DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation20548110.1155/2009/205481205481Research ArticlePermanence of a Discrete Periodic Volterra Model with Mutual InterferenceChenLijuan1ChenLiujuan2VecchioAntonia1College of Mathematics and Computer ScienceFuzhou UniversityFuzhou, Fujian 350002Chinafzu.edu.cn2Department of Mathematics and PhysicsFujian Institute of EducationFuzhou, Fujian 350001China200905032009200905122008120220092009Copyright © 2009This 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 discusses a discrete periodic Volterra model with mutual interference and Holling II type functional response. Firstly, sufficient conditions are obtained for the permanence of the system. After that, we give an example to show the feasibility of our main results.

1. Introduction

In 1971, Hassell introduced the concept of mutual interference between the predators and preys. Hassell  established a Volterra model with mutual interference as follows:x˙(t)=xg(x)φ(x)ym,y˙(t)=y(d+kφ(x)ym1q(y)), where m denote mutual interference constant and 0<m1.

Motivated by the works of Hassell , Wang and Zhu  considered the following Volterra model with mutual interference and Holling II type functional response: x˙(t)=x(t)(r1(t)b1(t)x(t))c1(t)x(t)k+x(t)ym(t),y˙(t)=y(t)(r2(t)b2(t)y(t))+c2(t)x(t)k+x(t)ym(t).Sufficient conditions which guarantee the existence, uniqueness, and global attractivity of positive periodic solution are obtained by employing Mawhin's continuation theorem and constructing suitable Lyapunov function.

On the other hand, it has been found that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations (see ). However, to the best of the author's knowledge, until today, there are still no scholars propose and study a discrete-time analogue of system (1.2). Therefore, the main purpose of this paper is to study the following discrete periodic Volterra model with mutual interference and Holling II type functional response:x(n+1)=x(n)exp{r1(n)b1(n)x(n)c1(n)k+x(n)ym(n)},y(n+1)=y(n)exp{r2(n)b2(n)y(n)+c2(n)x(n)k+x(n)ym1(n)}, where x(n) is the density of prey species at nth generation and y(n) is the density of predator species at nth generation. Also, r1(n),b1(n) denote the intrinsic growth rate and density-dependent coefficient of the prey, respectively, r2(n),b2(n) denote the death rate and density-dependent coefficient of the predator, respectively, c1(n) denote the capturing rate of the predator and c2(n) represent the transformation from preys to predators. Further, m is mutual interference constant and k is a positive constant. In this paper, we always assume that {ri(n)}, {bi(n)}, {ci(n)}, i=1,2, are positive T-periodic sequences and 0<m<1. Here, for convenience, we denote f¯=(1/T)n=0T1f(n), fM=supnIT{f(n)}, and fL=infnIT{f(n)}, where IT={0,1,2,,T1}.

This paper is organized as follows. In Section 2, we will introduce a definition and establish several useful lemmas. The permanence of system (1.3) is then studied in Section 3. In Section 4, we give an example to show the feasibility of our main results.

From the view point of biology, we only need to focus our discussion on the positive solution of system (1.3). So it is assumed that the initial conditions of (1.3) are of the formx(0)>0,y(0)>0.One can easily show that the solution of (1.3) with the initial condition (1.4) are defined and remain positive for all nN where N={0,1,2,}.

2. Preliminaries

In this section, we will introduce the definition of permanence and several useful lemmas.

Definition 2.1.

System (1.3) is said to be permanent if there exist positive constants x*,y*,x*,y*, which are independent of the solution of the system, such that for any positive solution (x(n),y(n)) of system (1.3) satisfiesx*lim infnx(n)lim supnx(n)x*,y*lim infny(n)lim supny(n)y*.

Lemma 2.2.

Assume that x(n) satisfiesx(n+1)x(n)exp{a(n)b(n)x(n)}nn0,where {a(n)} and {b(n)} are positive sequences, x(n0)>0 and n0N. Then, one haslim supnx(n)B,where B=exp(aM1)/bL.

Lemma 2.3.

Assume that x(n) satisfiesx(n+1)x(n)exp{a(n)b(n)x(n)}nn0,where {a(n)} and {b(n)} are positive sequences, x(n0)>0 and n0N. Also, lim supnx(n)B and bMB/aL>1. Then, one haslim infnx(n)A,where A=(aL/bM)exp(aLbMB).

Proof.

