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Dynamics of a stochastic non-autonomous predator-prey system with Beddington-DeAngelis functional response

Shuang Li1,2* and Xinan Zhang1

"Correspondence: oklishuang@yahoo.com.cn ' Schoolof Mathematics and Statistics, CentralChina Normal University, Wuhan, Hubei 430079, P.R. China

2Department of Mathematics, Xinxiang University, Xinxiang, Henan 453000, P.R. China

Abstract

A stochastic non-autonomous predator-prey system with Beddington-DeAngelis functional response is proposed, the existence of a global positive solution and stochastically ultimate boundedness are derived. Sufficient conditions for extinction, non-persistence in the mean, weak persistence in the mean and strong persistence in the mean are established. The global attractiveness of the solution is also considered. Finally, numerical simulations are carried out to support ourfindings. MSC: 92B05; 34F05; 60H10; 93E03

Keywords: predator-prey; non-autonomous; stochastic; Beddington-DeAngelis; persistence; extinction

1 Introduction

In population dynamics, the relationship between predator and prey plays an important role due to its universal existence. There are many significant functional responses in order to model various different situations. In fact, most of the functional responses are prey-dependent; however, the functional response should also be predator-dependent, especially when predators have to search for food. Beddington [1] and DeAngelis et al. [2] in 1975 first introduced the Beddington-DeAngelis type predator-prey model taking the form

du - - „, u2 fluv

dt - riU SlU p+yu+Sv' dv _ r „ ,,2 , /2UV

dv - r2v -g2v + p+YU+sV,

where u and v denote the population densities of prey and predator. Although the Beddington-DeAngelis functional response is similar to the Holling type-II functional, it can reflect mutual interference among predators. That is to say, this kind of functional response is affected by both predator and prey. In the last years, some experts have studied the system [3-7]. In the following, we introduce a non-autonomous predator-prey model with Beddington-DeAngelis functional response:

ft Spri

dJCt - x{ai(t) bl(t)x mi(t)+m2l(t)Xy+m3(t)y )' I - y(-a2(t) - b2(t)y + mi(t)+m2(t)tt+W3(t)y)

ringer

© 2013 Li and Zhang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

where x(t) and y(t) represent the population density of prey and predator at time t, respectively. fli(i) denotes the intrinsic growth rate of prey. bi(t) and b2(t) stand for the density-dependent coefficients of prey and predator, respectively. a2(t) is the death rate of predator. c1(t) is the capturing rate of predator, c2(t) represents the rate of conversion of nutrients into the reproduction of predator. For the other coefficients' biological representation, we refer the reader to [1] and [2].

On the other hand, population systems are often subject to environmental noise; therefore, it is important to study how the noise affects the population systems. In fact, stochastic population systems have been studied recently by many authors [8-17]. However, there are not many papers considering non-autonomous stochastic systems [18-23]. In this paper, considering the effect of environmental noise, we introduce stochastic perturbation into the intrinsic growth rate of prey and the death rate of predator in system (2) and assume the parameters a1(t) and a2(t) are disturbed to a1(t)+a(t) dB1(t), -a2(t)+p(t) dB2(t), respectively. Then corresponding to the deterministic system (2), we obtain the following stochastic system:

dx = X(a1(t) - b1(t)x - m1(t)+mC2(tt+m3(t)y) dt + a(t)xdB1(t), (3

dy = y(-a2(t) - b2 (t)y + m^^m^y ) dt + P (%dB2(t).

We assume all the coefficients are continuous bounded nonnegative functions on R+ = [0,+to).

Iff (t) is a continuous bounded function on R+, define fu = supf (t), f = inf f (t).

teR+ teR+

Throughout this paper, suppose that a2 > 0, mi > 0 (i = 1,2,3), bi > 0 (i = 1,2).

Definition 1 (1) The population x(t) is said to be non-persistent in the mean if (x>* = 0, where (f (t)> = \f0f (s) ds,f * = limsupt^+TOf (t),f* = liminft^+TOf (t).

(2) The population x(t) is said to be weakly persistent in the mean if (x>* > 0.

(3) The population x(t) is said to be strongly persistent in the mean if (x>* > 0.

Throughout this paper, unless otherwise specified, let (fi, F, P) be a complete probability space with a filtration {Ft}teR satisfying the usual conditions (i.e., it is right continuous and increasing and F0 contains all P-null sets), here Bi(t) (i = 1,2) is a standard Brownian motion defined on this probability space. In addition, R+ denotes {(x,y) e R2: x > 0,y > 0}.

Here we give the following auxiliary statements which are introduced in [24]. Consider the d-dimensional stochastic differential equation

dx(t) = f (x(t), t) dt + g(x(t), t) dB(t), t > t0.

Denote by C2,1(Rd x [t0, to); R+) the family of all nonnegative functions V(x, t) defined on Rd x [t0, to) such that they are continuously twice differentiable in x and once in t. The differential operator L of the above equation is defined by the formula

L=d + Yfi(x,t)d +1E(x, t)g(x, t)!i 92

dt dxi 2 /—'L Mj dxi dxi

i=1 i ij=1 i i

If L acts on a function V e C2,1 (Rd x [i0, to); R+ ), then

LV (x, t) = Vt (x, t) + Vx(x, t)f (x, t) + 2 trace^ (x, t) V«(x, %(x, t)]. 2 Global positive solution and stochastic boundedness

For a model of population dynamics, the first thing considered is whether the solution is globally existent and nonnegative, hence in this section we will show it. In order for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and the local Lipschitz condition [24]. The coefficients of model (3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, the solution of system (3) may explode at a finite time (cf. [24]). In the following, using the Lyapunov analysis method, we can prove that system (3) has a global positive solution.

