CC BY
0Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 426702, 11 pages http://dx.doi.org/10.1155/2014/426702
Research Article
Dynamical Behavior of a Stochastic Food-Chain System with Beddington-DeAngelis Functional Response
Yanming Ge and Zigen Song
College of Information Technology, Shanghai Ocean University, Shanghai 201306, China Correspondence should be addressed to Zigen Song; zigensong@163.com Received 11 March 2014; Revised 13 May 2014; Accepted 18 May 2014; Published 11 June 2014 Academic Editor: Yongli Song
Copyright © 2014 Y. Ge and Z. Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate a stochastic Food-Chain System dx(t) = [rl(t) - all(t)x - (al2(t)y/(1 + ftl(t)x + yl(t)y))]xdt + al (t)xdBl(t), dy(t) = [r2(t) - a2l(t)y + (a22(t)x/(l + ftl(t)x + yl(t)y)) - (a23(t)z/(1 + ft2(t)y + y2(t)z))]ydt + a2(t)ydB2(t), dz(t) = [-r3(i) + (a3l(t)y/(1 + ft2(t)y + y2(t)z)) - a32(t)z]zdt + a3(t)zdB3(t), where Bi(t), i = 1, 2, 3, is a standard Brownian motion. Firstly, the existence, the uniqueness, and the positivity of the solution are proved. Secondly, the stochastically ultimate boundedness of the system is investigated. Thirdly, the boundedness of moments and upper-growth rate of the solution are obtained. Then the global attractivity of the system is discussed. Finally, the main results are illustrated by several examples.
1. Introduction
In recent decades, a number of authors have studied the predator-prey type systems [1-8], especially the simple Food-Chain Systems [9-14]. These deterministic systems assume that all parameters in the models are deterministic, and some beautiful and valuable conclusions are obtained. Then it is well known that biology systems are often subject to environmental noise [15]. It is therefore useful to reveal how the noise affects these systems. Though there are various types of environmental noise, May [16] have claimed that the birth rates in the system should be stochastic. At the same time, the natural growth of many species vary with time, for example, due to the seasonality.
On the other hand, basing on some natural phenomenon and statistics in laboratory, the predator-dependent functional response should be considered in the predator-prey type systems. As we know, there are three classical predator-dependent functional responses: Hassell and Varley [17], Beddington et al. [18, 19], and Crowley and Martin [20]. Recently, Skalski and Gilliam [21] pointed out that these functional responses can provide better descriptions, and the Beddington-DeAngelis (B-D) type functional response performed even better in some cases.
Based on this background, in this paper we will discuss the following stochastic system:
dx (t) =
r1 (t)-au (t)x-
a12 (*) 7
l + ßi (t)x + Yi (t)y\
+ a1 (t) xdB1 (t),
dy(t) =
dz (t) =
r2 (t) - 021 (t) y +
a23 (t)z l+ß2 (t)y + 72 (t) Z ,
a3i (t) y
a22 (t) x
l + ßi (t)x + Yi (t)y
ydt + a2 (t) ydB2 (t),
-r3 (t) +
l+ ß2 (t)y + Ï2 (t) Z + a3 (t) zdB3 (t),
- ^32 (t) z
where Bi(t), i = l, 2,3, is a Brownian motion defined on a complete probability space (Q., F, P). All parametric functions are continuous and bounded functions on R+. x(t), y(t), and z(t) denote the population density of resource, consumer, and predator at time t, respectively. For convenience,
the model, consisting of (1), together with x(0) = x0 > 0, y(0) = J0 > 0, z(0) = z0 > 0, will be referred to as model SDE. On the other hand, for easy expression, if f(t) is a continuous bounded function on R+, we define f" = suPt^R f(t) and fl = infteR+ f(t). Throughout this paper, we
assume that rl >0, i = 1,2,3, and the others are similar. The notation K used in this paper expresses the positive constant, for convenience, it denotes different constant in different places, and they only differ in positive constant times.
The rest of the paper is organized as follows. In Section 2, we show that the system has a global positive solution. In Section 3, the stochastically ultimate boundedness of the system is acquired. In Section 4, we obtain the asymptotic moment estimation and the upper-growth rate of the solution. In Section 5, the global attractivity of SDE is investigated. In Section 6, we illustrate our main results through several numerical examples. Last but not least, conclusions are drawn in Section 7.
