This paper discusses a delayed discrete predator-prey system with general Holling-type
functional response and feedback controls. Firstly, sufficient conditions are
obtained for the permanence of the system. After that, under some additional conditions,
we show that the periodic solution of the system is global stable.

1. Introduction

The following predator-prey system with Holling-type
II functional response and delays x˙1(t)=x1(t)[r1(t)−a11(t)x1(t−τ1(t))−a12(t)x2(t)1+mx1(t)],x˙2(t)=x2(t)[−r2(t)−a21(t)x1(t−τ2(t))1+mx1(t−τ2(t))−a22(t)x2(t−τ3(t))], and some generalized systems of
general Holling-type functional response have been studied by many scholars (see
[1–3] and the references cited therein). It has
been found that the discrete time models governed by difference equations are
more appropriate than the continuous ones when the populations have
non-overlapping generations. Discrete time models can also provide efficient
computational models of continuous models for numerical simulations
(see [4–12]).
In [4], Yang considered the following
delayed discrete predator-prey system with general Holling-type functional
response: N1(k+1)=N1(k)exp{r(k)−b(k)N1θ(k−[τ1(t)])−α(k)N1p−1(k)1+mN1p(k)N2(k−[σ(t)])},N2(k+1)=N2(k)exp{−d(k)−a(k)N2(k−[τ2(t)])+β(k)N1p(k−[τ3(t)])1+mN1p(k−[τ3(t)])}. Sufficient conditions which
guarantee the existence of at least one positive periodic solution are obtained
by using the continuation theorem of coincidence degree theory. But Yang did
not consider the permanence and globally attractivity of system
(1.2), which are
two of the most important topics in the study of population dynamics.

On the other hand, as was pointed out by Huo and
Li [13], ecosystem in the real world is continuously distributed by unpredictable
forces which can result in changes in the biological parameters such as
survival rates. Of practical interest in ecology is the question of whether or
not an ecosystem can withstand those unpredictable disturbances which persist
for a finite period of time. In the language of control variables, we call the
disturbance functions as control variables (for more discussion on this
section, one could refer to [12–16] for more details). Though much works
dealt with the continuous time model. However, to the best of the author's
knowledge, up to this day, there are still no scholars that propose
and study the system (1.2) with feedback control. Therefore, the main purpose
of this paper is to study the following delayed discrete predator-prey system
with general Holling-type functional response and feedback control: x1(k+1)=x1(k)exp{r1(k)−b1(k)x1θ(k−τ1)−α1(k)x1p−1(k)1+mx1p(k)x2(k−τ3)−e1(k)u1(k)},x2(k+1)=x2(k)exp{−r2(k)−b2(k)x2(k−τ2)+α2(k)x1p(k−τ4)1+mx1p(k−τ4)−e2(k)u2(k)},Δu1(k)=−η1(k)u1(k)+q1(k)x1(k),Δu2(k)=−η2(k)u2(k)+q2(k)x2(k), where x1(k) is the density of prey species at kth generation, x2(k) is the density of predator species at kth generation, u1(k) and u2(k) are control variables. Also, r1(k),b1(k) denote the intrinsic growth rate and
density-dependent coefficient of the prey, respectively, r2(k),b2(k) denote the death rate and density-dependent
coefficient of the predator, α1(k) denote the capturing rate of the predator, α2(k)/α1(k) represent the rate of conversion of nutrients
into the reproduction of the predator. Further, τi(i=1,2,3,4) are nonnegative constants and m,p are positive constants. In this paper, we
always assume that {ri(k)},{bi(k)},{α1(k)},{ei(k)},{ηi(k)},{qi(k)},i=1,2, are bounded nonnegative sequences and 0<ηiL≤ηiM<1,i=1,2. Here, for any bounded sequence {a(k)},aM=supk∈N{a(k)}, and aL=infk∈N{a(k)}, where N={0,1,2,…}.