Theoreml For any initial value (x0, y0) e R+, there is a unique solution (x(t), y(t)) of system (3) ont > 0, and the solution will remain in R+ with probability 1.

Proof Since the coefficients of model (3) are locally Lipschitz continuous, for any given initial value (x0,y0) e R+, there is a unique local solution (x(t),y(t)) on t e [0, re), where xe is the explosion time [24]. To show the solution is global, we need to show that xe = to.

Let n0 > 0 be sufficiently large for x0 and y0 lying within the interval [1/n0, n0]. For each integer n > n0, we define the stopping times

%n = inf{ t e [0, Te):x(t) e (1/n, n) ory(t) e (1/n, n)},

where, throughout this paper, we set inf 0 = to (0 denotes the empty set). Obviously, Tn is increasing as n ^to. Let tto = limn^TO Tn, hence, tto < Te a.s. Now, we only need to show tto = to. If this statement is false, there is a pair of constants T >0 and e e (0,1) such that P{tto < T} > e. Consequently, there exists an integer n1 > n0 such that

P{Tn < T} > e, n > m. (4)

Define a C2-function V : R+ ^ R+ by

V(x,y) = (x -1 - logx) + (y -1 - logy).

The nonnegativity of this function can be seen from u - 1 - log u > 0, Vu >0. Applying Itô's formula, we get

dV =(x -1) (ai(t) - b1(t)x - —--c1Tt)y-—) dt + ^ dt

\ mi(t) + m2(t)x + m3(t)y/ 2

+<,-1)(-a2(t)-b2(t)y+mM+mjd'+f2rdt

+ (x - 1)a(t) dB1(t) + (y - 1)j8(t) dB2(t), (5)

LV = ai(t)x - bi(t)x2 -- a1(t) + b1(t)x +

,2 ci(t)xy

m1(t) + m2(t)x + m3(t)y

_ci(t)y_

m1(t) + m2(t)x + m3(t)y

+ ^ -a2(t)y- b2(t)y2 + C2(tfxy-j-r- + a2(t) + b2(t)y

2 m1(t) + m2(t)x + m3(t)y

C2(t)x p 2(t)

m1(t) + m2(t)x + m3(t)y 2

/x , / x 2 / x , / x ci(t)y a2(t)

< ai(t)x - bi(t)x - ai(t) + bi(t)x + ——-—-— + -—

m1 (t) + m2 (t)x + m3 (t)y 2

- a2(t)y - b2(t)y2 + C2(t}fxy-^ + a2(t) + b2(t)y + ^

m1 (t) + m2 (t)x + m3 (t)y 2

cu (au)2 cu (p u)2

< afc - blx2 - a1 + bux + -V + (—L - a2y - b'2y2 + -4y + au2 + buy +

m3 2 m2 2

< (au + bu)x - blx2 - a1 + +

cu (au)2 mi 2

+ ( bu + - a2 )y - b2y2+ au +

2 ml2 2 2 2

where K is a positive number. Substituting this inequality into Eq. (5), we see that

dV(x(t),y(t)) < Kdt+ (x - l)a(t) dBi(t) + (y - 1)p(t) dB2(t), which implies that

pTn AT pxnA..T pxnA..T

/ dV(x(t),y(t)) </ Kdt+ a(s)(x(s)-l) dBi(s) Jo Jo Jo

/ p (s)(y(s)-1) dB2(s),

where Tn A T = min(rn, T}. Taking the expectations of the above inequality leads to

EV(x(tn A T),y(tn A T)) < V(xo,yo) + KE(%n A T) < V(xo,yo) + KT. (6)

Let Qn = |rn < T} for n > n1, then by (4) we have P(^n) > e. Note that for every w e there is at least one of x(xn, w) and y(rn, w) equaling either n or n; therefore, V(x(vn, w),y(rn, w)) is no less than min{(n - 1 - logn), (n - 1 - log n))}. It then follows from (6) that

V(xo,yo) + KT > E[1Qn(w)V(x(tn),y(rn))]

> e min| (n -1 - logn),[ — - 1 - log — ) [,

where 1sn(®) is the indicator function of Qn, letting n — to, we have that

to > V(x0,y0) + KT = to

is a contradiction, then we must have tto = to. Therefore, the solution of system (3) will not explode at a finite time with probability 1. This completes the proof. □

Definition 2 The solution X(t) = (x(t),y(t)) of system (3) is said to be stochastically ultimately bounded if for any e e (0,1), there is a positive constant x (x (e)) such that for any initial value (x0,y0) e R+, the solution of (3) has the property that

limsupP{ |X(t)| > x} < e.

t—>to

Lemma 1 Let (x(t), y(t)) be a solution of system (3) with initial value (x0, y0) e R+, for all p >1, there exist K1(p) and K2(p) such that

E[xp(t)] < K1(p), E[yp(t)] < K2(p), t e [0, +to),

- au+1 n(au)2 - m+2p(p ) _

where K1(p) = max{/<1(p), (au+2T )p + e}, Kz(p) = max{«2(p), (^-)p + e}, K(p) (i =

b1 b2 1,2) and e are both positive constants.

