# Global Dynamics of Some 3 × 6 Systems of Exponential Difference Equations

Authors: A. Q. Khan; A. Sharif
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We study the global dynamics of a list of following3×6 systems of exponential difference equations: xn+1=α4+β4e-yn/γ4+yn-1, yn+1=α5+β5e-zn/γ5+zn-1, zn+1=α6+β6e-xn/γ6+xn-1, xn+1=α7+β7e-zn/γ7+zn-1, yn+1=α8+β8e-xn/γ8+xn-1, zn+1=α9+β9e-yn/γ9+yn-1, xn+1=α10+β10e-xn/γ10+xn-1, yn+1=α11+β11e-yn/γ11+zn-1, zn+1=α12+β12e-zn/γ12+yn-1, xn+1=α13+β13e-xn/γ13+zn-1, yn+1=α14+β14e-yn/γ14+yn-1, zn+1=α15+β15e-zn/γ15+xn-1, xn+1=α16+β16e-xn/γ16+yn-1, yn+1=α17+β17e-yn/γ17+xn-1,zn+1=α18+β18e-zn/γ18+zn-1, where αi,βi,γi,i=4,5,⋯,18 and the initial conditions x0,x-1,y0,y-1,z0,z-1 are arbitrary nonnegative real numbers. This proposed work is considerably extended and improve some existing results in the literature. Some numerical examples are given to not only verify our theoretical results but also provide different types of qualitative behavior of solution of these systems.


