Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 801812,14 pages doi:10.1155/2012/801812

Research Article

The Asymptotic Behavior of a Stochastic Predator-Prey System with Holling II Functional Response

Zhenwen Liu,1,2 Ningzhong Shi,1 Daqing Jiang,1 and Chunyan Ji1,3

1 School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

2 School of Science, Changchun University of Science and Technology, Changchun, Jilin 130022, China

3 Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, China

Correspondence should be addressed to Daqing Jiang, daqingjiang2010@hotmail.com Received 21 October 2012; Accepted 14 December 2012 Academic Editor: Ivanka Stamova

Copyright © 2012 Zhenwen Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We discuss a stochastic predator-prey system with Holling II functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we deduce the conditions that there is a stationary distribution of the system, which implies that the system is permanent. At last, we give the conditions for the system that is going to be extinct.

1. Introduction

One of the most popular predator-prey model is the one with Michaelis-Menten type (or Holling Type II) functional response [1, 2]:

where x(t) and y(t) are the population densities of prey and predator at time t, respectively. The constants a, b/a, a, ¡5, e, and k are positive constants that stand for prey intrinsic growth rate, carrying capacity, the maximum ingestion rate, half-saturation constant, predator death rate, and the conversion factor, respectively. This model exhibits the well-known but highly

controversial "paradox of enrichment" observed by Hairston et al. [3] and by Rosenzweig [4] which is rarely reported in nature. It is very important to study the existence and asymptotical stability of equilibria and limit cycle for autonomous predator-prey systems with Holling II functional response. If kaap > aep2 + kba + bep, then system (1.1) has a unique limit cycle which is stable. If aka > aep + be, then system (1.1) has a unique positive equilibrium:

e „ ka(aka - aep - be)

X " ka - ep' y ~ Çka - epf ' (L2)

which is a stable node or focus (see [5]).

However, countless organisms live in seasonally or diurnally forced environments. Hence, authors considered models with periodic ecological parameters or perturbations. For example, Liu and Chen [6] introduced periodic constant impulsive immigration of predator into system (1.1) and gave conditions for the system to be extinct and permanence, respectively. Zhang and Chen [7] studied a Holling II functional response food chain model with impulsive perturbations. Zhang et al. [8] further considered system (1.1) with periodic constant impulsive immigration of predator and periodic variation in the intrinsic growth rate of the prey.

On the other hand, the white noise is always present, and we cannot omit the influence of the white noise to the system. May [9] pointed out that due to continuous fluctuation in the environment, the birth rates, death rates, carrying capacity, competition coefficients, and all other parameters involved with the model exhibit random fluctuation to a great lesser extent, and as a result the equilibrium population distribution never attains a steady value, but fluctuates randomly around some average value. Many authors studied the effect of the stochastic perturbation to the predator-prey system with different functional responses, such as [10-14]. Therefore, in this paper, we also introduce stochastic perturbation system (1.1) and obtain the following stochastic system:

dx(t) = x(t)( a - bx(t) - ay}t] N ^ dt + oix(t)dBi(t),

1 + ßx(t) /

dy(t) = y(t) (-e + kfft)) dt + a2y(t)dB1(t),

where B1(t) and B2(t) are mutually independent Brownian motion with Bi(0) = B2(0) = 0, and of, °2 are intensities of the white noise.

The aim of this paper is to discuss the long time behavior of system (1.3). As the deterministic population models, we are also interested in the permanence and extinction of the system. The global stability of the positive equilibrium means that the system is permanence. But, for the stochastic system, there is no positive equilibrium. Hence, it is impossible that the solution of system (1.3) will tend to a fixed point. In this paper, we show that there is a stationary distribution of system (1.3) mainly according to the theory of Has'meminskii [15], if the white noise is small. While if the white noise is large, based on the techniques developed in [16,17], we prove that the predator population will die out a.s. and the prey population will either extinct or its distribution converges to a probability measure. It does not happen that both the prey population and the predator population in system (1.3) will die out, which is brought by large white noise, such as weather, epidemic

disease. From this point, we say the stochastic model is more realistic than the deterministic model.