This paper is organized as follows. In Section 2, we
will introduce some definition and establish several useful lemma. The
permanence of system (1.3) is then studied in Section 3. In
Section 4, based on
the permanence result, under the assumption that
all the delays are equal to zero and the coefficients of the system are
periodic sequences, we obtain a set of sufficient conditions which guarantee
the existence and stability of a unique globally attractive positive periodic
solution of the system.

By the biological meaning, we will 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 form xi(−k)≥0,ui(−k)≥0,k∈N∩(0,τ],xi(0)>0,ui(0)>0,i=1,2, where τ=max{τ1,τ2,τ3,τ4}.

One can easily show that the
solutions of (1.3) with the initial condition
(1.5)
are defined and remain positive for all k∈N.

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 xi*,ui*,xi*,ui*, which are independent of the solution of the
system, such that for any positive solution (x1(k),x2(k),u1(k),u2(k)) of system (1.3) satisfies xi*≤liminfk→∞xi(k)≤limsupk→∞xi(k)≤xi*,ui*≤liminfk→∞ui(k)≤limsupk→∞ui(k)≤ui*, for i=1,2.

Lemma 2.2.

Assume that x(k) satisfies x(k+1)≤x(k)exp{a(k)−b(k)xθ(k)}∀k≥k0, where {a(k)} and {b(k)} are positive sequences, x(k0)>0,θ is a positive constant, and k0∈N. Then one has limsupk→∞x(k)≤D, where D=(1/θbL)1/θexp(aM−1/θ).

Lemma 2.3.

Assume that x(k) satisfies x(k+1)≥x(k)exp{a(k)−b(k)xθ(k)}∀k≥k0, where {a(k)} and {b(k)} are positive sequences, x(k0)>0,θ is a positive constant, and k0∈N. Also, limsupk→∞x(k)≤D and bMDθ/aL>1. Then one has liminfk→∞x(k)≥C, where C=(aL/bM)1/θexp(aL−bMDθ).

Proof.

The proofs of Lemmas 2.2
and 2.3 are very similar to those of
[6, Propositions
2.1 and 2.2],
respectively. So we omit the detail here.

Lemma 2.4.

Assume that x(k) satisfies x(k+1)≤x(k)exp{a(k)−b(k)xθ(k−τ)}∀k≥k0>τ, where {a(k)} and {b(k)} are positive sequences, x(k0)>0,θ and τ are positive constants, and k0∈N. Then one has limsupk→∞x(k)≤B, where B=(1/θβL)1/θexp(aM−1/θ) and β(k)=b(k)exp{−θ∑i=k−τk−1a(i)}.

Proof.

From the above equation, one has x(k+1)≤x(k)exp{a(k)}∀k≥k0. Sequently we can easily obtain
that x(k−τ)≥x(k)exp{−∑i=k−τk−1a(i)}. So one has x(k+1)≤x(k)exp{a(k)−b(k)exp{−θ∑i=k−τk−1a(i)}xθ(k)}=x(k)exp{a(k)−β(k)xθ(k)}. By Lemma 2.2, we can complete
the proof of Lemma 2.4.

Lemma 2.5.

Assume that x(k) satisfies x(k+1)≥x(k)exp{a(k)−b(k)xθ(k−τ)}∀k≥k0>τ, where {a(k)} and {b(k)} are positive sequences, x(k0)>0,θ and τ are positive constants, and k0∈N. Also, limsupk→∞x(k)≤B and γMBθ/aL>1, where γ(k)=b(k)exp{−θ∑i=k−τk−1(a(i)−b(i)Bθ)}. Then one has liminfk→∞x(k)≥A, where A=(aL/γM)1/θexp(aL−γMBθ).

Proof.

From the above equation, one has x(k+1)≥x(k)exp{a(k)−b(k)Dθ}∀k≥k0. Sequently we can easily obtain
that x(k−τ)≤x(k)exp{−∑i=k−τk−1(a(i)−b(i)Dθ)}. So one has x(k+1)≥x(k)exp{a(k)−b(k)exp{−θ∑i=k−τk−1(a(i)−b(i)Dθ)}xθ(k)}=x(k)exp{a(k)−γ(k)xθ(k)}. By Lemma 2.3, we can complete
the proof of Lemma 2.5.