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## Body

## 1. Introduction

Discrete dynamical systems described by difference equations are more appropriate for population dynamics as compared to continuous ones. Population models involve exponential difference equations and their stability analysis though complicated, but interesting. The beginning of 21st century has witnessed an increasing interest in the population dynamics. Consequently, many works have been appeared on difference equations or systems of difference equations involving exponential terms (see [1–14] and reference cited therein). For instance, Metwallyet al. [1] have investigated the dynamics of the following second-order difference equation:(1)xn+1=α+βxn-1e-xn,which is the solution of the following logistic equation with piecewise constant arguments:(2)dxdt=ax1-xK,where α,β and initial conditions x-1,x0 are arbitrary nonnegative real numbers. Equation (1) may be viewed as a model in Mathematical Biology where α is immigration rate and β is the population growth rate. Moreover it is also pointed out in [2] that this model is suggested by the people from the Harvard school of public health, studying the population dynamics of single-species xn. Further Papaschinopouloset al. [2] and Papaschinopoulos and Schinas [3] produced nice results in this direction by investigating the dynamical properties like boundedness and persistence of positive solutions, existence of the unique positive equilibrium, and local and global asymptotic stability of two-species model described by systems of difference equations, which is natural extension of single-species population model depicted in (1).In [4], Groveet al. have investigated the global dynamics of the positive solution of the following difference equations:(3)xn+1=αxn+βxn-1e-xn,where α,β and initial conditions x-1,x0 are arbitrary nonnegative real numbers. This equation can be considered as a biological model, since it arises from models studying the amount of litter in perennial grassland (see [6]). After that Papaschinopouloset al. [5, 6] have studied the asymptotic behavior of the positive solution of two-species model which is also natural extension of single-species model represented in (3). In 2016, Wang and Feng [7] have investigated the dynamics of positive solution for the following difference equation which is naturally a new form of single-species model depicted in (1):(4)xn+1=α+βxne-xn-1,where α,β and initial conditions x-1,x0 are arbitrary nonnegative real numbers. In biological point of view α is immigration rate and β is population growth rate.Ozturket al. [8] have investigated the global asymptotic stability, boundedness and periodic nature of the following 2nd-order exponential difference equation:(5)yn+1=α+βe-ynγ+yn-1,n=0,1,⋯,where α, β, γ and y-1,y0 are arbitrary nonnegative numbers. Equation (5) is also viewed as a model in Mathematical Biology where α is immigration rate, β is population growth rate and γ is the carrying capacity. Later Papaschinopouloset al. [9] have investigated boundedness and persistence and local and global asymptotic behavior of two-species model which is natural extension of single-species model (5), represented by the