Proof Define V1(x) = xp for x e R+, where p >1. Applying Ito's formula leads to 1

dV1(x) = pxp 1 dx +-p(p - Vfxf 2(dx)2

a1(t) - b1(t)x -

C1(t)y

m1 (t) + m2 (t)x + m3 (t)y 2

+ -(p - 1)a2(t)

+ pxpa(t) dB1(t)

+ 1p(au)2 - b1x

dt + pxa(t) dB1(t).

Similarly, we have

dyp = pyp

-a2(t)-b2(t)y +

c2(t)x

m1(t) + m2(t)x + m3(t)y 2

+ 1(p -1)P 2(t)

+ py^P(t) dB2(t)

' ru 1

^^ + 2p(P f- b2y ml 2 2

dt + pyp P (t) dB2(t).

Taking expectation, we have

d£[xp(t)] dt

■p(au)2 E[xp(t)] - b1E[xp+1(t)]J

[xp(t)]p

< + 2p(a

E[xp(t^- b1 [E[xp(t)]]1+ ^^ - b1[E[xp(t)]]p

dE[yp(t)] dt

4 1 2 + ■

l ' 2 cu 1

p(pu)2 E[yp(t)] - b2E[yP+1(t)]j

p(pu)2 E[/(t)]- b2 [E[f (t)]]1+ p

: E[yP(t)]p{

m+2p(pu)2

- b2[E[/(t)]]p .

Therefore, by the comparison theorem, we get X + 2 p(au)2\ p

limsupE[xp(t)] < ( -—2Jp" ' ) , limsupE[/(t)] <

mc^r+2p(p u)\p

Thus, for a given constant e > o, there exists a T > o such that for all t > T,

i'u + 2p(au)2\ p

E[xp(t)] <( a1 + 2pr' ) + e, E[f{t)} <' m2

■^ + 2p(p u)\p

Together with the continuity of E[xp(t)] and E[yp(t)], there exist K1(p) > o, K2(p) > o such that E[xp(t)] < I<1(p), E[yp(t)] < K2(p) for t < T. Let

K1(p) = max] K1(p),

u + 2 p(au)2\ p

% + 2 p(p u)\p

K2(p)=ma^H2(p)W '2 then for all t e R+,

E[xp(t)] < K1(p), E[yp(t)] < K2(p).

Theorem 2 The solutions of system (3) with initial value (xo, yo) e R+ are stochastically ultimately bounded.

Proof If X = (x,y) e R2, its norm here is denoted by |X| = (x2 + y2)2, then |X(t)|p < 22 (|x(t)|p + |y(t)|p)

by Lemma 1, E|X(t)|p < K(p), t e (o, K(p) is dependent of (xo,yo) and defined by

K(p) = 22 (K1(p) + K2(p)). By virtue of Chebyshev's inequality, the above result is straightforward. □

3 Persistence in the mean and extinction

Lemma 2 The solutions of system (3) with initial value (xo, yo) e R+ have the following properties:

ln x(t) lny(t) limsup-< o, limsup-< o, a.s.

t—t t—t

Proof It follows from system (3) that

dx < x(a1(t) - b1(t)x) dt + a(t)xdB1(t),

c1(t) 1 1 (7)

dy < y(- b2(t)y) dt + p(t)ydB2(t).

dx = x(a1(t) - b1(t)^) dt + a(t)xdB1(t),

d^ = y(^ - fc(t)y) dt + p(t)ydB2(t),

where (x(t),y(t)) is a solution of system (8) with initial value x0 > 0 and y0 > 0. By the comparison theorem for stochastic differential equations, it is easy to have

x(t) < x(t), y(t) < y(t), a.s. t e [0, +to). (9)

By Lemma 3.4 in [20], it is easy to get the following result:

dx(t) = x(t)[(b(t) - au(t)x(t)) dt + a(t) dB(t)]

here b(t), a11(t) and a(t) are all nonnegative functions defined on R+. If a11 > 0, then limsupt—TO ^ < 1, a.s. Note that bi > 0 (i = 1,2), then it follows from Eqs. (8) and (9),

ln x(t) ln x(t) ^ lny(t) ^ lny(t) ^ limsup-< limsup-< 1, limsup-< limsup-< 1, a.s.

t—TO ln t t—TO ln t t—TO ln t t—TO ln t

In addition,

ln x(t) ln x(t) ln t ln t

limsup-= limsup-limsup — < limsup — = 0;

t—TO t t—TO ln t t—TO t t—TO t

therefore, it leads to limsupt—TO < 0, a.s. Similarly, we can have limsupt—TO < 0, a.s. □

Lemma 3 [25] Suppose thatx(t) e x R+,R+], where R+ := {a|a > 0,a e R}. (I) If there are positive constants X0, T and X > 0 such that

lnx(t) < Xt - X0 \ x(s) ds + V" PiBi(t) J° i=1

for t > T, where pi is a constant, 1 < i < n, then (x>* < X/X0, a.s. (i.e., almost surely). (II) If there are positive constants X0, T and X > 0 such that

ln x(t) > Xt - X0 \ x(s) ds PiBi(t) J° i=1

for t > T, where pi is a constant, 1 < i < n, then (x>* > X/X0, a.s.

