# Global Existence of the Cylindrically Symmetric Strong Solution to Compressible Navier-Stokes EquationsAcademic research paper on "Mathematics"

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## Academic research paper on topic "Global Existence of the Cylindrically Symmetric Strong Solution to Compressible Navier-Stokes Equations"

﻿4.2. Examples. The first 4 IFs are presented here. For

(a) p = 2:

si(x) = x- /au\'

f (1> (x)

which corresponds to Newton's IF of order 2;

(b) p = 3:

S3 (x) = x -

/(1) (X)

L 1 - (1/2) (/(2) (x) //(1) (x)) (/ (x) //(1) (x)) J '

which is Halley's IF of order 3 [15]; (c) p = 4:

S4 (x) = x-

/(1) (x) i-1/(2) (x) /(x)

2 /(1) (x) /(1) (x) /(2) (x) /(x)

/(1) (x) /(1) (x) 1 /(3) (x)

3! /(1) (x)

/(x) 2>1

[/(1) (x) >

(d) p = 5:

S5 (x) = x -

/(x) /(1) (x)

/(2) (x) /(x)

/(1) (x) /(1) (x)

1 /(3) (x)

x l 1-

/(1) (x)

/(x) [/(1) (x) J

3 /(2) (x) /(x)

2 /(1) (x) /(1) (x)

1 /(3) (x) 1 3 /(1) (x) + 4

1/(4) (x)

/(2) (x) L/(1) (x) J

/(x) L/(1) (x) J

4! /(1) (x) 5. Proof of the Main Result

/(x) 3>1

[/(1) (x) >

Since both processes Ep(x) and Sp(x) are of order p,following [9], the next result holds.

Lemma 1. Let a be a simple root of/(x); then

Ep (x) - Sp (x) = 0(f> (x))==o((/^]P).

-p VV 'p

From this lemma

/(1) (x)

Ap-2 (V-

rp-3 (S)

rp-2 (S)\ p

i=/(^)//(1)(x)

„ m rp-3 (S)

Ap- «'"vT®

?=/M//(1)M

/(x) )p-1 /(1) (x)J

The next step is to consider the following basic result about polynomial and rational approximations.

Lemma 2. Let us consider the expression

Ap (S) =

Tp-1 (S) rp (S)

-o(Sp+1),

where Ap(%) and rp(%) are polynomials of degree p such that A p(0) = 1 = rp(0).

(a) If rp_1(^) and rp(V) are given, there exists one and only one polynomial Ap(V) such that (36) holds.

(b) fTp^ (V) and A p (V) are given, there exists one and only one polynomial rp(V) such that (36) holds.

Proof. This result is based on the following identity:

rp-! (V) _ Tp-! (V

Ip (0 1-(1-rp (S))

= rP-1 (S)H1-rp (S))

= Tp_1 (S)l(1-rp (S)) +o(Sp+1),

and we would like to have

A p (S) = rp-1 (S)t(1-rp (S)) + o(Sp+1).

We knowthat A0 = yp_1>0 = Yp,o = 1. Moreover the coefficient of ;l on the left-hand side is \l and on the right-hand side is

+ an expression in terms of

yp-1,j for j = 0,■■■, U

Yp,j for j = °,...,l-1 (39)

for I = 1,..., p. This expression shows that if the yp's and the Yp-i's are given, we can obtain Xp's, and conversely if the Xp's and the yp-1 's are given, we can obtain the yp's. □

In view of these two lemmas we obtain the main result of this paper.

Theorem 3. Ep(x) and Sp(x) are related as follows.

(a) For Sp(x) given by (28), one can obtain the form (9) of Ep(x) by expanding the denominator in (28), multiplying, and truncating to keep powers of f(x)lf^Vl(x) up to p- 1.

(b) Since S2(x) = E2(x) = Nj(x), one can obtain recursively Sp(x) given by (28) from Ep(x) given by (9).