The rest of this paper is organized as follows. In Section 2, we show that there is a unique nonnegative solution of system (1.3). In Section 3, we show that there is a stationary distribution under small white noise. While in Section 4, we consider the situation when the white noise is large. We prove that the system will be extinct. Finally, we give an appendix containing the stationary distribution theory used in Section 3.

2. Existence and Uniqueness of the Nonnegative Solution

To investigate the dynamical behavior, the first concern is the global existence of the solutions. Hence in this section we show that the solution of system (1.3) is global and nonnegative. It is not difficult to check the uniqueness and global existence of solutions if the coefficients of the equation satisfy the linear growth condition and local Lipschitz condition (cf. [18]). However, the coefficients of system (1.3) do not satisfy the linear growth condition, but locally Lipschitz continuous, so the solution of system (1.3) may explode at a finite time. In this section, by changing variables, we first show that system (1.3) has a local solution, then show that this solution is global.

Theorem 2.1. For any initial value (x(0),y(0)) e R+, there is a unique solution (x(t),y(t)) of system (1.3) on t > 0, and the solution will remain in R with probability 1.

Proof. First, consider the following system, by changing variables, x(t) = eu(t), y(t) = ev(t),

It is clear that the coefficients of system (2.1) are locally Lipschitz continuous for the given initial value (logx(0), logy(0)) e R2 there is a unique local solution (u(t),v(t)) on t e [0,re), where Te is the explosion time (see [18]). Hence, by Ito formula, we know (eu(t),ev(t)), t e [0,Te) is a unique positive local solution of system (1.3). To show that this solution is global, we need to show that Te = œ a.s. Let m0 > 1 be sufficiently large so that x(0),y(0) all lie within the interval [1/m0,m0]. For each integer m > m0, define the stopping time:

Where, throughout this paper, we set inf 0 = to (as usual 0 denotes the empty set). Clearly, Tm is increasing as m ^ to. Set tto = limm^TOTm, whence tto < Te a.s. If we can show that tto = to a.s., then Te = to and (x(t),y(t)) e R+ a.s. for all t > 0. In other words, to complete the proof all we need to show is that tto = to a.s. If this statement is false, then there is a pair of constants T > 0 and e e (0,1) such that

P< T} > e.

Hence there is an integer m1 > m0 such that

P{Tm < T}> e Vm > mi. (2.4)

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

V(x,y) = (x - c - c log 0 + 1 (y - 1 - logy), (2.5)

where c is a positive constant to be determined later. The nonnegativity of this function can be seen from u - 1 - log u > 0, for all u> 0. Using Ito's formula, we get

dV := LVdt + 01 (x - c)dB1(t) + 02 (y - 1)dB2(t), (2.6)

ay \ co^ 1 ( / kay \ O

LV = (x - c)(a - bx - TTjx) + T2 +1 (y- -e+

1 + px/ ' 2 ' k ^ ^ V "1 + px/ ' 2k

co1 e o| e ,2 acy , _

= -ac + —- + - + —2 + (a + bc)x - -y - bx2 + -—(2./) 2 k 2k ' ky 1 + px

co2 e o2 2 / e \

< -ac + + k + 2k2 + (a + bc)x - bx2 - ( ^ ~ ac Jy.

Choose c = e/ak such that e/k - ac = 0, then

LV < -ac + + e + 02 + (a + bc)x - bx2 < K, (2.8)

2 k 2k v ; "

where K is a positive constant. Therefore

/"TmAT Tm AT

dV(x(t),y(t)) < Kdt

fTmAT 02

+ o1(x(s) - c)dB1(s) + — (y(s) - 1)dB2(s),

which implies that,

E[V(x(Tm A T),y(Tm A T))] < V(x(0),y(0)) + E Kdt < V(x(0),y(0)) + KT. (2.10)

Set Q,m = {rm < T} for m > m1, then by (2.4), we know that P(Qm) > e. Note that for every w e Qm, there is at least one of x(rm,w) and y(rm,w) equals either m or 1/m, then