The key method used in this paper is the analysis of Lyapunov functions. This Lyapunov function analysis for stochastic differential equations was developed by Khasmin-skii [22] and has been used by many authors [23-28].
2. Positive and Global Solutions
As the state of the SDE is the population density of species in the system at time t, it should be nonnegative. Moreover, in order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial data, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition [25]. However, the coefficients of each equation in SDE neither obey the linear growth condition nor local Lipschitz continuous. In this section, we show existence and uniqueness of the positive solution.
Lemma 1. For any initial value x0 > 0, y0 > 0, z0 > 0, system (SDE) has a unique positive local solution (x(t),y(t),z(t)) for t e [0, Te) almost surely (a.s.), where Te is the explosion time.
Proof. To begin with, consider the following equations:
du (t)
h (t)-an (t)eu{t) -
+ a1 (t) dB1 (t), dv (t)
b2 (t)-a21 (t)ev(t) +
a12 (*) e
1 + ß1 (t) eu(t) + y1 (t) e
a22 (*) e'
Ü23 (t) é
1 + ß2 (t) ev(t> + y2 (t)e
1 + ß1 (t) eu(t) + y1 (t) e
dt + a2 (t) dB2 (t),
dw (t)
-b3 (t) +
031 (t) e
1 + ß2 (t)ev(t) + y2 (t)e
-a32 (t) e
dt + a3 (t) dB3 (t)
on t > 0 with initial value u(0) = lnx0, v(0) = lnyQ, w(0) = ln zQ, where b1(t) = r1(t) - 0.5a^(t), b2(t) = r2(t) -0.5ol(t),b3(t) = r3(t)+0.5a^(t). Notice that the last equations' coefficients satisfy the local Lipschitz condition; thus, there is a unique solution (u(t), v(t),w(t)) ont e [0,Te). Therefore, it follows from Ito's formula that x(t) = eu(t\ y(t) = ev(\ z(t) = ew(t) is the unique positive local solution of SDE with initial value xQ > 0, yQ > 0, zQ > 0. □
Theorem 2. Consider system SDE, for any given initial value (xQ,yQ,zQ) e R+, and there is a unique solution (x(t),y(t),z(t)) ont > 0 and the solution will remain in R+ with probability 1, where R+ = {(x1,x2,x3) | xi > 0,i = 1,2, 3}.
Proof. From Lemma 1, to complete the proof we only need to show that Te = >x>. Hence, let kQ > 0 be sufficiently large for xQ, yQ, and zQ lying within the interval [k- , kQ]. For each integer k > kQ, define the stopping times:
Tk = inf ji e [0,Te] :x(t)i (k-1,k) or
y(t)i(k-1,k] or (3)
z(t)l(k-1,k)\.
Clearly, Tk is increasing as k ^ œ>. Set Tm = limk^mTk, from which Tm < Te. Now, we only need to prove Tm = <m. If this statement is false, there is a pair of constants T > 0 and e e (0,1) such that P{Tm < T} > e. Thus there exists an integer k1 > kQ such that
P{rk <T}>e, k>k1.
Define V(x, y,z) = x-1-lnx + y-1-lny + z-1-lnz. If (x(t), y(t), z(t)) e R3+, we have
dV (x, y, z)
= (x-1)
r1 (t) - an (t) x -
a12 (*) y
+ 0.5O-2 (t) dt
+ (y-1)
r2 (t) - 021 (t) y +
Ü23 (t) z 1+ß2 (t)y + 72 (*) z .
1+ß1 (t)x + Y1 (t)y\ a22 (*) x
1+ß1 (t)x + Y1 (t)y
dt + 0.5a2 (t) dt
+ (z-1)
-Г3 (t) +
a3i (t) У
1 + ß2 (t)y + y2 (t) z
-a.32 (t) z
dt + 0.5a2 (t)dt
+ (x-1)a1 (t) dB1 (t) + (у-1) o2 (t) dB2 (t) + (z-1)a3 (t) dB3 (t)
I 2 (и и \ 12 n г Г ил
-a11x + (r1 + a11)x+—]2 + 0.5[a1 ]
■ dt
I 2 I и и u22 \ U23 П гГ Ui2
-a21y + (rи +a^1 + -T )y+-r + a^aj
I 2 I U31 и \ и г, г[ ил2
-a32Z + ( ~ßT +a32)Z + r3 + 0.5[a3
• dt
+ (x-1)01 (t) dB1 (t) + (у-1) O2 (t) dB2 (t) + (z-1)a3 (t) dB3 (t) < Kdt + (x - 1) a1 (t) dB1 (t) + (y-1) o2 (t) dB2 (t) + (z- 1) a3 (t) dB3 (t).
