Global Stability of a Stage-Structured Predator-Prey Model with Stochastic Perturbation Academic research paper on "Mathematics"

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2014, Article ID 512817, 8 pages http://dx.doi.org/10.1155/2014/512817

Research Article

Global Stability of a Stage-Structured Predator-Prey Model with Stochastic Perturbation

Liu Yang and Shouming Zhong

School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 611731, China

Correspondence should be addressed to Liu Yang; ylazx@126.com

Received 14 October 2013; Accepted 4 January 2014; Published 24 February 2014

Academic Editor: Leonid Shaikhet

Copyright © 2014 L. Yang and S. Zhong. 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.

This paper is concerned with a new predator-prey model with stage structure on prey, in which the immature prey and the mature prey are preyed on by predator. We think that the model is more realistic and interesting than the one in which only the immature prey or the mature prey is consumed by predator. Our work shows that the stochastic model and its corresponding deterministic system have a unique global positive solution and the positive solution is global asymptotic stability for each model. If the positive equilibrium point of the deterministic system is globally stable, then the stochastic model will preserve the nice property provided that the noise is sufficiently small. Results are analyzed with the help of graphical illustrations.

1. Introduction

Within the past decades, the dynamic behavior between predator and their prey has received considerable interest due to their wide applications in ecology and mathematical ecology. There is a great deal of attention for predator-prey models from many scholars [1-9]. In most of the cases, the study is based on interactions between homogeneous populations. However, in the nature, most of the species must go through two life stages from birth to death. In [6-9], some stage-structured models of population growth consisting of immature and mature individuals were discussed. In particular, [9] considered a predator-prey model with two populations, that is, predator and their food prey. In their model, only prey species is divided into two life stages, the immature prey and the mature prey. And the predators only consume the immature prey species. In [10], Chinese fire-bellied newt is described as an example, which is unable to prey on the mature Rana chensinensis and can only prey on the immature one. So, they consider the following nonlinear ordinary differential equations:

dxi _ _2_ß_ß .

dx3 dt

dx 2 ^ ß

— -rx3 + kßiXxX3

In [11], the authors have studied the dynamical properties of deterministic model (1) and the stochastic behavior of the corresponding stochastic model (19), which was first introduced by Beretta et al. [12] and Shaikhet [13]. Then, they obtained stochastic stability condition in mean square sense by utilizing Lyapunov function.

On one hand, the predators functional response, that is, the rate of prey consumption by an average predator, is one of the important components which can impact the relationship between predator and prey in population dynamics. There are many functional responses such as Holling type [5], Beddington-DeAngelis type [7], and Watt type [14]. On the other hand, population is inevitably affected by environmental noise in nature [15, 16]. May [17] also showed that, due to environmental fluctuation, the birth rate, the death rate, competition coefficients, and other parameters usually show random fluctuation to a certain extent

that should be stochastic. Therefore, many authors have taken stochastic perturbation into deterministic models and shown the effect of environmental variability on population dynamics in mathematical ecology [18-26]. For example, [26] considered the following stochastic stage-structured predator-prey model:

dx1 = [rx2 - d1x1 - ax^ - bx1] dt - a1x1 (x1 - x*) dB1 (t);

dX-2 —

dx3 —

r J 2 a\2X2X3

^11 ^2

1 + mx2 + nx3 j

- o2x2 (x2 - x2)dB2 (t) ;

-dix, +

1 + mx2 + nx3

-°3X3 (x3 -x*3)dB3 (t).

In this paper, the authors mainly utilize Ito's formula, the theory of stochastic differential equations, and Lyapunov functions to investigate the global stability of the positive equilibrium of model (2).

Motivated by the above works, in this paper, we will consider the following stochastic stage-structured predator-prey model:

dx1 —

dx3 —

a13X1X3

YX2 d] X-l Xl

2 11 11 1 1 1 + mx1 +nx3\ - o1x1 (x1 - x2)dB1 (t) ;

7 J 2 ^23^2^3

1 2 2 22 2 a + ßx2 - o2x2 (x2 - x2)dB2 (t) ;

