Feng et al. Journal of Inequalities and Applications (2016) 2016:327 DOI 10.1186/s13660-016-1265-z

O Journal of Inequalities and Applications

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Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model

Tao Feng1,2, Xinzhu Meng1,2,3*, Lidan Liu1 and Shujing Gao2

CrossMark

"Correspondence: mxz721106@sdust.edu.cn 1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, P.R. China

2Key Laboratory of Jiangxi Province for NumericalSimulation and Emulation Techniques, Gannan Normal University, Ganzhou, 341000, P.R. China Fulllist of author information is available at the end of the article

Abstract

This paper formulates an infected predator-prey model with Beddington-DeAngelis functional response from a classical deterministic framework to a stochastic differential equation (SDE). First, we provide a global analysis including the global positive solution, stochastically ultimate boundedness, the persistence in mean, and extinction of the SDE system by using the technique of a series of inequalities. Second, by using Ito's formula and Lyapunov methods, we investigate the asymptotic behaviors around the equilibrium points of its deterministic system. The solution of the stochastic model has a unique stationary distribution, it also has the characteristics of ergodicity. Finally, we present a series of numerical simulations of these cases with respect to different noise disturbance coefficients to illustrate the performance of the theoretical results. The results show that if the intensity of the disturbance is sufficiently large, the persistence of the SDE model can be destroyed.

Keywords: stochastic eco-epidemiology model; Holder inequality and Chebyshev inequality; asymptotic behavior; persistence in mean; stationary distribution

ft Spri

ringer

1 Introduction

Mathematical inequalities play a large role in mathematics analysis and its application. Recently, the inequality technique was applied to impulsive differential systems [1, 2] and stochastic differential systems [3-5], thus some new results were obtained.

Predation can have far-reaching effects on biological communities. Thus many scientists have studied the interaction between predator and prey [6-10]. Interaction between predator and prey is hard to avoid being influenced by some factors. One of the most common factors is the disease. Therefore, there are many scholars who have studied the infected predator-prey systems [11-17]. For instance, Hadeler and Freedman [16] considered a predator-prey system with parasitic infection. They proved the epidemic threshold theorem for where there is coexistence of the predator with the uninfected prey. Han and Ma [15] analyzed four modifications of a predator-prey model to include an SIS or SIR parasitic infection. They obtained the thresholds and global stability results of the four systems.

Species may be subject to uncertain environmental disturbances, such as fluctuations of birth rate and death rate, food, habitat and water, etc. These phenomena can be de© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 InternationalLicense (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use,distribution, and reproduction in any medium, provided you give appropriate credit to the originalauthor(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

scribed by stochastic processes. Recently, the stochastic predator-prey systems have received much attention from scholars [18-21]. Zhang and Jiang [18] studied a stochastic three species eco-epidemiological system. They analyzed the stochastic stability and asymptotic behaviors around the equilibrium points of its deterministic model. Liu and Wang [19] considered a two-species non-autonomous predator-prey model with white noise. They obtained the sufficient criteria for extinction, non-persistence in the mean, and weak persistence in the mean.

The functional response of predator is a very important factor of predator-prey system, which reflects the average consumption rate of predator to prey. Therefore, many scholars prefer to study the predator-prey system with functional response [22-25]. For instance, Wang and Wei [22] explored a predator-prey system with strong Allee effect and an Ivlev-type functional response. Liu and Beretta [23] studied a predator-prey model with a Beddington-DeAngelis functional response. Some biologists have argued that in many instances, especially when predators have to hunt for food and, therefore, have to share or compete for food, the functional response in a prey-predator model should be predator-dependent. This view has been supported by some practical observations [26, 27]. Skalski and Gilliam [26] collected observation data from 19 predator-prey communities, they found that three predator-dependent functional responses (Crowley-Martin [28], Hassell-Varley [29] and Beddington-DeAngelis [30, 31]) were in agreement with the observation data, and in many instances, the Beddington-DeAngelis type looked better than the other two.

The Beddington-DeAngelis type functional response of per capita feeding rate can be expressed as follows:

F (x, y) = aXy , 1 + px + qy

where a (units: time-1) represents the effects of capture rate on the feeding rate, p (units: prey-1) denotes the effects of handling time on the feeding rate, q (units: predator-1) represents the magnitude of interference among predators. Compared with the Holling II functional response, the Beddington-DeAngelis type functional response has an additional term qy in the denominator. In other words, this type of functional response is affected by both predator and prey. Therefore, the effect of mutual interference on the dynamics of population is worth studying.

To the best of our knowledge, the research on global asymptotic behaviors of a stochastic infected predator-prey system with Beddington-DeAngelis has not gone very far yet. Therefore, according to a deterministic predator-prey model, this paper investigates the stationary distribution and ergodic property of a stochastic infected predator-prey with Beddington-DeAngelis and explores the influence of white noise on the persistence in mean and extinction of the predator-prey-disease system.

First of all, a deterministic predator-prey system is described in [32] by

X (t)=X(t)[b - anX (í)-I+p|^],

S(t)=S(t)[-c - a22S(t) + 1+a;S(t) - Him (1)

I(t)=I(t)[-d - a331 (t) + j0S(t)],

where X(t) is the population density of prey at time t, S(t) and I(t), respectively, stand for the densities of susceptible predator and infected predator at time t, b is the intrinsic growth rate of X(t), c is the natural mortality rate of S(t), d is the diseased death rate of I(t). an, a22, a33, respectively, stand for the density coefficients of X(t),S(t) and I(t). ai2

is the captured rate of X(t), — is the conversion rate from X(t) to S(t), j represents the

infection rate from S(t) to I(t), p, q > 0 are constant coefficients.

Second, the world is full of uncertainty and random phenomena, so species in the ecosystem may be subject to different forms of random interference. In this paper, we assume that the disturbance in the environment affects not only the rate of predation but also the infection rate of the disease, so that

a12 ^ a12 + 012-61, a21 ^ a21 + 021-61, j ^ j + oB2,

where B1(t) and B2(t) are standard Brownian motions, o122, o21, and o2 are the intensities of the Brownian motions. Taking into account the effects of random interference gives

The rest of this paper is organized as follows. In the next section, we consider the existence of a global positive solution and the stochastically ultimate boundedness of model (2). In Section 3, we study the global asymptotic behaviors of model (2) around the equilibrium points of its deterministic system. In addition, we explore the stationary distribution and ergodic property of model (2). In Section 4, we obtain the conditions for the persistence in mean and extinction of model (2). In the last section, we summarize our main results and give some numerical simulations.

Throughout this paper, let (fi, F, {F}t>0, P) be a complete probability space with a filtration {Ft}t>0 satisfying the usual conditions (i.e. it is increasing and right continuous while Fo contains all P-null sets). The function Bi(t) (i = 1,2) is a Brownian motion defined on the complete probability space fi. For an integrable function X(t) on [0, we define <X(t)> = 1 fQX(s) ds, <X(t)>* = liminf^+TO<X (t)>, <X(t)>* = limsup^+TO<X(t)>.

2 Global positive solution and stochastically ultimate boundedness 2.1 Global positive solution

Due to the physical meaning, variables 5(t), I(t), and Y(t) in model (2) should remain nonnegative for t > 0. We next prove that this is actually the case and, furthermore, the positive solution is unique.

Lemma 2.1 For any initial value (X(0), 5(0), I(0)) e R+, model (2) has a local unique positive solution (X(t), 5(t), I(t)) on t e [0, re), where xe is the explosion time.

