This paper discuss a discrete periodic Volterra model with mutual interference and Beddington-DeAngelis functional response. By using the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system. After that,we give an example to show the feasibility of our main result.

1. Introduction

In 1971, Hassell introduced the concept of mutual interference m(0<m≤1) and established a Volterra model with mutual interference as follows: (see [1])
ẋ=xg(x)-φ(x)ym,ẏ=y(-d+kφ(x)ym-1-q(y)).
Recently, Wang and Zhu [2] proposed following system:
ẋ(t)=x(t)(r1(t)-b1(t)x(t))-c1(t)x(t)k+x(t)ym(t),ẏ(t)=y(t)(-r2(t)-b2(t)y(t))+c2(t)x(t)k+x(t)ym(t).
Motivated by the works of Wang and Zhu [2], Lin and Chen [3] considered an almost periodic Volterra model with mutual interference and Beddington-DeAngelis function response as follows, which is the generalization of the model (1.2):
ẋ(t)=x(t)(r1(t)-b1(t)x(t))-k1(t)x(t)a(t)+d(t)x(t)+c(t)y(t)ym(t),ẏ(t)=(-r2(t)-b2(t)y(t))+k2(t)x(t)a(t)+d(t)x(t)+c(t)y(t)ym(t).
Sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained by applying the comparison theorem of the differential equation and constructing a suitable Lyapunov functional.

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 [4, 5]). This motivated us to propose and study the discrete analogous of predator-prey system (1.4): x(n+1)=x(n)exp{r1(n)-b1(n)x(n)-k1(n)a(n)+d(n)x(n)+c(n)y(n)ym(n)},y(n+1)=y(n)exp{-r2(n)-b2(n)y(n)+k2(n)x(n)a(n)+d(n)x(n)+c(n)y(n)ym-1(n)},
where x(n) is the density of prey species at nth generation, 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-dependant coefficient of the predator, respectively, k1(n) is the capturing rate of the predator, k2(n)/k1(n) is the rate of conversion of nutrients into the reproduction of the predator. Further, m is mutual interference constant. In this paper, we assume that all the coefficients a(n),d(n),c(n),ri(n),ki(n),bi(n),i=1,2, are all positive ω-periodic sequences and 0<m<1. Here, for convenience, we denote f¯=(1/ω)∑n=0ω-1f(n),fu=supn∈Iω{f(n)}, and fl=infn∈Iω{f(n)} where Iω={0,1,2,…,ω-1}.

The remaining part of this paper is organized as follows: in Section 2 we will introduce some definitions and establish several useful lemmas. The permanence of system (1.4) is then studied in Section 3. In Section 4, we give an example to show the feasibility of our main result.

By the biological meaning, we will focus our discussion on the positive solution of system (1.4). So it is assumed that the initial conditions of (1.4) are of the form
x(0)>0,y(0)>0.
One can easily show that the solution of (1.4) with the initial condition (1.5) 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 (1.4) is said to be permanent if there exist positive constants x*,y*,x*,y*, which are independent of the solution of system (1.4), such that for any position solution (x(n),y(n)) of system (1.4) satisfies
x*≤lim infn→+∞x(n)≤lim supn→+∞x(n)≤x*,y*≤lim infn→+∞y(n)≤lim supn→+∞x(n)≤y*.

Assume that {x(n)} satisfies x(n)>0 and
x(n+1)≤x(n)exp{a(n)-b(n)x(n)}
for n∈N, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants. Then
lim supn→+∞x(n)≤1blexp(au-1).

Assume that {x(n)} satisfies
x(n+1)≥x(n)exp{a(n)-b(n)x(n)},n≥N0.lim supn→+∞x(n)≤x* and x(N0)>0, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants and N0∈N. Then
lim infn→+∞x(n)≥alexp{al-bux*}bu.

The problem
x(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 ω-periodic sequences with a¯>0. Moreover, if b(n)=b is a constant and au<1, then bx(n)≤1 for n sufficiently large, where x(n) is any solution of (2.6).

Suppose that f:Z+×[0,+∞) and g:Z+×[0,+∞) with 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 of
x(n+1)=f(n,x(n)),u(n+1)=g(n,u(n)),
respectively, and x(0)≤u(0)(x(0)u(0)), then
x(n)≤u(n)(x(n)≥u(n))
for all n>0.

3. Permanence

In this section, we establish a permanent result for system (1.4).

Proposition 3.1.

If (H1):(1-m)r2u<1 holds, then for any positive solution (x(n),y(n)) of system (1.4), there exist positive constants x* and y*, which are independent of the solution of the system, such that
lim supn→+∞x(n)≤x*,lim supn→+∞y(n)≤y*.

Proof.

