (s)
920
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925 n t
=
y2i (s)[ri (s) − 0.5σi2 (s)] +
i=1
n n i=1 j=1,j̸=i
ri (s)yi (s)yj (s) ds
U 2 (y(s))
0 n n t
aij (s)yi (s)xj (s)
i=1 j=1
−
ds +
U (y(s))
0
n i=1
t 0
1 U (y(s))
σi (s)yi (s)dBi (s)
t n n ˇ u u l ≤ h +r t −a xj (s)ds + Qi (t )
0
where Qi (t ) = t −1 ln
t
1 0 U (y(s))
n
≤t
i=1
σi (s)yi (s)dBi (s). That is to say
xi (t ) − t −1 ln
i=1
−1
i=1
n
xi (0)
i=1
n
ln(1 + bik ) + hu + ru − t −1 aˇl
t n
i=1 0
0
i=1
xi (s)ds + t −1
n
Qi (t ).
(26)
i=1
Note that the quadratic variation of Qi (t ) is t
⟨Qi (t ), Qi (t )⟩ =
1 U2
0
(y(s))
2 )u t . σi2 (s)y2i (s)ds ≤ (σ
By the strong law of large numbers for martingales, we have lim Qi (t )/t = 0,
1 ≤ i ≤ n.
t →+∞
Thus for given ε > 0, there exists T such that for t ≥ T , t −1 ln
n
xi (0) ≤ ε/3,
t −1
i=1
t −1
n
Qi (t ) ≤ ε/3;
i =1
n
ln(1 + bik ) ≤ lim sup t −1 t →+∞
i=1 0
n
ln(1 + bik ) + ε/3 =: λ + ε/3.
i=1 0
In view of (26), we obtain ln
n
xi (t ) ≤ Ψ t − Γ
t n 0
i=1
xi (s)ds,
i =1
where
Ψ = λ + hu + ru + ε, Γ = aˇl . t n Let g (t ) = 0 i=1 xi (s)ds, we get ln(dg (t )/dt ) ≤ Ψ t − Γ g (t ),
t ≥ T.
In other words eΓ g (t ) (dg /dt ) ≤ eΨ t ,
t ≥ T.
Rewriting this inequality gives eΓ g (t ) ≤ eΓ g (T ) + Γ Ψ −1 eΨ t − Γ Ψ −1 eΨ T . Taking the logarithm of both sides results in g (t ) ≤ Γ
−1
ln Γ Ψ
−1 Ψ t
e
Γ g (T )
+e
−ΓΨ
−1 Ψ T
e
.
That is to say lim sup t −1 t →+∞
t n 0
i=1
xi (s)ds ≤ lim sup Γ −1 t −1 ln Γ Ψ −1 eΨ t + eΓ g (T ) − Γ Ψ −1 eΨ T t →+∞
.
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
921
By l’Hospital’s rule, we get lim sup t −1
t n
t →+∞
0
xi (s)ds ≤ Ψ /Γ .
i =1
Making use of the arbitrariness of ε results in lim sup t −1
t →+∞
t
|x(s)|ds ≤ lim sup t −1 t →+∞
0
t n 0
λ + hu + ru . aˇl
xi (s)ds ≤
i =1
Now let us turn to proving (25). Note that n
ln
n
yi (t ) − ln
i=1
n
yi (0) =
t
y2i (s)[ri (s) − 0.5σi2 (s)] +
i =1
i=1 j=1,j̸=i
ri (s)yi (s)yj (s) ds
U 2 (y(s))
0
i=1
n n
n n t
−
aij (s)
i =1 j =1
(1 + bjk )yi (s)yj (s)
0 < tk < s
ds +
U (y(s))
0
≥ hˇl /n2 − au
t n 0
xi (s)ds +
i=1
n
Qi (t )
i =1 n
Qi (t ).
i =1
In other words, we have shown that t
−1
ln
n
xi (t ) − t
−1
ln
i =1
n
xi (0) ≥ t
−1
n
ln(1 + bik ) + hˇl /n2 − au
i=1 0
i =1
t n 0
i =1
xi (s)ds + t −1
n
Qi (t ).
i=1
Then similar to the proof of (24), we can shown that
lim inf t −1 t →+∞
t n 0
hˇl /n2 + lim inf t −1 xi (s)ds ≥
t →+∞
n i=1 0
au
i=1
Then the required assertion follows from n|x| >
n
i=1
xi .
ln(1 + bik )
.
