# Periodic Property and Asymptotic Behavior for a Discrete Ratio-Dependent Food-Chain System with Delays

Authors: Changjin Xu; Peiluan Li; Maoxin Liao
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, a discrete ratio-dependent food-chain system with delay is investigated. By using Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, a set of sufficient conditions for the existence of positive periodic solutions and global asymptotic stability of the model are established.


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## Body

## 1. Introduction

The past decades have witnessed a great deal of interest in the periodic phenomena of predator-prey systems. For example, Zhang and Tian [1] investigated the multiple periodic solutions of a generalized predator-prey system with exploited terms. Zhang and Wang [2] analyzed the existence and global attractivity of a positive periodic solution for a generalized delayed prey-predator system. Li et al. [3] studied multiple positive periodic solutions of n species delay competition systems with harvesting terms. Ding et al. [4] made a detailed discussion on the periodic solution of a Gause-type predator-prey systems with impulse. Shen and Li [5] obtained a set of sufficient conditions for the existence of at least one strictly positive periodic solution and the uniqueness and global attractivity of positive periodic solution for an impulsive predator-prey model with dispersion and time delays. For more knowledge about the periodic solutions of predator-prey models, one can see [6–12]. For the papers mentioned above, it shall be pointed out that many investigations have been performed to analyze the dynamical behavior on biological species by using continuous or impulsive mathematical models [6–9]. It has been widely argued and accepted that difference equations often occur in numerous setting and forms, both in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamics, economics, biology, and other fields [13]. In recent years, Xu et al. [14] have studied the persistence and stability of the following ratio-dependent food-chain system with delay:(1)x˙1t=x1ta1−a11x1t−τ1−a12x2tm12x2t+x1t,x˙2t=x2t−a2+a21x1t−τ2m12x2t−τ2+x1t−τ2−a23x3tm23x3t+x2t,x˙3t=x3t−a3+a32x2t−τ3m23x3t−τ3+x2t−τ3,where x1t,x2t, and x3t denote densities of the prey, predator, and the top predator populations at time t, respectively, τ1≥0 is constant time delay due to negative feedback of the prey, and τ2≥0 and τ3≥0 are constant time delays due to gestation. aii=1,2,3,a11,a12,a21,a23,a32,m12, and m23 are all positive constants. In detail, one can see [14].In real life, many biological and environmental parameters do vary in time(for example, naturally subject to seasonal fluctuations). However, Xu et al. [14] did not involve the varying parameters of the food-chain model. To describe the object relationship between predator population and prey population, we modify system (1) as the following nonautonomous ratio-dependent food-chain system with varying delay:(2)x˙1t=x1ta1t−a11tx1t−τ1t−a12tx2tm12tx2t+x1t,x˙2t=x2t−a2t+a21tx1t−τ2tm12tx2t−τ2t+x1t−τ2t−a23tx3tm23tx3t+x2t,x˙3t=x3t−a3t+a32tx2t−τ3tm23tx3t−τ3t+x2t−τ3t.Many authors [15–18] argue that discrete time models governed by difference equations are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have nonoverlapping generations. Moreover, discrete time models can also provide efficient models of continuous ones for numerical simulation. In order to reveal the dynamic relationship of predator and prey and explain the stability law of both species by applying the computer simulations, we think that it is reasonable to study time ratio-dependent food-chain systems governed by difference equations. Following the lines of Wiener [19] and Fan and Wang [20], we obtain the discrete time analogue of system (2):(3)x1k+1=x1kexpa1k−a11kx1k−τ1k−a12kx2km12kx2k+x1k,x2k+1=x2kexp−a2k+a21kx1k−τ2km12kx2k−τ2k+x1k−τ2k−a23kx3km23kx3k+x2k,x3k+1=x3kexp−a3k+a32kx2k−τ3km23kx3k−τ3k+x2k−τ3k,where k=0,1,2,… and all the variables and parameters have the same biological meanings as those in system (1).The main task of this article is to discuss the dynamics of system (3). That is, applying Mawhin’s continuous theorem [21] to study the existence of positive periodic solutions of (3) and investigating the global asymptotical stability of system (3) by means of the method of Lyapunov function. The main innovation point lies in better applying computer simulation to explain the changing law of biological population.The remainder of the paper is organized as follows. In Section2, a easily verifiable sufficient condition for the existence of positive solutions of difference equations is obtained by the continuation theorem and priori estimations. The sufficient condition for the global asymptotical stability of system (3) when all the delays are zero is presented in Section 3. In Section 4, we give some computer simulations.