-^3X3 +

k1a13X1X3 + k2a23X2X3

( + ßx2

1 + mx1 + nx3

0.33X3

-°3X3 (x3 -xl)dB3 (t);

where x1 (t) denotes the population density of the immature prey, x2 (t) represents the population density of the mature prey, and x3(t) stands for the population size of predator. Bj(t) (i = 1,2,3) is a standard Brownian motion, which is defined on a complete probability space (Q, F, P) with a filtration {Ft}teR satisfying the usual conditions (i.e., right continuous and increasing while F0 contains all P-null sets). All parameters involved with the model are positive constants and can be interpreted in more detail: r is the birth rate of immature prey population, d1, d2, and d3 represent the death rate of immature prey population, mature prey population, and predator population, respectively, a11, a22, and a33 denote intraspecies competition rate of immature prey population, mature prey population, and predator population, respectively, b represents transformation rate from immature prey population to mature prey population, ai3 (i = 1,2) is the rate of predation, kt (i = 1,2) is fraction of prey

biomass converted into predator biomass, x1/(1+ mx1 + nx3) represents Beddington-DeAngelis functional response and x2/(a + px2) denotes Holling-II functional response, and (t2 (i = 1,2,3) is the intensity of the noise.

The initial condition of model (3) is any point in the biological meaning region R+ = {(x1,x2,x3) e R | x1 > 0,x2 > 0,x3 > 0}. (xf,x2,x3) is a positive equilibrium of model (3) which is the solution of the algebraic equations

Y&2 d1X^ 1^1 ~b%1 1

a13X1 X3

+ mx1 + nx3

-dix, +

1 j 2 ^23^2^3

0X1 d^X-^ $22 _ — 0;

1 22 22 2 a + ßx2

k1 &13X 1-^2^3 2

1 + mx1 + nx3 a + ßx2

a33X3 —

with initial conditions xt(0) > 0, i = 1,2, 3. Noting that if a1 = o2 = a3 = 0, then model (3) becomes the following corresponding deterministic stage-structured predator-prey systems:

^ — Y&2 d1 &11 ~bx1 1

+ m%1 + nx3

7 J 2 ^23^2 X3

dt 1 2 2 22 2 a + ßx2

dx3 dt

— -dix, +

Ka13X1X3 , k2a23X2X3

*13A1A3 ^2^-23^2^3 „ 2

+ „ a33x3.

1 + mx1 + nx3 a + ßx2

Therefore, in this paper, we only need to establish the sufficient conditions for global asymptotic stability of system (3). And, in this paper, we will also use Ito's formula, the theory of stochastic differential equations, and Lyapunov functions to study the global stability of the positive equilibrium point of stochastic system (3).

The paper is organized as follows. In Section 2, we study the existence and uniqueness of global positive solution of system (3). In Section 3, sufficient conditions for global asymptotic stability of system (3) are established. Then, we introduce some simulation figures to illustrate the main result in Section 4. In the last section, we give the conclusions.

2. Existence and Uniqueness of Solution

In this section, we will show that the solution of system (3) is positive and global. We give the following theorem.

Theorem 1. For any given initial value (x1(0), x2(0), x3(0)) e R+, there is a uniquesolution (x1(t),x2(t),x3(t)) to (3) on t > 0. Furthermore, with probability one, R+ are positive invariant for (3); that is, for all t > 0, (x1(t),x2(t),x3(t)) e R+ a.s., if (X1(0),X2(0),X3(0))eR3+.

Proof. Since the coefficients of (3) are locally Lipschitz continuous, there is a unique local solution (x1(t),x2(t),x3(t)) to (3) on t e [0, re), where Te is the explosion time [27, 28]. Therefore, to show that the solution is global, we only need

21 1 3

to show that Te = i a.s. We use the technique oflocalization [29, 30]. Let k0 > 0 be sufficiently large for xt(0) (i = 1,2, 3) lying within the interval [1/k0, k0]. Let us define a sequence of stopping time [29] for each integer k > k0 by

rk = inf {ie [0,Te):x, (t)t(^,k),i= 1,2,3}. (6)

The convention here is that the infimum of the empty set is 1. Since Tk is nondecreasing as k ^ i,weset rTO = limfc^TOrfc. Then, Tm < Te a.s. Now, we will show that rm = 1 a.s. If the statement is not right, then there exist T > 0 and e e (0,1) such that P[Tm < T} > e. Thus, by denoting i\ = [Tk < T}, there exists kt > k0 such that

sayby M. Thus, since (x1(tArk), x2(tATk),andx3(tATk)) e R+ and we consider (9), we have

dV (x1 (t),x2 (t),x3 (t))

№ 3 f

I Ui,-l I

3 rTATt

g, (xl,X2,x3)dBl (t).