4. Global stability In this section we will study the global stability of Eq. (2). Definition 3. Let x(t ) = (x1 (t ), . . . , xn (t ))T , z (t ) = (z1 (t ), . . . , zn (t ))T be two arbitrary solutions of Eq. (2) with initial values x(0), z (0) ∈ Rn+ respectively. If for every 1 ≤ i ≤ n, limt →+∞ |xi (t ) − zi (t )| = 0 a.s., then we say Eq. (2) is globally stable. To begin with, we prepare some useful lemmas. Lemma 6 (See e.g. [30]). Let X (t ) be an n-dimensional stochastic process on t ≥ 0. Suppose that there exist positive constants α, β, c such that E |X (t ) − X (s)|α ≤ c |t − s|1+β ,
0 ≤ s, t < ∞.
Then X (t ) has a continuous version, X˜ (t ). Moreover, almost every sample path of X˜ (t ) is locally but uniformly Hölder continuous with exponent ϑ < α/β . Lemma 7. Let Assumption 1 hold. If y(t ) = (y1 (t ), . . . , yn (t ))T is a solution of (4) with initial values y(0) ∈ Rn+ , then, almost every sample path of yi (t ) (1 ≤ i ≤ n) is uniformly continuous for t ≥ 0. q
Proof. By (9), there exists T > 0, such that E ( i=1 yi (t )) ≤ 1.5K1 (q) for all t ≥ T . Moreover, it follows from the continuity n q n q of E ( i=1 yi (t )) that there is a K2 (q) > 0 such that E ( i=1 yi (t )) ≤ K2 (q) for t ≤ T . Let K (q) = max{1.5K1 (q), K2 (q)}, then for all t ≥ 0,
n
E
n i=1
q yi
(t ) ≤ K (q).
922
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Clearly, Eq. (4) is equivalent to the following equation yi (t ) = yi (0) +
t
n
yi (s) ri (s) − 0
aij (s)
0 < tk < s
j=1
t
(1 + bjk )yj (s) ds +
σi (s)yi (s)dBi (s). 0
Therefore
q n E yi (t ) ri (t ) − aij (t ) (1 + bjk )yj (t ) j=1 0 < tk < t q n = E |yi (t )|q ri (t ) − aij (t ) (1 + bjk )yj (t ) j =1 0
≤ 0.5K (2q) + (n + 1)2q−2
i =1
max |ri |u
2q
1≤i≤n
+ n(au M )2q K (2q) =: G1 (q).
By the famous moment inequality for stochastic integrals (see e.g. [29, p. 39, Theorem 7.1]), we obtain that for 0 ≤ t1 ≤ t2 and q > 2,
E
t2
q/2 q q q(q − 1) 2 u (t2 − t1 )(q−2)/2 σi (s)yi (s)dBi (s) ≤ ((σ ) ) 2
t1
q(q − 1) 2 )u )q ≤ ((σ 2
t2
q
E (yi (s))ds
t1
q/2 (t2 − t1 )q/2 K (q).
Thus for 0 < t1 < t2 < ∞, t2 − t1 ≤ 1, 1/q + 1/p = 1, we can see that E (|yi (t2 ) − yi (t1 )|q )
q t2 n t2 yi (s) ri (s) − aij (s) (1 + bjk )yj (s) ds + σi (s)yi (s)dBi (s) =E t1 t 1 j =1 0
=2
q−1
q/p+1
(t2 − t1 )
G1 (q) + 2
q −1
q −1
≤2
q/ 2
(t2 − t1 )
(t2 − t1 )
q −1
≤2
q/ 2
(t2 − t1 )
q(q − 1) 2 )u )q ((σ 2
1+
q/2
+
q(q − 1) 2
q(q − 1)
q/2 (t2 − t1 )q/2 K (q)
q/2
2
G2 (q)
q/2 G2 (q),
2 )u )q K (q)}. Then Lemma 6 means that almost every sample path of y (t ) is uniformly Hölderwhere G2 (q) = max{G1 (q), ((σ i q −2 continuous with exponent ϑ ∈ (0, 2q ).
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
923
Lemma 8 (See e.g. [31]). Let f be a non-negative function defined on t ≥ 0 such that f is integrable on t ≥ 0 and is uniformly continuous on t ≥ 0. Then limt →+∞ f (t ) = 0. Theorem 9. If Assumption 1 holds and there exist positive constants α1 , . . . , αn and δ > 0 such that for 1 ≤ i ≤ n and t ≥ 0, n
αi aii (t ) −
αj aji (t ) ≥ δ,
(27)
j=1, j̸=i
then Eq. (2) is globally stable. Proof. Let x(t ) = (x1 (t ), . . . , xn (t ))T and z (t ) = (z1 (t ), . . . , zn (t ))T be two arbitrary solutions of Eq. (2) with initial values x(0), z (0) ∈ Rn+ respectively. Let yi (t ) be the solution of the following equation
dyi (t ) = yi (t ) ri (t ) −
n
aij (t )
(1 + bjk )yj (t ) dt + σi (t )yi (t )dBi (t ),
i = 1 , . . . , n;
y(0) = x(0)
i = 1 , . . . , n;
y(0) = z (0).