Lemma 2.6 is a direct corollary of [17, Theorem 6.2, page 125] by L. Wang and M. Q. Wang.

Lemma 2.6.

Consider the following
first-order difference equation: y(k+1)=Ay(k)+B,k=1,2…, where A,B are positive constants. Assuming A<1, for any solution {y(k)} of the above system, one has limk→∞y(k)=B1−A.

The following comparison theorem for the difference equation is of [17, Theorem 2.1, page 241] by L. Wang and M. Q. Wang.

Lemma 2.7.

Let k∈{k0,k0+1,…,k0+l,…},r≥0. For fixed k,g(k,r) is a nondecreasing function with respect to r, and for k≥k0, the following inequalities hold: y(k+1)≤g(k,y(k)),u(k+1)≥g(k,u(k)). If y(k0)≤u(k0), then y(k)≤u(k) for all k≥k0.

3. Permanence

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

Proposition 3.1.

In addition to (1.4), assume
further that

(H1)(α2(k)m−r2(k))L>0;

for any positive solution (x1(k),x2(k),u1(k),u2(k)) of system (1.3), one has limsupk→∞xi(k)≤xi*,limsupk→∞ui(k)≤ui*,i=1,2, where x1*=(1θβ1L)1/θexp(r1M−1θ),β1(k)=b1(k)exp{−θ∑i=k−τ1k−1r1(i)},x2*=1β2Lexp((α2(k)m−r2(k))M−1),β2(k)=b2(k)exp{−∑i=k−τ2k−1(α2(i)m−r2(i))},u1*=q1Mx1*η1L,u2*=q2Mx2*η2L.Proof.

Let (x1(k),x2(k),u1(k),u2(k)) be any positive solution of system (1.3), from
the first equation of (1.3), it follows that x1(k+1)≤x1(k)exp{r1(k)−b1(k)x1θ(k−τ1)}. By applying
Lemmas 2.4 and 2.7, we obtain limsupk→∞x1(k)≤x1*, where x1*=(1θβ1L)1/θexp(r1M−1θ),β1(k)=b1(k)exp{−θ∑i=k−τ1k−1r1(i)}. Similarly, from the second
equation of (1.3), it follows that x2(k+1)≤x2(k)exp{α2(k)m−r2(k)−b2(k)x2(k−τ2)}. Under the assumption (H1),
by applying Lemmas 2.4 and 2.7, we obtain limsupk→∞x2(k)≤x2*, where x2*=1β2Lexp((α2(k)m−r2(k))M−1),β2(k)=b2(k)exp{−∑i=k−τ2k−1(α2(i)m−r2(i))}.

For any positive constant ε small enough, it follows from (3.5) and (3.8)
that there exists large enough K1>τ such that x1(k)≤x1*+ε,x2(k)≤x2*+ε∀k≥K1. Then the third equation of (1.3)
leads to Δu1(k)≤−η1(k)u1(k)+q1(k)(x1*+ε). And so u1(k+1)≤(1−η1L)u1(k)+q1M(x1*+ε)∀k≥K1. By applying Lemmas 2.6 and
2.7,
it follows from (3.12) that limsupk→∞u1(k)≤q1M(x1*+ε)η1L. Setting ε→0 in the above inequality leads to limsupk→∞u1(k)≤u1*, where u1*=q1Mx1*/η1L. Similarly, we can obtain limsupk→∞u2(k)≤u2*, where u2*=q2Mx2*/η2L. Thus we complete the proof of Proposition 3.1.

Proposition 3.2.

In addition to (1.4), assume
further that

(H2)r1L−α1M(x1*)p−1x2*−e1Mu1*>0,

for any positive solution (x1(k),x2(k),u1(k),u2(k)) of system (1.3), there exist positive
constants xi*,ui*, such that (3.18)liminfk→∞xi(k)≥xi*,liminfk→∞ui(k)≥ui*,i=1,2.Proof.