Theorem 2.1 For any initial value (X(0), 5(0), I(0)) e R+, model (2) has a unique positive solution (X(t), 5(t),I(t)) e R+ ont > 0 with probability 1.

d5(t)=5(t)[-c - a225(t) +

+ 1+pX(t)+q5(t) dB1(t)-° 5(t)I (t) dB2(t),

dI(t) = I(t)[-d - a331 (t) + P 5(t)] dt + o 5(t)I (t) dB2(t).

Proof By Lemma 2.1, we only need to prove that Te = to a.s. To this end, let k0 > 0 be a sufficiently large constant such that X(0), 5(0) and 7(0) all lie in [k0]. For each k > k0 (k e N+), define the stopping time

Tk = infj t e [0, Te] :X(t) e ^k-, k^, S(t) e (k^ or I(t) e (J-, l<0

As is easy to see, Tk is a monotonically increasing function when k ^ to. Let tto = limk^TOTk, thus tto < Te a.s. Now we need to prove tto = to a.s., otherwise, there exist two constants T >0 and e e (0,1) such that P{tto < T} > e. Thus, there is an integer k1 > k0 such that

P{Tto< T} > e, k > k1. (3)

Define a C3-function V: R+ ^ R+,

V (X, S, I) = X -1-ln X + S -1-ln S +1 -1- ln I.

The non-negativity of the function V (X, S, I) can be seen by u -1-ln u > 0, u > 0. Applying Ito's formula to the stochastic differential system (2) yields

M n/^ °12(X -1)S o-21(S -1)X JD ^

dV = LV at--aBi(t) +-flBi(t)

1 + pX + qS 1 + pX + qS

- a (S - 1)IdB2(t) + a (I - 1)SdB2(t),

LV = (X-1) b - auX -

1 +pX + qS) 2(1 + pX + qS)2

+ (S -1) I -c - U22S +

1 + pX + qS

- PI) +

22 a221X2

2(1 + pX + qS)2

+ (I - 1)(-d - a33I + P S) + 1 a 2S2 + 1 a 2I2

= bX - auX2-. a12SX

1 + pX + qS

2 a21XS - cS - a22S +

- b + a11 X + :-—-- +

a122S2

1 +pX + qS 2(1 + pX + qS)2

1 + pX + qS

+ c + a22S--+ PI

1 + pX + qS

a2 2 a221X2

---— + ^a 2I2 - dI - a33I2 + d + a33I - PS + -a2S2

2(1 + pX + qS)2 2 2

auX2-(b + au)X - — - a1|

q 2q2 1

2 a21 a221

a22S -( a22 + yj S - C

- [a33I2 -(P + a33)I - d] + 2a2S +12).

d( 021X + S +I' J 021 x + S +1

dt \012

021v{i u v a12S

—X 1 + b - a11X--

012 V 1 + pX + qS

+1 [1 - d - a331 + j S]

< - 021 [anX2-(b + 1)X] -012

+ SI 1-c - a22S + —a21X--j I

1 22 1 + pX + qS j

a22S2-(1 + — IS

- (a3312-1)

^ i 021(b + 1)2 (1 + f)2 1 < 3 • max^ -

[ 4012au 4a22 4a33

where C0 is a positive constant. Then we have

021 X(t) + S(t) + I(t) <(021 X(0) + S(0) +1(oA e-t + Co(1 - e-t) 012 \ 012 /

< e"^^ + S(0) +1(0) + C0(et -1))

< ma^| 021 X(0) + S(0) +1(0), C0

limsup( 021 X(t)+S(t)+I(t) ) < C0.

t^TO \ °12 /

Therefore, we have

LV < -

a11X2 -(b + a11)X -

a12 012

q 2q2 - [a3312-(j + a33)I - d] + 02C2 < K0,

C2 / a21\c 0221

a22S - a22 + — S - c - —-

V p / 2p .

where K0 is a positive constant. So we have

^ A* 012 (X -1)S 021 (S -1)X

dV < K0 dt--dB1(t) +-dB1(t)

1 + pX + qS 1 + pX + qS

- 0 (S - 1)I dB2 (t) + 0 (I - 1)SdB2 (t).

Integrating (5) from 0 to Tk A T and taking expectation on both sides, we have

EV(X(rk A T), S(Tk A T),I(Tk A T)) < V(X(0), S(0),I(0)) + K0T.

Let Qk = {Tk < T}, from inequality (3) we can see that P(^k) > e. Note that, for every w e Qk, there exists at least one ofX(Tk,w), S(Tk,w),I(Tk, w) that equals either k or |.As a result, we have

V(X(Tk A T), S(Tk A T),I(Tk A T)) > (k -1 - ln k) A ^ 1 - 1 - ln . (7)

Applying equation (6) and equation (7), we get

V (X(0), S(0), I(0)) + K0T

> E[1Qk (w) V (X(Tk A T), S(ti A T), I (Tk A T))]

> e(k -1-ln k) A ^ 1-1 -ln i^,

where 1^k is the indicator function of Qk. When k ^ to, we have

to > V(X(0),5(0),7(0}) + K0T = to. This is a contradiction. So tto = to. □

2.2 Stochastically ultimate boundedness

Theorem 2.1 shows that R+ is the positive invariant set of model (2). Now we prove the stochastically ultimate boundedness of model (2).

Definition 2.1 Let (X(t), S(t), I(t)) be the solution of model (2) with initial value (X(0), 5(0),I(0)) e R+. If, for any e e (0,1), there exists a x(= X(«)) > 0 such that the solution of model (2) satisfies

limsup P{ |(X(t), S(t), I(t)) I > x} < e,

t—>to

then model (2) has stochastically ultimate boundedness.

Lemma 2.2 The following elementary inequality will be used frequently in the sequel.

(1) xr < 1 + r(x - 1),x > 0,1 > r > 0,

(2) n(1-p/2)A0|x|p < Y!=-ix?i < n(1-p/2)v0|x|p, where R+ := {x e Rn : xi > 0,1 < i < n}, n e R+,p > 0.

Theorem 2.2 Let (X (t), S(t), I(t)) be the solution of model (2) with initial value (X(0), S(0), I(0)) e R+, then (X(t), S(t),I(t)) is stochastically ultimate boundedness.

Proof Define

V(X,S,I)=X2 + S2 +12, (X(t),S(t),I(t)) e R+.

Applying Itô's formula to stochastic differential system (2) yields

, , a12X2 S , „ , a21S2 X , , ,

dV = LVdt - tt—--çT dBi(t) + —-ZT dBi(t)

2(1 + pX + qS) 2(1 + pX + qS)

- 1 a S 2 IdB2(t) + I —I 2 SdB2(t),

/ «12S ^ i -22x 2 s

LV = -X2 b - fluX -

2 V 11 1+PX + qS J 8(1+ pX + qS)2

1 1 / «21x \ ct,2s 2 x

+ - S2 -c - U22S + --21-- - pI) -■ 21

2 V 22 1+PX + qS ) 8(1+ pX + qS)

1 1 / . X 1 ^ 1 1 ^ 1

+ -12 (-d - fl33I + P S) - - a 2S 21 - - a 212 S

1 3 1 1 1 3 «21 11 3 11

<--«11X2 + - bX2 - - «22S2 + — S2 - -«3312 + -P12 S.