Let (x(n),y(n)) be any positive solution of system (1.4), from the first equation of (1.4), it follows that
x(n+1)≤x(n)exp{r1(n)-b1(n)x(n)}.
By applying Lemma 2.2, we obtain
lim supn→+∞x(n)≤x*,
where
x*=1b1lexp(r1u-1).
Denote P(n)=(1/y(n))1-m. Then from the second equation of (1.4), it follows that
P(n+1)=P(n)exp{(1-m)r2(n)+(1-m)b2(n)P(n)1-m-(1-m)k2(n)x(n)a(n)+d(n)x(n)+c(n)P(n)m-1P(n)},
which leads to
P(n+1)≥P(n)exp{(1-m)r2(n)-(1-m)k2uP(n)}.
Consider the following auxiliary equation:
J(n+1)=J(n)exp{(1-m)r2(n)-(1-m)k2uJ(n)}.
By Lemma 2.4, (3.7) has at least one positive ω-periodic solution and we denote one of them as J*(n). Now (H1) and Lemma 2.4 imply (1-m)K2uJ(n)≤1 for n sufficiently large, where J(n) is any solution of (3.7). Consider the following function:
g1(n,J)=Jexp{(1-m)r2(n)-(1-m)K2uJ}.
It is not difficult to see that g1(n,J) is nondecreasing with respect to the argument J. Then applying Lemma 2.5 to (3.6) and (3.7), we easily obtain that P(n)≥J*(n). So lim infn→+∞P(n)≥(J*(n))l, which together with that transformation P(n)=(1/y(n))1-m, produces
lim supn→+∞y(n)≤1(J*(n))l1-m=Δy*.
This ends the proof of Proposition 3.1.

Proposition 3.2.

Assume that
(H2):(r1(n)-k1(n)(y*)mal)l>0
hold, then for any positive solution (x(n),y(n)) of system (1.4), there exist positive constants x* and y*, which are independent of the solution of the system, such that
lim 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 (1.4). From (H2), there exists a small enough positive constant ε such that
((r1(n)-k1(n)(y*+ε)ma(n))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 (1.4), for n>N1, we have
x(n+1)≥x(n)exp{r1(n)-k1(n)(y*+ε)ma(n)-b1(n)x(n)}.
Let e(n,ε)=r1(n)-(k1(n)/a(n))(y*+ε)m, so the above inequality follows that
x(n+1)≥x(n)exp{e(n,ε)-b1(n)x(n)}.
From (3.12) and (3.15), by Lemma 2.3, we have
lim infn→+∞x(n)≥(e(n,ε))lb1uexp{(e(n,ε))l-bulx*}.
Setting ε→0 in the above inequality leads to
lim infn→+∞x(n)≥elb1uexp{el-b1ux*}≜x*,
where
e(n)=r1(n)-k1(n)a(n)(y*)m.
From above ε, there exists N2>N1 such that n≥N2, x(n)≥x*-ε. So from (3.5), we obtain that
P(n+1)≤P(n)exp{(1-m)(r2(n)+b2(n)(y*+ε))-(1-m)k2(n)(x*-ε)au+du(x*-ε)+cu(y*+ε)P(n)}.
Consider the following auxiliary equation:
L(n+1)=L(n)exp{(1-m)(r2(n)+b2(n)(y*+ε))-(1-m)k2(n)(x*-ε)au+du(x*-ε)+cu(y*+ε)L(n)}.
By Lemma 2.4, (3.20) has at least one positive ω-periodic solution and we denote one of them as L*(n).

Let
R(n)=lnP(n),W(n)=lnL*(n).
Then,
R(n+1)-R(n)≤(1-m)(r2(n)+b2(n)(y*+ε))-(1-m)k2(n)(x*-ε)au+du(x*-ε)+cu(y*+ε)exp{R(n)},W(n+1)-W(n)=(1-m)(r2(n)+b2(n)(y*+ε))-(1-m)k2(n)(x*-ε)au+du(x*-ε)+cu(y*+ε)exp{W(n)}.
Set
U(n)=R(n)-W(n).
Then
U(n+1)-U(n)≤-(1-m)k2(n)(x*-ε)au+du(x*-ε)+cu(y*+ε)exp{W(n)}[exp{U(n)}-1].
In the following we distinguish three cases.

Case 1.

{U(n)} is eventually positive. Then, from (3.24), we see that U(n+1)<U(n) for any sufficient large n. Hence, limn→+∞U(n)=0, which implies that
lim supn→+∞P(n)≤(L*(n))u.

Case 2.

{U(n)} is eventually negative. Then, from (3.23), we can also obtain (3.25).

