Scholarly article on topic 'Analysis of a stochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response'

Analysis of a stochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response Academic research paper on "Mathematics"

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Academic research paper on topic "Analysis of a stochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response"

Li and Wang Advances in Difference Equations (2015) 2015:224 DOI 10.1186/s13662-015-0448-0

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Analysis of a stochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response

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Shuang Li1,2* and Xiaopan Wang3

"Correspondence: oklishuang@163.com 1College of Mathematics and Information Science, Henan Normal University, Jianshe Road, Xinxiang, 453007, P.R. China 2Henan Engineering Laboratory for Big Data Statistical Analysis and OptimalControl, Schoolof Mathematics and Information Science, Henan NormalUniversity, Xinxiang,453007, P.R.China Fulllist of author information is available at the end of the article

Abstract

A predator-prey model with Beddington-DeAngelis functional response and disease in the predator population is proposed, corresponding to the deterministic system, a stochastic model is investigated with parameter perturbation. In Additional file 1, qualitative analysis of the deterministic system is considered. For the stochastic system, the existence of a global positive solution and an estimate of the solution are derived. Sufficient conditions of persistence in the mean or extinction for all the populations are obtained. In contrast to conditions of permanence for the deterministic system in Additional file 1, it shows that environmental stochastic perturbation can reduce the size of population to a certain extent. When the white noise is small, there is a stationary distribution. In addition, conditions of global stability for the deterministic system are also established from the above result. These results mean that the stochastic system has a similar property to the corresponding deterministic system when the white noise is small. Finally, numerical simulations are carried out to support our findings.

MSC: 92B05; 34F05; 60H10

Keywords: predator-prey; Beddington-DeAngelis functional response; stochastic; stationary distribution; stability; persistence; extinction

ft Spri

ringer

1 Introduction

Recently, epidemiological models have received much attention from scientists. Since the pioneering work of Kermack-Mckendrick, there have been many relevant papers [1-8], but only single-species is considered in these models. However, species does not exist alone; while species spreads the disease in the natural world, it also competes with other species for resource to exist, or is predated by their enemies. Therefore, it is more important to consider the effect of multi-species when we consider the dynamical behaviors of epidemiological models. There are not many papers [9-18] considering these two areas.

Due to its universal existence and importance, the dynamic relationship between the predator and the prey has been a dominant theme in ecology. The predator's functional response is one significant component of the predator-prey relationship. Generally, the classical Holling types I-III are only related to the density of the prey, and Hassell-Varley type [19], Beddington-DeAngelis type [20-23] as well as Crowley-Martin type [24] are functions of both the prey and the predator densities. A lot of data show that the func-

© 2015 Li and Wang; licensee Springer. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the originalauthor(s) and the source are credited.

tional response related to prey and predator densities performs much better. The classical predator-prey model with Beddington-DeAngelis type functional response is

\x'(t)=x(t)[b1-

{y<(t)=ym-b2 + j+mg^- «^(t)],

where x = x(t) and y = y(t) represent prey and predator densities at time t, all the coefficients are positive. We can refer to [20, 21, 25] for the biological representation of each coefficient in model (1).

In this paper, we first introduce a deterministic predator-prey model with disease in the predator and Beddington-DeAngelis functional response. We assume that the disease only spreads among the predator population based on the basic epidemiological model, namely the SI:

x'(t)=x(t)[r - «11 x(t)-T+mgn^]'

y1(t)=yi(t)[-d2 - «22yi(t) + l+mXife) - (2)

y2(t) =y2(t)[jSyi(t)-d3 - «33y2(t)].

Let x(t) denote the population density of prey, y1(t) and y2(t) represent the population density of the susceptible predator and the infected predator, respectively.

Model (2) is derived under the following assumptions: r is the intrinsic growth rate of the prey, a11 is the overcrowding rate of prey population, and a12 is the capturing rate of the predator. d2 is the death rate of the susceptible predator, a22 is the overcrowding rate of the susceptible predator. is the rate of conversion of nutrient into the reproduction of the predator. ¡3 is the transmission rate of the disease. We assume that infected predators do not have capturing ability and do not recover or become immune, d3 and a33 are the death rate and the overcrowding rate of the infected predator, all the coefficients are positive here. System (2) has four non-negative equilibria 0(0,0,0), E1(,0,0), the disease-free equilibrium E2(X,y1,0) and the positive equilibrium E* = (x*,y*,y*). E2 exists if

«21r > d2(a11 + mr). (3)

E* exists if

d3a22 nd3a11

«21 r >1 d2 + —3— 11 «11 + mr + —— I. (4)

Define the basic reproduction number R0 =-d «21r- ■ , obviously, E* exists

(d2 + )(«11 +mr+ —3L-11)

when R0 > 1. In addition, we can compute a useful result: y1 < y if R0 < 1.

In fact, all the populations in the natural world are inevitably affected by environmental white noise which is an important component in reality. Therefore, many stochastic models for single-species or multi-species have been developed [26-30]. In this paper, considering the effect of environmental noise, we introduce stochastic perturbation into some parameters. As we known, r is the intrinsic growth rate of preys, d2 and d3 are both death rates of susceptible predators and infected predators. In practice, these parameters can be estimated by an average value plus an error term. By the well-known central limit

theorem, we know the error term follows a normal distribution and sometimes depends on how much the current population sizes differ from the equilibrium state. Hence, we consider the perturbation as the following form [31-36]:

r ^ r + 0iBi(t), -d2 ^ -d2 + 02B2(t), -d3 ^ -d3 + a3B3(t),

where of (i = 1,2,3) is the intensity of noise and B;(t) (i = 1,2,3) is a standard Brownian motion. Corresponding to the deterministic model (2), a stochastic system has the following form:

dx(t) = x(t)[r- anx(t) - 1+;xg+g1W] dt + o1x(t)dB1(t),

dVi(t) =y1(t)[-d2 - a22yi(t) + ^j*)^ - M)] dt + 02j1(t) dB2(t), (5)

dy2(t) =y2(t)[^y1(t)-d3 - a33y2(t)] dt + ay^t) dB3(t).