2 2 2 2p 2 2

Applying the Holder inequality ab < p + q, p + q = 1 (p, q > 1), we have

1 13 2 3

12 S < -12 + - S 2.

Therefore,

1 1 ,, 1/ 2 „3 1/«21 - \ „1

LV < — «11X2 + -(b + 2)X2 - - ( «22 "P )S2 + -1^-21 + 2S2

1 ( 1 \ 3 1/1 1 1 \

- - i «33 - - Pi 12 + 12 - (X 2 + S 2 + 12)

< H0- V(X, S, I),

where H0 > 0 is a positive constant. Thus

, r / >n , a12X2S , , , a21S2X , „

dV < H0 - V(X, S, I)] dt - 12-- dB1(t) + 21-- dB1(t)

2(1 + pX + qS) 2(1 + pX + qS)

- 1a S 2 IdB2(t) + 1a 12 SdB2(t). Applying Ito's formula to efV(X, S, I) yields

d(eV (X, S, I )) = et[V(X, S, I) dt + dV (X, S, I)]

< etH0 dt + ê

-12X 2 S -21S 2 X

■ dBi(t) + —-— dBi(t)

2(1 + pX + qS)

2(1 + pX + qS)

- iff S 2 IdB2(t) + 1—12 SdB2(t)

So we have

etEV(X,S,I) < V(X(0),S(0),I(0)) + H0(et -l)

limsup EV(X, S, I) < H0.

Applying the second inequality of Lemma 2.2 and letting n = 3,p = j,we have

3?A0|(X(t),S(t),I(t))|2 < V(X,S,I). Thus, we obtain

limsupE|(X(t),S(t),I(t))|2 <H.

t—+ TO

Therefore, for any e > 0, set x = "2", applying the Chebyshev inequality, we have

P||(X(t),S(t),I(t))|>X) < E|(X(t),^),I(t))|2,

that is,

limsupP||(X(t),S(t),I(t))| > x) < e. □

3 Asymptotic behaviors

System (1) has three equilibrium points [32]: (i) when R0 = c^2^) < 1, system (1) has an equilibrium poto E1(K, 0,0); (ii) when Ro = ^^ > 1 and Ri = (c+ dy afa <

1, system (1) has another disease free equilibrium point E2(X,S,0); (iii) when R1 = . J*22"2lb, qdalu > 1 system (1) has a positive equilibrium point E3X*,S*,I*). For its

(c+ ¡5 )("11 +Pb+ ¡5 )

stochastic system (2), however, these equilibrium points do not exist.

In this section, we study the asymptotic behaviors of model (2) around the three equilibrium points El(K, 0,0), E2(X, S, 0), and E3(X*, S*,I*) of its deterministic model (l), respectively.

3.1 Asymptotic behaviors around the equilibrium point E1 of system (1)

When R0 < l, system (l) has an equilibrium point El(K, 0,0) = (b, 0,0), but it is not the equilibrium point of system (2). In this subsection, we study the asymptotic behaviors of system (2) around El(K, 0,0).

Theorem 3.1 Let (X (t), S(t), I(t)) be the solution of model (2) with initial value (X(0), S(0), I(0)) e R+. IfR0 < l and K = b < ^, then

limsup - [(X(0)-K)2+ S(0)2+12(0)] de < 2 t^œ t J0 2q2 Wl

where Wl = min(au, ^i^, ^^}.

l L 1-)' a2l a2l '

Proof Note that (K, 0,0) is the equilibrium point of system (1), where K = b.

Define

V(X,S,I)= ( X - K - Kln X I + —(S +1).

Applying Ito's formula to stochastic differential system (2) yields

^r a12(X - K)S a12a21SX

dV =LVdt - --n-- dB1(t) + —--— dB1(t),

1 + pX + qS

«21(1 + pX + qS)

b - «11X -

S -C - «22S +

LV = (X - K )

+ «12

= (X - K )

+ «12

< -«11 (X - k)2 +

1 + pX + qS «21X

—122KS2

1 + pX + qS

2(1 + pX + qS)2 - P I) -1 (d + «33I - P S)

b - «11 (X - K)-«uK -

1 + pX + qS

2(1 + pX + qS)2

SI -c - «22S +

1 + pX + qS

- PI -1 (d + «33I - P S)

«12KS 122KS2 «12 2 2

1-V-? 1V-«2--(CS + «22S + «33I2

1 + pX + qS 2(1 + pX + qS)2 «21 v '

< -au(X - K)2 + a, (K - ±)S + ^^ - S2 -

a21 2q2 a21

< -au(X - K)2 - S2 - I2 + ^

a21 a21 2q2

Integrating equation (8) from 0 to t, we obtain

«12«33 12 «21

V(t) - V(0) < -f t«11 (x(0)-K)2 de - «^ i s2

0 «21 0

(e ) de

12(e) de + —2Kt+M1(t),

a21 0 2q2

M1(t) =

—12(x(e) - k)S(e) «12—21S(e)x(e)

1 + pX(e ) + qS(e ) «21(1 + pX(e ) + qS(e )) _

dB1(e )

is a real-valued continuous local martingale. Thus

m1, m1>t r limsup-= limsup

t—+to t t—+to ^ j0

—12(x(e )-k )S(e ) «12—21S(e )x(e )

1 + pX (e ) + qS(e ) «21(1 + pX (e ) + qS(e )) _

2 22 —12 + «12—21 ,2 + „2 n2

< +TO.

q2 a221q2

Applying the strong law of large numbers, we obtain limt^+TO Mt^ = 0.

Dividing equation (9) by t and taking the limit superior, we have

limsup

t^œ t

aii(X(e) -O2 + S2(e) + I2(e) a21 a21

de < ~12J

1 C t o2 K

limsup-/ [an{X(0)-K)2+ S2(0)+I2(0)]dO < -f—. □

t^TO t jo 2q2 Wi

Corollary 3.1 From Theorem 3.1, when o12 = 0, we have

LV < -au(X - K)2 - S2 - a12a3/2 < o,

a21 a21

thus when R0 < 1 andK = b < ho/d, the equilibrium point E1(K, 0,0) of system (1) is globally asymptotically stable.

Remark 3.1 From Theorem 3.1, if the interference intensity is sufficiently small, the solution of model (2) will fluctuate around the equilibrium point E1(K, 0,0). Moreover, the fluctuation intensity is related with the disturbance intensity: the fluctuation intensity is positively correlated with the value of o12.

3.2 Asymptotic behaviors around the equilibrium point E2 of system (1)

When R0 > 1 and R1 < 1, system (1) has an equilibrium point E2(X,S, 0), but it is not the equilibrium point of system (2). In this subsection, we study the asymptotic behaviors of system (2) around E2(X,S, 0).

Theorem 3.2 Let (X (t), S(t), /(t)) be the solution of model (2) with initial value (X(0), S(0), I(0)) e R+. /fR0 > 1,R1 < 1 and a11q > a22p, then we have

limsup1 ft[(x(e) -X)2 + (S(e)-S)2 + I2(e)]de <

t^+œ t Jo W2

ai22x a12 (1 + Px) / 0-21S 0 Sn2

U-2 = -ô—I--=- I -T- +--

2 2q2 «21(1 + qS)\2p2 2 0

1V7 . [ «12P «12^22(1+ PX) fl12«33(1+ pX)l W2 = mim a11--,-=—,-=— k

( q «21(1 + qS) «21(1 + qS) J

Proof Noting that (X, S, 0) is the equilibrium point of system (1), thus

, — «12S — «21X

b - «11X--=-= =0, c + a22S--=-= = 0.

1 + pX + qS 1 + pX + qS

Define

V (X, S, I) = (x - X - X ln X ) + «12(1+ pX) (s - S - S ln S )

V X/ «21(1 + qS)\ S J

«12(1+ pX) «12(1+ pX)

:= V1 +-— V2 +-— V3.