The proofs of Lemmas 2.2 and 2.3 are very similar to that of [8, Lemmas 1 and 2], respectively. So, we omit the detail here.

The following Lemma 2.4 is Lemma 2.2 of Fan and Li .

Lemma 2.4.

The problemx(n+1)=x(n)exp{a(n)b(n)x(n)},with x(0)=x0>0 has at least one periodic positive solution x*(n) if both b:ZR+ and a:ZR are T-periodic sequences with a¯>0. Moreover, if b(n)=b is a constant and aM<1, then bx(n)1 for n sufficiently large, where x(n) is any solution of (2.6).

The following comparison theorem for the difference equation is Theorem 2.1 of L. Wang and M. Q. Wang [15, page 241].

Lemma 2.5.

Suppose that f:Z+×[0,+) and g:Z+×[0,+) with f(n,x)g(n,x) (f(n,x)g(n,x)) for nZ+ and x[0,+). Assume that g(n,x) is nondecreasing with respect to the argument x. If x(n) and u(n) are solutions ofx(n+1)=f(n,x(n)),u(n+1)=g(n,u(n)),respectively, and x(0)u(0) (x(0)u(0)), thenx(n)u(n),(x(n)u(n))for all n0.

3. Permanence

In this section, we establish a permanent result for system (1.3).

Proposition 3.1.

If (H1):(1m)r2M<1 holds, then for any positive solution (x(n),y(n)) of system (1.3), there exist positive constants x* and y*, which are independent of the solution of the system, such thatlim supnx(n)x*,lim supny(n)y*.

Proof.

Let (x(n),y(n)) be any positive solution of system (1.3), from the first equation of (1.3), it follows thatx(n+1)x(n)exp{r1(n)b1(n)x(n)}.By applying Lemma 2.2, we obtainlim supnx(n)x*,wherex*=1b1Lexp(r1M1).

Denote P(n)=(1/y(n))1m. Then, from the second equation of (1.3), it follows thatP(n+1)=P(n)exp{(1m)r2(n)+(1m)b2(n)P(n)1m(1m)c2(n)x(n)k+x(n)P(n)},which leads toP(n+1)P(n)exp{(1m)r2(n)(1m)c2MP(n)}.Consider the following auxiliary equation:Z(n+1)=Z(n)exp{(1m)r2(n)(1m)c2MZ(n)}.By Lemma 2.4, (3.7) has at least one positive T-periodic solution and we denote one of them as Z*(n). Now (H1) and Lemma 2.4 imply (1m)c2MZ(n)1 for n sufficiently large, where Z(n) is any solution of (3.7). Consider the following function:g(n,Z)=Zexp{(1m)r2(n)(1m)c2MZ}.It is not difficult to see that g(n,Z) is nondecreasing with respect to the argument Z. Then, applying Lemma 2.5 to (3.6) and (3.7), we easily obtain that P(n)Z*(n). So lim infnP(n)(Z*(n))L, which together with that transformation P(n)=(1/y(n))1m, produceslim supny(n)1(Z*(n))L1m    y*.Thus, we complete the proof of Proposition 3.1.

Proposition 3.2.

Assume that(H2):(r1(n)c1(n)k(y*)m)L>0holds, then for any positive solution (x(n),y(n)) of system (1.3), there exist positive constants x* and y*, which are independent of the solution of the system, such thatlim infnx(n)x*,lim infny(n)y*,where y* can be seen in Proposition 3.1.

Proof.

Let (x(n),y(n)) be any positive solution of system (1.3). From (H2), there exists a small enough positive constant ε such that(r1(n)c1(n)k(y*+ε)m)L>0.Also, according to Proposition 3.1, for above ε, there exists N1>0 such that for nN1,y(n)y*+ε.Then, from the first equation of (1.3), for nN1, we havex(n+1)x(n)exp{r1(n)c1(n)k(y*+ε)mb1(n)x(n)}.Let a1(n,ε)=r1(n)(c1(n)/k)(y*+ε)m, so the above inequality follows thatx(n+1)x(n)exp{a1(n,ε)b1(n)x(n)}.Because (a1(n,ε))L<r1L and b1L<b1M, we haveb1M(a1(n,ε))Lx*>b1Mr1Lexp(r1M1)b1L>1.Here, we use the fact exp(r1M1)>r1M. From (3.12) and (3.15), by Lemma 2.3, we havelim infnx(n)(a1(n,ε))Lb1Mexp{(a1(n,ε))Lb1Mx*}.Setting ε0 in the above inequality leads tolim infnx(n)a1Lb1Mexp{a1Lb1Mx*}x*,wherea1(n)=r1(n)c1(n)k(y*)m.For above ε, there exists N2>N1 such that for nN2,x(n)x*ε. So from (3.5), we obtain thatP(n+1)P(n)exp{(1m)(r2(n)+b2(n)(y*+ε))(1m)c2(n)(x*ε)k+x*εP(n)}.Consider the following auxiliary equation:W(n+1)=W(n)exp{(1m)(r2(n)+b2(n)(y*+ε))(1m)c2(n)(x*ε)k+x*εW(n)}.By Lemma 2.4, (3.21) has at least one positive T-periodic solution and we denote one of them as W*(n).