Throughout this paper, unless otherwise specified, let (fi, F, P) be a complete probability space with a filtration {Ft}tGR satisfying the usual conditions (i.e., it is right continuous and increasing and F0 contains all P-null sets).

The qualitative analysis of system (2) is in Additional file 1, here we mainly discuss the stochastic system. In the following section, we derive the existence of a positive solution of system (5), an estimate of the solution, and give the conditions of persistence in the mean or extinction for both populations. We also show that there exists a stationary distribution of the solution. As a result, conditions of global stability for model (2) are obtained.

2 Existence of the positive solution

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 [37].

Theorem 1 For any initial value x0 > 0, yi0 > 0 and y20 > 0, there is a unique solution (x(t), y1(t), y2(t)) of system (5) ont > 0, and the solution will remain inR+ with probability 1.

Proof Define a C2-function V: R+ ^ R+ by V(x,y1,y2) = (x -1 - logx) + (y1 -1 - logy1) + (y2 -1 - logy2), by a similar way of the proof in Theorem 2.1 of [34], Theorem 2.1 of [35] and Lemma 2 of [36], we can have the required assertion. □

Though we cannot get an explicit solution for model (5), an estimate of positive solution of (5) can be derived, we firstly show a very useful lemma derived from [38]. Consider the equation

dN(t) = N(t)[(a(t) - b(t)N(t)) dt + a(t) dB(t)]. (6)

Lemma 1 Assume that a(t), b(t) and a(t) are bounded continuous functions defined on [0, a(t) > 0 and b(t) > 0. Then there exists a unique continuous positive solution N(t) for any initial value N(0) = N0 > 0, which is global and represented by

exp{/0t [a(s) - ds + a(s) dB(s)} t > 0

1/N0+ /0 b(s) exp{/0s[a(r)- a2r ] dt + a(t) dB(r)} ds' > .

Since the solution is positive, we have dx(t) < x(t)[r - a11x(t)] dt + a1x(t)dB1(t) from system (5), let

exp(r—2")t+a1B1(t)

*(t) =-p-a2-,

1/X0 + «11/0 e-p(r-+)s+a1B1(s) ds by Lemma 1, it is easy to see that $(t) is the unique solution of the following equation:

Jd$(t) = $(t)(r - «n$(t)) dt + a1$(t) dB1(t), |$(0)= X0.

The comparison theorem for stochastic equations yields x(t) < $(t), t > 0, a.s. Besides,

dyi(t) < yi(t) Obviously, by Lemma 1,

-d2 - fl22yi(t) + — m

dt + a2y1(t) dB2(t).

¡an ^

( ammt-d2-ajr)t+o2B2(t)

^1(t) = — a

^ + «22/0 exP( «m1-d2- a2)s+a2B2(s) ds

is the solution to the equation

j d^(t) = *1(t)(«21 - d2 - «22^1«) dt + a2^1(t) dB2(t), |^1(0)= y10,

and y1(t) < ^1(t), t > 0, a.s. On the other hand,

dx(t) > x(t)

r - a11 x(t) - ■ n

dt + aix(t) dB1(t),

similarly, we can get x(t) > 0(t), t > 0, a.s., where

exp(r- «2- al)t+a1B1(t)

m =-e-P-^-

1/X0 + «11/0 e-p(r-«T-ar)s+a1B1(s) ds is the solution of

j (t) = 0(t)(r - «f - «110(t)) dt + a^(t) dB1(t),

|^(0) = X0.

It is easy to see that dy2(t) < y2(t)[3^1(t)-d3 - «33y2(t)] dt + a3y2(t) dB3(t), we also have y2(t) < *2(t), t > 0, a.s., and

e-pj/t(№(s)-d3- |) ds + a3 dB3(s)} ^2(t) = —-;-;-2-

y2- + a33 /Jexp{/0s(^1(t)-d3 - -2) dr + a3 dB3(r)}ds

is the solution of

| d^(t) = ^2(t)(№(t) - d3 - «33(t)) dt + a3*2(t) dB3(t), l*2(0)= y20-

By system (5), we have dyi(t) > yi(t)[-d2 - «2^i(t) + i+mffiff^t) - №(t)] dt + 02 x yi(t) dB2(t). Therefore, we derive y1(t) > ^i(t), t > 0, a.s., where

, M exp(/Q(i+m:2(lf+(ii(s) - d2 - №(s) - j) ds + 02 dB2(s)}

¥i(t) =-2-

yL + «22/0 exp^O (l+mHi&T) - d2 - №(T) - dT + 02 dB2(T)} ds

is a solution of

J d^i(t) = ^i(t)(i+mftfei) - d2 - №(t) - «22^i(t)) dt + a2fi(t) dB2(t), I ^i(O) = yio.

We can also get y2(t) > ^2(t), t > 0, a.s., where

expj/0(^^i(s) - d3 - ds + 03 dB3(s)}

fi(t) =

is a solution of

+ «33 /0 exp{/0s(^i(t) - d3 - dr + 03 )} ds

I d^(t) = fo(t)(fifi(t) - d3 - «33^2(t)) dt + 03 ^2 (t) dB3(t), [ ^2(0) = y20,

then we derive the following theorem.

Theorem2 Assume (x(t), yi(t), y2(t)) ok t > 0 is the positive solution of system (5) for initial value x0 > 0, yi0 > 0 andy20 > 0, then there exist functions $(t), 0(t), ty(t), ^(t) (i = i, 2), defined as above, such that

4>(t) < x(t) < $(t), fi(t) < yi(t) < (i = i, 2), t > 0,a.s.

3 Persistence in the mean and extinction

In order to consider the conditions of persistence in the mean and extinction for the prey and predator population, at first, we give two useful lemmas. Applying Ito's formula to system (5) yields

dlnx(t) = [r - 202 - aiix(t) - i+mXty+ny^)] dt + a dBi(t),

dlnyi(t) = [-d2 - 2a22 - a22yi(t) + i+mX^i)X+(nyi(t) - ^(t)] dt + a2 dB2(t), (8)

dlny2(t) = [yi(t) - d3 - 2a32 - a33y2(t)] dt + 03 dB3(t).