## 2. Existence of Positive Periodic Solutions

For convenience and simplicity in the following discussion, we always use the following notations throughout the paper:(4)Iω≔0,1,2,…,ω−1,f¯≔1ω∑k=0ω−1fk,fL≔mink∈Zfk,fM≔maxk∈Zfk,where fk is an ω−periodic sequence of real numbers defined for k∈Z. Let Z denote the integer number, R denote the real number, R+ denote the nonnegative real number, and R3 denote the three-dimensional real vector.We always assume thatH1aii=1,2,3,a11,a12,a21,a23,a32,m12,m23:Z⟶R+ are ω periodic, i.e.,(5)aik+ω=aiki=1,2,3,a11k+ω=a11k,a12k+ω=a12k,a21k+ω=a21k,a23k+ω=a23k,m12k+ω=m12k,m23k+ω=m23k,for anyk∈Z.In order to explore the existence of positive periodic solutions of (3) and for the reader’s convenience, we shall first introduce a few concepts and results without proof, borrowing from Gaines and Mawhin [21].LetXandY be normed vector spaces, L:DomL⊂X⟶Y is a linear mapping, and N:X⟶Y is a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL=codimImL<+∞ and ImL is closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projectors P:X⟶X and Q:Y⟶Y such that ImP=KerLand ImL=KerQ=ImI−Q, It follows that LDomL∩KerP:I−PX⟶ImL is invertible. We denote the inverse of that map by KP. If Ω is an open-bounded subset of X, the mapping N will be called L−compact on Ω¯ if QNΩ¯ is bounded and KPI−QN:Ω¯⟶X is compact. Since ImQ is isomorphic to KerL, there exist isomorphisms J:ImQ⟶KerL.Lemma 1.
(see [21], continuation theorem). Let L be a Fredholm mapping of index zero, and let N be L− compact on Ω¯. Suppose(a)
For eachλ∈0,1, every solution x of Lx=λNx is such that x∉∂Ω(b)
QNx≠0 for each x∈KerL∩∂Ω, and degJQN,Ω∩KerL,0≠0
Then, the equationLx=Nx has at least one solution lying in DomL∩Ω¯.Lemma 2.
(see [20]). Let g:Z⟶R be ω periodic, i.e., gk+ω=gk; then, for any fixed k1,k2∈Iω and any k∈Z, one has(6)gk≤gk1+∑s=0ω−1gs+1−gs,gk≥gk2−∑s=0ω−1gs+1−gs.Lemma 3.
Assume thata¯2<a¯21 and a¯3<a¯32; then, the system algebraic equations(7)a¯1−a¯11v1=0,a¯2−1ω∑k=0ω−1a21kv1m12kv2+v1=0,a¯3−1ω∑k=0ω−1a32kv2m23kv3+v2=0,have a unique solution v1∗,v2∗,v3∗ with vi∗>0,i=1,2,3.Proof.
Obviously,v1=a¯1/a¯11>0. Substituting v1=a¯1/a¯11 into the second equation of system (7) and simplifying, we obtain(8)a¯2−1ω∑k=0ω−1a21km12ka¯11/a¯1v2+1=0.
In the following, we define the function:(9)fθ=a¯2−1ω∑k=0ω−1a21km12ka¯11/a¯1θ+1,θ≥0.
It is easy to see that(10)f0=a¯2−1ω∑k=0ω−1a21k=a¯2−a¯21<0,(11)limθ⟶+∞fθ=a¯2>0.