In the following, we give the result about weak persistence in the mean and extinction of the prey and predator population. Applying Ito's formula to Eq. (3) leads to

d ln x = (Ai(i) - - bi(t)x--—-C1(t)y-—) dt + a(t) dBl(t) (10)

\ - m1(t) + m-(t)x + m3(t)y /

d Iny = (-a-(t)-b-(t)y + mi{t) + m2^x + m3(t)^dt + P(t) dB-(t). (11)

Let ri(t) = Ai(t) - r-(t) = -a-(t) - then <r->* < 0. For the prey population x(t) of system (3), we have

Theorem 3 (i) If <r1>* < 0, then the prey population x(t) will go to extinction a.s.

(ii) If <r1>* = 0, then the prey population x(t) will be non-persistent in the mean a.s.

(iii) If <r1>* > 0, then the prey population x(t) will be weakly persistent in the mean a.s.

(iv) If <r1>* - < m^t >* > 0, then the prey population x(t) will be strongly persistent in the mean a.s.

Proof (i) It follows from (10) that

C1(s)y(s)

ln x(t)-ln x0 = I J0

r1(s) - b1(s)x(s) -

m1(s) + m-(s)x(s) + m3(s)y(s) _ + i a(s) dB1(s), (1-)

ln x(t)-ln x0 < /0 r1(s) ds + /0 a(s) dB1(s), set M1(t) = f0 a(s) dB1(s), it is a martingale whose quadratic variation is <M1, M1>t = /^i numbers for martingale yields

quadratic variation is <M1, M1>t = /0 a-(s) ds < (au)-t. Making use of the strong law of large

r M1(t)

lim -= 0, a.s. (13)

1 P ^V^

- r1(s) ds + —(14)

ln x(t) - ln x0 1 f M1(t)

Taking superior limit on both sides of inequality (14)leadsto lim supt—^nr^ < <r1>* <0, we can see that limt—x(t) = 0. (ii) By Eq. (1-), we have

lnx(t)-lnx0 <<n> - b1(x(t)) + ^ (15)

It follows from the property of superior limit and (13) that for arbitrary s >0, there exists

-and ^ < -

T >0 such that <r1> < <r1>* + - and < - for all t > T. Substituting these inequalities

into (15) yields

lnx(t)-lnx0 < ^<r1>* + 01 - b[f x(s) ds + St < (<r1>* + s)t - b[f x(s) ds,

when <ri>* = 0, then

x(t) , .

ln-< st - bx0 / -as.

xo ' --

fx® ds.

By bl > 0 and Lemma 3, we have <x(t)>* < 4-, by virtue of the arbitrariness of s, <x(t)>* < 0.

Since the solution of system (3) is nonnegative, it is easy to have <x(t)>* = 0, that is to say, the prey population x(t) is non-persistent in the mean a.s.

(iii) We only need to show that there exists a constant ¡xl>0 such that for any solution (x(t),y(t)) of system (3) with initial value (x0,y0) e R+, <x(t)>* > ¡xl>0 a.s. Otherwise, for arbitrary sl > 0, there exists a solution (x(t),y(t)) with positive initial value x0 > 0 andy0 > 0 such that P{<x(t)>* < sl} > 0.

Let sl be sufficiently small so that

<rl>* - bUsl>0, <r2>* + -2Sl<0. (l6)

It follows from Eq. (ll) that

<<2> - b2y (t)) + 4 (x (t)) + ^ (lT)

t 2 ml t

here M2(t)= ft p(s) dB2(s), it also has

M2(t) , ,

lim —2(-) = 0. (l8)

t^TO t

By virtue of (lT), it leads to [--1 ln ^(t)]* < <T2>* + -2T si <0, thus

lim y(t) = 0. (19)

On the other hand, it follows from Eq. (12) that

ln x(t)-ln x0 cU ^ —l(t)

—-0 > <ri> - bU{x(t)) - -lJ{y(t)) + —(A

t l ml t

Taking the superior limit to the above inequality and making use of (13), (16) and (19), we have [t-1 lnx(t)]* > <rl>* - bUsl > 0, that is to say, we have shown P{[t-1 lnx(t)]* >0} > 0, this is a contradiction to Lemma 2. Therefore, <x(t)>* > 0, the prey population x(t) will be weakly persistent in the mean a.s.

(iv) By Eq. (10), we get

dlnx > ( ri(t) - bi(t)x - ^ dt + a(t) dBi(t). \ m3(t)J

It is easy to have

lnx(t) - lnx0 . / ci(t) \ ( ^ —i(t)

—t— > <ri>- mm)-b^x(t)+"1T.

If (rib - (md>* > there exists sufficiently small e >0 such that (n>* - <md>* - e > 0. It follows from the property of superior limit, interior limit and (13) that for positive number e, there exists a T >0 such that

e / Cl(i) \ / dit) \* e M1(t) e

<r1> ><r1>* toôH m3(t)/+з, mt>-3

for all t > T. Then

lnx(t) — lnx0 e I c1(t) \* e e

t 3 \m3(t)l 3 11 ' 3

- (<1>- ^ m§))' - e) - *?('"».