Lemma 2 The solution (x(t), yi(t), y2(t)) of system (5) for any initial value (x0, yi0, y20) e R+ satisfies the following inequalities:

ln x(t) . ln yi(t^ . / ln yi(t) ln y2(t) \ limsup-< 0, limsup-< 0, limsupl -+- I < 0.

t—t t—t t—tt

Proof It follows from Eq. (8) that

dlnx(t) < [r - 2a12 - a11x(t)] dt + a1 dB1(t),

dlny1(t) < [am1 - d2 - 2a2 - fl22y!(t)] dt + a2 dB2(t),

dlny2(t) = [py1(t) - d3 - 2a32 - «33y2(t)] dt + a3 dB3(t). Consider the following system:

d ln u(t) = [r - 2a12 - a11 u(t)] dt + a1 dB1(t),

d ln V1 (t) = [«21 - d2-2 a| - «22V1W] dt + a2 dB2(t), (9)

_ d ln V2 (t) = [fiV1(t) - d3 - 2a32 - «33 V2(t)] dt + a3 dB3(t),

with initial value (x0,y1o,y2o) e R+. By the comparison theorem for stochastic differential equations, we have

x(t) < u(t), yi(t) < Vi(t) (i = 1,2), t e [0, a.s.

In fact, by virtue of Theorem 3.3 and Corollary 3.4 in [39], we can get

ln «(t) . ln V1(t) limsup-< 0, limsup-< 0.

t—t t—t

Hence, limsupt—TO l-nX— < 0, limsupt—TO lnytl(t) < 0. In the following, we show that limsupt—TO(^ < 0. Applying Ito's formula to e-p(t) ln vi(t) (i = 1,2) results in

e-p(t) ln v1(t) = lny10 + /0 e-p(s)[ln v1(s) + «21 - d2 - 2a2 - «22^(s)] ds

+M1(t), (10)

exp(t) ln V2(t) = lny20 + /0 exp(s)[ln V2(s) + pV1(s)-d3 - 2 a32 - «33 V2(s)] ds + M2(t),

where Mi(t) = /0 ai e-p(s) dBi(s) (i = 1,2) is a real-valued continuous local martingale with quadratic form <Mi(t),Mi(t)> = /0 a2 e-p(2s) ds.

By virtue of the exponential martingale inequality of [40], for any positive constants T, S and n, we have

Pj sup

lo<t<r

Mi(t) - 2Mi(t),Mi(t))

> ^ < exp(-5rç).

Choosing T = y k, S = e-p(-y k), n = & e-p(y k) ln k gives that

Pj sup

lo<t<y k

Mi(t) - exp(-Yk) lMi(t), Mdt))

> 0 exp(y k) ln H < k-0,

where k e W, & >1 and y > 1. It follows from the Borel-Cantelli lemma that there exists c ^ (i = 1,2) with P(^) = 1 such that for any w e ni, an integer ki = ki(o>) satisfying

Mi(t) < exp(2 Yk)iMi(t),Mi(t)) + 0 exp(yk) lnk

for all 0 < t < yk and k > ki(w) can be found. Now let = P|2=1 Ui, clearly, P(^0) = 1. Moreover, let k0(w) = ma-jki(w), i = 1,2}, then for any w e it follows from Eq. (10) that

e-p(t) ln v1(t) < lny10 + /0 e-p(s)[ln v1(s) + «21 - d2 - 2 a2 - «22V1(s)] ds

rt / \ a2

+ /0 exp(s) Or exp(s - y k) ds + 0 exp(y k) ln k, exp(t) ln V2(t) < lny2o + /g exp(s)[ln V2(s) + PV1(s) - d3 - 2a32 - «33V2(s)] ds

- «33V2(s)] as

+ /g exp (s)exp (s - y k) ds + 0 exp ( y k) ln k for all 0 < t < yk and k > ki(w). Then exp(t) [ln v1(t) + ln v2(t)]

< lny10 + lny20 + i exp(s)

ln V1(s)-(«22- P )v1(s) + — m

2a"2 + ln V2(s) - «33V2(s) - d3 - 2<

f t 3 a2 + / e-p(s) 2_) — e-p(s- yk)ds

0 i=2 2

+ 20 e-p(y k) ln k.

It is easy to see that for any 0 < s < yk and (u(s), v1(s), v2(s)) e R+, there exists a constant A independent of k such that

ln V1(s) - («22 - P)v1(s) + «21 - d2 - 1 a22 + ln V2(s) - «33 V2(s) - d3 - 1 a32

m 2 2 2

+ J2 exp(s - Y k) < A.

Hence, it follows that for all 0 < t < y k, with k > k0(«), we have

exp(t)[ln v1(t) + ln v2(t^ < lny10 + lny20 W A exp(s) ds + 20 exp(yk) lnk.

ln v1(t) + ln v2(t) < exp(-t)[lny10 + lny20] + A [1 -exp(-t)] + 20 exp(-t) exp(yk) ln k.

Consequently, for y (k -1) < t < y k and k > k0(«), it follows that ln V1(t) ln v2(t) 1 A[1 - exp(-t)]

—t— + —t— < t exp(-t)[lny10 + lny20] +-1-

20 exp(-y (k -1)) exp(y k) ln k

Now let k ^ +œ, then t ^ we have

limsup/+ \ < 0.

t^^ t t )

Therefore,

limsup(^ + ^ < o.

m t t •- d

t^TO \ t t

Definition 1

(1) The population x(t) is said to be non-persistent in the mean if <x(t)>* = 0, where f (t)> = \ fof (s) ds,f * = limsuPt^+c»/(t),f* = liminft^+TO/(t).

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

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

Lemma 3 [41] 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

ln x(t) < kt - ko / x(s) ds + V PiBi(t)

Jo i=i

for t > T, where fai 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

lnx(t) > kt - ko / x(s) ds + V faBi(t)

Jo i=i

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

Theorem 3 For the prey population, we have:

(i) If r < 2 of, then the prey population x(t) will go to extinction a.s.