Let (x1(k),x2(k),u1(k),u2(k)) be any positive solution of system (1.3). From (H2) and (H3),
there exists a small enough positive constant ε such that (3.19)r1L−α1M(x1*+ε)p−1(x2*+ε)−e1M(u1*+ε)>0,(3.20)−r2M+α2L(x1*−ε)p1+m(x1*−ε)p−e2M(u2*+ε)>0. Also, according to Proposition
3.1, for the above ε,
there exists K2>K1 such that for k≥K2,(3.21)x1(k)≤x1*+ε,x2(k)≤x2*+ε,u1(k)≤u1*+ε,u2(k)≤u2*+ε. Then from the first equation of
(1.3), one has (3.22)x1(k+1)≥x1(k)exp{r1(k)−b1(k)x1θ(k−τ1)−α1(k)x1p−1(k)x2(k−τ3)−e1(k)u1(k)},≥x1(k)exp{r1(k)−α1(k)(x1*+ε)p−1(x2*+ε)−e1(k)(u1*+ε)−b1(k)x1θ(k−τ1)}. Let a1(k,ε)=r1(k)−α1(k)(x1*+ε)p−1(x2*+ε)−e1(k)(u1*+ε), so the above inequality follows that (3.23)x1(k+1)≥x1(k)exp{a1(k,ε)−b1(k)x1θ(k−τ1)}. Consequently, let γ1(k,ε)=b1(k)exp{−θ∑i=k−τ1k−1(a1(i,ε)−b1(i)x1*θ)}. Because γ1M>β1L, one has (3.24)γ1Ma1L(x1*)θ=γ1Ma1Lexp(θr1M−1)θβ1L>1. Here we use the fact that
exp(θr1M−1)>θr1M>θa1M>θa1L. From (3.19) and (3.23), by
Lemmas 2.5 and 2.7,
one has (3.25)liminfk→∞x1(k)≥(a1L(ε)γ1M(ε))1/θexp{a1L(ε)−γ1M(ε)(x1*)θ}. Setting ε→0 in the above inequality leads to (3.26)liminfk→∞x1(k)≥x1*, where (3.27)x1*=(a1Lγ1M)1/θexp{a1L−γ1M(x1*)θ},a1(k)=r1(k)−α1(k)(x1*)p−1x2*−e1(k)u1*, and γ1(k)=b1(k)exp{−θ∑i=k−τ1k−1(a1(i)−b1(i)x1*θ)}.

Similarly, from the second equation of (1.3), one has (3.28)x2(k+1)≥x2(k)exp{−r2(k)+α2(k)(x1*−ε)p1+m(x1*−ε)p−e2(k)(u2*+ε)−b2(k)x2(k−τ2)}. Let a2(k,ε)=−r2(k)+α2(k)(x1*−ε)p/(1+m(x1*−ε)p)−e2(k)(u2*+ε), so the above inequality leads to (3.29)x2(k+1)≥x2(k)exp{a2(k,ε)−b2(k)x2(k−τ2)}. Consequently, let γ2(k,ε)=b2(k)exp{−∑i=k−τ2k−1(a2(i,ε)−b2(i)x2*)}. Because γ2M>β2L, one has (3.30)γ2Ma2Lx2*=γ2Ma2Lexp{(α2(k)/m−r2(k))M−1}β2L>1. Here we use the fact that
exp{(α2(k)/m−r2(k))M−1}>(α2(k)/m−r2(k))M>a2M>a2L. From (3.20) and (3.29), by
Lemmas 2.5 and 2.7,
one has (3.31)liminfk→∞x2(k)≥a2L(ε)γ2M(ε)exp{a2L(ε)−γ2M(ε)x2*}. Setting ε→0 in the above inequality leads to (3.32)liminfk→∞x2(k)≥x2*, where (3.33)x2*=a2Lγ2Mexp{a2L−γ2Mx2*},a2(k)=−r2(k)+α2(k)x1*p1+mx1*p−e2(k)u2*,γ2(k)=b2(k)exp{−∑i=k−τ2k−1(a2(i)−b2(i)x2*)}. Then the third equation of (1.3) leads to (3.34)Δu1(k)≥−η1(k)u1(k)+q1(k)(x1*−ε). And so (3.35)u1(k+1)≥(1−η1M)u1(k)+q1L(x1*−ε)∀k≥K2.