By virtue of Lemma 3 and the arbitrariness of e, we have (x(t))* - <ri>* ^>* >0.

In other words, the prey population x(t) is strongly persistent in the mean a.s. □

Remark 1 The results of Theorem 3 illustrate that <r1>* is the threshold between weak persistence in the mean and extinction. Note r1(t) = a1(t) - if > a1(t), then the prey population will be extinct, no matter whether there are predators. However, the prey population will survive when not considering environmental noise. This indicates that when the density of environmental noise is larger than the intrinsic growth rate of prey, it will cause the extinction of prey population. Therefore, it is more suitable to take into account stochastic perturbation in the systems. Here we can also find that the condition (iv) implies condition (iii), that is to say, the prey population must be weakly persistent in the mean when it is strongly persistent in the mean.

For the predator population, we have the following result.

Theorem 4 (i) If (b^*^* + (m^)*^ >* < 0, then the predator population y(t) will go to extinction a.s.

(ii) If (b1)*(r2 >* + (m2f|)*(r1)* = °, then the predator population y(t) will be non-persistent in the mean a.s.

(iii) If (r2>* + ( mi(t)+m22t)X(t)+m3(t)y(t) > * > 0 then the predator population y(t) wiU be weakly persistent in the mean a.s. where (X(t),y(t)) is the solution of Eq. (8) with initial value (x°,y°) e R+.

(iv) If (r2 >* + (>* > °, then the predator population y(t) has a superior bound in time

<r2)* + < >* \ * ^ v vm2(t) '

average, that is, (y(t)>* <--m

Proof (i) If (ri>* < 0, then it follows from Theorem 3 that (x(t)>* = 0. By Eq. (11),

lny(t) - lny° , cu , . M2(t)

---< (r2> + —r{x(t)) + ;

t m1 t

therefore, [t-1 lny(t)]* < (r2>* < 0, then lim^TOy(t) = 0.

Now, if <ri>* > 0, it follows from the property of superior limit, interior limit and (13) that for sufficiently small e, there exists a T >0 such that

lnx(t) —lnx0 w Mi(t) -1-- <ri> - (bi)*(x(t)) + ——

- <ri>* + --(bi),{x(t))+-

for all t > T. Applying Lemma 3 and the arbitrariness of e yield

(x(t)f - Ix. (0)

Substituting the above inequality into (ii) gives

[t-1 lny(t)X - <r2>* + |

I c2(t)

-<*>• ml) (x<«

C2(tn *<ri>*

- <r2> + <-mm) (bi, (2i)

r i (bi)*<r2>*+(m§)*<ri>* ln ------^—<a

which means limt^TOy(t) = 0 a.s.

(ii) In the case (i), we have shown that if <ri>* - 0, then limt^TOy(t) = 0, therefore, <y(t)>* = 0. Now, we will prove that <y(t)>* = 0 is still valid when <ri>* > 0. Otherwise, if <y(t)>* > 0, then it follows from Lemma 2 that [t—i lny(t)]* = 0. Making use of (2i), one can see that

0 = ln y(t)]*-<r2>* + ( m§) *(x(t)\*. (2)

On the other hand, for arbitrary e >0, there exists a T >0 such that

, . , .* e / C2(t) \ /C2(t)\ *. \* e M2(t) e

<r2> < <r2> + 3, [mmx(t)j < 1 -mm)xt) +3, m- < 3

for all t > T. Substituting these inequalities into (ii) yields lny(t) — lny0 j , / C2(t) \ ( \ M2(t)

—t—- <r2>+\-mm I—{b2(t)y(t)]

-<2>*+3+(ml )*(x(t)\*+3—(b2)*(y(t)\+3

- <2 > * + e + ( m§j) (x(t)Y-(bMy(t)).

<r2)*+e+( mi|)*<*(t)}*

Then an application of Lemma 3 and (22) results in <y(t)>* ---. Byvirtue

of (20) and the arbitrariness of e,

w - {r,nbt ;;rfl*<ri>*=0, (bi)*(b2)*

which is a contradiction to our assumption, therefore, <y( t)>* = 0 a.s.

(iii) In the following, we need to show that <y( t)>* > 0 a.s. Otherwise, for arbitrary e2 > 0, there exists a solution (x(t),y(t)) of system (3) with positive initial value (xo,y0) e R+ such that P{<y(t)>* < e2} > 0. Let e2 be sufficiently small so that

* / c2(t)x(t) \* / 2c1c'u \

<r2> +\mi(t) + m2(t)x(t) + m3(t)y(t)l >{b + b^)e2 (23)

It follows from (ii) that

ln y(t)—ln y0 = <r2> + / ^ „ \ — (b2(t)y(t)\ + M2(t)

mi(t) + m2(t)x (t) + m3(t)y (t) / t

c2 (t)x(t) c2 (t)x(t)

mi(t) + m2(t)x (t) + m3(t)y (t) mi(t) + m2(t)x (t) + m3(t)y (t) /

Here (x(t),y/(t)) is the solution of model (8) with initial value (x0,y0) e R+ and x(t) - x(t), y(t) - y(t), a.s. for t e [0, +ro). Because of

c2(t)x (t) c2(t)x (t)

mi(t) + m2(t)x (t) + m3(t)y(t) mi(t) + m2(t)ee (t) + m3(t)y(t)