(ii) If r = 2 o2, then the prey population x(t) is non-persistent in the mean a.s.

(iii) If r >2 o12, then the prey population x(t) is weakly persistent in the mean a.s.

(iv) If r >2 o12 + 02, then the prey population x(t) is strongly persistent in the mean a.s.

Proof It follows from the first equation of system (5) that

dx(t) < x(t) [r - anx(t)] dt + oix(t) dB1(t),

the right-hand side is a logistic system, because of the comparison theorem, Theorem 2, Theorem 7 and Theorem 8 in [42], we can get consequences (i) and (ii). (iii) We need to show that there exists a constant p >0 such that for any solution of

system (5) with initial value (x0,y10,y20) e R+ satisfying <x(t)>* > p >0. Now we assume

that the contrast is true, let e1 > 0 sufficiently small such that (-d2 - °2r) + a21e1 < 0, (r - 2o12) - a11e1 > 0, then for e1 > 0, there exists a solution (x,y1,y2) with initial value (x0,yi0,y20) e R+ such that P{<x(t)>* < ei} > 0. By virtue of system (8), we have

ln yi(t)-ln yi0 / o2\ B2(t) < -d2 - -2 + a2i x(t) + 02-.

We also have limt^+TO B^ = 0, thus limsupt^+TO lnytl(t) < (-d2 - + a21e1 < 0, then limt^+TO yi(t) = 0.

It follows from system (8) that

d ln x(t) >

r -2a2 - aux(t) - a12yi(t)

dt + a1 dB1(t).

Integrating both sides from [0, t] and multiplying by ^ ,we obtain Inx(t)-lnxo / 1 2\ |- B1(t)

-1-- I ■r - 2ai j - (t)/ - + 01 ——'

then limsupt^+TO — (r- 2ff12)-fl11e1 > 0, which contradicts Lemma 2. Therefore, our assumption is false, <x(t)>* > 0, the prey population x(t) will be weakly persistent in the mean a.s. (iv) It is easy to get

dx(t) > x(t)

r - aH | - a11x(t)

dt + aix(t) dB1(t).

Similarly, by the comparison theorem, Theorem 2 and Theorem 10 in [42], result (iv) is obtained. □

Remark 1 By Theorem 3, we find that r - 2of is the threshold between weak persistence in the mean and extinction for the prey population. If 2of > r, then the prey population will be extinct in the future, no matter whether the predator exists. It implies that environmental random perturbation plays a very important role in the biological system.

Theorem 4 For the predator population, we have:

(i) If a21(r - 2 of) < a11(d2 + -y), then the susceptible predator population y1(t) will go to extinction a.s.

(ii) If a21(r - 2 of) = a11(d2 + Of), then the susceptible predator population y1(t) is

non-persistent in the mean a.s.

(iii) If ¡3 a21(r - 2 o12) < ¡¡a11(d2 + "2L) + a11a22(d3 + Or), then the infected predator

population y2(t) will go to extinction a.s.

(iv) If ¡3 a21(r - 2 o12) = ¡3 a11(d2 + 02) + a11 a22(d3 + Of), then the infected predator population y2(t) is non-persistent in the mean as.

Proof (i) If r < 2of, then it follows from Theorem 3 that <x(t)>* = 0. By the second equation of system (8), we have

lny1(t)-lny10 / o-22\ i uw B2(t)

-< -d2 - -2 + aa x(t) + 02-. (12)

Hence, [t-1 lny1(t)]* < (-d2 - 02r)< 0, then lim^+TOy1(t) = 0. Now we consider that if r > 2o12, it follows from the first equation of system (8) that

ln x(t) - ln X0 / 1 A 1 ,A\ B1(t)

-1-< I r - 201 j - an(x(t)) + 01 t •

Applying Lemma 3 leads to

(x(t)f < . (13)

Substituting the above inequality into (12) gives

r i, , ct22\ ( a2i(--2ai2)-au(^2 + -y) n

t 1 lny^t)] < l-d2--±)+ a2i(x(t)) <-2-^ < 0,

\ 2 / an

which implies that limt^+TOy1(t) = 0 a.s.

(ii) Assume (y1(i)>* > 0, then it follows from Lemma 2 that [t-1 lny{\* = 0. Making use of (14), we can see

0 = [r1 lny!(t)f < (y-d2 - + a21(x(t))*. (15)

On the other hand, for sufficiently small e3 > 0, there exists T3 > 0 such that for all t > T3, a21(x(t)> < a21(x(t)>* + S3. Substituting these inequalities into the second equation of system (8) yields that

lny1(t)-lny10 J O^V _ L.t+W. _ B2(t)

t < - y J + «21\x(t)) - a22[yi(t)) + 0-2- t

< (çd2 - 0-) + «21 (*(t))* + e3 - «22^1^)) + 02—y^, then application of Lemma 3 and (15) results in

(yi(t)) * <

(-¿2 - 02L)+«21<x(t)>* + £3 «22

Condition «21(r -2 o12) = «n(d2 + Of) means that r > 2 o12, because of (13) and the arbitrariness of £3,

(y1(t))*< «21(r -2 °12)-a11(d2 + 02) =0, (16)

«11«22

which is a contradiction to our assumption, therefore, <y1(t)>* = 0 a.s.

(iii) If «21(r - 2o12) < a11(d2 + Or), then from (i) and (ii), we get <y1(t)>* = 0. Hence, it follows from the third equation of system (8) that

ln y2(t)-ln y20 a( , 12 ( u^ B3(t)

-1-= Ply>1(t)} - ¿3 - 2 032 - a33y(t)) + 03 t • (17)

Taking superior limit leads to [t-1 lny2(t)]* < -d3 - 2of < 0, then limt^+œy2(t) = 0 a.s.

If «21(r - 2o12) > a11(d2 + 02), r > 2o12 must be verified, thus by the proof of (ii), we have (16), that is,

( , «21 (r - 2012) - «11 (¿2 + 02 )

(y1(t)} <-2-—.