By applying Lemmas 2.6 and
2.7, it follows from (3.35)
that (3.36)limsupk→∞u1(k)≥q1L(x1*−ε)η1M. Setting ε→0 in the above inequality leads to (3.37)limsupk→∞u1(k)≥u1*, where u1*=q1Lx1*/η1M. Similarly, we can obtain (3.38)limsupk→∞u2(k)≤u2*, where u2*=q2Lx2*/η2M. Thus we complete the proof of Proposition 3.2.

Theorem 3.3.

In addition to (1.4), assume further that (H1),(H2), and (H3) hold, then system (1.3) is permanent.

It should be noticed that, from the proofs of
Propositions 3.1 and
3.2, we know that under the conditions of Theorem 3.3, the
set Ω={(x1,x2,u1,u2)∣xi*≤xi≤xi*,ui*≤ui≤ui*,i=1,2} is an invariant set of system (1.3).

4. Existence and Stability of a Periodic Solution

In this
section, we consider the stability property of system (1.3) under the
assumption τi=0(i=1,2,3,4), that is, we consider the following system: (4.1)x1(k+1)=x1(k)exp{r1(k)−b1(k)x1θ(k)−α1(k)x1p−1(k)1+mx1p(k)x2(k)−e1(k)u1(k)},x2(k+1)=x2(k)exp{−r2(k)−b2(k)x2(k)+α2(k)x1p(k)1+mx1p(k)−e2(k)u2(k)},Δu1(k)=−η1(k)u1(k)+q1(k)x1(k),Δu2(k)=−η2(k)u2(k)+q2(k)x2(k), which are similar to system
(1.3) but do not include delays. In this section, we always assume that {ri(k)}, {bi(k)}, {α1(k)}, {ei(k)}, {ηi(k)}, {qi(k)} are bounded nonnegative periodic sequences
with a common period ω and satisfy (4.2)0<ηi(k)<1,k∈N∩[0,ω],i=1,2. Also it is assumed that the
initial conditions of (4.1) are of the form (4.3)xi(0)>0,ui(0)>0,i=1,2. Using a similar way, under some
conditions, we can obtain the permanence of system (4.1). As above, still let xi* and ui*,i=1,2, be the upper bound of {xi(k)} and {ui(k)},xi* and let ui*,i=1,2, be the lower bound of {xi(k)} and {ui(k)}, where xi*,ui*,xi*,
and ui* are independent of the solution of system
(4.1). Our first result concerns with the existence of a periodic solution.

Theorem 4.1.

In addition to (4.2), assume further that (H1),(H2), and (H3) hold, then system (4.1) has a periodic
solution denoted by {x¯1(k),x¯2(k),u¯1(k),u¯2(k)}.

Proof.

Let Ω={(x1,x2,u1,u2)∣xi*≤xi≤xi*,ui*≤ui≤ui*,i=1,2},Ω is an invariant set of system (4.1). Thus, we
can define a mapping F on Ω by (4.4)F(x1(0),x2(0),u1(0),u2(0))=(x1(ω),x2(ω),u1(ω),u2(ω)) for (x1(0),x2(0),u1(0),u2(0))∈Ω.

Obviously, F depends continuously on (x1(0),x2(0),u1(0),u2(0)). Thus F is continuous and maps a compact set Ω into itself. Therefore, F has a fixed point (x¯1,x¯2,u¯1,u¯2). It is easy to see that the solution {x¯1(k),x¯2(k),u¯1(k),u¯2(k)} passing through (x¯1,x¯2,u¯1,u¯2) is a periodic solution of system (4.1). This
completes the proof.