V(x(rm), y(rm)) > ^m - c - c log A ^m - c + c log(cm)^

1 1/1 \ A- (m - 1 - log m) A — l--1 + log m I.

k v ° ' k\m /

It then follows from (2.4) and (2.10) that

V(x(0),y(0)) + KT > E[1am{w)V(x(rm),y(rm))]

> e(m - c - c log A ^m ~ c + c log(cm)^ A k (m - 1 - logm)

a 1 (— -1+log m\ k\m /

(2.12)

(2.11)

where 1am(w) is the indicator function of Q.m. Letting m ^ œ leads to the contradiction that œ > V(x(0),y(0)) + KT = œ. So we must therefore have Tœ = œ a.s. □

3. Permanence

There is no equilibrium of system (1.3). Hence we cannot show the permanence of the system by proving the stability of the positive equilibrium as the deterministic system. In this section we show that there is a stationary distribution of system (1.3).

Remark 3.1. Theorem 2.1 shows that there exists a unique positive solution (x(t),y(t)) of system (1.3) with any initial value (x(0),y(0)) e R+. From the proof of Theorem 2.1, we obtain that LV < K. Define V = V + K, then LV < V, and it is clear that % = inf(x,y)er2\DkV(x,y) —> go as k — to, where Dk = (1/k,k) x (1/k,k). Hence by Remark 2 of Theorem 4.1 of Has'meminskii, 1980, page 86 in [15], we obtain that the solution (x(t),y(t)) is a homogeneous Markov process in R+.

Theorem 3.2. If aef + be < aka < aef + bka/f and o1 > 0, o2 > 0 such that o| < kax*/(1 + fix*) and

(2 +,2x-)x-o2 + (i-+ix + ^ (

<min{1(b" T^^lfe - iM P1)

where (x*, y*) is the positive equilibrium of system (1.1) and l2 is defined as in the proof. Then system (1.3) has a stationary ergodic solution.

Proof. Since aka > aep + be, then there is a positive equilibrium (x*,y*) of system (1.1), and

a = bx* +

1 + ßx*

1 + ßx*

Vi(x,y) = (x - x - x log x*) + h(y - y*- y* log yy^

where l1 is a positive constant to be determined later. Let L be the generating operator of system (1.3). Then

tt, / , ay \ x*ct2 / kax \ "iy*a2

LVi = (x - x)(a - bx - l+y + V+« y- y\-e + ï+Tx) + "V

= (x - x*)

-b(x - x*) -

1 + ßx

+ M y - y*)

ka(x - x*)

(y - y*) +

"1y*a2

(1 + ßx*) (1 + ßx)

(x - x*)

(1 + ßx*) (1 + ßx) 2

(b - aîr)x - x-)2 - 1+a*( 1 - î-^)(x - x-)(y - y»+^+

x*a2 "ly*^

Choose l1 = (1 + px*)/k such that 1 - l1k/(1 + px*) = 0 and yields

aßy* \ 2 x*oi2 (1 + ßx*)y*a,2 LVi <- b - „ ßy )(x - x*)2 + —1 + ^-ß'y 2

1 " v 1 + ßx* r 2 2k

V2( x,y) =

(x - o?) + k(y - y*)

Note that

(x - x*) + k(y - y*)

ax - bx2 - k;y^dt + a1xdB1(t) + kydB2(t)

, , *\2 y*(x - x*) - x*(y - y*)

-bx(x - x ) + a--—--

v ; 1+ ßx*

+ a1xdB1(t) + kydB2(t),

Abstract and Applied Analysis then

(x - x*) + k(y - V*)

, , n2 V*(x - x*) - x*{V - V*)

-bx(x - x ) + a--—-—--1

v ; 1 + ßx*

2 a2 2 IT x + W2 V

= - bx(x - x*) +

ay* by*

1 + ßx* ' k

2 b 2 ax

(x - x ) - b(x - x )y - k^T+ßxy

k( 1 + ßx*) 1 + ßx* k

Cr: ~2

2 2 1^2, a2 2

1 + ßx k ay

^ + a2) (x - x* )2 -

k 1 + ßx k2

- i) (V - V f

k 1 + ßx 1 + ßx k

- ^ )(x - x*) (y - V) + a\(x*)2 + kf (y*)2,

where L is also the generating operator of system (1.3). Note that ay* ax* bx*

k 1 + ßx 1 + ßx k

(x - x*)(y - y*)