following exponential systems of difference equations: (6)xn+1=α+βe-ynγ+yn-1,yn+1=δ+ϵe-xnς+xn-1,xn+1=α+βe-ynγ+xn-1,yn+1=δ+ϵe-xnς+yn-1,xn+1=α+βe-xnγ+yn-1,yn+1=δ+ϵe-ynς+xn-1,where α,β,γ,δ,ϵ,ς and initial conditions x-1,x0,y-1,y0 are nonnegative real numbers. Motivated from aforementioned studies, our aim in this paper is to investigate the global dynamics of three-species models, which is natural extension of single-species model studied by Ozturket al. [8], represented by the following list of 3×6 systems of exponential difference equations:(7)xn+1=α4+β4e-ynγ4+yn-1,yn+1=α5+β5e-znγ5+zn-1,zn+1=α6+β6e-xnγ6+xn-1,(8)xn+1=α7+β7e-znγ7+zn-1,yn+1=α8+β8e-xnγ8+xn-1,zn+1=α9+β9e-ynγ9+yn-1,(9)xn+1=α10+β10e-xnγ10+xn-1,yn+1=α11+β11e-ynγ11+zn-1,zn+1=α12+β12e-znγ12+yn-1,(10)xn+1=α13+β13e-xnγ13+zn-1,yn+1=α14+β14e-ynγ14+yn-1,zn+1=α15+β15e-znγ15+xn-1,(11)xn+1=α16+β16e-xnγ16+yn-1,yn+1=α17+β17e-ynγ17+xn-1,zn+1=α18+β18e-znγ18+zn-1,where αi,βi,γi,i=4,5,⋯,18 and the initial conditions x0,x-1,y0,y-1,z0,z-1 are arbitrary nonnegative real numbers. Equations (7) to (11) may be viewed as discrete-time population models in Mathematical Biology and biological parameters αi,βi,γi,i=4,5,⋯,18 can be interpretations as shown in Table 1.Table 1
Parameters of the discrete-time model (7) to (11) along their Biological interpretations.
Parameters Biological interpretations α i , i = 4,7 , 10,13,16 Immigration rate of speciesxn α i , i = 5,8 , 11,14,17 Immigration rate of speciesyn α i , i = 6,9 , 12,15,18 Immigration rate of specieszn β i , i = 4,7 , 10,13,16 Growth or reproductive rate of speciesxn β i , i = 5,8 , 11,14,17 Growth or reproductive rate of speciesyn β i , i = 6,9 , 12,15,18 Growth or reproductive rate of specieszn γ i , i = 4,7 , 10,13,16 Carrying capacity of speciesxn γ i , i = 5,8 , 11,14,17 Carrying capacity of speciesyn γ i , i = 6,9 , 12,15,18 Carrying capacity of speciesznThe rest of the paper is organized as follows: In Section2, we investigate the global dynamics and rate of convergence of system (7). This includes boundeness and persistence, existence and uniqueness of positive equilibrium point, local and global stability about the unique positive equilibrium point and rate of convergence that converge to its unique positive equilibrium point of the system. Sections 3, 4, 5 and 6 include same analysis for system (8), (9), (10) and (11), respectively. Section 7 is about the numerical simulations that verify our theoretical results. A brief conclusion is given in Section 8.


## 2. On the System:xn+1=α4+β4e-yn/γ4+yn-1, yn+1=α5+β5e-zn/γ5+zn-1, zn+1=α6+β6e-xn/γ6+xn-1

In this section we will study the main results for system (7).


### 2.1. Boundedness and Persistence

The following theorem shows that every positive solution{(xn,yn,zn)}n=-1∞ of system (7) is bounded and persists.Theorem 1.
Every positive solution{(xn,yn,zn)}n=-1∞ of system (7) is bounded and persists.Proof.
If{(xn,yn,zn)}n=-1∞is an arbitrary solution of (7), then(12)xn≤α4+β4γ4=U4,yn≤α5+β5γ5=U5,zn≤α6+β6γ6=U6,n=0,1,⋯.In addition from (7) and (12), one gets(13)xn≥α4+β4e-α5+β5/γ5γ4+α5+β5/γ5=L4,yn≥α5+β5e-α6+β6/γ6γ5+α6+β6/γ6=L5,zn≥α6+β6e-α4+β4/γ4γ6+α4+β4/γ4=L6,n=2,3,⋯.Hence from (12) and (13), one gets (14)L4≤xn≤U4,L5≤yn≤U5,L6≤zn≤U6,n=3,4,⋯.


