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 [1] established a Volterra model with
mutual interference as follows:x˙(t)=xg(x)−φ(x)ym,y˙(t)=y(−d+kφ(x)ym−1−q(y)), where m denote mutual interference constant and 0<m≤1.

Motivated by the works of Hassell [1], Wang and Zhu [2]
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 [3–15]). 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)ym−1(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=0T−1f(n), fM=supn∈IT{f(n)},
and fL=infn∈IT{f(n)}, where IT={0,1,2,…,T−1}.

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 n∈N 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 infn→∞x(n)≤lim supn→∞x(n)≤x*,y*≤lim infn→∞y(n)≤lim supn→∞y(n)≤y*.

Lemma 2.2.

Assume that x(n) satisfiesx(n+1)≤x(n)exp{a(n)−b(n)x(n)}∀n≥n0,where {a(n)} and {b(n)} are positive sequences, x(n0)>0 and n0∈N. Then, one haslim supn→∞x(n)≤B,where B=exp(aM−1)/bL.

Lemma 2.3.

Assume that x(n) satisfiesx(n+1)≥x(n)exp{a(n)−b(n)x(n)}∀n≥n0,where {a(n)} and {b(n)} are positive sequences, x(n0)>0 and n0∈N. Also, lim supn→∞x(n)≤B and bMB/aL>1. Then, one haslim infn→∞x(n)≥A,where A=(aL/bM)exp(aL−bMB).

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 [12].

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:Z→R+ and a:Z→R 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 n∈Z+ 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 n≥0.

3. Permanence

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

Proposition 3.1.

If (H1):(1−m)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 supn→∞x(n)≤x*,lim supn→∞y(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 supn→∞x(n)≤x*,wherex*=1b1Lexp(r1M−1).

Denote P(n)=(1/y(n))1−m.
Then, from the second equation of (1.3), it follows thatP(n+1)=P(n)exp{(1−m)r2(n)+(1−m)b2(n)P(n)1−m−(1−m)c2(n)x(n)k+x(n)P(n)},which leads toP(n+1)≥P(n)exp{(1−m)r2(n)−(1−m)c2MP(n)}.Consider the following auxiliary
equation:Z(n+1)=Z(n)exp{(1−m)r2(n)−(1−m)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 (1−m)c2MZ(n)≤1 for n sufficiently large, where Z(n) is any solution of (3.7). Consider the
following function:g(n,Z)=Zexp{(1−m)r2(n)−(1−m)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 infn→∞P(n)≥(Z*(n))L, which together with that transformation P(n)=(1/y(n))1−m, produceslim supn→∞y(n)≤1(Z*(n))L1−m≜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 infn→∞x(n)≥x*,lim infn→∞y(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 n≥N1,y(n)≤y*+ε.Then, from the first equation of
(1.3), for n≥N1, we havex(n+1)≥x(n)exp{r1(n)−c1(n)k(y*+ε)m−b1(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(r1M−1)b1L>1.Here, we use the fact exp(r1M−1)>r1M. From (3.12) and (3.15), by Lemma 2.3, we havelim infn→∞x(n)≥(a1(n,ε))Lb1Mexp{(a1(n,ε))L−b1Mx*}.Setting ε→0 in the above inequality leads tolim infn→∞x(n)≥a1Lb1Mexp{a1L−b1Mx*}≜x*,wherea1(n)=r1(n)−c1(n)k(y*)m.For above ε,
there exists N2>N1 such that for n≥N2,x(n)≥x*−ε. So from (3.5), we obtain thatP(n+1)≤P(n)exp{(1−m)(r2(n)+b2(n)(y*+ε))−(1−m)c2(n)(x*−ε)k+x*−εP(n)}.Consider the following auxiliary
equation:W(n+1)=W(n)exp{(1−m)(r2(n)+b2(n)(y*+ε))−(1−m)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).

SetU(n)=R(n)−Y(n).Then,U(n+1)−U(n)≤−(1−m)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, limn→∞U(n)=0, which implies thatlim supn→∞P(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,t∈N) 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 supn→∞U(n)=lim sups→∞U(ns1). From (3.25), and U(ns1−1)<0, we can obtainU(ns1)≤(1−m)c2(ns1)(x*−ε)k+x*−εexp{Y(ns1)}[1−exp{U(ns1)}],≤(1−m)c2M(x*−ε)k+x*−ε(W*(n))M.

From (3.22) and (3.24), we easily obtainlim supn→∞P(n)≤(W*(n))Mexp{(1−m)c2M(x*−ε)k+x*−ε(W*(n))M}.Setting ε→0 in the above inequality leads tolim supn→∞P(n)≤(W*(n))Mexp{(1−m)c2Mx*k+x*(W*(n))M}≜P*,which together with that
transformation P(n)=(1/y(n))1−m, we havelim infn→∞y(n)≥1P*1−m≜y*.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*≤x≤x*,y*≤y≤y*} 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)−0.6y−0.4(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)L≈0.4197>0,(1−m)(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).

HassellM. P.Density-dependence in single-species populationsWangK.ZhuY.Global attractivity of positive periodic solution for a Volterra modelChenF.Permanence of a discrete N-species cooperation system with time delays and feedback controlsChenF.Permanence of a discrete n-species food-chain system with time delaysChenF.WuL.LiZ.Permanence and global attractivity of the discrete Gilpin-Ayala type population modelChenF.Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systemsFanM.WangK.Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey systemYangX.Uniform persistence and periodic solutions for a discrete predator-prey system with delaysLiY.Positive periodic solutions of discrete Lotka-Volterra competition systems with state dependent and distributed delaysChenY.ZhouZ.Stable periodic solution of a discrete periodic Lotka-Volterra competition systemZhouZ.ZouX.Stable periodic solutions in a discrete periodic logistic equationFanY.-H.LiW.-T.Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional responseHuoH.-F.LiW.-T.Existence and global stability of periodic solutions of a discrete ratio-dependent food chain model with delayLiY.ZhuL.Existence of positive periodic solutions for difference equations with feedback controlWangL.WangM. Q.