(ay*/(k(1 + ßx*)) - ax*/(1 + ßx*) - bx*/k)2 * 2 " 2(ax*/(k(1 + ßx*)) - a\/k2) (x - x )

1/ ax* af\ ( >N2

2 k 1 + ßx k2

- li (y - y r

= &(x - x )2 + K Ä - (y - y )2'

LV2 <(-OL- + b-f + a2 + 6)(x - x)2 - U- (y - y*)2 2"\1 + ßx* k 1 y 2\ k( 1 + ßx*) k2) ^

+ a?(x* )2 + g (y* )2

(3.10)

Now define

V(x,y) = Vi( x,y) + Z2V2( x,y),

(3.11)

where l2 is a positive constant to be determined later. Then

aPV* aV* , bV* , „2 ,

LV < - b --M Jn + of + 6 (x - x*)

" v 1 + px* \ 1 + px* k 1 ' 'v ;

ax oA (y - y)2 + Q+l2X-)x-Of+

(3.12)

2 V k( 1 + px*) k2 JK ' \2 2 / 1 \ 2

Choose I2 > 0 such that (b - apy*/(1 + px*) -12(ay*/(1 + px*) + by*/k + o2 + 6)) = (1/2)(b -apy* /(1 + px*)), then it follows from (3.12) that

lv <-k b - ayx* )(x - x* )2 -1( kciaxpxT -1) (y - y )2

+ ( 2 + ),o2 + (^ + f) ^

(3.13)

Note that

1 + 2 + 1 + px + l2y y o22

(2 +l2x )x o + H" + —) —

. (W apy* \ *)2 h( ax* o2\ A

< b - T+px ;(x )'2(, w+p*) - -2)(y } I,

then the ellipsoid

-U h-OpyL

2\ 1 + px*

2 l2 (x - x ) - -

* 2 x o2

ax o22 2

kCTTpxj - (y - y }

1 + px

(3.15)

lies entirely in R+. We can take U to be a neighborhood of the ellipsoid with U c El = R+, so that for (x,y) e U \ El, LV < -C (C is a positive constant), which implies condition (B.2) in Lemma A.1 is satisfied. Hence the solution (x(t),y(t)) is recurrent in the domain U, which together with Lemma A.3 and Remark 3.1 implies that (x(t),y(t)) is recurrent in any bounded domain D c R+. Besides, for all D, there is an M = min{o2x2/o^y2/ (x,y) e D} > 0 such that

xk-jte-j = o2x2^ + 022y2(g > m|(2| all x e D, ( e R2, (3.16)

which implies that condition (B.1) is also satisfied. Therefore, system (1.3) has a stable a stationary distribution ¡(-) and it is ergodic. □

Abstract and Applied Analysis Note that

dxp = pxp(a - bx - i + + pa1xpdB1(t) + !p(p - 1)a^xpdt

^ +(3.17)

< pxp(^a + pa1 - bx^)dt + pa1xpdB1(t),

dt \ 1

Pai ) £[xP] - b(£[xP])(p+1)/p.

t ^ 00

a + pa!/l\ p

(3.18)

dE[xp] < + £[xP] - bE{xp+1] < p(^a +

Hence by comparison theorem, we get

lim supE [xp (t)] <(" r~1'~ ) , (3.19)

by which together with the continuity of E [xp (t)], we have that there exists a positive constant K = K(p) such that

E[xp(t)] < K(p). (3.20)

By Doob's martingale inequality, together with the (3.20), for 6 > 0, we have tJ x(t) A E[xp (n6)] K(p)

T: cn-sup^Hr > 6) < "<n6nis < ntOk, p>1 (321)

In view of the well-known Borel-Cantelli lemma, we see that for almost all w £ Q,

sup < 6 (3.22)