«11«22

At this time, making use of (16), we obtain (t-1 lny2(t))*< P(yn(t))* -

2 2 P«21 (r - 2of) - P«11 d + Of) - aila22(d3 + °t)

«11 «22

then limt^+TOy2(t) = 0 as desired.

2 2 2 (iv) If p«21 (r - 2of) = P«11 (d + i2) + «n«22(d3 + Of), then «21 (r — 2of) > «11(^2 + Op)

and r > 2 of. By the properties of superior limit, for sufficiently small £4 > 0, there exists a constant T4>0 such that for all t > T4, <y1(t)> < <y1(t)>* + £4.

Substituting (16) and the above results into (17) yields

ln y2 (t) — ln y20 P «21 (r -2 °f)-P«n(d2 + Of) of , , ---<-+ £4 — «3 — — — «33 (y2(t))

t «11 «22 2

+ 03" t

22 P «21 (r - 2 °f) - P«11(d2 + 02)- «11 «22(d3 + Of)

- «3^y2(t^ + 03

«11«22 B3(t) t '

By Lemma 3, <y2(t)>* — P«21(r-2^-W^-«11«22^0f) + £4] = Considering the arbitrariness of £4, we have <y2(t)>* — 0 . Notice the positivity of the solution (x(t),y1(t), y2(t)), it is easy to get <y2(t)>* = 0 a . s. □

Remark 2 Observing conditions (i) and (iii) of Theorem 4, we can see that if condition (i) is true, (iii) must be verified. That is to say, if the susceptible predator population goes to extinction, the infected predator population will also die out, which is consistent with the reality. Though we have some difficulties to research persistence for the predator population now, we can consider it in another way in Section 5.

Remark 3 It follows from Theorems 3,4 and Theorem A.4 in Additional file 1 that the so-

satisfies x(L)/ r-201 „ ^ 1„2 if„ ^ 1„2 «12.

-2°i2-if . When of :

«11 1

and K in Theorem A.4, this is consistent with our expectations. Besides, if «21(r - 2of) >

0 2 0 2 0 2

« (d + 02) y (t)\* < «21 (r-2o2 )-«11 № + 02) , (t), * < P«21 (r-2o12)-P«11 (d2 + )-«11 «22 №+ °h) if

11( 2 2 ), <y1( )> — «11 «22 . <y2( )> — «11 «22«33

P«21(r - 2o12) > P«11(d2 + o2) + «11 «22(d3 + ^ ). These results are all the same conclusions with K1 and K2 in Theorem A.4 when of = 0 (i = 1,2,3). Furthermore, we find that the upper bound and lower bound of the solution for the stochastic system are smaller than those for the deterministic system. It means environmental random perturbation can reduce the size of the population to a certain extent.

lution of stochastic system (5) satisfies <x(t)>* < «2 1 when r > 2 of. If r > 2 of + «•2, then r—1O 2 — «12 „ 11 _ <x(t)>* > —2« n . When of = 0, the upper bound and lower bound are the same with K

im A /1 fKic ic /^Anciff Anf TIT1 fl"! Anr nvnActltiAnC RAri/'lAC if ,•"»,_ (!/» ~O

o 2 o 2 o 2 _ 1 O i r!J__, , Prtm( v — 1 O P mi (fin J__2 ^ _/7i 1 n^r^ I rl^i__3 '

4 The long time behavior of solution

For system (2), by analyzing the characteristic equation of four equilibria, we can easily get sufficient conditions of local stability for these equilibria. At this time, we know 0(0,0,0) is saddle, not stable. If a21r < d2(a11 + mr), then E1 is locally asymptotically stable, and the disease-free equilibrium E2 does not exist. When r > a^, a21r > d2(a11 + mr) and R0 <1 are verified, the disease-free equilibrium E2(x,yi,0) is locally asymptotically stable, E* does not exist; when R0 > 1 and r > a2, the positive equilibrium E* is locally asymptotically stable.

For stochastic system (5), E1 and E2 are no longer equilibria, but in this section we can study the asymptotic behavior of solution around them. Meanwhile, the conditions of global asymptotic behavior for E1 and E2 are derived.

Theorem 5 If a21r < a11 d2, then for any given initial value (x0, y1o, y2o) e R+, the solution X(t) = (x(t), y1(t), y2(t)) of system (5) has the property

here x = min{a21a11, a12a11, a12a33}.

Proof Define a function V(t) = c1(x - ^ - ^ log ^) + c2y1 + c3y2, where ci (i = 1,2,3) are positive constants to be determined later. Then the function V(t) is positive definite, and

, «21X „

+ C2y1 -¿2 - «22y1 + ---№2

1 + mx + ny1

+ C2y1 -¿2 - «22y1 +

fiy2 dt + C202y1 dB2(t)

+ C3y2[^y1 - ¿3 - «33y2] dt + C303y2 dB3(t).

+ C2y1 -¿2- «22y1 +

Py2 + C3y2 [^y1 - ¿3 - «33y2]

1 + mx + ny1

C2d2y1 - C2«22y2 +

C2«21Xy1

C2&y1y2 + C3^y1y2 - C3d3y2 - C3«33y2.

1 + mx + ny1

Let C1 = «21, C2 = C3 = «12, then

C2d2y1 - C2«22y2 - C3d3y2 - C3«33y2

< -C1au ( x - — ) - ( C2d2 - C1a12 — m

a11 a11

2 j 2 C1ff12r

- C2a22yi - C3d3y2 - C3a33y2 + —-.

If a11d2 > a21r, thus

( r V 2 2 C1a,2r

LV < -C1au x--- C2a22y2 - C3a33y^ + —-,

a11 1 2 2a11

therefore,

dV < -

C1 «11\ x - — J + C2«22y2 + C3«33y2 «11 1 2

, C1012r

dt +-dt

- — )01 dB1(t) + C202y1 dB2(t) + C303y2 dB3(t).

+ ca x

Integrating both sides of the above inequality from 0 to t, then taking expectations, yields

0 < E[V (t)] < V (0) - /if E J0

x(s) — ^ + y1(s)2+ y2(s)2 «11

C1°12r

as +-1,

which leads to

limsup- i E

1 f [/x(s)-^) + y1(s)2+ y2(s)2 «11

ds < C1°12r

here x = min{a21a11, a12a11, a12a33}. The result is straightforward. □

When ai = 0 (i = 1,2,3), system (5) becomes system (2). By Theorem 5, we know the stability of equilibrium E1.