Then, it follows from the zero-point theorem and monotonicity offθ that there exists a unique v2∗>0 such that fv2∗=0. Similarly, substituting v2∗ into the third equation of system (7) and simplifying, we have(12)a¯3−1ω∑k=0ω−1a32km23k/v2∗v3+1=0.
We define the function(13)gγ=a¯3−1ω∑k=0ω−1a32km23k/v2∗γ+1,γ≥0.
Clearly,(14)g0=a¯3−1ω∑k=0ω−1a32k=a¯3−a¯32<0,(15)limθ⟶+∞gγ=a¯3>0.
Then, it follows from the zero-point theorem and monotonicity ofgγ that there exists a unique v3∗>0 such that gv3∗=0. The proof is complete.
Define(16)l3=z=zk:zk=z1k,z2k,z3kT∈R3,k∈Z.
Letlω⊂l3 denote the subspace of all ω periodic sequences equipped with the usual supremum norm ⋅, i.e., z=z1k+z2k+z3k for any z=zk:k∈Z∈lω. It is easy to show that lω is a finite-dimensional Banach space.
Let(17)l0ω=z=zk∈lω:∑k=0ω−1zk=0,lcω=z=zk∈lω:zk=h∈R3,k∈Z.
Then, it follows thatl0ω and lcω are both closed linear subspaces of lω and(18)lω=l0ω+lcω,dimlcω=3.
In the following, we will ready to establish our result.Theorem 1.
LetS1 be defined by (37). Under condition (H1), suppose that the following conditions(19)H2a¯1>a12m12¯,a¯1exp−S1>a¯11,H3a¯2<a¯21,a¯3<a¯32,hold; then, system (3) has at least an ω positive periodic solution.Proof.
First, we make the change of variablesxit=expuiti=1,2,3; then, (3) can be reformulated as(20)u1k+1−u1k=f1u1,u2,u3,u2k+1−u2k=f2u1,u2,u3,u3k+1−u3k=f3u1,u2,u3,where(21)f1u1,u2,u3=a1k−a11kexpu1k−τ1k−a12kexpu2km12kexpu2k+expu1k,f2u1,u2,u3=−a2k+a21kexpu1k−τ2km12kexpu2k−τ2k+expu1k−τ2k−a23kexpu3km23kexpu3k+expu2k,f3u1,u2,u3=−a3k+a32kexpu2k−τ3km23kexpu3k−τ3k+expu2k−τ3k.
LetX=Y=lω,(22)Luk=uk+1−uk=u1k+1−u1ku2k+1−u2ku3k+1−u3k,Nuk=f1u1,u2,u3f2u1,u2,u3f3u1,u2,u3,where u∈X,k∈Z. Then, it is trivial to see that L is a bounded linear operator and(23)KerL=lcω,ImL=l0ω,(24)dim KerL=3=codimImL.
Then, it follows thatL is a Fredholm mapping of index zero. Define(25)Pu=1ω∑s=0ω−1us,u∈X,Qz=1ω∑s=0ω−1zs,z∈Y.
It is not difficult to show thatP and Q are continuous projectors such that(26)ImP=KerL,ImL=KerQ=ImI−Q.
Furthermore, the generalized inverse (toL) kP:ImL⟶KerP∩DomL exists and is given by(27)KPz=∑s=0ω−1zs−1ω∑s=0ω−1ω−szs.
Obviously,QN and KPI−QN are continuous. Since X is a finite-dimensional Banach space, using the Ascoli–Arzela theorem, it is not difficult to show that KPI−QNΩ¯¯ is compact for any open-bounded set Ω⊂X. Moreover, QNΩ¯ is bounded. Thus, N is L−compact on Ω¯ with any open-bounded set Ω⊂X.
Now, we are at the point to search for an appropriate open-bounded subsetΩ for the application of the continuation theorem. Corresponding to the operator equation Lu=λNu,λ∈0,1, we have(28)u1k+1−u1k=λf1u1,u2,u3,u2k+1−u2k=λf2u1,u2,u3,u3k+1−u3k=λf3u1,u2,u3.