«21(1 + qS) «21(1 + qS)

«12(1+ pX)

+-— I

«21(1 + qS)

Applying Itô's formula to stochastic differential system (2) yields

, , —11 (X - X)S , ,,

dV1 = LV1 dt - - 1U y ' dB1(t), 1 + pX + qS

LV1 = (X - X ) = (X - X ) = (X - X )

b - «11X -

1 + pX + qS _

—122XS2

2(1 + pX + qS)2

b - «11 (X - X)-«uX -

1 + pX + qS _

2(1 + pX + qS)2

-«11 (X - X) + «12

pS(X - X)-(S - S)(1+ pX) (1 + pX + qS)(1 + pX + qS)

—¿XS2

2(1 + pX + qS)2

= - «11 (X - X)2 +

«12pS(X - X )2

__«12(1+ pX )(S - S)(X - X )

(1 + pX + qS)(1 + pX + qS) (1 + pX + qS)(1 + pX + qS)

—¿XS2

2(1 + pX + qS)2 '

Similarly,

dV2 = LV2 dt + —21(S S)X dB1(t) - — (S - S)IdB2(t), 1 + pX + qS

LV2 = (S - S) = (S - S) = (S - S)

-c - «22S n - PI

1 + pX + qS -c - «22 (S - S) - «22S +

—221SX2

+ —-12 2(1 + pX + qS)2 2

1 + pX + qS

—ni SX2 — S o 21 ■ + — I2

2(1 + pX + qS)2 2

' ^ (X - X)(1 + qS)-qX(S - S) " -«22 (S - S) + «21 --=--- - PI

(1 + pX + qS)(1 + pX + qS)

—221SX2

2(1 + pX + qS)2 2

= - «22 (S - S)2 + «21

- «21

qX(S - S)2

(1 + qS)(X - X)(S - S) (1 + pX + qS)(1 + pX + qS) (1 + pX + qS)(1 + pX + qS)

- PI (S - S) +

2(1 + pX + qS)

—2S 2 +-12.

Also, we have

dV3 = I (t)[-d - a^I + ß S] dt + a SI dB2(t).

dV = LVdt -^ - X)S dBi(t) + ai^d^ l+pX + qS a2i(l + qS)

- a (S - S)I dB2 (t) + a SI dB2 (t)

a2i(S - S)X l + pX + qS

dBi(t)

fl/ fl/ al2(l+ pX)fl/ al2(l+ PX)fl/

LV = LVl +-— LV2 +-— LV3

a2l(l + qS) a2i(l + qS)

2 ai2pS(X - X)2

= - au(X - X)2 +

__ai2(l+ pX )(S - S)(X - X)

(l + pX + qS)(l + pX + qS) (l + pX + qS)(l + pX + qS)

ai2(l + pX)

2(l + pX + qS)2 a2i(l + qS)

qX(S - S)2

- a2l-=-=-

(l + pX + qS)(l + pX + qS)

+1 (-d - a331 + ß S)

-a22(S - S)2 + a2i

- ß I(S - S)+ a2lSX2

(l + qS)(X - X)(S - S) (l+ pX + qS)(l+ pX + qS)

2(l + pX + qS)2 2

- al2p\(X - X)2 - al2a22 (l + pX) (S - S)2 - al2a33(l + pX) ^ V q J a2i(l + qS) a2i(l + qS)

ai2(l+ pX) -

+-=-(ßS - d)I +

a2i(l + qS)

al2XS2

2(l + pX + qS)2

ai2(l + pXU a22SX2 a2S 2 + a2i(l + qS) \2(l + pX + qS)2 + ~I

Since ßS < d, thus

LV < -1 aii - ^^ )(X-X)2 - ai2a22(l+ pX) (S-S)2 -

ai2p q

a2i(l + qS)

ai2a33(l + pX) 2 a2i(l + qS)

al2lX + al2(l + pX) /ai22S + a2SC2\ 2q2 + a2i(l + qSU2p2 +2 0

Integrating both sides of equation (10) from 0 to t yields

v(t)- V(0) < Jo^ -^11- (X(0)- X)2

a12a22(1 + pX) ^2 a12a33(1 + pX) 2

a2i(l + qS)

-(S(0) - S)

a2i(l + qS)

al2X + ai2(l + pX) / a22LS + a2S ^ a2i(l + qSA2p2 + ^" 0

t + M2(t)+M3(t),

M2(t) =

—12(x (e )-x)s(e ) + «12(1+ pX) —2l(s(e )-S)x(e ) " 1 + pX(e ) + qS(e ) + «21(1 + qS) 1 + pX(e ) + qS(e )

dB1(e )

M3(t) =

t «12—(1 + pX)SI(e ) 0 «21(1 + qS)

dB2(e)

are real-valued continuous local martingales. Thus

M2, M2>t lim sup-

t—+TO t

1 (t "

= lim sup - I

t—+TO t Jq

—12(X (e )-x )S(e ) + «12(1+ pX) —2l(s(e )-S)x(e ) " 1 + pX(e ) + qS(e ) + «21(1 + qS) 1 + pX(e ) + qS(e ) J

—122 «12—21(1 + px)2

JV1 + «12 — 21(

q2 + _2 n2(

L q2 «21p2(1 + qS)2 J

<M3, M3>, 1 lim sup-= lim sup -

t—+TO t t—+ TO t

«12—(1+ pX)SI(e )

«21(1 + qS)

C2 < C0

«22—2s2(1+ pX) (1 + qS)2

< +TO.

Applying the strong law of large numbers, we have limt—+TO ^^ = 0 (i = 2,3). Dividing equation (11) by t and taking the limit superior, we have

lim sup 1 Tiï«11- «^ (X(e )-X)

t—+TO t Jq L\ q /

2 «12«22 (1 + pX)

+ ----- -_ • (S(e2

t—+to t jq l\ q ' «21(1 + qS)

«12«33 (1 + pX )

«21(1 + qS)

^i 2(e )

122X «12 (1 + pX) ( —21S -2^2 de < —-—1--=-1 —— +--cn

" 2q2 «21(1 + qS)\2p2 2 0

limsup1 i [(x(e)-x)2 + (S(e)-5)2+ I2(e)] de

t—+to t Jq

t—>+to t J0

^ ' ' — - W2 Corollary 3.2 From Theorem 3.2, when a12 = a21 = a = 0, we have

LV < - «11-«2p) (X - X)2-«12«22(1+ pX) (S - S)2-«12«33,(1+ pX) 12 < 0,

«21(1 + qS)

«21(1 + qS)

thus when a11q > a22p, R0 > 1 andR1 < 1 hold, the equilibrium point E2(X, S, 0) of system (1) is globally asymptotically stable.

Remark 3.2 From Theorem 3.2, if the interference intensity is sufficiently small, the solution of model (2) will fluctuates around the equilibrium point E2(X, S, 0). Moreover, the fluctuation intensity is related with the disturbance intensity: the fluctuation intensity is positively correlated with the value of a12, a21 and a.

3.3 Asymptotic behaviors around the positive equilibrium point E3 of system (1)

When R1 > 1, system (1) has a positive equilibrium point E3(X*,S*,I*), but it is not the equilibrium point of model (2). Now, we explore the asymptotic behaviors of system (2) around E3(X*, S*, I *).

X(t) is a temporally homogeneous Markov process in E¡, which is given by the stochastic differential equation

dX(t) = b(X) dt am(x) dBm(t),

where E¡ c Rl represents a l-dimensional Euclidean space. The diffusion matrix of X(t) is given by

A(x) = (fly(x)), aij (x) = £ am (x)a'm (x).