### 2.2. Existence of Invariant Set for Solutions

Theorem 2.
Ifxn,yn,znn=-1∞ is a positive solution of system (7), then [L4,U4]×[L5,U5]×[L6,U6] is invariant set for system (7), where L4=α4+β4e-α5+β5/γ5/γ4+α5+β5/γ5, U4=α4+β4/γ4, L5=α5+β5e-α6+β6/γ6/γ5+α6+β6/γ6, U5=α5+β5/γ5, L6=α6+β6e-α4+β4/γ4/γ6+α4+β4/γ4 and U6=α6+β6/γ6.Proof.
For any positive solution{(xn,yn,zn)}n=-1∞ of system (7) with initial conditions x0,x-1∈[L4,U4],y0,y-1∈[L5,U5] andz0,z-1∈L6,U6, one has from (7)(15)x1=α4+β4e-y0γ4+y-1≤α4+β4γ4,and (16)x1=α4+β4e-y0γ4+y-1≥α4+β4e-α5+β5/γ5γ4+α5+β5/γ5.Also(17)y1=α5+β5e-z0γ5+z-1≤α5+β5γ5,and(18)y1=α5+β5e-z0γ5+z-1≥α5+β5e-α6+β6/γ6γ5+α6+β6/γ6.Finally, (19)z1=α6+β6e-x0γ6+x-1≤α6+β6γ6,and(20)z1=α6+β6e-x0γ6+x-1≥α6+β6e-α4+β4/γ4γ6+α4+β4/γ4.Hence x1∈[L4,U4],y1∈[L5,U5] and z1∈L6,U6. Similarly one can show that if xk∈[L4,U4],yk∈[L5,U5] and zk∈L6,U6, then xk+1∈[L4,U4],yk+1∈[L5,U5] and zk+1∈L6,U6.


### 2.3. Existence and Uniqueness of Positive Equilibrium Point

In the following we will study the existence and uniqueness of positive equilibrium point of system (7).Theorem 3.
If(21)-α5+β5e-L61+γ5+U6/γ5+L62α4+β4e-α5+β5e-U6/γ5+U61+α5+β5e-L6/γ5+L6+γ4β6eΓ1+U6γ4+α5+β5e-L6/γ5+L6α4+β4e-α5+β5e-L6/γ5+L6+γ6α5+β5e-U6/γ5+U6+γ4<1,where (22)Γ1=-α4+β4e-α5+β5e-L6/γ5+L6γ4+α5+β5e-L6/γ5+L6,then system (7) has a unique positive equilibrium point (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6].Proof.
Consider the following system of equations:(23)x=α4+β4e-yγ4+y,y=α5+β5e-zγ5+z,z=α6+β6e-xγ6+x.From (23) set(24)y=gz=α5+β5e-zγ5+z, and (25)x=fy=α4+β4e-yγ4+y.Denote (26)Fz=α6+β6e-kzγ6+kz-z,where(27)kz=x=fgz,=fα5+β5e-zγ5+z,=α4+β4e-α5+β5e-z/γ5+zγ4+α5+β5e-z/γ5+z,and z∈L6,U6. We claim that Fz=0 has a unique solution in [L6,U6]. From (26) and (27), one gets(28)F′z=-k′zβ6e-kzγ6+kz+α6+β6e-kzγ6+kz2-1.(29)k′z=β5γ5e-z+β5ze-z+α5+β5e-z/γ5+z2α4+β4e-α5+β5e-z/γ5+z1+α5+β5e-z/γ5+z+γ4α5+β5e-z/γ5+z+γ42.If z-∈[L6,U6] is a solution of Fz=0 then from (26) and (27), one gets (30)α6+β6e-kz-=z-γ6+kz-,where (31)kz-=α4+β4e-α5+β5e-z-/γ5+z-γ4+α5+β5e-z-/γ5+z-.In view of (29), (30), and (31), (28) becomes(32)F′z=-β5γ5e-z-+β5ze-z-+α5+β5e-z-/γ5+z-2α4+β4e-α5+β5e-z-/γ5+z-1+α5+β5e-z-/γ5+z-+γ4α5+β5e-z-/γ5+z-+γ42β6e-α4+β4e-α5+β5e-z-/γ5+z-/γ4+α5+β5e-z-/γ5+z-+z-α4+β4e-α5+β5e-z-/γ5+z-/γ4+α5+β5e-z-/γ5+z-+γ6-1.After some manipulation, equation (32) reduces into the following form:(33)F′z=-β5γ5e-z-+β5ze-z-+α5+β5e-z-/γ5+z-2α4+β4e-α5+β5e-z-/γ5+z-1+α5+β5e-z-/γ5+z-+γ4β6e-α4+β4e-α5+β5e-z-/γ5+z-/γ4+α5+β5e-z-/γ5+z-+z-α5+β5e-z-/γ5+z-+γ4α4+β4e-α5+β5e-z-/γ5+z-+γ6γ4+α5+β5e-z-/γ5+z--1,≤-α5+β5e-L61+γ5+U6/γ5+L62α4+β4e-α5+β5e-U6/γ5+U61+α5+β5e-L6/γ5+L6+γ4β6eΓ1+U6γ4+α5+β5e-L6/γ5+L6α4+β4e-α5+β5e-L6/γ5+L6+γ6α5+β5e-U6/γ5+U6+γ4-1,where Γ1=-α4+β4e-α5+β5e-L6/γ5+L6/γ4+α5+β5e-L6/γ5+L6. Now assuming that (21) along with (22) hold, then from (33) we get F′z<0. Hence F(z)=0 has a unique positive solution in z-∈[L6,U6].