Integrating both sides of the above inequality from 0 to Tk AT and then taking the expectations lead to
EV(x(rk AT),y(rk AT),z(rk AT)) <V(x0,y0,z0) + KT.
Set i\ = [Tk < T}; then it follows from inequality (4) that P{Qk} > e. Clearly, for every w e Qk, there is x(Tk,«), y{Tk,w), or z{Tk,w) which equals either k or k-1; therefore, V(x(Tk, y(Tk, z(Tk, &)) is no less than
in [k-1- ln k,k-1 - 1+ ln k\.
Then an application of (6) results in V (x0, y0, z0) + KT>E [lntV (x (Tk), y (Tk), Z (Tk))] > min {k- 1- ln k,k-1 - 1 + ln k\,
where 1n is the indicator function of Qk. Letting k ^ >x> leads to the contradiction
<x> > V (x0, y0, z0) + KT >
(9) □
3. Stochastically Ultimate Boundedness
In the previous section, we have showed that the solutions of SDE will remain in the positive cone R+ forever. However, this nonexplosion property in a population dynamical system
is often not good enough while the property of ultimate boundedness is more desired. Let us discuss it in this section. We begin with giving the definition of stochastically ultimate boundedness.
Definition 3. The SDE is said to be stochastically ultimate bounded if for any e e (0,1), there is a positive constant H = H(e) such that for any initial value, the solution of SDE has the property that
lim sup P-j \(x (t), y (t), z (t))\
t — L
= ^X^iti+y^iti+z^it') >н} <£.
Theorem 4. For any 0 e (0,1) there is a positive constant H = H(d) such that for any initial value, the solution of SDE has the property that
limsup E\(x (t), y (t), z (t))f < H. (11)
In particular, the SDE is stochastically ultimate bounded. Proof. Define
\ 8 8 8 V (x,y,z) = X + y + z .
Applying Itos formula gives
dV (x, y, z)
a12 (t)y
= вх°
Г1 (t)-an (t)x-
1 + ß1 (t)x + Y1 (t)y\
+ 0.5в (в - 1) xea1; (t)dt
Г2 (t) - Ü21 (t) у +
Ü23 (t) Z
1+ß2 (fy + У2 (f Z -
Ü22 (t) X
1 + ß1 (t)x + Y1 (t)y
+ 0.5в (в - 1) yal (t) dt
-Г3 (t) +
031 (t) У
- ^32 (t) z
1+ß2 (t)y + Y2 (fz + 0.5d(e-1)zeol (t)dt
+ вхва1 (t) dB1 (t) + вува2 (t) dB2 (t) + dzea3 (t) dB3 (t) < вхв [0-1 +г'и - alnx + 0.5в(а'и)2] dt
a~1 и l a22 n га/ и\2
в + г2 - апу + -г- + °.5в((Г2 ) Р1
О-1 + f - a32Z + 0.59(0"Г
- V (x, у, z) dt + вхва1 (t) dB1 (t) + вува2 (t) dB2 (t) + вгвa3 (t) dB3 (t)
< Kdt - V (x, y, z) dt + dxea1 (t) dB1 (t) + 0ydo2 (t) dB2 (t) + dzeo3 (t) dB3 (t).
Using Itôs formula again we have d(eV (x, y, z))
= et [V (x, y, z) dt + dv (x, y, z)] < e Kdt + e [dx9o1 (t) dB1 (t) + dyeo2 (t) dB2 (t) + dz9o3 (t)dB3 (f)].