Suppose thatuk=u1k,u2k,u3kT∈X is an arbitrary solution of system (28) for a certain λ∈0,1; summing both sides of (28) from 0 to ω−1 with respect to k, respectively, we obtain(29)∑k=0ω−1a11kexpu1k−τ1k+a12kexpu2km12kexpu2k+expu1k=a¯1ω,∑k=0ω−1a21kexpu1k−τ2km12kexpu2k−τ2k+expu1k−τ2k−a23kexpu3km23kexpu3k+expu2k=a¯2ω,∑k=0ω−1a32kexpu2k−τ3km23kexpu3k−τ3k+expu2k−τ3k=a¯3ω.
In view of the hypothesis thatu=uk∈X, there exist ξi,ηi∈Iωi=1,2,3 such that(30)uiξi=mink∈Iωuik,uiηi=maxk∈Iωuik.
It follows from (28) and (29) that(31)∑k=0ω−1u1k+1−u1k≤λ∑k=0ω−1a1k−a11kexpu1k−τ1k−a12kexpu2km12kexpu2k+expu1k<2a¯1ω,(32)∑k=0ω−1u2k+1−u2k≤λ∑k=0ω−1−a2k+a21kexpu1k−τ2km12kexpu2k−τ2k+expu1k−τ2k−a23ku3km23kexpu3k+expu2k<2a¯2ω,(33)∑k=0ω−1u3k+1−u3k≤λ∑k=0ω−1−a3k+a32kexpu2k−τ3km23kexpu3k−τ3k+expu2k−τ3k<2a¯3ω.
By the first equation of (29), we have(34)∑k=0ω−1a11kexpu1k−τ1k<a¯1ω,∑k=0ω−1a11kexpu1k−τ1k+a12km12k≥a¯1ω,which leads to(35)u1ξ1<lna¯1a¯11,u1η1>lna¯1−a12/m12¯a¯11.By (31) and (35) and Lemma 2, we obtain(36)u1k≤uξ1+∑s=0ω−1u1s+1−u1s≤lna¯1a¯11+2a¯1ω≔M1,u1k≥u1η1−∑s=0ω−1u1s+1−u1s≥lna¯1−a12/m12¯a¯11−2a¯1ω≔m1.
Thus,(37)maxk∈Iωu1k<maxm1,M1≔S1.
In view of the second equation of (29) and (37), it is easy to obtain(38)∑k=0ω−1a21kexpu1k−τ2km12kexpu2k−τ2k>a¯2ω,then(39)∑k=0ω−1a21kexpS1m12kexpu2ξ2>a¯2ω.
Thus,(40)u2ξ2<lna21/m12¯expS1a¯2.
From the first equation of (29) and (37), we obtain(41)∑k=0ω−1a11kexpS1+a12kexpu2η2exp−S1>a¯1ω.
Then,(42)u2η2>lna¯1exp−S1−a¯11a¯12.
It follows from (40) and (42) and Lemma 2 that(43)u2k≤uξ2+∑s=0ω−1u2s+1−u2s≤lna21/m12¯expS1a¯2+2a¯2ω≔M2,(44)u2k≥u2η2−∑s=0ω−1u2s+1−u2s≥lna¯1exp−S1−a¯11a¯12−2a¯2ω≔m2.
Thus,(45)maxk∈Iωu2k<maxm2,M2≔S2.
By the third equation of (29), we obtain(46)∑k=0ω−1a32kexpu2k−τ3km23kexpu3k−τ3k>a¯3ω,which leads to(47)∑k=0ω−1a32kexpS2m23kexpu3ξ3>a¯3ω.
Then,(48)u3ξ3<lna32/m23¯expS2a¯3.
By the third equation of (29), we also obtain(49)∑k=0ω−1a32kexp−S2m23Mexpu3η3+exp−S2<a¯3ω.
Thus, we obtain(50)u3η3>lna¯32−a¯3exp−S2a¯3m23M.
From (48) and (50) and Lemma 2, we derive(51)u3k≤uξ3+∑s=0ω−1u3s+1−u3s≤lna32/m23¯expS2a¯3+2a¯3ω≔M3,u3k≥u3η3−∑s=0ω−1u2s+1−u2s≥lna¯32−a¯3exp−S2a¯3m23M−2a¯3ω≔m3.