P(Q.k)>e Vk>kv

Consider the following function: V(x1 ,x2,x3 ) = 2 £i=1 ( -

Taking expectations yields

EV(xx (TArk),x2 (TArk),x3 (TArk))

<V(x1 (0),x2 (0),x3 (0))+ I Mdt

< V (x1 (0), x2 (0), x3 (0)) + MT.

On the other hand, for every w e Qfc, either x1(rk,w) or x2(rk, w) or x3(rk, w) equals either k or 1/k. Then,

dV(xi, X2, X3) = f (xi,x2,x3)dt -^g, (xvX2,X3)dBi (t),

1-0.5 lnx). It is clear to see that V e C2(R+,R+). If V (x1 (TArk,w),x2 (TArk,w),x3 (T A rk, w)) (x1(t),x2(t),x3(t)) e R+ ,byusingIto formula, weget _

> min - Vk-1-0.5 lnk,^1-1- 0.5 ln (1)

We therefore get from (7) that

EV (Xi (t A Tk) ,X2 (tA Tk) , X3 (tA Tk))

> E [1nkV (x1 (t A rk) ,X2 (tA Tk) ,X3 (tA Tk))]

f (x1, x2, X3)

(V*1-1)

> e min

Vk-1- 0.5 lnk, 0.5 ln (1)

- bx1 -

rX^ $1^1 ^n X

+ j (-Vxl + 2) (x1 -x»)

1 + mx1 + nx3 j

where 1Qk is the indicator function of Qfc. Then, it follows from (11) that

V(x1 (0),x2 (0),x3 (0)) + MT

> e min

Vk-1- 0.5lnk, ^k-1-0.5ln(k)

(Vr2 -2)

b%1 &22^2

2 023X2X3

+ -J (-V^2 +2) (x2 -x¡)2,

(V^3-1)

Letting k ^ m leads to the contradiction m > V(x1(0),x2(0),x3(0)) + MT = m. Therefore, rm = m a.s. Then, re = m a.s. and (x1(t),x2(t),x3(t)) e R3+ a.s. This completes the proof of Theorem 1. □

3. Globally Asymptotically Stable

For the sake of convenience, denote

k1O13X1X3 k2O23 ^2^3 2

&3X3 + i + ~ O33X'.

1 + mx1 + nx3 a + ?x2

+ J (-V*~3 +2) (x3 - xlf

gi (x1,x2,x3) = °i (Vxi -1)(xi -x»).

A = —1

» 2 *

mo13x3 °1X1

1 + mx» + nx»

a (a + fix») 2

x* (1 + mx») rk1 (1 + nx*»)

Since all coefficients of system (2) are positive constants, it is easy to see from (9)thatthe function f(x1 ,x2,x3) is bounded,

^ = 023X» + k2 023Xl (1 + mXl )

rk1 (1 + nx*») (a + fix»)'

Theorem 2. If

as well as

A<0, C<0

4BC -D2 >0

Then, the positive equilibrium position (x\ ,x*2 ,x3) of model (3) is globally asymptotically stable with probability one; that is, for any positive initial data (x1(0),x2(0),x3(0)), the solution of system (3) has the property that

lim X: (t) = x* (i = 1,2, 3)

almost surely.

Proof. From the stability theory of stochastic differential equations, we only need to find a suitable Lyapunov function V(z) satisfying LV(z) < 0 and the identity holds if and only if z = z* [30], where z = z(t) is the solution of the following stochastic differential equation:

dz (t) = f(t,z (t)) dt + g (t, z (t)) dB (t), (19)

z* is the positive equilibrium position of (19), and

LV (z) = Vz (z) f (t, z) + 0.5 trace [gT (t, z) Vzz (z) g (t, z) ].