Proof. (a) Indeed, if Sp(x) is given, which means we know rp-3(£) and r^-2(£), we can write

g (x) x f(x) SP (x) — x f(D {X)

t,=f(x)/f(l)(x)

— x -

f(l) (x)

f(1) (x)

Tp-3 (S)

l-(l- Tp-2 (H))

(,=f(x)/f1)(x)

Tp-3 (t)l{l- Tp-2 (Ï))

Z=f(x)/fW(x)

— Ep (x) + 0(f (x)),

which follows from part (a) of Lemma 1.

(b) As already observed, E2(x) = S2(x) = Nf(x). If, for p > 2, we have Sp-1(x), and we know rp-4(^) and r^_3(£), and Ep(x), and we know also A p-2(%), then we can determine rp-2(%) from part (b) of Lemma 1. Consequently we obtain

Sp (x).

The computation of the polynomials A p(Ç) and rp(^), and their coefficients X's and the yp's, can be done explicitely using (11) for the Ap's and (22) and (26) for the yp s. The verification of the link between the two Schroder's processes has already been done using symbolic computation up to order 20 [5, 6].

Acknowledgment

Thisworkhas been financially supportedbyanindividualdis-

covery grant from NSERC (Natural Sciences and Engineering

References

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[2] E. Schröder, "On infinitely many algorithms for solving equations," Tech. Rep. TR-92-121, TR-2990, Institute for advanced Computer Studies, University of Maryland, 1992, Translated by G. W. Stewart.

[3] B. Kalantari, I. Kalantari, and R. Zaare-Nahandi, "A basic family of iteration functions for polynomial root finding and its characterizations," Journal of Computational and Applied Mathematics, vol. 80, no. 2, pp. 209-226,1997.

[4] B. Kalantari, Polynomial Root-Finding and Polynomiography, World Scientific, Singapore, 2009.

[5] L. Petkovic and M. Petkovic, "The link between Schröders iteration methods of the first and second kind," Novi Sad Journal of Mathematics, vol. 38, no. 3, pp. 55-63, 2008.

[6] M. S. Petkovic, L. D. Petkovic, and D. Herceg, "On Schröders families of root-finding methods," Journal of Computational and Applied Mathematics, vol. 233, no. 8, pp. 1755-1762, 2010.

[7] B. Kalantari, "Polynomial root-finding methods whose basins of attraction approximate Voronoi diagram," Discrete & Computational Geometry, vol. 46, no. 1, pp. 187-203, 2011.

[8] F. Dubeau, "On comparisons of Chebyshev-Halley iteration functions based on their asymptotic constants," International Journal of Pure and Applied Mathematics, vol. 58, pp. 965-981, 2013.

[9] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall, Englewood Cliffs, NJ, USA, 1964.

[10] A. S. Householder, Principles of Numerical Analysis, McGraw-Hill, Columbus, NY, USA, 1953.

[11] E. Bodewig, "On types of convergence and on the behavior of approximations in the neighborhood of a multiple root of an equation," Quarterly of Applied Mathematics, vol. 7, pp. 325-333, 1949.

[12] M. Shub and S. Smale, "Computational complexity. On the geometry of polynomials and a theory of cost. I," Annales Scientifiques de l'Ecole Normale Superieure, vol. 18, no. 1, pp. 107142, 1985.

[13] M. Petkovic and D. Herceg, "On rediscovered iteration methods for solving equations," Journal of Computational and Applied Mathematics, vol. 107, no. 2, pp. 275-284,1999.

[14] B. Kalantari and J. Gerlach, "Newton's method and generation of a determinantal family of iteration functions," Journal of Computational and Applied Mathematics,vol. 116, no. 1, pp. 195200, 2000.