Integrating the above inequality and then taking expectations on both sides, we can see that
e EV (x, y,z)<V (x0, y0, z0) + K(e - l) . (15)
This implies immediately that
EV (x, y, z) < K. (16)
On the other hand,
\(x (t),y(t),z(t))f < 30-56 max {xe,ye,ze}
< 305eV(x,y,z),
and then we have
limsupE\(x(t),y(t),z(t))\9 <3056K (18)
which yields the required assertion by setting H = 30'seK. Especially, let d = 0.5; then
limsupE\(x(t),y(t),z(t))\0'5 <H. (19)
Now, for any e e (0,1), let H = H ¡e2, by Chebyshev's inequality,
P{\(x(t),y(t),z(t))\>H}<
Hence,
E^\(x(t),y(t),z(t))\
P{\(x(t),y(t),z(t))\>H}< — = e. (21) In other word, SDE is stochastically ultimate bounded. □
4. Boundedness of Moments and Upper-Growth Rate Estimation
Theorems 2 and 4 show that the solutions of SDE will remain in a subset of R+. Now let us construct different types of Lyapunov functions to investigate how the solutions behave in R+.
Theorem 5. Consider system SDE for any positive numbers p1,p2, p3; 01,d2, Q3, there exists a positive number K2 such that, for any initial value (x0, y0, z0) e R+, the solution ofSDE obeys
E [£ (pixdl (s) + p2ye2 (s) + p3z03 (s)) ds] limsup-< K.
t — m t
Proof. Define
V (x, y, z) = p1xe1 + p2y°2 + p3z6}. (23)
Applying Itos formula gives dV (x, y, z)
a12 (V y
= PldlX '
r1 (t)-an (t)x-
i+ßl (t)x + Yl (t)y\
+ 0.5p1d1 (d1 - i)xtilo\ (t) dt
+ P2°2y
V2 (t) - Ü21 (t) y +
__a23 (t) Z
i+ß2 (t)y + Y2 (t)z\
a22 (*) X
i + ßi (t)x + Yi (t)y
+ 0.5P2Ö2 (02 - i)y92(t)dt
+ Psd3
-r3 (t) +
a31 (t) y
i+ ß2 (t)y + Ï2 (t) Z
- a32 (t) Z
+ 0.5p3d3 (d3 - i)z03o2 (t)dt
+ P1Ö1X01IJ1 (t) dB1 (t) + P2Ö2y 2O2 (t) dB2 (t) + p3d3zdlo3 (t) dB3 (t) < p1d1xdl [0-1 +r" - a11x + 0.5d1(o")2] dt
+ P2d2y
+ P383
d-1 +ru2 -^ y+-22 + 0.592(o2) P1
^2 +if - a32Z + 0.563(03) ßl
- V (x, y, z) dt + p1d1x°1 o1 (t) dB1 (t) + P2®2y02(*) dB2 (t) + p3d3z03o3 (t) dB3 (t) < Kdt - V (x, y, z) dt + p1d1x°1 o1 (t) dB1 (t) + P2®2y02(*) dB2 (t) + p3d3z03o3 (t) dB3 (t).
Integrating the above inequality and then taking expectations on both sides, we can see that
EV (x, y,z) + E\ V (x, y, z)ds < Kt + V (x0, y0, z0). J0
Together with EV(x, y, z) > 0, this inequality shows that
e\ V(x, y, z)ds<Kt + V (x0, y0, z0). (26) Jo
From this inequality, we obtain the desired assertion. □
This theorem, together with Theorem 2, denotes 0 < E[(1/t) x(s)ds] < K, in other word, the average value of x(t) on time and "space" is bounded. It is similar to y(t) and z(t).
Theorem 6. For any positive numbers a1, a2, and a3, solutions of SDE satisfy
ln [x* f* Za3 ] , x
lim sup-< ai + a2 + a3 a.s. (27)
t^rn ln t
Proof. Define
V1 (x, y, z) = a1 ln x + a2 ln y + a3 ln z. (28) Applying Itos formula gives dV1 (x, y, z)
b1 (t) - aii (t) x -
a-2 (t)y
h (t) - a2i (t) y +
Ü23 (t) Z
1 + ß- (t)x + n (t)y\
Ü22 (t) X
1+ßi (t)x + yi (t)y
\+ß2 (t)y + Y2 (t) Z .
Ü3- (t) y
-b3 (t) +
- &32 (t) Z
' 1 ß M 32
l+ß2 (t)y + Y2 (*) Z
+ aiai (t) dBi (t) + a2a2 (t) dB2 (t) + a3a3 (t) dB3 (t).