For x e R,, define

i=i 2 h

q rl X: - X;

q (1 + mx*) kx (1 + nx*)

We can rewrite (3) as follows:

dx, — (X2 ) x2 X* )] dt

dux, X—) d-t

(1 + mx*) (x3 - x*) - mx* (xx -x*)

$13X1 T dt

(1 + mx, + nx3) (1 + mx- + nx*)

- o1x1 (x, - xi—)dB1 (t);

b * * dX2 — — \ X— ) X1 (X2 )] ^^

2 (,X2 ) ^^ - a23X2

(a + ßx*) (X3 -x*)- ßx* (x2 -x*)

(a + ßx2) (a + ßx

- o2x2 (x2 - x*)dB2 (t);

dX3 — &33X3 X3 ) ^^

(1 + nx*-) (%1 - X—) - nx\ (X3 - X—) + 3^3 7 77 r

(1 + mx, + nx3) (1 + mx* + nx* )

OC (X2 - X* ) + K2CI2-1X-1—,---r—---r dt

2 23 3 (a + ßx2) (a + ßx3*)

-°3X3 (x3 -x3*)dB3 (t).

Applying Ito formula to model (23), we can get that LV(x)

(x2 X*) (xl X\) X-i

an(x, x*)

(1 + mx*) (x3 - x**) - mx* (x, - x*)

(1 + mx, + nx3) (1 + mx* + nx*)

/ n ^2

X(X1-Xl)+—^(X1-Xl )

(x2 X*) . X2

(x! X1 ) „ (x2 X*)

a22(x2 X*)

(a + ßX*) (x3 -X*)- ßX* (x2 - X*]

(a + ßx2) (a + ßx*)

/ * \ ^2X2 ( X (X2 - X2 ) + - (X2 - X2 )

x* (1 + mx*) rkx (1 + nx*)

- a33(x3 - x*) +

(1 + nx*) (xx - x*) - nx* (x3 - X*) (1 + mx1 + nx3) (1 + mx* + nx*)

X (X3 ) + ^2^23

X (X2 X2 ) (X3 X* )

(a + ßx2) (a + ßx*)

"3'~3 t ^2

+ — (x3 -x3 )

= (X± - X*) (X2 - X2)

x2 ( *\2 _ a1 /• _ ^2

2t ^2 2r

(x, - x^2

r(l + mx1 + nx3) (1 + mx* + nx»)

2_ X» (l + mxD X(Xl X2) rk, (l+nx*3)

2 * a3 x3

(X, X») + (X, X»)(X2 X2) (x2 X2)

f *\2 a23X2

(X3- X*)

I X2 X2 I I Xr3 X3 !

^ (*2- x2)2

°2 (X2 ) f 2x2 | _ ^ ) +

2b y 2 2 b(a + fix2) (a + fix»,

k2a23x* (l + mx*) rk, (l + nx») (a + fix**)

2 2 I I 3 3 I

X (X2 X* )

2 x* (1+mx*)

rk, (l + nx*)

Let \Z _ Z*\ = (1%! _ x*\, \x2 _ x*\, \x3 _ x* |) ; then we

2* o^x

x ( a33 _

33 \ 3 ) (x3 x*)

na13(x*)2 (l + mx*) (x3 _ x*)2 r(l + nx*) (l + mx, + nx3) (l + mx* + nx*)

'2A 0 0

LV(x)<-\Z_Z^[ ( 0 2B D )\Z_Z"\. (25) 2 \ 0 D 2C

a23X* (x2 x*)(x3 X*)

b(a + fix2)

ak2a23x* (l + mx*) (x2 _ x*) (x3 _ x*)

rk, (l + nx*) (a + fix2) (a + fix*)

'-(Xi _X*)_

C1 / — (x2 x2)

Clearly, the conditions of Theorem 2 and the above inequality denote LV(x) < 0 along all trajectories in R+ except (x*,xl,xl). Then, we get the desired assertion immediately.

For deterministic system (5), we have that the following theorem holds. q

Theorem 3. If

l + mx* + nx*

x (xi _x*)2

a23fix*

a(a + fix* ) 2

(x2 x*)

2 x* (l+mx*)

rk, (l + \

(x3 x*)

mau x3

l + mx* + nx*

a(a + fix* )

4a33x*x* (l + mx*) brk, (l + nx*)

a(a + fix* )

(l + ^^X* )

ba ' rk,(l + nx*)(a + fix**)

+ a23X2 I _ *i I _ * I

yC2 ^C^ ^Co

ba 1 2 211 3 31 k2a23x* (l + mx*) rk, (l + nx*) (a + fix**)

2 2 I I 3 3 I

mau x3

l + mx* + nx*

then the equilibrium position (x*,x*,x*) of system (5) is globally asymptotically stable.