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— x -

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 720283, 10 pages http://dx.doi.org/10.1155/2014/720283

Research Article

Persistence and Nonpersistence of a Predator Prey System with Stochastic Perturbation

Haihong Li,1,2 Daqing Jiang,1 Fuzhong Cong,2,3 and Haixia Li4

1 College of Science, China University of Petroleum (East China), Qingdao 266580, China

2 Department of Basic Courses, Air Force Aviation University, Changchun, Jilin 130022, China

3 School of Mathematics, Jilin University, Changchun, Jilin 130024, China

4 School of Business, Northeast Normal University, Changchun, Jilin 130024, China

Correspondence should be addressed to Daqing Jiang; daqingjiang2010@hotmail.com Received 13 December 2013; Accepted 25 February 2014; Published 3 April 2014 Academic Editor: Jifeng Chu

Copyright © 2014 Haihong Li 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 analyze a predator prey model with stochastic perturbation. First, we show that this system has a unique positive solution. Then, we deduce conditions that the system is persistent in time average. Furthermore, we show the conditions that there is a stationary distribution of the system which implies that the system is permanent. After that, conditions for the system going extinct in probability are established. At last, numerical simulations are carried out to support our results.

1. Introduction

Recently, the dynamic relationship between predator and prey has been one of the dominant themes in both ecology and mathematical ecology due to its universal importance. Especially, the predator prey model is the typical representative. Thereby it significantly changed the biology and the understanding of the existence and development of the basic law and has made the model become the research hot spot. One of the most famous models for population dynamics is the Lotka-Volterra predator prey system which has received plenty of attention and has been studied extensively; we refer the reader to [1-3] for details. Specially persistence and extinction of this model are interesting topics.

The predator prey model is described as follows:

x(t) = rx(t)(l-^)-cx(t)y(t) y (t) = -^y (t ) + mcx (t) y (t),

where x(t), y(t) denote the population densities of the species at time t. The parameters r, K, c, ^, m, are positive constants that stand for prey intrinsic growth rate, carrying capacity,

the maximum ingestion rate, predator death rate, and the conversion factor, respectively. From a biological viewpoint, we not only require the positive solution of the system but also require its unexploded property in any finite time and stability. We know that system (1) has a unique positive equilibrium (x*,y*) which is a stable node or focus if the following condition holds, mcK > ^:

rmcK - r^ mc2K

and the system (1) has a unique limit cycle which is stable (see [4]).

However, population dynamics in the real world is inevitably affected by environmental noise (see, e.g., [5-7]). Parameters involved in the system are not absolute constants; they always fluctuate around some average values. The deterministic models assume that parameters in the systems are deterministic irrespective of environmental fluctuations which impose some limitations in mathematical modeling of ecological systems. So we cannot omit the influence of the noise on the system. Recently many authors have discussed population systems subject to white noise (see, e.g., [8-12]). May (see, e.g., [13]) pointed out that due to continuous

fluctuation in the environment, the birth rates, death rates, saturated rate, competition coefficients, and all other parameters involved in the model exhibit random fluctuation to some extent, and as a result the equilibrium population distribution never attains a steady value but fluctuates randomly around some average value. Sometimes, large amplitude fluctuation in population will lead to the extinction of certain species, which does not happen in deterministic models.

Therefore, Lotka-Volterra predator prey models in random environments are becoming more and more popular. Ji et al. [14, 15] investigated the asymptotic behavior of the stochastic predator prey system with perturbation. Liu and Chen [16] introduced periodic constant impulsive immigration of predator into predator prey system and gave conditions for the system to be extinct and permanent.

In this paper, we introduce the white noise into the intrinsic growth rate and predator death rate of system (1); that is, r ^ r + a1 B1 (t), ^ ^ ^ + a2B2(t);then, weobtain the following stochastic system:

x(t) = rx(t)(l-^)-cx(t)y(t) + ^x(t) (K-x(t))B1 (t), y (t) = -W (t) + mcx (t) y (t) - o2y (t) B2 (t),

where Bt(t) (i = 1,2) are independent white noises with Bj(0) = 0, a2 > 0 (i = 1,2) representing the intensities of the noise.