0 < tk < t
j=1
and let y˜ i (t ) be the solution of the following equation
dyi (t ) = yi (t ) ri (t ) −
n
aij (t )
(1 + bjk )yj (t ) dt + σi (t )yi (t )dBi (t ),
0 < tk < t
j=1
Then for every 1 ≤ i ≤ n, one can see that
xi (t ) =
(1 + bik )yi (t ),
zi (t ) =
0
Define V¯ (t ) =
(1 + bik )˜yi (t ).
0
n
i =1
d+ V¯ (t ) =
αi | ln yi (t ) − ln y˜ i (t )|. By Itô’s formula,
n
sgn(yi (t ) − y˜ i (t ))d(ln yi (t ) − ln y˜ i (t ))
i=1
=
n
sgn(yi (t ) − y˜ i (t )) −
i=1
≤−
aij (t )
n
αi aii (t )
0
n
αi aii (t ) −
n n
n
n
αj aji (t ) |xi (t ) − zi (t )|dt
αi aii (t ) −
n
αj aji (t )
j=1, j̸=i
n
0
|yi (t ) − y˜ i (t )|dt .
i=1
By integrating and then taking the expectation, we have V¯ (t ) ≤ V¯ (0) − mδ
n i=1
t
|yi (s) − y˜ i (s)|ds.
0
That is to say, n i =1
αj aij (t )|xj (t ) − zj (t )|dt
j=1, j̸=i
i =1
V¯ (t ) + mδ
αj aij (t )
i=1 j=1, j̸=i
n
≤ −δ m
n n i=1 j=1, j̸=i
αi aii (t )|xi (t ) − zi (t )|dt +
i=1
=−
(1 + bjk )(yj (t ) − y˜ j (t )) dt
(1 + bik )|yi (t ) − y˜ i (t )|dt +
i=1
=−
0 < tk < t
j=1
i=1
≤−
n
t
|yi (s) − y˜ i (s)|ds ≤ V¯ (0) < ∞. 0
(1 + bik )|yi (t ) − y˜ i (t )|dt
0 < tk < t
(1 + bjk )|yj (t ) − y˜ j (t )|dt
924
M. Liu, K. Wang / Mathematical and Computer Modelling 57 (2013) 909–925
Noting that V¯ (t ) ≥ 0, then |yi (t ) − y˜ i (t )| ∈ L1 [0, ∞). Consequently, by Lemmas 7 and 8, we get limt →+∞ |yi (t ) − y˜ i (t )| = 0 a.s. Consequently lim |xi (t ) − zi (t )| = lim
t →+∞
This completes the proof.
t →+∞
0 < tk < t
(1 + bik )|yi (t ) − y˜ i (t )| ≤ M lim |yi (t ) − y˜ i (t )| = 0, t →+∞
a.s.
5. Concluding remarks This paper is concerned with an n-species stochastic non-autonomous Lotka–Volterra competitive system with impulsive effects. We first establish the sufficient conditions for stochastic permanence and extinction. Then we estimate the limit of the average in time of the sample paths of solutions. Finally, we study the global stability. Our key contributions are the following. (A) This article deals with the stochastic n-species population system with impulsive effects, while most of the existing results (see, e.g. [3–12]) are concerned with the deterministic case although [28] considers the stochastic single-species case. (B) Most of the existing papers (see, e.g. [5,8,9]) suppose that the coefficients are periodic, while we here do not have this restriction. That is to say, our model covers more practical situations. (C) Comparing with deterministic results [3–12], our work reveals that if the noise is sufficiently small, then the stochastic model keeps many desired properties that the associated deterministic system owns, for example, permanence. However, if the noise is sufficiently large, the stochastic noises may change the properties of the system greatly. For example, if the noise is sufficiently large, the solution to the associated stochastic system is extinct with probability one, although the solution to the original deterministic model may be persistent. Some interesting topics deserve further investigation. One may investigate some realistic but complex systems, for example, the stochastic model with Markovian switching (see e.g. [21,23]). It is also interesting to study what happens if aij is stochastic. Acknowledgments The authors thank the editor and reviewers for their important and valuable comments. The authors also thank the NSFC of PR China (Nos 11126219, 11171081 and 11171056). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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