Thus,(52)maxk∈Iωu3k<maxm3,M3≔S3.
Obviously,Sii=1,2,3 are independent of the choice of λ∈0,1. Take S=maxS1,S2,S3+S0, where S0 is taken sufficiently large such that lnv1∗+lnv2∗+lnv3∗<S0, where v1∗,v2∗,v3∗T is the unique positive solution of (7).
Now, we have proved that any solutionu=uk=u1k,u2k,u3kT of (28) in X satisfies u<S,k∈Z.
LetΩ≔u=uk∈X:u<S; then, it is easy to see that Ω is an open-bounded set in X and verifies requirement (a) of Lemma 2. When u∈∂Ω∩KerL,u=u1,u2,u3T is a constant vector in R3 with u=u1+u2+u3=S. Then,(53)QNy=a¯1−a¯11expu1−1ω∑k=0ω−1a21kexpu1m12kexpu2+expu1−a¯2+1ω∑k=0ω−1a21kexpu1m12kexpu2+expu1−1ω∑k=0ω−1a23kexpu3m23kexpu3+expu3−a¯3+1ω∑k=0ω−1a32kexpu2m23kexpu3+expu2≠0.
Now, let us consider homotopicϕu1,u2,u3,μ=μQNu+1−μGu,μ∈0,1, where(54)Gz=1ω∑k=0ω−1a¯1−a¯11expu1−a¯2+1ω∑k=0ω−1a21kexpu1m12kexpu2+expu1−a¯3+1ω∑k=0ω−1a32kexpu2m23kexpu3+expu2.
In fact, whenϕu1,u2,u3,0=Gu and ϕu1,u2,u3,1=QNu, then ϕ is a homptopic mapping. Letting J be the identity mapping and by direct calculation, we obtain(55)degJQNu1,u2,u3T;Ω∩kerL;0=degQNu1,u2,u3T;Ω∩kerL;0=degϕu1,u2,u3,1;Ω∩kerL;0=degϕu1,u2,u3,0;Ω∩kerL;0=signdetχ1100χ21χ2200χ32χ33,where(56)χ11=−a¯11expu1∗<0,χ21=1ω∑k=0ω−1a21km12kexpu1∗+u2∗m12kexpu2∗+expu1∗2>0,χ22=−1ω∑k=0ω−1a21km12kexpu1∗+u2∗m12kexpu2∗+expu1∗2<0,χ32=1ω∑k=0ω−1a32km23kexpu2∗+u3∗m23kexpu3∗+expu2∗2>0,χ33=−1ω∑k=0ω−1a32km23kexpu2∗+u3∗m23kexpu3∗+expu2∗2<0.
Thus,(57)degJQNu1,u2,u3T;Ω∩kerL;0=signχ11χ22χ33=−1≠0.
By now, we have proved thatΩ verifies all requirements of Lemma 2; then, it follows that Lu=Nu has at least one solution in DomL∩Ω¯, namely, (20) has at least one ω periodic solution in DomL∩Ω¯, say u∗=u∗k=u1∗k,u2∗k,u3∗kT; then, it follows that expu1∗k,exp{u2∗k,expu3∗kT is an ω periodic solution of system (3) with strictly positive components. The proof is complete.


## 3. Global Asymptotic Stability

In this section, we will present sufficient conditions for the globally asymptotical stability of system (3) when all the delays are zero.Theorem 2.
LetA1∗,A2∗, and A3∗ are defined by (67), (68), and (69), respectively. Assume that (H1)–(H3) are satisfied. Furthermore, suppose that there exist positive constants η,δ1,δ2, and δ3 such that Ai∗>0i=1,2,3 and(58)i1−a11kexpM1−a12kexpM2m12kexpm2+expm1+a12kexpm1+m2m12kexpM2+expM12≥η,iiexpm2−a21kexpM1+M2−a23kexpM2+M3m12kexpM2+expM1−a21ka23kexpM1+M2+M3m12kexpm2+expm12≥η,iiim23km23kexpM3+expM2≤1.