LetR(n)=ln(P(n)),Y(n)=ln(W*(n)).

Then,R(n+1)R(n)(1m)(r2(n)+b2(n)(y*+ε))(1m)c2(n)(x*ε)k+x*εexp{R(n)},Y(n+1)Y(n)=(1m)(r2(n)+b2(n)(y*+ε))(1m)c2(n)(x*ε)k+x*εexp{Y(n)}.

SetU(n)=R(n)Y(n).Then,U(n+1)U(n)(1m)c2(n)(x*ε)k+x*εexp{Y(n)}[exp{U(n)}1].In the following we distinguish three cases.

Case 1.

{U(n)} is eventually positive. Then, from (3.25), we see that U(n+1)<U(n) for any sufficiently large n. Hence, limnU(n)=0, which implies thatlim supnP(n)(W*(n))M.

Case 2.

{U(n)} is eventually negative. Then, from (3.24), we can also obtain (3.26).

Case 3.

{U(n)} oscillates about zero. In this case, we let {U(nst)}(s,tN) be the positive semicycle of {U(n)}, where U(ns1) denotes the first element of the sth positive semicycle of {U(n)}. From (3.25), we know that U(n+1)<U(n) if U(n)>0. Hence, lim supnU(n)=lim supsU(ns1). From (3.25), and U(ns11)<0, we can obtainU(ns1)(1m)c2(ns1)(x*ε)k+x*εexp{Y(ns1)}[1exp{U(ns1)}],(1m)c2M(x*ε)k+x*ε(W*(n))M.

From (3.22) and (3.24), we easily obtainlim supnP(n)(W*(n))Mexp{(1m)c2M(x*ε)k+x*ε(W*(n))M}.Setting ε0 in the above inequality leads tolim supnP(n)(W*(n))Mexp{(1m)c2Mx*k+x*(W*(n))M}P*,which together with that transformation P(n)=(1/y(n))1m, we havelim infny(n)1P*1my*.Thus, we complete the proof of Proposition 3.2.

Theorem 3.3.

Assume that (H1) and (H2) hold, then system (1.3) is permanent.

It should be noticed that, from the proofs of Propositions 3.1 and 3.2, one knows that under the conditions of Theorem 3.3, the set Ω={(x,y)x*xx*,y*yy*} is an invariant set of system (1.3).

4. Example

In this section, we give an example to show the feasibility of our main result.

Example 4.1.

Consider the following system x(n+1)=x(n)exp{0.7+0.1sin(n)0.7x(n)0.4y0.6(n)2.4+x(n)},y(n+1)=y(n)exp{(0.8+0.1cos(n))(1.1+0.1sin(n))x(n)0.6y0.4(n)2.4+x(n)},where r1(n)=0.7+0.1sin(n), b1(n)=0.7, c1(n)=0.4, r2(n)=0.8+0.1cos(n), b2(n)=1.1+0.1sin(n), c2(n)=0.6,m=0.6,k=2.4.

By simple computation, we have y*1.1405. Thus, one could easily see that(r1(n)c1(n)k(y*)m)L0.4197>0,(1m)(r2(n))M=0.36<1.Clearly, conditions (H1) and (H2) are satisfied, then system (1.3) is permanent.

Figure 1 shows the dynamics behavior of system (1.3).

Dynamics behavior of system (1.3) with initial condition (x(0),y(0))=(0.6,0.03).

x

y

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions which greatly improve the presentation of the paper. This work was supported by the Program for New Century Excellent Talents in Fujian Province University (NCETFJ) and the Foundation of Fujian Education Bureau (JB08028).

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