### 2.4. Local and Global Asymptotic Stability

Theorem 4.
For local stability about(x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7), the following statements hold:
(i) Equilibrium(x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7) is locally asymptotically stable if(34)
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, (ii) Equilibrium (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7) is an unstable if(35)β4e-U4-U5γ4+U5γ6+U4γ5+U6β5α6+β6e-U4γ6+U4+β6α5+β5e-U6γ5+U6+β5e-U6α4+β4e-U5γ5+U6γ6+U4γ4+U5β6e-U4γ4+U5+α6+β6e-U4γ6+U4γ4+U5+α5+β5e-U6γ6+U4γ4+U5γ5+U62β4e-U5α6+β6e-U4γ6+U4+β6e-U4α4+β4e-U5γ4+U5+1γ4+U5γ6+U4γ5+U6β4β5β6e-U4-U5-U6+α4+β4e-U5α5+β5e-U6α6+β6e-U4γ4+U5γ6+U4γ5+U6>1.Proof.
If (x-,y-,z-) is equilibrium point of system (7), then(36)x=α4+β4e-y-γ4+y-,y=α5+β5e-z-γ5+z-,z=α6+β6e-x-γ6+x-.In order to construct the corresponding linearized form of system (7), we consider the following transformation:(37)xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1,where(38)f=α4+β4e-ynγ4+yn-1,f1=xn,g=α5+β5e-znγ5+zn-1,g1=yn,h=α6+β6e-xnγ6+xn-1,h1=zn.The Jacobian matrix FJx-,y-,z- about (x-,y-,z-) under the transformation (37) is given by(39)FJx-,y-,z-=00A1A2001000000000B1B2001000C1C20000000010,where(40)A1=-β4e-y-γ4+y-,A2=-α4+β4e-y-γ4+y-2,B1=-β5e-z-γ5+z-,B2=-α5+β5e-z-γ5+z-2,C1=-β6e-x-γ6+x-,C2=-α6+β6e-x-γ6+x-2.The characteristic equation of FJx-,y-,z- about (x-,y-,z-) is given by(41)λ6+μ1λ3+μ2λ2+μ3λ+μ4=0,where(42)μ1=-A1B1C1,μ2=-A1B1C2+A1B2C1+A2B1C1,μ3=-A1B2C2+A2B1C2+A2B2C1,μ4=-A2B2C2.Now(43)∑i=14μi=β4β5β6e-x--y--z-γ4+y-γ6+x-γ5+z-+β4β5e-x--y-α6+β6e-x-γ4+y-γ5+z-γ6+x-2+β4β6e-x--y-α5+β5e-z-γ4+y-γ6+x-γ5+z-2+β5β6e-x--z-α4+β4e-y-γ5+z-γ6+x-γ4+y-2+β4e-y-α6+β6e-x-α5+β5e-z-γ4+y-γ6+x-2γ5+z-2+β5e-z-α4+β4e-y-α6+β6e-x-γ5+z-γ6+x-2γ4+y-2+β6e-x-α4+β4e-y-α5+β5e-z-γ6+x-γ5+z-2γ4+y-2+α4+β4e-y-α6+β6e-x-α5+β5e-z-γ5+z-2γ4+y-2γ6+x-2,≤β4β5β6e-L4-L5-L6γ4+L5γ6+L4γ5+L6+β4β5e-L4-L5α6+β6e-L4γ4+L5γ5+L6γ6+L42+β4β6e-L4-L5α5+β5e-L6γ4+L5γ6+L4γ5+L62+β5β6e-L4-L6α4+β4e-L5γ5+L6γ6+L4γ4+L52+β4e-L5α6+β6e-L4α5+β5e-L6γ4+L5γ5+L62γ6+L42β5e-L6α4+β4e-L5α6+β6e-L4γ5+L6γ4+L52γ6+L42+β6e-L4α4+β4e-L5α5+β5e-L6γ6+L4γ4+L52γ5+L62+α4+β4e-L5α6+β6e-L6α5+β5e-L4γ5+L62γ4+L52γ6+L42,=β4e-L4-L5γ4+L5γ6+L4γ5+L6β5α6+β6e-L4γ6+L4+β6α5+β5e-L6γ5+L6+β5e-L6α4+β4e-L5γ5+L6γ6+L4γ4+L5β6e-L4γ4+L5+α6+β6e-L4γ6+L4γ4+L5+α5+β5e-L6γ6+L4γ4+L5γ5+L62β4e-L5α6+β6e-L4γ6+L4+β6e-L4α4+β4e-L5γ4+L5+1γ4+L5γ6+L4γ5+L6β4β5β6e-L4-L5-L6+α4+β4e-L5α5+β5e-L6α6+β6e-L4γ4+L5γ6+L4γ5+L6<1.Assuming condition (34) holds, then from (43) one gets ∑i=14μi<1 and hence Theorem 1.5 of [15] implies that equilibrium (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7) is locally asymptotically stable.