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 (3.24), we know that U(n+1)<U(n) if U(n)>0. Hence, lim supn→+∞U(n)=lim supn→+∞U(ns1). From (3.24), and U(ns1-1)<0, we can obtain
U(ns1)≤(1-m)k2(ns1-1)(x*-ε)au+du(x*-ε)+cu(y*+ε)exp{W(ns1)}[1-exp{U(ns1)}],≤(1-m)k2u(x*-ε)au+du(x*-ε)+cu(y*+ε)(L*(n))u.
From (3.21) and (3.23), we easily obtain
lim supn→+∞P(n)≤(L*(n))uexp{(1-m)k2u(x*-ε)au+du(x*-ε)+cu(y*+ε)(L*(n))u}.
Setting ε→0 in the above inequality leads to
lim supn→+∞P(n)≤(L*(n))uexp{(1-m)k2ux*au+dux*+cuy*(L*(n))u}≜P*.
which together with that transformation P(n)=(1/y(n))1-m, we have
lim 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 (1.4) is permanent.

It should be noticed that, from the proof 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 (1.4).

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+0.1cos(n)-0.7x(n)-0.4y0.6(n)2+x(n)+y(n)},y(n+1)=y(n)exp{-0.8-0.1sin(n)-(1.1+0.1cos(n))y(n)+0.6x(n)y-0.4(n)2+x(n)+y(n)},
where m=0.6,r1(n)=0.7+0.1cos(n),b1(n)=0.7,k1(n)=0.4,a(n)=2,d(n)=1,c(n)=1,r2(n)=0.8+0.1sin(n),b2(n)=1.1+0.1cos(n),k2(n)=0.6.

By simple computation, we have y*≈1.1405.Thus,one could easily see that
(1-m)(r2(n))u≈0.36<1,(r1(n)-k1(n)al(y*)m)l≈0.3836>0.

Clearly, conditions (H1) and (H2) are satisfied,then system (4.1) is permanent.

Figure 1 shows the dynamics behavior of system (4.1).

Dynamics behavior of system (4.1) with initial condition (x(0),y(0))=(0.8,0.02).

Example 4.2.

Consider the following system:
x(n+1)=x(n)exp{0.2+0.1cos(n)-0.7x(n)-0.4y0.8(n)2+x(n)+y(n)},y(n+1)=y(n)exp{-0.8-0.4sin(n)-(0.8+0.1cos(n))y(n)+0.6x(n)y-0.2(n)2+x(n)+y(n)},
where m=0.8,r1(n)=0.2+0.1cos(n),b1(n)=0.7,k1(n)=0.4,a(n)=2,d(n)=1,c(n)=1,r2(n)=0.8+0.4sin(n),b2(n)=0.8+0.1cos(n),k2(n)=0.6.

By simple computation, we have y*≈37.3346. Thus, one could easily see that
(1-m)(r2(n))u≈0.24<1,(r1(n)-k1(n)al(y*)m)l≈-3.5201<0.

Clearly, condition (H1) is satisfied and condition (H2) is not satisfied, but the system (4.3) is permanent. It shows that conditions (H1) and (H2) are sufficient for the system (1.4) but not necessary.

Figure 2 shows the dynamics behavior of system (4.3).

Dynamics behavior of system (4.3) with initial condition (x(0),y(0))=(0.4,0.01).

Example 4.3.

Consider the following system:
x(n+1)=x(n)exp{0.2+0.1cos(n)-0.7x(n)-2y0.8(n)2+x(n)+y(n)},y(n+1)=y(n)exp{-0.8-0.4sin(n)-(0.8+0.1cos(n))y(n)+3x(n)y-0.2(n)2+x(n)+y(n)},
where m=0.8,r1(n)=0.2+0.1cos(n),b1(n)=0.7,k1(n)=2,a(n)=2,d(n)=1,c(n)=1,r2(n)=0.8+0.4sin(n),b2(n)=0.8+0.1cos(n),k2(n)=3.

By simple computation, we have y*≈116670. Thus, one could easily see that
(1-m)(r2(n))u≈0.24<1,(r1(n)-k1(n)al(y*)m)l≈-11313<0.

Clearly,condition (H1) is satisfied and (H2) is not satisfied, then the species x is extinct and the species y is permanent.

Figure 3 shows the dynamics behavior of system (4.5).

Dynamics behavior of system (4.5) with initial condition (x(0),y(0))=(0.8,0.02).

Acknowledgment

The work is supported by the Technology Innovation Platform Project of Fujian Province (2009J1007).

HasselM. P.Density dependence in single-species populationWangK.ZhuY. L.Global attractivity of positive periodic solution for a Volterra modelLinX.ChenF.Almost periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional responseChenL. J.ChenL. J.Permanence of a discrete periodic Volterra model with mutual interferenceApanasovB.XieX.Discrete actions on nilpotent Lie groups and negatively curved spacesChenF. D.Permanence and global stability of nonautonomous Lotka-Volterra system with predator-prey and deviating argumentsFanY.-H.LiW.-T.Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional responseWangL.WangM. Q.