Corollary 1 Ifa21r < a11 d2, the equilibrium E1( , 0,0) of system (2) is globally asymptotiCally stable.

Remark 4 It is not difficult to find that the solution of stochastic system (5) fluctuates

around equilibrium E1(-L-,0,0) of system (2) when E1 is globally asymptotically stable.

The intensity of fluctuation is relevant to ct2. The smaller ct2 is, the weaker the fluctuation is.

Theorem 6 Assume a21r > d2(a11 + mr), a11 > m(r - a11xc) and R0 < 1, for anygiven initial value (x0, y10, y20) e R+, the solution X(t) = (x(t), y1(t), y2(t)) of system (5) has the property

c1xc ct12 C2y1a22

1 / m ^ M2^, C1xxo12 limsup- I E[||X(s)-E2|| Jds <—— + t^+œ t J0 2/1

t^+TO t J0 2x 2x

where C1 = a21(1 + nj)1), c2 = c3 = a12(1 + mx) and¡x = min{[a11 -m(r-a11^)]C1,C2a22,C3a33}. Proof Define a C2 function V: R+ ^ R+ by

V (t) = d(x - x - x log x^J + C^y1 - y1 - log y^j + C3y2,

where ct >0(i = 1,2,3) are constants to be chosen later. Make system (5) into the following form:

dx=x[-«11(x -x)+«12 71+m—++n—^mLm^dt+°1xdB1(t), dy1=y1[-«22(y1-y1)+«21(x-^mx^-mm*^- p ^ dt+O2 y1 ^^

dy2 = y2[Py1 - d3 - «33y2] dt + O3y2 dB3(t). From Ito's formula, we compute

c1«12my1(x - x)2

-d«n(x - x) + - „

(1 + mx + ny1)(1 + mx + ny1)

C1«12(1 + mX)(x - X)(y1 - y0 '

(1 + mx + ny1)(1 + mx + ny1) -C2«22(y1 - y1)2 +

dt + c1 o1(x - x) dB1(t)

+ C1x012 dt + f „ „ !.. i,\2 , c2«21(1 + ny1)(x -x)(y1-yO

C2«21«x(y1- y1)^ , _ C2y1022n

■- C2Py2(y1-y1) +

(1 + mx + ny1)(1 + mx + ny1) dt

(1 + mx + ny1)(1 + mx + ny1) 2

+ C2(y1 -y1)o2 dB2(t) + [ C3y2(Py1 - d3 - «33y2^ dt + C303y2 dB3(t).

Make c1 = «21(1 + ny1), c2 = «12(1 + mx), therefore

t(r - «11x)(x -1 + mx + ny1

, ^2 C1 m(r - «ux)(x -x)2 2 LV — -C1«u(x - x)2 +----C2 «22 (y1 - y1)2

2 c1xo12 c2y1 o22

- c2Py1y2 + c2Py1y2 + c3Py1y2 - c3d3y2 - c3«33y2 + + -2-.

Let c2 = c3, then

LV < -c1«11(x -x)2 + c1 m(r - «11x)(x -x)2 - c2«22(y1 -y1)2 / - x 2 c1xo12 c2y1o22

- (c3d3 - c2Py1)y2 - c3«33y2 + + -2-.

When R0 < 1, we have d3 - Py1 > 0, thus

r - 2 c1xo12 c2y1o2!

LV < -c1 [«11- m(r - «ux)J(x -x) - c2«22(y1-y1) - c3«33y>2 + 2 +—2—.

If «11 > m(r - «11x), set ¡i = min{[«11 - m(r - «11 x)]c1, c2«22, c3«33}, integrating both sides from 0 to t, taking expectations leads to

0 < EV(t) < V(0) - ¡fQtEr(x(s) -x)2 + (y1(s)-y1)2 + y2(s)] ds + (^^ + ^^ t. Dividing both sides by t and letting t ^ +to, we get

1 f t 2 2 limsup- I £[(x(s)-x) + (y1(^)-yO + y2(5)]ds

t^+œ t Jo

^ C1xo12 + C2y1022

t^+TO t J0 2fi 2fi

This completes the theorem. □

When oi = 0 (i = 1,2,3), it is easy to get the following.

Corollary 2 Assume a21r > d2(a11 + mr), the disease-free equilibrium E2(x, y1,0) of system (2) is globally asymptotically stable when a11 > m(r - a11^) and R0 < 1.

Remark 5 The solution of stochastic system (5) fluctuates around the disease-free equilibrium E2(x, yy1,0) of deterministic system (2) when E2 is globally asymptotically stable. The values of o12 and o| determine the extent of fluctuations.

Remark 6 According to y1 = (1+mX)(r-a"*}, we have x > - when r > The condi-

a21-nr+nanx a\\ nan n

tion a11 > m(r - anX) in Theorem 6 is equivalent to X > ^ - mm, thus, if ma12 < na11, the condition a11 > m(r - anX) must be verified. Therefore, the condition a11 > m(r - a11^) in Theorem 6 can be replaced by the condition a12 < min{nr} which is easier to verify.

5 Stationary distribution

Before giving the main theorems, we first give a lemma [43].

Let X(t) be a homogeneous Markov process in El (El denotes Euclidean l-space) described by the stochastic equation

dX(t) = b(X) dt + J2 or (X) dBr(t). (18)

The diffusion matrix is

A(x) = (a^x)), aij(x) = ^ olr(x)o'r (x).

Assumption B There exists a bounded domain U c E¡ with regular boundary r, having the following properties: (B.l) In the domain U and some neighborhood thereof, the smallest eigenvalue of the

diffusion matrix A(x) is bounded away from zero. (B.2) If x e Ei\U, the mean time t at which a path issuing from x reaches the set U is finite, and supxe^ExT < to for every compact subset K c El.