The aim of this paper is to discuss the long time behavior of system (3). We have mentioned that (x*,y*) is the positive equilibrium of system (1). But when it suffers stochastic perturbations, there is no positive equilibrium. Hence, it is impossible that the solution of system (3) will tend to a fixed point. In this paper, we show that system (3) is persistent in time average. Furthermore, under certain conditions, we prove that the population of system (3) will die out in probability which will not happen in deterministic system and could reveal that large white noise may lead to extinction.

The rest of this paper is organized as follows. In Section 2, we show that there is a unique nonnegative solution of system (3). In Section 3, we show that system (3) is persistent in time average, while in Section 4 we consider three situations when the population of the system will be extinct. In Section 5, numerical simulations are carried out to support our results.

Throughout this paper, unless otherwise specified, let (Q, [Ft}t>0, P) be a complete probability space with a filtration [Ft}t>0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets). Let R+ denote the positive cone of R2; namely, R+ = {x = (x1,x2) e R2 : xt > 0, i = 1, 2}, R+ = {x = (x1,x2) eR2 : xt > 0, i = 1,2}.

2. Existence and Uniqueness of the Nonnegative Solution

To investigate the dynamical behavior, first, we should concern whether the solution is global existence. Moreover, for

a population model, we should also consider whether the solution is nonnegative. Hence, in this section we show that the solution of system (3) is global and nonnegative. As we have known, in order for a stochastic differential equation to have a unique global (i.e., no explosion at a finite time) solution with any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (see, e.g., [17]). It is easy to see that the coefficients of system (3) are locally Lipschitz continuous, so system (3) has a local solution. By Lyapunov analysis method, we show the global existence of this solution.

By the classical comparison theorem of stochastic differential equations, we could get the following.

Lemma 1. Let (x(t), y(t)) be a positive solution of system (3) with (x(0), y(0)) e R+. Then, we have

x(t) <X(t), y(t)<Y(t), a.s.,

where (X(t),Y(t)) are solutions of the following stochastic differential equations:

X(t) =rX(t)(l-^) + ^ X (t) (K-X (t)) È1 (t),

X(0) = x(0),

Y (t) = -yiY (t) + mcX (t) Y (t) - o2Y (t) B2 (t),

Y(0) = y(0).

Consider the stochastic logistic equation

/ Y \ ¡X

dN (t) = N(t) (r- -N(t))dt + -(K-N (t)) dB (t) L V K / K

r,K > 0. (6)

Jiang et al. [18] studied system (6) and obtained the following result.

Lemma 2. There exists a unique continuous positive solution 0 < N(t) < K to system (6) for any initial value N(0) = N0 with 0 <N0 < K.Ifr > a2¡2, then

lim N (t) = K, a.s.

From Lemmas 1 and 2, it is easy to get the following result.

Lemma 3. Let (x(t), y(t)) be a positive solution of system (3) with 0 < x(0) < K. Then, we have

0 < x(t) < K. a.s.

Theorem 4. For any initial value |(x(0),y(0)) e R+,x(0) e (0, K)|, there is a unique solution (x(t), y(t)) of system (3) on t > 0, and the solution will remain in R+ with probability 1.

Proof. It is clear that the coefficients of system (3) are locally Lipschitz continuous for the given initial value |(x(0), y(0)) e R+, x(0) e (0,K)}. So there is a unique local solution (x(t), y(t)) on te [0, re), where re is the explosion time (see, e.g., [17]). To show this solution is global, we need to show that re = m a.s. Let k0 >1 be sufficiently large so that x(0) and y(0) all lie within the interval [1/k0,k0]. For each integer k>k0, define the stopping time

rm = inf {te [0,Te): min [x(t),y(t)}

< ^ or max [x (t), y (t)} > k} .