Now, we study the globally stability property of the
periodic solution obtained in Theorem 4.1.

Theorem 4.2.

In addition to the conditions of Theorem 4.1, if
system (4.1) satisfies (4.5)λ1=max{|1−θb1Lx1*θ|,|1−θb1M(x1*)θ−α1MW1x1*|}+α1MW2x2*+e1M<1,(4.6)λ2=max{|1−b2Lx2*|,|1−b2Mx2*|}+α2MW3x1*+e2M<1,(4.7)λ3=1−η1L+q1Mx1*<1,(4.8)λ4=1−η2L+q2Mx2*<1, where the definition of Wi,i=1,2,3 can be seen in the following proof, then the ω-periodic solution (x¯1(k),x¯2(k),u¯1(k),u¯2(k)) obtained in Theorem 4.1 is globally attractive.

Proof.

Assume that (x1(k),x2(k),u1(k),u2(k)) is any positive solution of system (4.1), let (4.9)xi(k)=x¯i(k)exp{yi(k)},ui(k)=u¯i(k)+vi(k),i=1,2. To complete the proof, it
suffices to show that (4.10)limk→∞yi(k)=0,limk→∞vi(k)=0,i=1,2. Since (4.11)y1(k+1)=y1(k)−b1(k)x¯1θ(k)(exp{θy1(k)}−1)−e1(k)v1(k)−α1(k)[x¯1p−1(k)exp{(p−1)y1(k)}1+mx¯1p(k)exp{py1(k)}x¯2(k)exp{y2(k)}−x¯1p−1(k)1+mx¯1p(k)x¯2(k)]=y1(k)−b1(k)x¯1θ(k)exp{ξ1(k)θy1(k)}θy1(k)−e1(k)v1(k)−α1(k)[f1′(ξ2′(k),x2(k))x¯1(k)exp{ξ4(k)y1(k)}y1(k)+f2′(x¯1(k),ξ3′(k))x¯2(k)exp{ξ5(k)y2(k)}y2(k)], where (4.12)f(x,y)=xp−1y1+mxp,ξ2′(k)=x¯1(k)+ξ2(k)(x1(k)−x¯1(k)),ξ3′(k)=x¯2(k)+ξ3(k)(x2(k)−x¯2(k)), and ξi(k)∈(0,1) for i=1,2,3,4,5. Because of the boundedness of {x¯1(k)}, {x¯2(k)}, {x1(k)}, {x2(k)}, |f1′(ξ2′(k),x2(k))|, |f2′(x¯1(k),ξ3′(k))| are bounded, where f1′ and f2′ mean the partial derivation of the function f(x,y).
Let |f1′(ξ2′(k),x2(k))|<W1 and |f2′(x¯1(k),ξ3′(k))|<W2.

Similarly, we get (4.13)y2(k+1)=y2(k)−b2(k)x¯2(k)(exp{y2(k)}−1)+α2(k)[x¯1p(k)exp{py1(k)}1+mx¯1p(k)exp{py1(k)}−x¯1p(k)1+mx¯1p(k)]−e2(k)v2(k)=y2(k)−b2(k)x¯2(k)exp{ξ5(k)y2(k)}y2(k)+α2(k)g′(ξ6′(k))x¯1(k)(exp{y1(k)}−1)−e2(k)v2(k)=y2(k)−b2(k)x¯2(k)exp{ξ5(k)y2(k)}y2(k)+α2(k)g′(ξ6′(k))x¯1(k)exp{ξ4(k)y1(k)}y1(k)−e2(k)v2(k), where (4.14)ξ6′(k)=x¯1(k)+ξ6(k)(x1(k)−x¯1(k)),ξ6(k)∈(0,1). Because of the boundedness of {x¯1(k)},{x¯2(k)},{x1(k)},{x2(k)},g′(ξ6′(k)) is bounded, where g(x)=xp/(1+mxp) and g′ means the derivation of the function g(x).
Let |g′(ξ6′(k))|<W3.