Assumption 3.1 ([33]) Assume that there is a bounded domain U c E¡ with regular boundary, satisfying the following conditions:

(1) In the domain U and some of its neighbors, the minimum eigenvalue of the diffusion matrix A(x) is nonzero.

(2) When x e E¡\U, the mean time t at which a path starting from x to the set U is limited, and supxeHExT < to for every compact subset H c E¡.

Lemma 3.1 ([33]) When Assumption 3.1 holds, the Markov process X (t) has a stationary distribution fi(-) with density in El. Letf (x) be a function integrable with respect to the measure ¡x, where x e El, then, for any Borel set B c El, we have

lim P(t, x, B) = ¡(B)

Px\y\—m¿/ f(x(t))dt = jff(x)¡(dx)j = 1.

Theorem 3.3 Let (X(t), S(t), I (t)) be the solution of model (2) with initial value (X(0), S(0), I(0)) e R+. If a11q > a12p and R1 > 1 hold, then

limsup1 f [(X(0) -X*)2 + (S(0) -S*)2 + (I(0)-I*)2] d0 <

t—t Jo W3

a^X* a12(1+ pX*) /a21S* Cga2 . )\ a21(1 + qS*)\2p2~ + 2 {S +I})

W3 = mim a11 -

a12p a12a22(1+ pX*) a12a33(1 + pX*)

q ' a21(1 + qS*) ' a21(1 + qS*) Proof Noting that (X*, S*, I*) is the equilibrium point of system (1), thus

b - a11X* -

1+pX*+qS*

-c - a22S* + j+0qs* - PI* = 0,

PS* - d - a33I* = 0.

Define

V(X,S,I) = (X -X* - X* ln + a12((1 + pXVS - S* - S* ln S V X*/ a21(1 + qS*) \ S*

+ a12(l+ pX) / - I* - I * m I

a21(1 + qS*) \ I*

a12(1+ pX *) a12(1 + pX*)

:= V1 +-V2 +-V3.

a21(1 + qS*) a21(1 + qS*)

Applying Ito's formula to the stochastic differential system (2) yields

a12(X - X*)S

dV1 = LV1 dt -

1 + pX + qS

dB1(t),

LV1 = (X - X*)

b - a11X -

a22X * S2

1 + pX + qS = (X - X*) b - an(X -X*) - auX* -

2(1 + pX + qS)2 a12S

1 + pX + qS_

a122X *S2

2(1 + pX + qS)2

= (X - X*)

-au(X - X*) + a12

pS*(X - X*) -(S - S*)(1 + pX*) (1 + pX* + qS*)(1 + pX + qS)

a2X*S2

2(1 + pX + qS)2 = -au(X - X*)2 + ■

a12pS*(X - X *)2

a22X *S2

(1+ pX* + qS*)(1 + pX + qS) 2(1+ pX + qS)2 a12(1 + pX *)(S - S*)(X - X *)

(1 + pX* + qS*)(1 + pX + qS)'

Similarly,

a (S - S*)X

dV2 = LV2 dt + 21( - )^ dB1(t) - a(S - S*)IdB2(t), 1 + pX + qS v '

LV2 = (S - S*) = (S - S*)

-c - a22S +--ß I

1 + pX + qS

a 2 ç*

S*X2 1

2(1 + pX + qS)2 2

+ - a 2S*I2

-c - ^ s - s*)- a22S* + i+p~X+qS - ^1 -1 *)- ß I*

a2lS*X2 + 1 a 2S*I2

2(1 + pX + qS)2 2

= (S - S*) -a22(S - S*) + a21(X - X*)(1 + qS*)-qX*(S - S*) - ß(I -1*

(1 + pX* + qS* )(1 + pX + qS)

a221S*X2 1

+ - a 2S*I2

2(1 + pX + qS)2 2 = - a (S - S*)2 + a (1 + qS*)(X - X*)(S - S*) a2^ ' + a21(1+ pX* + qS*)(1 + pX + qS)

- ß(I -1*) (S - S*)

qX*(S - S*)2

a2 S*X2 1

(1+ pX* + qS*)(1 + pX + qS) 2(1+ pX + qS)2 2

+ - a 2S*I2.

Also, we have

dV- = LV3 dt + a (I -1*)SdB^i),

LVi = (I -1*) [-d - a-31 + ßS] + 2a2I*S2

= (I - I*)[-d - a-3 (I -1*) - aiiI* + ß (S - S*) + ßS*] + 2a2I*S2 = -a-3 (I -1*)2 + ß (S - S*)(I -1*) + 1 a 2I * S2.

Then we have

, a12(X - X*)S , a12(1+ pX*) dV = LVdt - , ( „-)s dB1(t)+ 12( p

1 + pX + qS 1 a2i(1 + qS*) - a (S - S*)IdB2 (t) + a (I -1*)SdB2 (t)

a21(S - S*)X _ 1 + pX + qS

dB1(t)

V = LV1 + OS^ LV2 + îs^ LVi

a21(1 + qS*) î21(1 + qS*)

2 a12pS*(X - X*)2

= - an(X - X*) +

a2 X * S2

(1+ pX* + qS*)(1 + pX + qS) 2(1+ pX + qS)2

î12(1 + pX*)(S - S*)(X - X*) î12(1 + pX*)

(1+ pX* + qS*)(1 + pX + qS) a2i{1 + qS*)

(1 + qS*)(X - X*)(S - S*)

+ a21 T.-TZ-Z-TT:-—--T - a21 -

-Î22(S - S*)2- ai^I - I*)2 qX* (S - S*)2

(1 + pX* + qS*)(1 + pX + qS) (1 + pX* + qS*)(1 + pX + qS)

a221S*X2 a2S* 2 a 2I* 2'

—-21-TTT + -12 + -S2

2(1+ pX + qS)2 2 2

_ . «12p\(v ^)2 «12«22(1 + pX*) ( )2 «12«33(1+ pX*) ( ^2 < - «11-—J (X - X) - «21(1 + qS*) (S - S) - «21(1+ qS*) (I - 7)

—22X * «12(1+ pX *)

2q2 a21(1 + qS*) It is easy to see that, for any

$ < min j ^a11 - -^^X*,

the ellipsoid

—2 S* C2—2

■(S* +1 *)

«1^\X* «12«22(1 + pX*)S* «12«33(1 + pX*)I* q J , «21(1 + qS*) , «21(1 + qS*) h

- «11 - (x-X*)2 - «12«22(1+ pX*) (S-s*)2 1 11 q r ' «21(1 + qS*) V '

«12«33 (1 + pX *)

«21(1 + qS*)

(I - 7*) 2 + $ = 0

lies entirely in R+. Let U to be any neighborhood of the ellipsoid with U ç E3= R+, thus for any x e U\Ei, we have LV < -M (M is a positive constant). Therefore, condition (2) in Assumption 3.1 is satisfied. Moreover, there exists a G = min{ —^x^ —22x2, —32x2, (x1,x2,x3) e U} >0 such that

J2\J2«ik (x)«jk (x)j tëi = —i^i + —22x2f22 + > G||£ II2

i,;=1 \ k=1

for all x e U, % e R3, which means condition (1) in Assumption 3.1 is satisfied. Therefore, the stochastic model (2) has a unique stationary distribution ¡(-), it also has the ergodic property.