_ c2(t)m3(t)x (y — y) — c2(t)mi(t)(^ — x) — c2(t)m3(t)y (x — x)

(mi(t) + m2(t)x (t) + m3(t)y (t))(mi(t) + m2(t)x (t) + m3(t)y(t))

>_c2(t)m3(t)x (y — y)_

" (mi(t) + m2(t)x(t) + m3(t)y)(t))(mi(t) + m2(t)x(t) + m3(t)y(t))

— c2(t)mi(t) (x. — ee)— c2(t)m3(t)y x — x)

m2(t) mi(t)m3(t)y

2c2(t)

(x — x)

ln y(t) — ln y0 > <2> + ) — (b2(t)y(t)) + M^

t \mi(t) + m2(t)xx (t) + m3(t)y(t)

—(mi <x <«—x <D

^ . . /_c2(t)x(t)_\ (, M2(t)

> <r2> + (mi(t) + m2(\)x(i) + m3(t)y(t)l— ^2(t)y(t^ + "7"

(x (t) — x (t))\. (24)

Consider the Lyapunov function V2(t) = | lnx(t) - lnxx(t)|, then V2(t) is a positive function on R+, by Ito's formula, (8) and (10), we have

d+V2(t) =

-b1(t)(x (t)-xx (t)) ■ C1(t)^(t)

—1f^(t) - b1(x(t)-x(t))

m1(t) + m2(t)x (t) + m3(t)y(t) _ dt.

Integrating from 0 to t and dividing by t on both sides of the inequality yield

^MJM < I m - b1(x W-x (t)).

Owing to > 0, it leads to

bKx(t)-x(t)) (t)

—u V2(0)

mi " '' t

here V2(0) = 0, then (x(t) - xx(t)> <—hj (y(t)>. Substituting the above inequality into (24),

we have

lny(t) - lny0 > ( > +1 C2(t)x(t)

t \m1(t) + m2(t)xx (t) + m3(t)y(t)

- <«* «>+H2 ^

Taking superior limit of the above inequality, we get

^W > (r2>" + (mMyJ ~(bu + bCfe |e2>0,

which contradicts Lemma 2, then (y(t)>* > 0 a.s., that is to say, the predator population y(t) is weakly persistent in the mean a.s. (iv) It follows from Eq. (11) that

dlny < ^(t) - b2(t)y + dt + p(t) dB2(t),

that is to say,

lny(t) - lny0 ( C2(t) \ I( , M2(t)

—t— < (r2>+( mt)l- b2m+~iT'

The following proof is similar to the proof of (iv) in Theorem 3, here we omit it. □

Remark 2 From the proof of Theorem 4, we can observe that if (r1>* < 0, then (b1)*(r2>" + (tn(t))* (r1>* < 0 is straightforward. It shows that if the prey population goes to extinction, the predator population will also go to extinction, which is consistent with the reality. In the other case, (r1>* > 0, (b1)*(r2>" + (mi|)))"(r1>" < 0, which means the prey population

will survive, but the predator population will go to extinction. Notice r2(t) = -a2(t)- p2(t), this phenomenon may be caused by the large death rate of predator a2(t) or the noise density p 2(t).

4 Global attractiveness of the solution

In this section, we give sufficient conditions of global attractiveness.

Definition 3 System (3) is said to be globally attractive if lim 1x1 (t) - x2(t) I = lim Mt) -y2(t) I = 0 a.s.,

t^-ro1 1 t^TO1 1

where (x1(t),y1(t)) and (x2(t),y2(t)) are two arbitrary positive solutions of system (3) with initial values (xw,yw) e R+ and (x20,y20) e R+.

To give the result of global attractiveness, we show some lemmas first.

Lemma 4 [26] Suppose that an n-dimensional stochastic process X(t) on t > 0 satisfies the condition

E\X(t)-X(s)\a1 < c\t -sj1+a2, 0 < s, t < to

for some positive constants a1, a2 and c. Then there exists a continuous modification X (t) of X(t) which has the property thatfor every û e (0, a2/a1), there is a positive random variable h(oji) such that

P\ o :

0<|t-s|<h(o),0<s,t<œ

\X(t,o)-X(t,o)\ < 2 , _1

\t - s\û

In other words, almost every sample path ofX (t) is locally but uniformly Holder continuous with an exponent $.

Lemma 5 Let (x(t), y(t)) be a solution of system (3) ont > 0 with initial value (x0, y0) e R+, then almost every sample path of (x(t), y(t)) is uniformly continuous.

Proof Equation (3) is equivalent to the following stochastic integral equation:

c1(s)y(s)

x(t) _ x0 + / x(s)

a1(s) - b1(s)x(s) -

m1(s) + m2(s)x(s) + m-i(s)y(s)

W a(s)x(s) dB1(s).