V2 (x, y, z) = e6tV1 (x, y, z),
where d > 0, and by the Itos formula, we have
dV2 (x, y, z) = 0e6tV1 (x, y, z)
bi (t) - aii (t) x -
a-2 (t) y
1 + ß- (t)x + Yi (t)y\
+ a2 e
h (t) - 02i (*) y +
a22 (t) x
1 + ßi (t)x + Yi (t)y
Ü23 (t) Z
+ a3 e
\+ß2 (t)y + Y2 (*) Z .
O31 (t) y
-b3 (t) +-„ ^--a32 (t) z
3W l+ß2 (t)y + Y2 (t)z 32W
+ ai eeta1 (t) dB, (t) + a2eeta2 (t) dB2 (t)
+ a3 eeta3 (t) dB3 (t).
Therefore, V2 (x,y,z)
= v2 (x0' yo> z0) + d\ e6sVi (x, y, z) ¡0
C „es \u „ („)„ Oi2 (s)y
+ ai I e Jo
+ a2 I e
bi (s) - aii (s) x -
h (S) - 02i (S) y +
1 + ßi (s)x + Yi (s) y.
022 (S) X 1+ßi (s)x + Yi (s) y
023 (s) Z
l+ß2 (S) y + Y2 (S) Z .
+ a, I e
-hi (s) +
a3i (s) y
- &32 (s) Z
3 l+ß2 (S) y + Y2 (s)z ^
+ a1 I e6sa1 (s) dB1 (s) + a2 I e6sa2 (s) dB2 (s)
+ a3 I e6sa3 (s) dB3 (s).
Define
Mt (t)=ie9sai (s)dBt (s), i= 1,2,3.
Notice that the quadratic variation of Mt(t) is
(Mt (t),M, (t))= fe29s a2 (s) ds.
Let e e (0,1) and A > 1. In view of the exponential martingale inequality, for every k > 1 and i = 1,2,3, we have
sup [Mi (t) - 0.5ee-ek (Mt (t), Mt (i))]
Xeek ln k
An application of Borel-Cantelli lemma yields that, for almost any a e O, there is a random integer k0 = k0(w) such that, for k > k0,
Xeek ln k
sup [Mi (t) - 0.5ee-ek (M, (t) >Mi (i))] <
0<t<k e
Mt (t) < 0.5ее-вк l e^o] (s) ds + Jo
Хевк ln к
for 0 < t < k, k > k0 a.s. Substituting the above inequality into (32), we have
Vi (x, y,z)
< v2 (x0> Уо, zo) + o\ ев% (x, y, z)
+ ai le
+ a2 le
b1 (s) - an (s) x -+ 0.5a! (s) eee(s-k)
h (S) - 021 (s) У + __Ol3 (s)z
1+ßi (s)y + Yi (s)z
0\2 (S) У
l+ßi (s) x + y1 (s) y ds
О22 (S) X
1 + ßi (s)x + Yi (s) y
2 /Л rJ(s-k)
+ a-, le
+ 0.5a2 (s) ее
-h (s) +
a-i (s)y
l + ß2 (s) У + У2 (s) z
- a32 (s) z + 0.5a- (s) еев( kk) + (^ + a2 + a3)
Xeek ln к
< v2 (x0, Уо, zo) + K (евt - l) + (ai +a2 + a-)
Xeek ln к
In other words, we have shown that ln [x" ya2]
< e etV2 (xo, Уо, Zo)+K + (a.1 +02 + a-)
Thus, for all k>k0, k - 1 < t < k,we have ln [x"1 y"2] <V1{xo,y2,z2)+ K
Хев(к-*) ln к
ев* ln t ln t (a1 +a2 + a,) (\ев{к-) ln k/e) ln (k-l) .
ln [x"1 y"2] Xe limsup---< (aj + a2 + a3)-. (41)
t^rn ln t £
Letting e ^ 1, X ^ 1, and d ^ 0, we can obtain the required assertion. □
Based on this theorem, obviously, the upper-growth rate is linear.
5. Global Attractivity
In this section, we turn to establishing sufficient criteria for the global attractivity of system SDE. Firstly, we prepare some useful lemmas.