Then, the positiveω-periodic solution of system (3) is globally asymptotically stable.Proof.
In view of Theorem2, there exists a positive periodic solution x1∗k,x2∗k,x3∗k of system (3). We prove below that it is uniformly asymptotically stable. First, we introduce the change of variables as follows:(59)N1k=x1k−x1∗k,N2k=x2k−x2∗k,N3k=x3k−x3∗k.
Then, it follows from (3) that(60)N1k+1=x1k+1−x1∗k+1=x1kexpa1k−a11kx1k−a12kx2km12kx2k+x1k,−x1∗kexpa1k−a11kx1∗k−a12kx2∗km12kx2∗k+x1∗k=x1kexp−a11kN1k−a12kx2km12kx2k+x1k−x2∗km12kx2∗k+x1∗k−x1∗kx1∗k+1x1∗k,=1−a11kx1∗k−a12kx2∗km12kx2∗k+x1∗k+a12kx1∗kx2∗km12kx2∗k+x1∗k2N1kx1∗k−a12km12kx2∗k+x1∗kN2k+ρ1x1∗k+1,(61)N2k+1=x2k+1−x2∗k+1=x2kexp−a2k+a21kx1km12kx2k+x1k−a23kx3km23kx3k+x2k−x2∗kexp−a2k+a21kx1∗km12kx2∗k+x1∗k−a23kx3∗km23kx3∗k+x2∗k=x2kexpa21kx1km12kx2k+x1k−a21kx1∗km12kx2∗k+x1∗k−a23kx3km23kx3k+x2k−a23kx3∗km23kx3∗k+x2∗k−x2∗kx2∗k+1x2∗k=a21km12kx2∗k+x1∗k−a21kx1∗km12kx2∗k+x1∗k2N1k+x2∗k1−a21kx1∗k−a23kx3∗km12kx2∗k+x1∗k−a21ka23kx1∗kx2∗kx3∗km12kx2∗k+x1∗k2N2kx2∗k+a21km23kx3∗km12kx2∗k+x1∗k2−m23km23kx3∗k+x3∗kN3k+ρ2x2∗k+1,(62)N3k+1=x3k+1−x3∗k+1=x3kexp−a3k+a32kx2km23kx3k+x2k−x3∗kexp−a3k+a32kx2∗km23kx3∗k+x2∗k=x3kexpa32kx2km23kx3k+x2k−x2∗km23kx3∗k+x2∗k−x3∗kx3∗k+1x3∗k,=a32k−a32kx2∗km23kx3∗k+x2∗kN2k+a32kx2∗kx3∗km23kx3∗k+x2∗k−a32kx2∗kx3∗kkm23km23kx3∗k+x2∗k2N3kx3∗k+ρ3x3∗k+1,where ρi/Ni=1,2,3 converges, uniformly with respect to k∈Z+, to zero as N⟶0.
Define Lyapunov function as follows:(63)VNk=δ1N1kx1∗k+δ2N2kx2∗k+δ3N3kx3∗k,where δ1,δ2, and δ3 are positive constants given in (67)–(69), respectively, and satisfy Ai∗>0i=1,2,3. Calculating the difference of V along the solution of systems (60)–(62) and using (i), (ii), and (iii), we have(64)ΔV=δ1N1k+1x1∗k+1−N1kx1∗k+δ2N2k+1x2∗k+1−N2kx2∗k+δ3N3k+1x3∗k+1−N3kx3∗k≤−δ1a11k+a12kx2∗kx1∗km12kx2∗k+x1∗k−a12kx2∗km12kx2∗k+x1∗k2N1k+δ1a12km12kx2∗k+x1∗kN2k−δ2a21ka23kx1∗kx3∗km12kx2∗k+x1∗k2−1−a21kx1∗k−a23kx3∗km12kx2∗k+x1∗kN2k+δ2a21km12kx2∗k+x1∗k−a21kx1∗km12kx2∗k+x1∗k2N1k+δ2a21km23kx3∗km12kx2∗k+x1∗k2−m23km23kx3∗k+x3∗kN3k−δ3a32kx2∗kkm23km23kx3∗k+x2∗k2−a32kx2∗km23kx3∗k+x2∗kN3k+δ3a32k−a32kx2∗km23kx3∗k+x2∗kN2k+∑i=13δiρi−A1N1k−A2N2k−A3N3k+∑i=13δiρi,where(65)A1=δ1a11k+a12kx2∗kx1∗km12kx2∗k+x1∗k−a12kx2∗km12kx2∗k+x1∗k2+δ2a21km12kx2∗k+x1∗k−a21kx1∗km12kx2∗k+x1∗k2,A2=δ2a21ka23kx1∗kx3∗km12kx2∗k+x1∗k2−1−a21kx1∗k−a23kx3∗km12kx2∗k+x1∗k+δ1a12km12kx2∗k+x1∗k+δ3a32k−a32kx2∗km23kx3∗k+x2∗k,A3=δ3a32kx2∗kkm23km23kx3∗k+x2∗k2−a32kx2∗km23kx3∗k+x2∗k+δ2a21km23kx3∗km12kx2∗k+x1∗k2−m23km23kx3∗k+x2∗k.