Lemma 4 [43] If Assumption B holds, then the Markov process X(t) has a stationary distribution fi(-). Letf (■) be a function integrable with respect to the measure ¡x. Then

Px[Um T jT f (X(t)) dt = f (x)x(dx)} = 1

for all x e El .

Remark 7 To validate (B.1), it suffices to prove that F is uniformly elliptical in U, where Fu = b(x) ■ ux + [tr(A(x)uxx)]/2, that is, there is a positive number M such that Xj aij (x)fáj > M|f |2, x e U, f e Rk (see p.103 of [44]). To verify (B.2), it is sufficient to show that there exists some neighborhood U and a non-negative C2-function V such that for any x e El\U, LV is negative. (For details, we refer to p.1163 of [45].)

Theorem 7 Assume Ro > 1 is satisfied, an > m(r - anx*), and w < min{a21(an - m(r -aux*))(1 + ny*)(x*)2,ai2a22(1 + mx*)(y*)2,ai2a33(1 + mx*)(y*)2}, where

a21x*of(1 + ny*) a12y*a22(1 + mx*) a12y2a32(1 + mx*)

w =-+-+-,

then there is a stationary distribution ¡x(-) for system (5) and it has ergodicproperty. Proof System (5) can be written as the form of system (18),

fx(i)\/ x(i)(r - a11x{i)-T+m^) \ yi(t)(-d2- a22y1(i) + I-^7i) - ßrti))

yi(i) Vy2(i)/

1+mx(i)+nyi(i)

\ y2(i)(ßyi(i)-d3- a33y2(i)) /

( aix(i)\ 0 0

+ 0 dBi(t) + 02 yi(i) dB2(i) + 0 dB 3(i),

I 0 ) 0 \03y2(i)/

and the diffusion matrix is

/a2x2 0 0

A = 0 22 y2 0

0 0 22

Define

V(x,yi,y2) = ci[ x -x* -x* log ) + C2( yi-yl -yi logyi

+ c^ y2-y2-y2logyi ).

where ci (i = 1,2,3) are positive constants to be determined. System (5) can be rewritten

dx = [-xau(x - x ) + ai2xmy+mmx^y^my+^]di + ^ixdBi(i),

dyi = [-a22yi(yi- yî) - ßyi(y2- y2) + a2iyi ^o+mn^^-^] di

+ a2yi dB2(i), dy2 = [ßy2(yi - yi) - a33y2(y2 - y2)] di + 0^2 dB3(i).

If (x,y1,y2) e R+, applying Ito's formula to system (19) gives

LV = ci

dx + —- (dx)2

yi - yi yi

dyi + ^ (dyi)2

yi 2yi2

y2 2 2y22 2

, .,2 ai2myi(x - x1)2- ai2(yi- y2)(x - x*)(mx* + i) x*ai2 -au(x - x) +-^-—-i—:---+ i

(i + mx + nyi)(i + mx2 + nyi)

-a2^yi-yi)2 - ß(yi -yi2)(y2 -y2)

+ «21 (x -x*)(1 + «yi)(yi -y*) - a2inx*(yi -y*)2 y*a2 (1 + mx + ny1)(1 + mx* + ny*) 2

Pfa - y*)(y2 - y*) - «33 (y2 - y2)2 +y2a3

= -c1«1^x - x*) 2 +

c1a12my*(x - x*)2 (1 + mx + ny1)(1 + mx* + ny* )

c1 a12(y1 - y*)(x - x*)(mx* +1) c1x*a-

- C2«22(y1- yl)2

(1 + mx + ny1)(1 + mx* + ny*) 2

C2«21(x - x*)(1 + ny*)(y1 - y*)

- C2^(y1- yi*) (y2- y2)

(1 + mx + ny1)(1 + mx* + ny*)

C2«21 nx*(y1- y*)2 + C2y1*a2!

(1 + mx + ny1)(1 + mx* + ny*) 2

+ c3^1 -y*) (y2 -y2) - C3«33{y2 -y*)2 + C3y-L.

Here, let c1 = «21(1 + ny*), c2 = c3 = «12(1 + mx*), therefore,

T,r ( ^2 C1«12my*(x - x*)2 (

LV < -C1«1Wx - x*) + --^---- - C2«22{y1 -yD

v ' (1 + mx + ny1)(1 + mx* + ny*) v '

( *)2 C1x*a12 C2y*a22 C3y2a32 - c3«33{y2- y*) + ——

x - x * 2

^ *) «21«12my1:(1 + ny*)

«11«21 ( 1 + nyj------—

v u (1 + mx* + ny*)

- «12«22 (1 + mx*) y - y*)2 - «12«3^1 + mx*) (y2 - y*)2

«21x*a12(1 + ny*) a12y1a2!(1 + mx*) «12y*a32(1 + mx* ) + 2 + 2 + 2

= -[«n«21(1 + ny*) - «21m(r - «nx*)(1 + ny*)](x -x*)2

- «12«22(1 + mx*) y - y*)2 - «12«3^1 + mx*) (y2 - y*)2

«21x*a12(1 + ny*) a12y1a2!(1 + mx*) «12y*a32(1 + mx* ) + 2 + 2 + 2 .

When w < min{«21(«11 - m(r - «11x*))(1 + ny*)(x*)2, «12«22(1 + mx*)(y*)2, «12«33(1 + mx*)(y2)2}, the ellipsoid

-«21 («11 - m(r - «nx*)) (1 + ny*) (x - x*)2 - «12«22(1 + mx*) (y - y*)2 - «12«3^1 + mx*) y -y2)2 + w = 0

lies entirely in R+. We can take U to be a neighborhood of the ellipsoid with U ç E3= R+, so for (x,y1,y2) e E3\U, LV < -K (K is a positive constant), which implies that condition (B.2) in Lemma 4 is satisfied. Besides, there is M = min{a12x2, a^y"2, a32y2, (x,y1,y2) e U} >0 such that

J2«HMi = a2x2§12 + a22y2^22 + a3y2f3 >M||£||2 i,j=1

for all (x,yl,y2) e U, % e R3, which implies that condition (B.l) in Lemma 4 is satisfied.