Throughout this paper, we set inf 0 = ot (as usual 0 denotes the empty set). Clearly, Tk is increasing as k ^ ot. Set rTO = limfc^TOTfc; then, rTO < Te a.s. If we can show that rTO = ot a.s., then re = ot 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 Tm = ot a.s. If this statement is false, then there is a pair of constants T > 0 and e e (0,1) such that

P[Tm <T}>£. (10)

Hence, there is an integer kt > k0 such that

P{rk <T}>e Vk>kv (11)

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

V (x, y) = (x-a-a log —) + — (y-1- log y), (12) V a J m

where a 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+-1 (x- a) (K - x) dBl (t)

+^ (y-l)dB2 (t), m

fr \ aa, i

LV=(x-a)[r--x-cyj + 2^2 (K-x)2

2 2 2 ao, u a2 / ar — ao, = -ar + —1 + — + ^ + ( r+-1 - c Ix

2 m 2m

■ V-it-

K-2K2 )X~-(rn-aC)y.

Choose a = ^/mc such that ^/m - ac = 0, together with Lemma 3; then,

22 aa, u o-s LV < -ar +--1 + - + -22 m 2m

( ar - aa2 \ ( r aa2 \ 9

+ (r +-1 -c)x-(---V )x2 <M,

V K ) VK 2K2

where M is a positive constant. Therefore,

dV(x(t),y(t))

Mdt+\ -1 (x- — ) (K-x) dB, (s) Jo Jo K \ mcJ

+--2 (y(s)-l)dB2 (s),

E[V(x(rk AT),y(rk AT))] <V(x(0),y(0))

rtk AT

+ E\ Kdt <V(x(0),y (0)) + MT.

Set i\ = i^k < T] for k > kx; then, by (11), we know that P(i\) > e. Note that for every to e Qk, there is at least one of x(Tk, to) and y(Tk, to) equals either k or 1/k; then,

v(x(Jk),y(Jk)) > [k-a-a log ^

a(^ - a + a log (ak))

A-(k-1-log k)

Al{l-1+ log k) It then follows from (11) and (16) that

V (x (0), y (0)) + KT > E [^V (x (rk), y (rfc))]

> e ( k-a-a log -

A^-a + a log (ak)) (18)

A-(k-1- log k) Al{1-1 + log k

where 1nk(a) is the indicator function of Qk. Letting k ^ m leads to the contradiction that m > V(x(0),y(0))+MT = m. So we must, therefore, have rm = m a.s. □

3. Permanence

There is no equilibrium of system (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 first show that this system is persistent in mean.

3.1. Persistent in Time Average. L. S. Chen and J. Chen in [19] proposed the definition of persistence in mean for the

deterministic system. Here, we also use this definition for the stochastic system.

Definition 5. System (3) is said to be persistent in mean if

lim inf- I y (s) ds > 0,

tirn t Jo

Lemma 6 (Xia et al. [20, Lemma 17]). Let f e C([0, +<m) x &,(0,+<m)), F e C([0,+ot) x Q.,R). If there exist positive constants X0, X, such that

log f(t)>Xt-X0 \ f(s)ds + F(t), t>0 a.s. (20)

and limt^m(F(t)/t) = 0 a.s., then

1 f* X

liming I f(s)ds>—, a.s. (21)

t Jo X0

Assumption 7. We have

(r- y ) Kmc - r(u + ) > 0.

Theorem 8. If Assumption 7 is satisfied, then the solution (x(t), y(t)) of system (3) with any initial value |(x(0), y(0)) e R+, x(0) e (0, K)} has the following property:

1 rt (r-(a21/2))Kmc-r(^ + (a21/2)) liming I y (s) ds > -----—-—---—

t^™ t J0 Kmc2

> 0, a.s.