From (64), we have(66)ΔV≤−A1∗N1k−A2∗N2k−A3∗N3k+∑i=13δiρi,where(67)A1∗=δ1a11k+a12kexpm2expM1m12kexpM2+expM1−a12kexpM2m12kexpm2+expm12+δ2a21km12kexpm2+expm1−a21kexpm1m12kexpM2+expM12,(68)A2∗=δ2a21ka23kexpm1+m3m12kexpM2+expM12−1−a21kexpm1−a23kexpm3m12kexpM2+expM1+δ1a12km12kexpm2+expm1+δ3a32k−a32kexpm2m23kexpM3+expM2,(69)A3∗=δ3a32kexpm2m23km23kexpM3+expM22−a32kexpM2m23kexpm3+expm2+δ2a21km23kexpM3m12kexpm2+expm12−m23km23kexpm3+expm2.
Sinceρi/Ni=1,2,3 converges, uniformly with respect to k∈Z+, to zero as N⟶0, it follows from conditions (i),(ii), and (iii) that there is a positive constant φ such that, if k is sufficiently large and N<φ, then(70)ΔV≤−η3N1k+N2k+N3k=−η3N.
It follows from Freedman [22] that the trivial solutions of (60)–(62) is uniformly asymptotically stable. Thus, the solution x∗=x∗k=x1∗k,x2∗k,x3∗kT of (3) is uniformly asymptotically stable. According to Wang and Lu [23], we can conclude that the positive periodic solution of (3) is globally asymptotically stable. The proof ends.


## 4. Numerical Example

Given a discrete model as follows:(71)x1k+1=x1kexpa1k−a11kx1k−τ1k−a12kx2km12kx2k+x1k,x2k+1=x2kexp−a2k+a21kx1k−τ2km12kx2k−τ2k+x1k−τ2k−a23kx3km23kx3k+x2k,x3k+1=x3kexp−a3k+a32kx2k−τ3km23kx3k−τ3k+x2k−τ3k,where a1k=0.8,a2k=0.3,a3k=0.4,a11k=0.3+0.2sinkπ,a12k=0.2+0.2sinkπ,a21k=0.8+0.4sinkπ,a23k=0.6+0.4sinkπ,a32k=0.9+0.2sinkπ,m12k=0.4+0.4sinkπ,m23k=0.5+0.3sinkπ,andτ1k=τ2k=τ3k=1. Then, a¯1=0.8,a12/m12¯=0.5,S1=0.5823,a¯11=0.3,a¯2=0.3,a¯21=0.8,a¯3=0.4,canda¯32=0.9. Thus, (H1)–(H3) hold. Therefore, system (71) has at least a positive two-periodic solution, which is shown in Figure 1.Figure 1
The time histories oft−x1, t−x2, and t−x3, where the red line represents x1, the blue line represents x2, and the green line represents x3.



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*Source: 9464532-2020-07-23.xml*