(n-1)6<t<n6 t

holds for all but finitely many n. Hence there exists an n0(w), for all w £ Q excluding a P-null set, for which (3.22) holds whenever n > n0. Consequently, letting 6 ^ 0, we have, for almost all w,

lim = 0. (3.23)

t ^tt t

By the ergodic property, for any given constant m> 0, we have

lim1 f (xp(s) A m)ds = f (zp A mVW(dz1,dz2), a.s. (3.24)

t — TO t Jo JR++X '

On the other hand, by dominated convergence theorem and (3.20), we get 1t

lim 1 f (xp(s) A m)ds = lim 1 f E[xp(s) A m]ds < K(p), (3.25)

: — to t J0 t — to t J0

which together with (3.24) implies

j (zPi Am)^(dz1,dz2) < K(p). (3.26)

Letting m —> to, we get

zp^(dz1 ,dz2) < K(p). (3.27)

That is to say, the function f1 (z) = zvx is integrable with respect to the measure ¡. Therefore, by ergodicity property again, we get

lim1 i xp(s)ds = f ¿¡¡(dz1,dz2), a.s. (3.28)

t — to t J0 Jr++

Besides,

= a (x(s)ds - b (x2(s)ds - a f x+^ds + ^ itx(s)dB1(s). (3.29) t t Jo t Jo t Jo 1 + Px(s) t Jo

Let M1(t) = /0 x(s)dB1(s) which is a martingale with M1 (0) = 0 and

(^Ti MM\) 1 ft i

limsup--- = lim - x2(s)ds = z\i(dz1,dz2) < to,

t^® t t—to t Jo JR++

(3.30)

then by the strong law of large numbers, we get

lim —^ = lim 1 f x1 (s)dB1 (s) = 0, (3.31)

t — to t t — to t Jq

which together with (3.23) and (3.28) implies (3.29) that

1 (l x(s)y(s) , a f i . b f 2 , .

t Jq1 + px(s) a)Rl a)Rl 1

lim . „ , ,

t^œ t Jq 1 + f)x(s)

(3.32)

Hence from these arguments, we get the following result. Theorem 3.3. Assume the same conditions as in Theorem 3.2. Then one has

lim1 I xp(s)ds = I zP^(dzi,dz2), t Jq JR++

lim1 f x(s)y(s) ds = — f z1u(dz1,dz2) - b f z1u(dz t Jq 1 + 6x(s) a)R++ a)R++ 1

__ -ds = t^œ t J q 1 + ¡5x(s) a JR.

z1,dz2).

(3.33)

4. Extinction

In this section, we show the situation when system (1.3) will be extinct.

Casel. (a < of/2). Obviously,

dx < x(a - bx)dt + o1xdB1(t),

then when a < o2 /2,

lim x(t) = 0, a.s.

That is to say, for all 0 < e1 < e + of/2, there exist T1 = T\(w) and a set Q.e1 such that P(Q^) > 1 - e1 and kax(t) < e1 for t > T1 and w e ^e1. Then

-ey(t)dt + o2y(t)dB2(t) < dy(t) < y(t)(-e + e1)dt + o2y(t)dB2(t),

and so

lim y(t) = 0, a.s.

Case 2. (a>o\/2, e + of/2 >ka/f). Note that

dy(t) < y(t) ^-e + ja^dt + 02y(t)dB2(t),

then if e + o\/2 > ka/ß, we have

lim y(i) = 0, a.s. (4.6)

In this situation, for all 0 < e2 < a - o\/2, there exist T2 = T2(w) and a set Q.ei such that P(Q.e2) > 1 - e2 and ay(t) < e2 for t > T2 and w e ^e2. Then

x(t)(a - bx(t) - e2)dt + oix(t)dBi(t) < dx(t) < x(t)(a - bx(t))dt + oix(t)dBi(t), (4.7)

o2? \ / o22 \

a - 2 - bx(t) - e2 Jdt + oidBi(t) < dx(t) < ( a - -1 - bx(t) Jdt + oidBi(t). (4.8)

Consider the following equation:

dO(i) =(a - O- - be®(i)^dt + oidBi(t).