Remark 4. From Theorems 2 and 3, we can see that if the positive equilibrium of the deterministic system is globally stable and the noise perturbation is not very large, then the stochastic system will keep the nice property.

/ ^2 x2 x(xi _X1) _ T

a23fix3

a(a + fix* ) 2

4. Numerical Simulations

In this section, we will utilize the Milstein method mentioned in Higham [31] to consolidate the analytical findings.

Figure 1: Solution of system (3) with initial conditions Xj(0) = x2(0) = x3(0) = 0-2 and r = 0.6, = 0.2, all = 0.3, b = 0.8, a13 = 0.6, d2 = 0.3, a22 = 0.2, a23 = 0.3, i3 = 0.1, fc1 = 5/6, fc2 = 0.8, a33 = 0.2, At = 0.01, m = 0.3, « = 0.1, a = 2,and ^ = 1. (a) Oj = o2 = o3 = 0. (b) of = 1.9, of = 0.6, and o2 = 1.9. (c) o2 = 20, of = 20, and of = 20.

Here, we consider the discretization equations of model (3) as follows:

(fc+ 1) = *1 (fc +

rx2 (fc) - (fc) - anxf (fc)

«13*3 (fc)

- (fc) -

1 + mx1 (fc) + nx3 (fc)

x(^1 (fc)-x;)2 $ - 1] Ai, (fc+ 1)

= X2 (fc) +

(fc) - d2x2(fc)

fl23X3 (fc)

- «22*2 (fc) -

a + (fc)_

- ^2*2 (fc) (*2 (fc) - *2*)

- a^ (fc) (*1 (fc) - x2) VÂÎ4 + -2(fc)

+ -2-*2 (fc) (*2 (fc)-x2)2 [^ - 1] Ai.

x3 (k+1)

= X3 (к) +

-d3x3 (k) +

Ml 3X1 (k)

1 + тхг (к) + nx3 (к)

k2a23X2 (k)

&33X3 (k)

a + /Зх2 (k) - 0^3X3 (к) (X3 (k) - x*3) VAt(k

+ -3 (k)(X3 (k)-xlf [?k - l] Ai.

where %к, цк, and (к = 1,2,...) are the Gaussian random variables which follow N(0,1).

Set r = 0.6, d1 = 0.2, a11 = 0.3, b = 0.8, a13 = 0.6, d2 = 0.3, a32 = 0.2, a23 = 0.3, d3 = 0.1, k1 = 5/6, k2 = 0.8, a33 = 0.2, a = 2, p = 1, m = 0.3, n = 0.1, and At = 0.01. From Figure 1, we can get that the equilibrium is x1 = 0.2137, x2 = 0.4184, and x3 = 0.2003. The only difference between the conditions of Figures 1(a), 1(b), and 1(c) is the values of a1, o2, and a3. In Figure 1(a), we suppose that a1 = o2 = a3 = 0. By Theorem 3, the equilibrium point (x3,x3,x3) of deterministic system (5) is globally asymptotically stable. Figure 1(a) confirms this. In Figure 1(b), we choose = 1.9, af = 0.6, and o\ = 1.9. From Theorem 2, we can easily get that the equilibrium position (x3 ,x3,x3) of stochastic system (3) is globally asymptotically stable. By Figure 1(c), we can see that if we choose o\ = 20, = 20, and o\ = 20, these values violate conditions (19); all the species will die out. That is to say, if the conditions of Theorem 2 are not satisfied, the positive equilibrium point (x3,x3,x3) may be no longer globally asymptotically stable.

5. Conclusions

In this paper, we investigated two stage-structured predator-prey systems: deterministic one and stochastic one. In the models, we suppose that both immature prey and mature prey are consumed by predator. The model is more realistic and complicated than the one in which only the immature prey or mature prey is preyed on by predator. For each system, we established the sufficient conditions for global asymptotic stability. From the results and simulation figures, we can see that if the positive equilibrium position of the corresponding deterministic model is globally stable and the noise is sufficiently small, then the stochastic system will preserve the nice property. The result is useful and important for ecological balance. Up to our knowledge, the present work is the first attempt to study such stochastic model with stage structure on prey.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee and editor for their valuable comments and suggestions that greatly improved the presentation of this paper. This work was supported by the Program for New Century Excellent Talents in University (NCET-10-0097), the NSFC Tianyuan Foundation (Grant no. 11226256), and the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY13A010010).

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