Letf (s) _ x(s)[a1(s) - b1(s)x(s) - m1(s)+m2l()ï(ï)+m3(s)y(s)g(s) _ a(s)x(s), notice that

Ef (t)\p _ E

x( a1(t) - b1(t)x -

c1(t)y

a1(t) - b1(t)x -

m1(t) + m2(t)x + m3(t)y c1(t)y

m1(t) + m2(t)x + m3(t)y

i 9„ i

- - E\x\2p + - E

i 2„ i

- - E\x\2p + - E

- 2 1 1 2

ai(t) — bi(t)x —

ci(t)y

aU + bUx + —-j-y

mi(t) + m2(t)x + m3(t)y

- iE\X\2p + -32P—1

- 2 \ \ 2

2p + (w )2pFi-v\2p+ fi

(at)2p + (%) E\x\

- 2^i(2P)+ 2

[at)2p + (bt)2pKi(2p) ■

=: F (p),

E|g(t)|p = E|«(t)x(t)|p - (a")pE\x\p - (a")pKi(p) =: G(p).

Moreover, in view of the moment inequality for stochastic integrals in [24, p.39], one can obtain that for 0 - ti - t2 and p >2,

/2 g(s) dBi

p(p — i)

p(p — i)

(t2 —

E|g(s)|pds

(t2— ti)2 G(p).

Then for 0 < ti < t2 < to, t2 — - i, l + l = i, we have

E|x(t2)— x(ti)|p

2p—iE

fV (s) ds + it2 g(s) dBi(

f 02 p f 02

/ /(s) ds + 2p—iE / g(s) dB ti | | ti

- 2p—i i it2 iqd^ "e^ /(s)|p d^ + 2p—i

p(p — i)

(t2— ti)2 G(p)

- 2p—i(t2 — ti) *F(p)(t2 — ti) + 2p

p(p — i)

(t2— ti)2 G(p)

- 2p—i(t2 — ti)pF(p) + 2p—

p(p — i)

- 2p—i(t2 — ti)2 (t2 — ti)2 +

p(p — i)

(t2— ti)2 G(p) M(p)

- 2p—i(t2 — ti)2 i +

p(p — i)

where M(p) = F(p) + G(p), it follows from Lemma 4 that almost every sample path of x(t) is locally but uniformly Holder continuous with an exponent & for & e (0, p—2) and therefore almost every sample path of x(t) is uniformly continuous on t e R+. In the same way, we can demonstrate that almost every sample path of y(t) is uniformly continuous. □

Lemma 6 [27] Letf be a nonnegativefunction defined on R+ such thatf is integrable and is uniformly continuous, then limt^TOf (t) = 0.

Theorem 5 System (3) is globally attractive if

bi - __

Proof For (x1(t),y1(t)) and (x2(t),y2(t)), any two positive solutions ofsystem (3) with initial values (x10,y10) e R+ and (x20,y20) e R+ define

V(t) _ \lnX1(t)-lnX2(t)\ + \lny1(t)-lny2(t)\,

then V(t) is a continuous positive function on t > 0. A direct calculation of the right differential d+V(t) of V(t), by (10) and (11) and Itô's formula,

d+V(t) = sgn(xl - x2)

+ sgn(yi- y2)

dxl (dxl)2 xl

dx2 (dx2)21 X2

dyl (dyl

dy2 - (dy2)2 L y2 2y2 J

= \ -bi(t) sgn(xi - X2)(Xi - X2) - b2(t) sgn(yi - y2)(yi - y2)

Ci(t)mi(t)(yi - y2) + Ci(t)m2(t)[x2(yi - y2) -y2(xi - x2)]

- sgn(xi- x2)

(mi(t) + m2(t)xi + m3(t)yi)(mi(t) + m2(t)x2 + m3(t)y2)

+ sgn(y - ) C2(t)mi(t)(xi -x2)-C2(t)m3(t)[x2(yi -y2) -y2(xi -X2)] j dt (mi(t) + m2(t)xi + m3(t)yi)(mi(t) + m2(t)x2 + m3(t)y2) J

< |-bi(t)|xi- x2| - b2(t)|yi- y2|

+ Ci(t)mi(t)|yi -y2| + Ci(t)m2(t)x2|yi -y2| + Cl(t)m2(t)y2|Xl -X2| (mi(t) + m2(t)xi + m3(t)yi)(mi(t) + m2(t)x2 + m3(t)y2) C2(t)mi(t)|Xi - X2 | - C2(t)m3(t)X2|yi - y2 | + C2(t)m3(t)y2|Xi - X2 | (mi(t) + m2(t)xi + m3(t)yi)(mi(t) + m2(t)x2 + m3(t)y2)

< { -bi (t) |Xi - X2 | - b2 (t) |yi - y2 | + —^TT |yi - y2 | + -Cl(7) |yi - y2 |

Ci(t)m2(t) C2(t) C2 (t)

+ ——-TT |xl - x2| + —— |xl - x2| + —— |xl - x21 j dt

mi(t)m3(t)

. i > /a 2C2(t) Ci(t)m2(t)\ ( 2Ci(t)\ 1

<H bi(t) - mw- mmiwmw j|Xi - X2| Hb2(t)|yi - y2||dt

< M bi-2Cu - «

|Xl- X2| -( b2-2Cl )|yi-y2|[ dt.

mi mim3 ml

Integrating both sides leads to

m < v(0) - /7b[ - ^ - 44) M^s) + (b2 - |*(s) -y2(5)| ds.