Lemma 7. Let (x(t),y(t),z(t)) be a solution to SDE with initial value (x0 ,y0,z0) e R+. Then for all p > 1, there exist three constants related to p: Lt(p) for i = 1,2,3, such that
limsupE[xp (f)] < L1 (p),
t — Ж
limsupE[yp (f)] < L2 (p),
t —> ^O
limsupE[zp (i)] < L3 (p).
Proof. Define W(y) = yp for y e R+. Making use of Itos formula we have
dW (y)
Г2 (t) - an (t) у +
О22 (t) x
l + ßi (t)x + Yi (t)y
a23 (*) z . n r ! - л \ -2,
1+ß2 (t)y + У2 (t) Z
+ 0.5(p-l)a22 (t)
+ pa2 (t) ypdB2 (t).
Applying Itos formula again to e*W(y) and then taking expectations on both sides, we can see that
E[etyp (t)]
<Уо +P
-i и L и22 п г Л и\2
р +Ги -a2iy+-r +°5р(аи)
<yp + рК (e* -l),
which is the desired assertion of limsupt^TO£[y^(i)j < L2(p), and the other two inequalities can be proved similarly. □
Lemma 7 tellsusthatthereisaT > 0,suchthat E(xp(t)) < 1.5L 1(p), E(yp(t)) < 1.5L2(p), and E(zp(t)) < 1.5L3(p) for all t > T. At the same time, it follows from the continuity of E(xp(t)), E(yp(t)), and E(zp(t)) that there exist L 1(p) > 0, L2(p) > 0, and L3(p) > 0 such that E(xp(t)) < L 1(p), E(yp(t)) < L2(p), and E(zp(t)) < L3(p) for all 0 < t < T. Denote L(p) = max|1.5L 1(p),TL 1 (p)}, H(p) = max|1.5L2(p),L2(p)}, and G(p) = max|1.5L3(p),L3(p)}. Then for all t > 0,
E(xp (t))<L(p), E(yp (t))<H(p), E(zp (t))<G(p).
Lemma 8 (see [29]). Supposethatann-dimensionalstochastic process X(t) on t > 0 satisfies the condition
E\X (t) - X (s)f1 <c\t-s\1+li2, 0<s,t<rn, (46)
for some positive constants p1t p2, and c. Then there exists a continuous modification X(t) of X(t) which has the property that for every v e (0, ^2/^1) there is a positive random variable h(w) such that
\X(t,to)-X(t, to) I
to : sup -
0<|f-s|<fe(a>)>0<s,f<œ> = 1.
|f-s|t>
In other words, almost every sample path of X(t) is locally but uniformly Holder continuous with exponent v.
Lemma 9. Let (x(t), y(t), z(t)) be a solution ofSDE on t > 0 with initial value (x0, y0, z0) e R+; then almost every sample path of (x(t), y(t), z(t)) is uniformly continuous on t > 0.
Proof. The second equation of SDE is equivalent to the following stochastic integral equation:
= 70 +1 ®2
Í y(s)
\ a2 (s)y(s)dB2 (s)
r2 (S) - a2l (S) 7 +
a23 (s) z
a22 (S) X
1 + ßi (s)x + Yi (s) y
1+ ß2 (S) y + Ï2 (S) Z .
We estimate
r2 (S) - a2\ (S) 7 +
a23 (s) z
a22 (s) x
1+ßi (s) x + y1 (s) y
1+ß2 (S) y + Ï2 (S) Z .
<0.5%(s)|
+ 0.5E
r2 (s) - Ü21 (s) y +
023 (s) Z
a22 (s)x
1+ßi (s)x + Yi (s) y
1+ß2 (S) y + Y2 (S) Z
H(2p) + 42p-1
(r1)2P + (aU2i)2PH(2p)
22 ßl
= :.C(P).
On the other hand, as an application of the moment inequality for stochastic integrals [30], we see that for 0 < t1 < t2 and p > 2,
E\\ <J2 (s)y(s)dB2 (s) \}t1
< [(o22)u]F[0.5p (p - l)F>(t2 - 11)°-5(p-2) E\y (s)\pds <[(au2)2]P[0.5P(P-l)r?(t2-t1)°5pH(P).