Also, one has (4.15)v1(k+1)=(1−η1(k))v1(k)+q1(k)x¯1(k)(exp{y1(k)}−1)=(1−η1(k))v1(k)+q1(k)x¯1(k)exp{ξ4(k)y1(k)}y1(k),v2(k+1)=(1−η2(k))v2(k)+q2(k)x¯2(k){(exp{y2(k)}−1})=(1−η2(k))v2(k)+q2(k)x¯2(k)exp{ξ5(k)y2(k)}y2(k).

In view of (4.5)–(4.8), we can choose a ε>0 such that (4.16)λ1ε=max{|1−θb1L(x1*−ε)θ|,|1−θb1M(x1*+ε)θ−α1MW1(x1*+ε)|}+α1MW2(x2*+ε)+e1M<1,λ2ε=max{|1−b2L(x2*−ε)|,|1−b2M(x2*+ε)|}+α2MW3(x1*+ε)+e2M<1,λ3ε=1−η1L+q1M(x1*+ε)<1,λ4ε=1−η2L+q2M(x2*+ε)<1. Also, from Propositions 3.1
and
3.2, there exist K3>K2 such that (4.17)xi*−ε≤xi(k),xi*(k)≤xi*+ε∀k≥K3,i=1,2. Then from (4.11), for k>K3, one has (4.18)|y1(k+1)|≤max{|1−θb1L(x1*−ε)θ|,|1−θb1M(x1*+ε)θ−α1MW1(x1*+ε)|}⋅|y1(k)|+α1MW2(x2*+ε)|y2(k)|+e1M|v1(k)|. So from (4.13), for k>K3, one has (4.19)|y2(k+1)|≤max{|1−b2L(x2*−ε)|,|1−b2M(x2*+ε)|}|y2(k)|+α2MW3(x1*+ε)|y1(k)|+e2M|v2(k)|. Also, for k>K3, one has (4.20)|v1(k+1)|≤(1−η1L)|v1(k)|+q1M(x1*+ε)|y1(k)|,(4.21)|v2(k+1)|≤(1−η2L)|v2(k)|+q2M(x2*+ε)|y2(k)|. Let λ=max{λ1ε,λ2ε,λ3ε,λ4ε}, then 0<λ<1. In view of (4.18)–(4.21), one has (4.22)max{|y1(k+1)|,|y2(k+1)|,|v1(k+1)|,|v2(k+1)|}≤λmax{|y1(k)|,|y2(k)|,|v1(k)|,|v2(k)|} for k>K3. This implies (4.23)max{|y1(k)|,|y2(k)|,|v1(k)|,|v2(k)|}≤λk−K3max{|y1(K3)|,|y2(K3)|,|v1(K3)|,|v2(K3)|}. Therefore (4.24)limk→∞yi(k)=0,limk→∞vi(k)=0,i=1,2. This completes the proof.

5. Examples

The following
two examples show the feasibility of our main results.

Example 5.1.

Consider system (1.3) with (5.1)r1(k)=0.14+0.01cos(k),b1(k)=0.1,α1(k)=0.001,e1(k)=0.03+0.01sin(k),r2(k)=0.18+0.02cos(2k),b2(k)=1.8+0.1sin(k),α2(k)=1.4,e2(k)=0.008+0.002sin(k),η1(k)=0.7,q1(k)=0.2+0.1sin(k),η2(k)=0.8,q2(k)=0.1,τ1=τ2=τ3=τ4=1,p=1.3,θ=1.2,m=0.8, for all k∈N. One can easily see that (5.2)x1*≈3.7945,x1*≈0.2882,x2*≈5.2037,u1*≈1.6262,u2*≈0.6505, which means that (5.3)r1L−α1M(x1*)p−1x2*−e1Mu1*≈0.0872,(5.4)−r2M+α2Lx1*p1+mx1*p−e2Mu2*≈0.0333. Also, one has (5.5)(α2(k)m−r2(k))L≈1.55. Inequalities (5.3)–(5.5) show
that (H1)–(H3) are fulfilled. From Theorem 3.3, system (1.3)
is permanent. Figure 1 is the numeric simulation of the solution of system
(1.3) with initial condition (x1(p),x2(p),u1(p),u2(p))=(1.4,0.8,0.3,0.03)(P=−1,0).