Therefore, the stochastic system (5) has a stationary distribution ¡(-) and it is ergodic.

Without considering the random fluctuations of environment, that is say, ai (i = l, 2,3) = 0, according to the proof of Theorem 7, it is easy to get the following conclusion.

Corollary 3 Assume R0 > l, when all > m(r - allx*), the positive equilibrium E*(x*, y*, y2) of system (2) is globally asymptotically stable.

6 Numerical simulation

In order to confirm the results of Sections 3 and 5, we numerically simulate the solution of stochastic system (5). For system (5), we use the Milstein method mentioned in Higham [46] to substantiate the analytical findings. We consider the following discrete equations:

Xk+l = Xk + Xk[(r - allXk - l+mxk+nylk) At + alSl^^/Xt + 2al2(s2k - l)At],

yl,k+l = yl,k + yl,k[(-¿2 - a22yl,k + l+maX!k1+nylk - Py2,k)At + a2S2k\TXt

+ 2a22(s22k -1) At], y2,k+l = y2,k + y2,k[(Pyl,k - ¿3 - a33y2,k) At + a3S3kVXt + 2a32(s2k - l) At],

where time increment At = 0.l > 0, and sik (i = l, 2,3) are N(0, l) distributed independent random variables which can be generated numerically by pseudo-random number generators.

Here, let r = 0.8, an = 0.2, a^ = 0.8, m = n = 0.5, ¿2 = 0.2, a22 = 0.2, a2l = 0.6, P = 0.4, ¿3 = 0.2, a33 = 0.l, X0 = 0.6, yl0 = 0.4, y20 = 0.2, then E*(x*,y*,y*) = (3.040l, 0.6872, 0.7488). At first, in the absence of noise, in view of Corollary 3,theequilibriumE*(x*, y*, y*) of deterministic system (2) is globally asymptotically stable, Figure l confirms it. We choose al = 0.08, a2 = 0.02, a3 = 0.05, then (4) is satisfied, all > m(r-allxf), w ^ 0.0l, and min{a2l(all -m(r-allxf))(l + nyf)(xf)2, al2a22(l + mxf)(yf)2, al2a33(l + mx*)(yf)2} ^ 0.ll3. Hence, the conditions of Theorem 7 are verified, there is a stationary distribution for system (5). Figure l shows that the solution of (5) is fluctuating around the positive equilibrium E* of deterministic system (2) in a small neighborhood. In the last figure from Figure l, we can see that all the solutions of system (5) are around E*, which illustrates that there is a stationary distribution for system (5).

Suffering large density of white noise, we can refer to Figure 2 which also simulates system (5). Here, we choose r = 0.4, al = l, a2 = 0.8, a3 = 0.5, the other parameters are the same, the conditions of Theorem 7 are not verified, then by Theorems 3 and 4, all the populations of stochastic system (5) will become extinct, which does not happen in the corresponding deterministic system (2). Figure 2 shows it.

7 Conclusions

A stochastic model corresponding to a predator-prey model with Beddington-DeAngelis functional response and disease in the predator population is investigated. We show that system (5) has a unique global positive solution as this is essential in any population dynamics model. We also give an estimate of the solution by the comparison theorem. The threshold between persistence in the mean and extinction for prey population is given. Sufficient conditions of extinction for both the susceptible predator and the infected

IN 0.5

Figure 1 Solutions of systems (2) and (5). The black lines represent solutions of deterministic system (2), red lines are solutions of stochastic system (5), the last figure of Figure 1 is the population distribution of system (5) around E*.

1.4 1.2

x(t) - y1(t) ---y2(t) 6 x(t) - y1 (t) ---y2(t)

1 1 5 ■

0.8 4 ■

0.6 3 ■

0.4 2

0 C X J

500 1000 0 t 500 1000 t

Figure 2 Solutions of systems (2) and (5). The red line represents the prey population, the purple line and

black line are susceptible predator and infected predator population, respectively. The left figure is the

solution of deterministic system (2), the right one is the solution of stochastic system (5).

predator are obtained. Furthermore, sufficient conditions of permanence for deterministic system (2) are derived in Additional file 1, which can give us a contrast between stochastic system (5) and its corresponding deterministic system (2). This shows that environmental perturbation will make the reduced population size. There is a stationary distribution for system (5) when the environmental noise is very small, we can consider it as stability in stochastic sense. By the way, conditions of global stability of system (2) can be established. Numerical simulations illustrate that if the positive equilibrium of the deterministic system is globally stable, then the stochastic model will preserve this nice property provided

the noise is sufficiently small, but it is not true when the noise is large. All these consequences imply that the environmental white noise has an important effect on biological systems; therefore, it is more realistic and suitable to include random effects in the models.

Some interesting questions deserve further investigation. Here, we cannot get the condition of persistence for the predator population at present. In fact, there are some difficulties that cannot be overcome at present, we leave them for future research. Moreover, it is interesting to study other parameters perturbed by the environmental noise.

Additional material

Additional file 1: Appendix. In the Appendix, the local and global stability of equilibria for system (2) are discussed, the condition of permanence is also derived, we can compare these results with stochastic system (5), it shows that the environmental random perturbation plays an important role, it can not be neglected.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

SL conceived of the study, drafted the manuscript and participated in the sequence alignment. XW performed the

statistical analysis and helped to draft the manuscript. All authors read and approved the final manuscript.

Author details

1 College of Mathematics and Information Science, Henan Normal University, Jianshe Road, Xinxiang, 453007, P.R. China.

2Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and

Information Science, Henan Normal University, Xinxiang, 453007, P.R. China. 3College of Xinlian, Henan Normal University,

Jianshe Road, Xinxiang, 453007, P.R. China.

Acknowledgements

Shuang Li would like to thank Professor Xinan Zhang, her teacher, for his helpful comments which improved this work.

This work is supported by Science and Technology Search Key Project of Education Department of Henan Province in

China (No. 14A110019) and Foundation for Ph.D. of Henan Normal University (No. qd13043).

Received: 10 October 2014 Accepted: 19 March 2015 Published online: 21 July 2015

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