Then for 0 < t1 < t2, t2 -t1 < 1, p-1 + q-1 = 1, we obtain
E\y(t2)-y(h)\P
°2 (s)y(s) dB2 (s)
+ Í2 y(s)
T2 (s) - 021 (s) y
Ü22 (S) X
1 + ßi (s)x + Yi (s) y
a23 (s) z
^IP y (S) \Jt,
1+ß2 (S) y + Y2 (S) Z . r2 (s) - a2i (s) y
a22 (s) x
1 + ßi (s)x + Yi (s) y
a23 (s) z
1+ß2 (S) y + Y2 (S) Z .
, I f^ P
+ 2p-iE\\ o2 (s)y(s)dB2 (s) <2p-i(t2 -ti)PC(p)
+ 2p-i[(a«)2]P[0.5p(p-1)rp(t2-ti)0-5pH(p) <2p-i(t2-ti)05p [1+[0.5P(P-1)]0-5p ]K.
Then it follows from Lemma 8 that almost every sample path of y(t) is locally but uniformly Holder-continuous with exponent v for every v e (0, (p - 2)/2p) and therefore almost every sample path of y(t) is uniformly continuous on t > 0. Similarly, we can show that almost every sample path of x(t) and z(t) is uniformly continuous on t > 0. □
Lemma 10 (see [31]). Let f be a nonnegative function defined on R+ = [0, >x) such that f is integrable on R+ and is uniformly continuous on R+. Then limt^mf(t) = 0.
Now, we give our main result of this section.
Definition 11 (see [32]). Let A1 (t) = (xx(t),y1(t),z1(t)) and A2(t) = (x2(t),y2(t),z2(t)) be two arbitrary solutions of system SDE with initial values A 1(0), A2(0) e R+. If
lim \x! (t) - x2 (i)| = lim \yi (t) - ^
f — œ 1 t — œ
= lim Iz! (t) - Z2 (i)| = 0, a.s.,
then we say that the system is globally attractive. Theorem 12. If A := alu - a22 - a"2p"/Yi > 0, B := a21 - a"2 -
«31 - a23^2/r2 - a22r"/^l > 0 C := a32 - a23 - a3lYl/P2 > 0
then system SDE is globally attractive. Proof. Define
V (t) = J ln Xi (t) - lnx2 (i)| + |ln yi (t) - ln y2 (i)| + |lnz1 (t) - ln z2 (i)|,
then it is a continuous and positive function on t > 0.A direct calculation of the right differential of V(t), and then applying Ito's formula, we have
d+V(t)
= sgn (Xi - X2)
X{-fl" (t)(X1 -X2 )+ 1+ß1 (1)1
«12 (*) yi Xi + Yi (*) yi
«12(*) y2
1 + ßi (t) X2 + Yi (t) y2.
+ sgn (yi -y2)
x {-«2i (t) (yi - y2) +1 *22;(t)Xl () [ 1+ßi (t) Xi + Yi (t) yi
«22 (t) X2 «23 (*) zi
1 + ßi (t) X2 + Yi (t) y2 1 + ß2 (t) yi + Y2 (*) Zi
«23 (*) Z2
1 + ß2 (*) y2 + Y2 (*) Z2 + sgn (Zi -Z2)
X { -«32 (t) (Zi -Z2)+1
«3i (t) yi
+ ß2 (*) yi + Y2 (*) Zi
«3i (t) y2
1 + ß2 (t) y2 + Y2 (V Z2
10 20 30 40 50 60 70 80 90 100 t
Figure 1: The curve of \n\xyz]/lni for initial values (x0,yO,zO2 (0.6,0.5,0.5) and At = 0.01.
< «ii |Xi X21 «2i |yi y21 «32 lZi ß"
u Iii I w I I
+ «i2— lXi -X2 l+«i2 |yi - y21 Yi
+ «22 \Xi X2\+«22pl lyi ßi
w /2 1 I w I I
+ «23 T |yi -y2| +«23 lZi -Z2 I
, _« Y2 I _ I , _«
+ «3i ß lZi Z21 + «3i ß2
|y1 — y2 |
= -A\xi -X2I -B\yi -y2\-C\zi -Z2 Integrating both sides gives
V(t)<V(0)
\ [A|X1 -X2\+B\yi -y2\+C\Zi -Z2\]dS.
Therefore,
V(t)+ \ [A\Xi -X2\+B\yi -y2\+C\Zi - Z2I ] ds JO
<V(0) < m.