Figure 1

Dynamics
behaviors of system (1.3) with initial condition (x1(p),x2(p),u1(p),u2(p))=(1.4,0.8,0.3,0.03)(P=−1,0).

Example 5.2.

Consider system (4.1) with (5.6)r1(k)=0.13+0.02cos(k),b1(k)=0.1,α1(k)=0.01,e1(k)=0.03+0.01sin(k),r2(k)=0.16+0.02cos(k),b2(k)=0.7,α2(k)=0.6,e2(k)=0.015+0.005sin(k),η1(k)=0.7,q1(k)=0.2+0.1sin(k),η2(k)=0.8,q2(k)=0.2+0.1sin(k), for all k∈N. One can easily see that (5.7)x1*≈1.2808,x1*≈0.8195,x2*≈0.9672,x2*≈0.1157,u1*≈0.5489,u2*≈0.3627, which means that (5.8)r1L−α1M(x1*)p−1x2*−e1Mu1*≈0.0801,(5.9)−r2M+α2Lx1*p1+mx1*p−e2Mu2*≈0.1387. Also, one has (5.10)(α2(k)m−r2(k))L≈0.5700. Inequalities (5.8)–(5.10) show
that (H1)–(H3) are fulfilled. We can obtain that (5.11)W1≈0.7258,W2≈0.6630,W3≈0.075, which means that
(5.12)λ1≈0.8861,λ2≈0.9966,λ3≈0.6842,λ4≈0.4902. So (4.5)–(4.8) are fulfilled.
From Theorem 4.2, system (4.1) is globally attractive.
Figure 2 is the numeric
simulation of the solution of system (4.1) with initial condition (x1(0),x2(0),u1(0),u2(0))=(0.9,0.7,0.6,0.05),(1.2,0.5,0.3,0.2), and (1,0.2,0.25,0.1).

Figure 2

Dynamics
behaviors of system (4.1) with initial values (x1(0),x2(0),u1(0),u2(0)) = (0.9, 0.7, 0.6, 0.05), (1.2, 0.5, 0.3, 0.2),
and (1, 0.2, 0.25, 0.1).

Acknowledgment

This work was supported by the Foundation of Fujian Education Bureau (JB05042).

1HollingC. S.The components of predation as revealed by a study of small mammal predation of the European
pine sawfly2HollingC. S.Some characteristics of simple types of predation and parasitism3FanM.WangK.Global existence of positive periodic solutions of a predator-prey system with Holling type II functional response4YangL.Periodic solutions of a delayed discrete predator-prey system with general functional response5FanM.WangK.Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system6ChenF.WuL.LiZ.Permanence and global attractivity of the discrete Gilpin-Ayala type population model7LiY.Positive periodic solutions of discrete Lotka-Volterra competition systems with state dependent and distributed delays8ChenY.ZhouZ.Stable periodic solution of a discrete periodic Lotka-Volterra competition system9ZhouZ.ZouX.Stable periodic solutions in a discrete periodic logistic equation10FanY.-H.LiW.-T.Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response11HuoH.-F.LiW.-T.Existence and global stability of periodic solutions of a discrete ratio-dependent food chain model with delay12LiY.ZhuL.Existence of positive periodic solutions for difference equations with feedback control13HuoH.-F.LiW.-T.Positive periodic solutions of a class of delay differential system with feedback control14LiY.Positive periodic solutions for a periodic neutral differential equation with feedback control15ChenF.Positive periodic solutions of neutral Lotka-Volterra system with feedback control16ChenF.The permanence and global attractivity of Lotka-Volterra competition system with feedback controls17WangL.WangM. Q.