Proof. According to Ito's formula, the system (3) is changed into

d log x (t) = r - kX (t) - cy (t)

--K(K - x (t)) +K (K-x(t))dB1 (t),

d log y(t) = -p —— + mcx (t) - a2dB2 (t);

d (logx (t) + Kmc logy(t))

r (H+?l)-cy(t)--K (K-x(t))2

+ 0l (K-x(t))dB1 (t)--p-dB2(t) (25)

/ „2 \ 2 r f a2\a1

> ( r--( u +

1 Kmc V 2 2

-cy(t) + ^ (K-x (t)) dBi (t) - ^dB2 (t). K Kmc

After that

log x(t)- log x(0) r log y(t)- log y(0)

>ir-^(u+°l)-°l)-ckK^ (26)

Kmc\ 2 ) 2 ) t

^ I (K-x (s)) dB1 (s) f dB2 (s);

Ct J0 Kmct J0

Kt J0 Kmct j0

besides, from Lemma 3, it is clear that

limsup—g—— < 0,

where M1(t) = fQ(K - x(s))dB1(s) and M2(t) = fQdB2(s) are martingale with Mt(0) = 0 (i = 1,2), and from Lemma 3 we get

(M1,M1) 1 2 2

lim sup-, = lim sup- I (K - x (s)) ds < K ;

Urn t Urn t Jo

then, by strong law of large numbers, we know that limt^m(Mt/t) = 0 (i=1,2). Hence,

lim ((r - (r/Kmc) (u + (a22/2)) - (a\/2)) t

+ (a7jK) M1 (t) - (raJKmc) M2 (t) ) x (t)-1

2 2 (29)

r a a1

, u + -Kmc V 2 2

With Lemma 6 and Assumption 7 we could get

lim - I y (s)

imt Jo

r - (a\/2) - (r/Kmc) (u + (a^/2)) c

(r - (a\/2)) Kmc -r(u+ (a^/2)) Kmc2

3.2. Stationary Distribution and Ergodicity for System (3). In this section we show there is a stationary distribution of system (3).

Theorem 9. Let (x(t), y(t)) be the solution of system (3) with any initial value |(x(0), y(0))

^ < min|mcK, rmcK/af} and ax > 0, a2 > 0, such that af < Kr/x* and

(1+lxt)xta2 + (

< min -

-1 / * 2

1 r x a.

2 K K2

L-(u-al)(f)2,(K-x)2

where (x*,y*) isthepositive equilibrium ofsystem (1) and I is defined as in the proof, then system (3) exists as a stationary distribution and it is ergodic.

Proof. Since ^ < mcK, then there is a positive equilibrium (X, y*) of system (1), and

r = — X + cy K y

U = mcx .

Define

V (x,y) = (x-x -x* log XX)

+ — (y-y*-y* logy ) ,

and let L be the generating operator of system (3). Then, r

LV1 = (x- x*)(r - K-x - cy)

x*a? o — , *w . y'o%

(k-x)2+—(y-fu-,+mcx)+ym

= (x-x*) [-K(x-x*)-c(y-y*)

+ xa (K-X-(X-X*))2

* „2

(* \ / * \ y a 2 x - x )(y - y )+--

^ r *s2 x

<--(x - x ) + -t1

K( ) K2

((K-X)2 + (x-x*)2 ) +

)(x-x•)■

* 2 * 2 x a, _2 y a?

+ x-è(K-x) +~2m

*\2\ . y a2

Define

V2 (x,y) =

(x-x)+-(y-y)

Note that

r (x - x) - K ((x - x*)2 + 2x* (x - x*))

(y-f)dt

+ —x (K - x) dB1 (t) - —ydB2 (t) K m

2rx \ f *\ r f ^2 u / *\

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

K )( ) K( ) m y y

+ —x (K - x) dB, (t) - — ydB2 (t) ; K m

t * \ — t *

(x-x )+--(y-y

r / ^2 u / *

-K(x-x) -m(y-y

< I r -

/x-x ) + (---x--

m K m m

x(x-x*)(y- y)- — (y-y)

/ *\2 f r 2r * p

< r(x - x ) + (---x--

% m Km

x(x- X) (y - f) - A (y - y*f

+a [(x - x*)2+(x*)2][(y- y*f+(y*)2]

( 2\r *\2 u a2 ( *\2

= (r + ai)(x-x ) (y-y )

(r 2r * *\/ * \

+ (---x--)(x-x )(y-y )