Integrating equation (12) from 0 to t on both sides yields

V(t) - V(0) <

- a11 -

«12 «33(1 + pX *) «21(1 + qS*)

«12p) (x(e )-x *)2- «««f+|p (S(e )-s*)2

(i(e )-i*

—X «12(1 + px*)/—^* c2—2 *

_ 2q2 + «21(1 + qS*) V 2p2 + 2 1 + ' + M4(t)+M5(t),

M4(t) =

—12(X (e )-x *)S(e ) + «12(1+ pX*) —21(S(e )-s*)x(e ) " 1 + pX(e ) + qS(e ) + «21(1 + qS*) 1 + pX(e ) + qS(e )

dB1(e )

M5(t) =

t «12 (1 + pX*) «21(1 + qS*)

[-— (S(e ) -S*) I (e ) + — (I (e )-i *) S(e )] dB2(e )

are real-valued continuous local martingales.

limsup

(M4, M4)t t

= limsup - I

t^+œ t Jo

1 ^r«12(1 + pX*) cr21(S(0)-S*)X(e) a12(X(Û)-X*)S(e)n2

«2i (1 + qS*) 1+pX(e ) + qS(e ) 1+pX(e )+qS(e )

d -4^(1 + pX*)2 _ q2 + «21p2(1 + qS*)2

limsup

(M5, M5)t t

: limsup 1 (c S*/(e )-cI* S(e ))2 de < 2C2c 2(S* + I*) <+œ.

t^+œ t Jo

Applying the strong law of large numbers, we have limt—^^ = 0 (i = 4,5). Dividing equation (13) by t and taking the limit superior, we have

limsup - I

t^+œ t Jo

«11-f) <X<e )-X')2 + Z-jj+g?- ^ >"S')2

+ -12-3f(1+ pX • ) (/(e )_/.

-21(1 + qS*) c2X* -12(1+ pX*)

^S* C2c2

2q2 -21(1 + qS*)

S* + I*

limsup1 / [(x(e) -X*)2 + (S(e) _S*)2 + (/(e)_/*)2] de < ^

t^+œ t Jo W3

(14) □

Corollary 3.3 From Theorem 3.3, when a12 = a21 = a = 0, we have

JVC L a12P W Y*^ a12a22(1 + PX*) , ,2 a12a33(1+ pX *) , *x2

IV < ^n-—J (X - X) - a21(1 + qS*) (S - S) - <221(1 + qS*) ^ -

Thus when a11q > a22p and R1 > 1 hold, the positive equilibrium point E3(X*, S*, I*) of system (1) is globally asymptotically stable.

Remark 3.3 From Theorem 3.3, if the interference intensity is sufficiently small, the solution of model (2) will fluctuates around the equilibrium point E3(X*, S*,I*). Moreover, the fluctuation intensity is related with the disturbance intensity: the fluctuation intensity is positively correlated with the value of a12, a21 and a.

Remark 3.4 If the conditions in Theorem 3.3 are hold, then the solution of model (2) has a unique stationary distribution, it also has the ergodic property.

4 Persistence in mean and extinction

When we consider a biological population system, persistence in mean and extinction are two very important properties. In this section, we investigate the persistence in mean and extinction of system (2).

Since there is no equilibrium point in system (2), we cannot determine the persistence of system (2) by proving the stability of the equilibrium point as a deterministic system.

Definition 4.1 ([5]) The definition of persistence in mean and extinction are given as follows:

(1) The species X(t) is said to be in extinction if limt—+TOX(t) = 0.

(2) The species X(t) is said to be in persistence in mean if limt—+TO (X(t)>* > 0.

Lemma 4.1 ([34]) LetX(t) e C(Q x [0,+to),R+). (1) If there exist T > 0, X0 > 0, X, ni, when t > T, we have

f t ' lnX(t) < kt - ko I X(s) ds + V" HiB(t) «.s.,

Jq i=1

<X>*< k «.s., if k > 0;

limt —+to X(t) = 0 «.s., if k < 0.

(2) If there exist T > 0, k0 > 0, k >0, ni, when t > T, we h«ve

f t ' lnX(t) > kt - ko / X(s) ds + V" niB(t) «.s.,

then <X>* > k «.s. 4.1 Persistence in mean

Theorem 4.1 Let (X (t), S(t), 7(t)) be the solution of model (2) with initi«l v«lue (X(0), S(0), I (0)) e R+. Model (2) h«s persistence in me«n if conditions «11 q > «12p, R1 > 1, «nd

! • ,v* W3 „* W3 f* W3 q = max{ —12, —21, —} < mi^ Xy —-, S* / —,IM —

U0 U0 U0

hold, th«t is,

liminf1 i X(e) de > 0, liminf1 i S(e) de > 0, liminf1 i I(e) de > 0,

t—+TO t Jo t—+TO t Jo t—+TO t Jo

Uo=V + «21(1 + qS* A^p2 + 2

X* «12(1+ pX*)( S* CQ(S* +1*)

U3 «nd W3 «re defined in Theorem 3.3.

Proof Applying equation (14) in Theorem 3.3 we have

limsupt—1/0(X(0) -X*)2 < f, limsupt—1 /0(S(0) -S*)2 < W3,

t—+TO t Jo v

limsupt—1 /0(I(0)-I*)2 < |3.

Applying the inequality 2a2 - 2ab < a2 + (a - b)2 to X(t), we have

X* (X - X*)2

X >-----.

" 2 2X *

Therefore

a2X* a12(1+ pX*)

2q2 a21(1 + qS*)

a221S* + C0a

■(S* +1 *)

X* a12(1 + pX*) ( S* C2(S* +1*)

_2q2 a21(1 + qS*)\2p2

= Q2UO.

When q < X\/ —, we have

1 f , , , X* 1 /'t liming X(0)d0 >--limsup-

t—t Jo 2 t—t Jo

1 /,t (X(0)-X*)

t—+TO t Jo

2 2W3X * X* a 2Uo

T - 2W3X*

Similarly, when q < s\j — ,we have

1 T ^ , S* 1 ^ (S(0)-S*)2 liming I S(0)d0 >--limsup- I

t—t Jo 2 t—t Jo

2 2W3S* S* a 2Uo

2 2W3S*

When q < i\j — ,we have

1 r ^ , I* 1 r

liminf- I(0)d0 >--limsup-

t—t Jo 2 t—t Jo

1 ^ (I(0 )-I*)2

2 2W3I *

I* —2UQ > Y - 2 W37*

>0. □

Remark 4.1 From Theorem 4.1, when R1 > 1, a11q > a12p and the intensity of random disturbance is sufficiently small, system (2) will persistence in mean. This shows that biological populations can resist a small environmental disturbance to maintain the original persistence.

4.2 Extinction

Extinction and persistence in mean are closely related, so we also concern ourselves with the situation of population extinction. In this subsection, we point out the conditions of predator extinction.

Theorem 4.2 Let (X(t), S(t), I (t)) be the solution of model (2) with initi«l v«lue (X(0), S(0), I (0)) e R+. If one of the following conditions holds:

(1) —21 > max{-^=, ^«21p},

(2) R* = «ï-i <1—21 <V«1?-

pc 2p2 c

lim X(t) = —,

t—+TO «11

lim S(t) = 0,

t—+ TO

lim 7(t) = 0.

t—+ TO

Proof Applying Ito's formula to the second equation of stochastic differential system (2) yields

d ln S(t) =

22 221X2

2 2(1 + pX + qS)2

- «22S +

1 + pX + qS

1 + pX + qS

dB1(t)- —IdB2(t)

-c - «22S -

«21 y

2 \ 1 + pX + qS —21 / 2—

—21X 1 + pX + qS

dB1(t)- —IdB2(t).