If a> of/2, (4.9) has the density g*(Z) such that

10*gm(Z) =(ai - ^ - bneZ )g*(Ç). (4.10)

Therefore from (4.8) and the arbitrary of e2, we get that the distribution of log x(t) converges weakly to the probability measure with density g„. Thus, from (4.10), we obtain that the distribution of x(t) converges weakly to the probability measure with density f*(£,) = CoZ2(a-CT2/2)/CT2-1e-2bZ/CT2, where Co = (2b/o2 )2(a-o1/2)/o1 /r(2(a - o\/2)/o\). Besides, from the ergodic theorem and (4.10), it follows that

1 (l r !■ rj a - o1/2 a - o1/2 lim- x(s)ds = ezg*(i)dZ = -r1-g*(Z)di =-r1- a.s. (4.11)

^ ^ * 0 * —CO •'—CO b b

Therefore, by the above arguments, we obtain the following.

Theorem 4.1. Let (x(t),y(t)) be the solution of system (1.3) with the initial value (x(0), y(0)) e Rl-Then,

(i) if a <o2/2, then

lim x(t) = 0, lim y(t) = 0, a.s., (4.12)

t^^ t^^

(ii) if a > o\/2,e + of/2 > ka/p, then the distribution of x(t) converges weakly to the probability measure with density f*(Z) = C0Z2(a-61/2)/6i-1e-2bZ/61, where Co = (2b/a2)2(a-61/2)/6/r(2(a - a2/2)/a2), and

1 (l a - 62/2 lim- x(s)ds =--1—, lim y(t) = 0, a.s. (4.13)

t ^^ t Jo b t ^TO

Appendix

For the completeness of the paper, in this section, we list some theories about stationary distribution (see [15]).

Let X(t) be a homogeneous Markov process in El (El denotes euclidean l-space) described by

dX(t) = b(X)dt + ^gr(X)dBr(t). (A.1)

The diffusion matrix is A(x) = (aij(x)), aij(x) = ^k=1 g'r(x)gj(x).

Assumption B. There exists a bounded domain U c El with regular boundary r, having the following properties.

(B.1) In the domain U and some neighbourhood thereof, the smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero.

(B.2) If x e El \ U, the mean time t at which a path issuing from x reaches the set U is finite, and supxeKExT < to for every compact subset K c El.

Lemma A.1 (see [15]). If(B) holds, then the Markov process X(t) has a stationary distribution ¡i(-). Let f (■) be a function integrable with respect to the measure ¡a. Then Px (lim^ f (X(t))dt =

\Eif (x)A(dx)} = 1 for all x e El.

Remark A.2. The proof is given in [15]. Exactly, the existence of stationary distribution with density is referred to Theorem 4.1, Page 119, and Lemma 9.4, Page 138, in [4]. The weak convergence and the ergodicity is obtained in Theorem 5.1, Page 121, and Theorem 7.1, Page 130, in [4].

To validate (B.1), it suffices to prove that F is uniformly elliptical in any bounded domain D, where Fu = b(x) ■ Ux + (1/2)tr(A(x)u xx); that is, there is a positive number M such that Xtky=1 aij(x)Uj > M\l\2, x e D, £ e Rk (see Chapter 3, Page 103 of [19] and Rayleigh's principle in [20, Chapter 6, Page 349]). To verify (B.2), it is sufficient to show that there exists some neighborhood U and a nonnegative C2-function such that and for any El \ U,LV is negative (for details refer to [21, Page 1163]).

Lemma A.3. Let X(t) be a regular temporally homogeneous Markov process in El. IfX(t) is recurrent relative to some bounded domain U, then it is recurrent relative to any nonempty domain in El.

Acknowledgments

The work was supported by the Ministry of Education of China (no. 109051), the Ph.D. Programs Foundation of Ministry of China (no. 200918), NSFC of China (no. 10971021), and

the Graduate Innovative Research Project of NENU (no. 09SSXT117), Youth Fund of Jiangsu

Province (no. BK2012208), and the Tian Yuan Special Funds of the National Natural Science

Foundation of China (no. 11226205).

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