V m Km m) 7 '

(x-x*) + — (y-y*)

I r 2 u \

( rx--x--y)dt

+ —x (K - x) dB, (t) - —ydB2 (t) K m

r(x-x*)--(x - (x*)2) (y - y)

+ —x (K - x) dB, (t) - —ydB2 (t) K m

2, a2 / * n2

+ a1 (x ) +— (y ) ,

where L is also the generating operator of system (3). Note that

r 2r * *\/ * \

---x--¡(x-x ) (y - y )

m Km m)

(r/m - (2r/Km) x* - u/m)2 . *.2 < ---——--(x-x )

2((u-a2)/m2)

1 u - o0 + - 1 2

)(y-y)2

LV2 < (r + o\ +5) (x-x)2

Now define

22 U-02 \( *\2 2, 02 f *\2

V(x,y) = V1 (x,y) + W2 (x,y),

lies entirely in D0 = {(x, y) e R+ \ 0 < x < K}. We can take U to be a neighborhood of the ellipsoid with U C El = D0, so that for x e U \ El, LV < -C (C is a positive constant), which implies that condition (B.2) in Lemma 3.2 of [21] is satisfied. Hence, the solution (x(t), y(t)) is recurrent in the domain U, which together with Lemma 3.3 and Remark 3.3 of [21] imply that (x(t), y(t)) is recurrent in any bounded domain D c D0. Besides, for VD, there is a M = min{(x2af /K)(K - x)2, o\y2, (x, y) eD}> 0, such that

2 2 x o

(K-xftl

>m\Ç2\ VxeD,ÇeR2,

where I is a positive constant to be determined later. Then,

LV < -

(r-^o-l(r + o\ + S))(x-x*)2

(^°)(y-/)2 + (±+lx*)x*oï (41)

+ ( 1 + ly* ) y* o2

2 m J m

Choose / > 0 such that ((r/K)-(x* a2t/K2) -l(r + a\ + 5)) (1/2)((r/K) - (x*a2/K2)). Then, it follows from (47) that

LV<-l-(r--xl)(x-x

2\K K2

-)(x-xf

)(y-y)2+Ci+"')*'ot

--V? (42)

which implies that condition (B.l) in Lemma 3.2 of [21] is also satisfied. Therefore, system (3) has a stationary distribution p(-) and it is ergodic. □

From Lemma 3, with the initial value 0 < x(0) < K, we have the property

0 < x(t) < K a.s.

Therefore, by ergodicity property, we know that function f(z) = zp is integrable with respect to the measure u, and

lim 1 [ xp (s)ds= [ zpu(dz1 ,dz2), a.s. (47)

t^^t Jo J^j

Hence, from these arguments, we get the following result.

Theorem 10. Assume the same conditions as in Theorem 9. Then, we have

lim 1 [ xp (s)ds= [ zpu (dz,, dz2), a.s. (48) t^^t Jo Jb+

Note that

(l+lx*^)x*o\ + (

1 + lf\f°: 2 m m

< min -

-i / * 2 1 r x o-

2\K K2

1 )(xl2 ,

) (y*^' (K - x*))

Then, the ellipsoid

-, / * 2 11 r x a1

2 K K2

)(*-'■)-îtë y S)

,1 J A * 2 (1 ly

+ ( - + lx )x o, + ^ + — I-2 = 0

4. Extinction

In this section, we show the situation when the population of system (3) will be extinct. Before we give the result, we should do some prepare work. We first introduce a result on the Feller's test (see, e.g., [22]).

Let I = (l, r), ->x> < r < +ot. Consider the following one-dimensional time-homogeneous stochastic differential equation:

dXt = p (Xt) dt + a (Xt) dBt, X0 = x. (49)

Assume that the coefficients a : I ^ R, p : I ^ R satisfy the following conditions:

(1) o2 (x) > 0; Vxel,

ix+e 1 + u(y) (2) ^x el, 3e>0, \ —< Jx-c o2 (y)

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