Case I. When —21 > max{^2=^«21p}, inequality (16) takes its maximum value on the

interval [0, ¿]at «21, so we have

d ln S(t) <

-c - «22S + „

22 2 2

dt + ——X— dB1(t) - —IdB2(t). 1 + pX + qS

Integrating (16) from 0 to t and dividing it by t, we get

—21X(e)

1 S(t) - ln ——

t S(0)

Applying Lemma 4.1, we obtain

tln 575) < «t"c - «2^ S(t8+

1 + pX(e ) + qS(e )

dB1(e )-t"M — I(e ) dB2(e ).

lim S(t) = 0.

t—+TO

Case II. When R* = pr - ^pc < 1 and a21 < v^ip' inequality (16) takes its maximum value on the interval [0, i] at i, so we have

l 'pj p'

d ln S(t) <

a21 a21 c

--TT? - C - fl22S

dt + —a21X— dBi(t) - aIdB2(t). 1 + pX + qS

Integrating (16) from 0 to t and dividing it by t, we obtain

-ln^ < — - - c - a22{S(t)) + t1 i --dBi(e)

1 S(t) a2i f O2iX(0)

- ln —-—- <---- - c - a22S(t) + t -—- „ ,

t S(0) " p 2p2 22M)/ J0 1+ pX(0) + qS(e)

- t 1 / a I (0) dB2(0)

= cf- - -1) - a22(S(t)) + t-1 f11 )S(0) dB1(0)

\cp 2cp2 / Jo 1 + pX(0) + qS(0)

- t 1 f a I (0) dB2(0)

= c(Ä* -1) - a22(S(t)) + t1 i -

t a21X(0)

0 1+pX(0) + qS(0)

dB1(0)

- t-1 / a I (0) dB2{0). 0

Applying Lemma 4.1, we obtain

lim S(t) = 0.

Applying Ito's formula to the third equation of stochastic differential system (2), one has

d ln I (t) =

, a 2S2 -d---a331 + ß S

dt + a SdB2(t).

Since limt^+TOS(t) = 0, there is an arbitrarily small constant e >0 such that when t > T,we have - + PS < e, thus

/ a 2 S2 \ ln I (t) = i -d - a33I + P S--— \dt + a SdB2(t)

< (e - d - a33l) dt + aSdB2(t). (17)

Integrating equation (17) from 0 to t and dividing it by t yields

1 I (t) ft

- ln < e - d - a33(!(t)) + t1 aS(0) dB2(0).

t 1 (0) J0

Applying Lemma 4.1 and the arbitrariness of e, we obtain

lim I(t) = 0.

t—+TO

«12S 122S2 12S

Similarly,

d ln X(t) = b - auX - --12-e - ^TT^-dt -1-12-c dB1(t).

\ 1 + pX + qS 2(1 + pX + qS) v 1 + pX + qS

Since limt^+TO S(t) = 0, there is an arbitrarily small constant e >0 such that when t > T,

we have ,—S—5 < e, thus

1+pX+qS

d ln X(t) > (b - anX - a12e - a12,e2) dt--^^-dB1(t).

v ' 1 + pX + qS

Integrating the above equation from 0 to t and dividing it by t, one has 1, X(t) 2 2 .-1 ft °12S(V)

ln —> b - «12S - —f2e2 - «11 X(t) -t-1 -dB1(e).

t"*X(0)_" ~12" 12 - J0 1+pX (0 ) + qS(0)'

Applying Lemma 4.1 and the arbitrariness of e, we obtain b

lim X(t) > —. (8)

t ^+to a11

On the other hand,

dlnX(t) =(b - anX - a12S „ - ^^) dt - g12S „ dB1(t)

1+pX + qS 2(1 + pX + qS)2 / 1+pX + qS

< (b - a 11X) dt - g"S dB1(t). (19)

1 + pX + qS

Integrating equation (19) from 0 to t and dividing it by t, we have

1 In X| <b - «11 (X(t)) -1-1 f11 >) dB1(0).

t X (0) J0 1+ pX(0) + qS(0)

Applying Lemma 4.1, we obtain

lim X(t) < —. (0)

t ^+to a11

From (18) and (20), we have b

lim X(t) = —. □

t ^+to a11

Remark 4.2 From Theorem 4.2, if the intensity of random disturbance is sufficiently large or R* <1 and a12 < ^Ja21p, the predator population will be extinct.

5 Conclusions and numerical simulations

This paper investigates a stochastic infected predator-prey model with Beddington-

DeAngelis functional response. The existence of a global positive solution of model (2) is

first proved, then we show the stochastically ultimate boundedness of the solution. In addi-

tion, by using the Lyapunov method and Itô's formula, we study the asymptotic properties

■X(t) ■S(t) ■I(t)

0 10 20

Figure 1 Time sequence diagram and phase portrait of model (2). (a) The deterministic model; (b) the Brownian motion model with a12 = a21 = a = 0.5; (c) phase portrait: the red o is corresponding to the deterministic model, while the blue o represents the stochastic model.

and stationary distribution of the solution of model (2) around the equilibrium points of its deterministic. At last, we discuss the persistence in mean and extinction of model (2). The biological significance of the result shows that the external environment disturbance may have a certain effect on the stability of the biological system: the population's ability to adapt to the environment is limited. If the disturbance in the environment is small enough, the stability of the population will not be destroyed; if large disturbances occur in the environment, it may lead to the extinction of species.

We next give some numerical simulations to support our results. We consider the following discrete equations:

'= X„ + X„[b - anX„ - j+X+sn] A - T+XXk A ,

• Sn+1 = + [-C - «22Sn + 1+pXXk - At + lOXXtSk A W1k - aSnInAW2k, In+1= In + InW Sn - d - a33In]At + a SJnAW2k,

where At = 0.01, AWik = W(tk+1) - W(tk) obeys the Gaussian distribution N(0, At).

In Figure 1, we choose X(0) = 2, S(0) = 2,I(0) = 2, b = 1,c = 0.4, d = 0.2, an = 0.8, «12 = 0.55, «21 = 0.2, «22 = 0.1, «33 = 0.1, W = 0.2,p = 1,q = 1, and step size At = 0.001.

Under this condition,

E1 = (K, 0,0) = (1.25,0,0), R0 = 0.76, K = — = 1.25 < — = 2.

«11 «21

The numerical simulation of Figure 1 is consistent with our conclusion in Theorem 3.1.

In Figure 2, we choose X(0) = 2, S(0) = 2,I(0) =2, b = 0.6, c = 0.2, d = 0.3, «11 = 0.8, «12 = 0.6, «21 = 0.8, «22 = 0.1, «33 = 0.4, W = 0.2,p = 1,q = 1, and step size At = 0.001.

Under this condition,

E2 = (X,S, 0) = (0.6,0.4,0), «11 q = 0.8 > «22P = 0.1,

R0 = 5 > 1, R1 = 0.53 < 1.

In Figure 2(a), we choose a12 = a21 = a = 0.1, thus

1 ft —9 2 U2

limsup- / [(X(0)-X) + (S(0)-S) +12(0)] de <-2 = 11.8648.

t^+TO t J0 W2

Figure 2 Time sequence diagram and phase portrait of model (2). (a)-(c) are a Brownian motion model with a12 = a21 = a = 0.1,0.2,0.4, respectively. (d)-(f) are phase portraits of (a)-(c), respectively. The red o is corresponding to the deterministic model, while the blue o represents the stochastic model.