Making use of V(t) > 0 and the known conditions result in that
|x1 (t)-X2 (i)| e L1 \0,ot), |yi (t)-y2 (i)| eL1 \0,m), |z1 (t)-Z2 (i)| e L1 \0,m).
Then the required assertion follows from Lemmas 9 and 10 immediately. □
- |x1(i)-x2(i)|
- lyi (t) - y2 (t)l
- |Zi(t)-Z2(t)|
Figure 2: The curves of \x1(t) - x2(t)\, \y1(t) - y2(t)\ and \z1(t) - z2(t)\ for initial values (^(0), ^(0), 2^(0)) = (0.8,0.8,0.8) and (x2(0),y2(0),z2(0)) = (0.1,0.1,0.1), At = 0.01.
6. Numerical Results
We present numerical experiments in this section to show how the proposed model works in the constructive examples. The results enhance the readers to understand the theoretical conclusions from the practical applications.
Here, we use the Milstein method [33] to construct the discretization equation of SDE. That is,
xk+1 = xk
r1 (kAt) - a11 (kAt)xk
ai2 (kAt) yk
l + ßi (kAt)xk + Yi (kAt)yk\ 2
+ o1 (kAt) xk VÂtÇk + 0.5a2 (kAt) x2k (tf, - 1) At,
yk+i = yk
r2 (kAt) - an (kAt) yk
a22 (kAt) xk
1 + ßi (kAt)xk + Yi (kAt)yk Ü23 (kAt) zk
1 + ß2 (kAt) yk + Y2 (kAt) zk J ■ 02 (kAt) ykVAtnk + 0.501 (kAt) yk (nl - 1) At,
zk+1 = zk
-r3 (kAt) +
a3i (kAt) yk
1 + ß2 (kAt) yk + Y2 (kAt) zk
-a32 (kAt) zk
At + a3 (kAt) zk VAtÇk
0.5(72 (kAt) z2 (% -1)At,
where i^k, qk, and %k, k = 1,.. .,n, are the Gaussian random variables. We obtain the following results.
As we pointed out in Theorem 6, limsupt^TO ln [x"1 y"2z"3]/lnt < a1 + a2 + a3 a.s., and Figure 1 confirms this. Here we choose r1(t) = 9 + sin(i), r2(t) = 5 + sin(i), r3(t) = 0.1 + 0.05 sin(t), a11(t) = a2l(t) = 1 + 0.1 sin(f), al2(t) = a22(t) = 0.4 + 0.2 sin(t), a23(t) = 0.1 + 0.05 sin(f), a31 = 5 + 0.05 sin(t), a32 (t) = 0.5 + 0.1 sin(t), p1(t) = y1(t) = 1, p2(t) = y2(t) = 2, o1(t) = o2(t) = o3(t) =1 + 0.1 sin(i), and ai = a2 = a3 = 1.
Figure 2 confirms that the system is globally attractive (Theorem 12). Here r1(t) = 1.1 + sin(t), r2(t) = 0.1 + 0.05sin(t), r3(t) = 2 + 0.1 sin(t), a11(t) = a21 (t) = 10.1 + 0.1 sin(t), a12(t) = a22(t) = a23(t) = a31(t) = 0.8 + 0.2sin(t), a^2(t) = 4 + sin(i), 01 (t) = Y1(t) = 1, p2(t) = y2(t) = 10, a1(t) = a2(t) = 2 + 0.1 sin(i), a3(t) = 0.4 + 0.1 sin(i).
The initial values are (x1(0),y1(0),z1(0)) = (0.8,0.8, 0.8) and (x2(0),y2(0),z2(0)) = (0.1,0.1,0.1). Then A =8, B = 6, C = 1, and the assumptions of Theorem 12 are satisfied.
7. Conclusions
This paper studies a stochastic nonautonomous food-chain system with Beddington-DeAngelis functional response. This system retains some conventional properties of stochastic systems. That is, it has a unique global positive solution for any initial conditions, the moments of the solutions are bounded, and the solutions are global attractive and stochastically ultimate bounded.
Nevertheless, there are rooms to continue working on this issue. It is interesting to study other parameters; for example, a^ (i) are stochastic. The permanence and extinction of the system and the stability in distribution deserve to be investigated too.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research is supported by the National Natural Science Foundation of China under Grant no. 11302126.
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