Jo\ ml m[m3/ \ ml/'

Consequently,

i W 1cu cumu\ i 2cu\

V(t) + U - -f - -m |«i(s)-«2(s)I + bl2 - -i |ji(5)-y2(s) I ds < V(0) < TO.

Jo \ mi m\m3) V mf /

It then follows from V(t) > 0 and (25) that

|*i(i)-*-(i)| eLf[0, +to), |yi(t)-y-(t)| e^[0,+to). Then from Lemmas 5 and 6, we get the desired assertion. □

5 Numerical simulation

In this section, to support the main results in our paper, some simulation figures are introduced. Here we use the Milstein method mentioned in Higham [28] to simulate stochastic equations, considering the following discrete equations:

xk+i = xk + xk

ai(kAt)-bi(kAt)xk - ci(kAt)yk

mi(kAt) + m2(kAt)xk + m3(kAt)yk_ (kAt)xkVAt^k + a (A)xk(Hk -1)At,

yk+i = yk + yk

-a-(kAt) - b-(kAt)yk +

c-(kAt)xk

mi(kAt) + m2(kAt)xk + m3(kAt)yk _

■ /}(kAt)yk4Atnk + P (2At)yk- i) At,

Hk and nk (k = i,2,...,n) are the Gaussian random variables W(0,i). Let ai(t) = 0.4 + O.Oi sin t, bi(t) = 0.7 + 0.0i sin t, ci(t) = 0.3 + 0.02 sin t, c2(t) = 0.i + 0.02 sin t, «2(t) = 0.2 + 0.05 sin t, b2(t) = 0.2 + 0.0i sin t, mi(t) = 0.2 + 0.04sin t, m2(t) = 0.5 + 0.05 sin t, m3(t) = 0.3 + 0.05 sin t with initial value (x0,y0) = (0.5,0.6). In Figure i, we choose = 0.5 + 0.02 sin t, ^n = 0.3 + 0.02 sin t, then it is easy to have <ri>* = -0.i < 0. In view of Theorems 3 and 4, both prey population x and predator population y go to extinction. Figure i confirms this. In Figure 2, we choose ^ = 0.2 + 0.02 sin t, = 0.4 + 0.02 sin t, the other parameters are the same. At this time, we get <ri>* = 0.2 > 0 and (bi)^<r2>* + (m|)))*<ri>* = -0.3i4 < 0. By virtue of Theorems 3 and 4, the prey population x is weakly persistent in the mean and the predator population y goes to extinction, which is shown in Figure 2.

In Figure 3, we choose ci(t) = 0.i + 0.02 sin t, c2(t) = 0.2 + 0.02 sin t, mi = 0.i + 0.04 sin t, ^ = 0.02 + 0.02 sin t, ^ = 0.02 + 0.02 sin t, the other parameters are the same; then

<ri>* = 038 > 0 to <r2> + <mM+m%MtL3Vm>* > 0 « ^^ to verify at present Rg-ure 3 illustrates that the situation that both prey population x and predator population y are weakly persistent in the mean exists.

6 Conclusions

Owing to theoretical and practical importance, the predator-prey system with Bed-dington-DeAngelis functional response has received great attention and has been studied extensively, but for stochastic non-autonomous case, there is none. Here, we consider a stochastic non-autonomous predator-prey system with Beddington-DeAngelis functional

" X(t) - Y(t)

Figure 1 Both populations go to extinction. Solutions of system (3) for a2(t) = 1 +0.04 sin t,

p2(t) = 0.6 + 0.04 sin t, with step size At = 0.05 > 0. Prey and predator populations are represented by red line

and blue line with a star, respectively.

response. Firstly, we show that the solution of system (3) is globally positive and stochastically ultimately bounded. Sufficient conditions for extinction, non-persistence in the mean, weak persistence in the mean and strong persistence in the mean are obtained. The threshold between weak persistence and extinction for prey population is established. We also show that the solution of system (3) is globally attractive under some sufficient conditions. These results are useful to estimate the risk of extinction of species in the system. Besides, global attractiveness means that all the species in the community can coexist.

Figure 3 Both populations are weakly persistent in the mean. Solutions of system (3) for

a2(t) = 0.04 + 0.04 sin t, f>2(t) = 0.04 + 0.04 sin t, with step size At = 0.1 > 0. Prey and predator populations are

represented by red line and blue dotted line, respectively.

There are still some interesting questions deserved to be study. For example, here the condition of the weak persistence in the mean for predator is only a sufficient condition, which is not so ideal. Maybe we can give the threshold between weak persistence in the mean and extinction for predator in the future. Moreover, we can consider colored noise in the models owing to sudden environmental changes caused by seasons or other reasons.

Competing interests

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

SL proposed the modeland completed the main part of the manuscript. XZ checked allthe results and polished the language. Allauthors read and approved the finalmanuscript.

Acknowledgements

This work was supported by NationalScience Foundation of China (Nos. 11071275 and 11228104) and by the Special Fund for Basic Scientific Research of CentralColleges (CCNU10B01005).

Received: 3 August 2012 Accepted: 3 January 2013 Published: 22 January 2013

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doi:10.1186/1687-1847-2013-19

Cite this article as: Li and Zhang: Dynamics of a stochastic non-autonomous predator-prey system with Beddington-DeAngelis functional response. Advances in Difference Equations 2013 2013:19.

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