In Figure 2(b), we choose a12 = a2i = a = 0.2, thus

1 ft — 1 1 — limsup i / [(X(0)-X) + (S(0)-S) +12(0)] d0 < —- = 47.4592.

t—+to t Jo

^ > - W2

In Figure 2(c), we choose a12 = a21 = a = 0.4, thus

1 Ct — 2 2 —

limsup - [(X(0)-X) + (S(0)-S) + /2(0)] d0 < —2 = 189.8369.

t—+ TO t Jo W

Figure 2 shows that the solution of model (2) fluctuates around the equilibrium E2(0.6,0.4,0). In addition, the fluctuation intensity is related with the disturbance intensity: with the increase of a12, a21, a, the fluctuation intensity is also increasing. These all meet the conditions of Theorem 3.2.

In Figure 3, we choose X(0) = 2, S(0) = 2,1(0) = 2, b = 1,c = 0.1, d = 0.1, an = 0.5, «12 = 0.3, «21 = 1, a22 = 0.1, a33 = 0.2, fi = 0.5,p = 1, q = 1, and step size At = 0.001.

Under this condition,

E3 = (X*, S*, I*) = (1.9085,0.5231,0.8071), a11q = 0.5>«22p = 0.3, R1 = 5.21 > 1.

In Figure 3(a), we choose a12 = a21 = a = 0.03, thus

limsup1 i [(X(0)-X*)2 + (S(0)-S*)2 + (I(0)-I*)2] d0 < —3 = 8.1131.

t—>+to t J0 W3

■ X(t) of model (1)

■ S(t) of model (1)

■ I(t) of model (1) X(t) of model (2)

■ S(t) of model (2) I(t) of model (2)

■ X(t) of model (1)

■ S(t) of model (1)

■ I(t) of model (1) X(t) of model (2)

■ S(t)ofmodel(2) I(t) of model (2)

■ X(t) of model (1)

■ S(t) of model (1) ■I(t) of model (1)

X(t) of model (2)

■ S(t)ofmodel(2) I<t) of model (2)

Figure 3 Time sequence diagram and phase portrait of model (2). (a) The Brownian motion model with a12 = a21 = a = 0.03,0.06,0.1, respectively. (d)-(f) are phase portraits of (a)-(c), respectively. The red o is corresponding to the deterministic model, while the blue o represents the stochastic model.

In Figure 3(b), we choose a12 = a21 = a = 0.06, thus

limsup1 i [(x(e)-x*)2 + (S(e)-s*)2 + (i(e)-i*)2] de < U3 = 22.4523.

t—> + t J0 W3

In Figure 3(c), we choose a12 = a21 = a = 0.1, thus

limsup1 f [(x(e)-x*)2 + (s(e)-s*)2 + (i(e)-i*)2] de < — = 90.1453.

t—+TO t J0 W3

Figure 3 shows that the solution of model (2) fluctuates around E3(1.9085,0.5231, 0.8071). In addition, the fluctuation intensity is related with the disturbance intensity: with the increase of a12, a21 and a, the fluctuation intensity is also increasing. These all meet the conditions of Theorem 3.2.

In Figure 4, we choose X(0) = 2, S(0) = 2,I(0) = 2, b = 1,c = 0.1, d = 0.1, «11 = 0.5, «12 = 0.3, «21 = 1, «22 = 0.1, «33 = 0.2, W = 0.5,p = 1, q = 1, and step size At = 0.001. Figure 4 shows that the solution of model (2) fluctuates up and down in a small neighborhood. According to the density functions in Figure 4(b)-(d), we see that there is a stationary distribution. This is in line with our conclusions.

In Figure 5, we choose X(0) = 2, S(0) = 2,I(0) = 2, b = 1,c = 0.1, d = 0.1, «11 = 0.5, «12 = 0.3, «21 = 0.1, «22 = 0.1, «33 = 0.2, W = 0.5,p = 0.5, q = 0.5, and step size At =0.001.

In this condition,

E3 = (X*, S*, I*) = (1.8793,0.4339,0.5847), «11q = 0.25 > «22p = 0.15, R1 = 3.4 > 1.

X(t) of model (2) ■ S(t) of model (2) I(t) of model (2)

0.3 0.4 0.5 0.6 0.7 The density function of S(t)

1500 2000

1.9 1.95 2

The density function of X(t)

0.6 0.7 0.8 0.9

The density function of I(t)

Figure 4 Time sequence diagram and density function of model (2) with a12 = a21 = a = 0.01. (a) Time sequence diagram; (b)-(d) the density functions of X(t), 5(f), /(f), respectively.

X(t) of model (1) -S(t) of model (1) I(t) of model (1)

X(t) of model (2) ■ S(t) of model (2) I(t) of model (2)

1 X(t) of model (2) -S(t) of model (2) I(t) of model (2)

Figure 5 Persistence in mean and extinction of model (2). (a) The deterministic model; (b) persistence in mean of model (2); (c) extinction of model (2).

In Figure 5(b), we choose o12 = 021 = 0 = 0.006. In this case,

p = max{oi2,021,0} = 0.006 <min|X\—,S\ —,I\ — \ =0.0064, p 1 y U, y uq , y U0\ ,

which satisfies the conditions in Theorem 4.1. Figure 5(b) shows that X(i),S(i),I(t) have persistence in mean, this is in line with our conclusion in Theorem 4.1. In Figure 5(c), we choose oi2 = 0 = 0.5 and

{a12 _1

, VaaP f = 0.223, v 2c J

which satisfies the conditions in Theorem 4.2. Figure 5(c) shows that S(t),I(t) are extinct and

lim X(t) = — = 2,

t —+to a11

this is in line with our conclusion in Theorem 4.2. To sum up, we have the following conclusions: I. Asymptotic behaviors

(1) When R0 < 1 and Ka21 < c, the solution of model (2) is fluctuating around E1. Therefore, the intensity of the fluctuation is positively correlated with a12.

(2) When R0 > 1,R1 < 1 and a11 q > a22p, the solution of model (2) is fluctuating around E2. Therefore, the intensity of the fluctuation is positively correlated with a12,a21 and a.

(3) When R1 > 1 and a11 q > a22p, the solution of model (2) is fluctuating around E3. Therefore, the intensity of the fluctuation is positively correlated with a12, a21 and a. When the interference intensity is sufficient small, the solution of model (2) has a unique stationary distribution, it also has the ergodic property.

II. Persistence in mean and extinction

(1) When R1 > 1, a11 q > a12p and

q = max{a12,a21, a} < mm[X*^p—3, S*^—3,I*^—|}, the solution of model (2) can have persistence in mean.

(2) When a21 > max{-^= ,^/a21 p} or R* <1 and a21 < ^a21p, the predator of model (2) can be extinct.

Competing interests

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

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details

1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, P.R. China. 2Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou, 341000, P.R. China. 3State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, 266590, P.R. China.

Acknowledgements

The authors would like to thank Dr. Tonghua Zhang, who helped them during the writing of this paper. This work was supported by National Natural Science Foundation of China (11371230,11501331,11561004), the SDUST Research Fund (2014TDJH102), Shandong Provincial Natural Science Foundation, China (ZR2015AQ001, BS2015SF002), Joint Innovative Center for Safe And Effective Mining Technology and Equipment of Coal Resources, the Open Foundation of the Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, China, SDUST Innovation Fund for Graduate Students (No. SDKDYC170225).

Received: 30 June 2016 Accepted: 29 November 2016 Published online: 22 December 2016 References

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