Proof (ii). Using same manipulations as in the proof of (i), we have(44)∑i=14μi=β4β5β6e-x--y--z-γ4+y-γ6+x-γ5+z-+β4β5e-x--y-α6+β6e-x-γ4+y-γ5+z-γ6+x-2+β4β6e-x--y-α5+β5e-z-γ4+y-γ6+x-γ5+z-2+β5β6e-x--z-α4+β4e-y-γ5+z-γ6+x-γ4+y-2+β4e-y-α6+β6e-x-α5+β5e-z-γ4+y-γ6+x-2γ5+z-2+β5e-z-α4+β4e-y-α6+β6e-x-γ5+z-γ6+x-2γ4+y-2+β6e-x-α4+β4e-y-α5+β5e-z-γ6+x-γ5+z-2γ4+y-2+α4+β4e-y-α6+β6e-x-α5+β5e-z-γ5+z-2γ4+y-2γ6+x-2,≤β4β5β6e-U4-U5-U6γ4+U5γ6+U4γ5+U6+β4β5e-U4-U5α6+β6e-U4γ4+U5γ5+U6γ6+U42+β4β6e-U4-U5α5+β5e-U6γ4+U5γ6+U4γ5+U62+β5β6e-U4-U6α4+β4e-U5γ5+U6γ6+U4γ4+U52+β4e-U5α6+β6e-U4α5+β5e-U6γ4+U5γ5+U62γ6+U42+β5e-U6α4+β4e-U5α6+β6e-U4γ5+U6γ4+U52γ6+U42+β6e-U4α4+β4e-U5α5+β5e-U6γ6+U4γ4+U52γ5+U62+α4+β4e-U5α6+β6e-U6α5+β5e-U4γ5+U62γ4+U52γ6+U42,=β4e-U4-U5γ4+U5γ6+U4γ5+U6β5α6+β6e-U4γ6+U4+β6α5+β5e-U6γ5+U6+β5e-U6α4+β4e-U5γ5+U6γ6+U4γ4+U5β6e-U4γ4+U5+α6+β6e-U4γ6+U4γ4+U5+α5+β5e-U6γ6+U4γ4+U5γ5+U62β4e-U5α6+β6e-U4γ6+U4+β6e-U4α4+β4e-U5γ4+U5+1γ4+U5γ6+U4γ5+U6β4β5β6e-U4-U5-U6+α4+β4e-U5α5+β5e-U6α6+β6e-U4γ4+U5γ6+U4γ5+U6>1.Assume that (35) hold, then from (44) we get ∑i=14μi>1. Hence by Theorem 1.5 of [15], (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of (7) is unstable.Hereafter we will study the global dynamics of system (7) about the unique positive equilibrium point (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6]. This is main component in the theory of dynamical systems based on the knowledge of its present state. In recent years it is challenging task to determine the global dynamics for higher-order difference equations or systems of difference equations. So, in this paper we study the global dynamics of (7) about the (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] by Lyapunov stability theory because semiconjugacy and weak contraction cannot be used to analyze global dynamics of system (7). Here we construct the discrete-time Lyapunov function: Ωn=x-xn/x--1-ln⁡xn/x-+y-yn/y--1-ln⁡yn/y-+z-zn/z--1-ln⁡zn/z- whose nonnegativity follows from inequality: x-1-ln⁡x≥0,∀x>0 (see [12–14]).Theorem 5.
Equilibrium(x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of (7) is globally asymptotically stable if(45)α4+β4e-L5<x-γ4+L5,α5+β5e-L6<y-γ5+L6,α6+β6e-L4<z-γ6+L4.Proof.
Consider the following discrete-time Lyapunov function:(46)Ωn=x-xnx--1-ln⁡xnx-+y-yny--1-ln⁡yny-+z-znz--1-ln⁡znz-,where the nonnegativity of Ωn follows from the following inequality:(47)x-1-ln⁡x≥0,∀x>0.Moreover, we have (48)-ln⁡xn+1xn=ln⁡xnxn+1=ln⁡1-1-xnxn+1,≤1-xnxn+1≤-xn+1-xnxn+1,-ln⁡yn+1yn=ln⁡ynyn+1=ln⁡1-1-ynyn+1,≤1-ynyn+1≤-yn+1-ynyn+1,-ln⁡zn+1zn=ln⁡znzn+1=ln⁡1-1-znzn+1,≤1-znzn+1≤-zn+1-znzn+1.Now,(49)Ωn+1-Ωn=x-xn+1x--1-ln⁡xn+1x-+y-yn+1y--1-ln⁡yn+1y-+z-zn+1z--1-ln⁡zn+1z--x-xnx--1-ln⁡xnx--y-yny--1-ln⁡yny--z-znz--1-ln⁡znz-,=xn+1-xn+yn+1-yn+zn+1-zn-x-ln⁡xn+1xn-y-ln⁡yn+1yn-z-ln⁡zn+1zn,Ωn+1-Ωn≤xn+1-xn+yn+1-yn+zn+1-zn-x-ln⁡xn+1xn-y-ln⁡yn+1yn-z-ln⁡zn+1zn.Using (48) in (49) and assume that (45) hold then one gets(50)Ωn+1-Ωn≤xn+1-xn+yn+1-yn+zn+1-zn-x-xn+1-xnxn+1-y-yn+1-ynyn+1-z-zn+1-znzn+1,=xn+1-xn1-x-xn+1+yn+1-yn1-y-yn+1+zn+1-zn1-z-zn+1,=xn+1-xn1-x-γ4+yn-1α4+β4e-yn+yn+1-yn1-y-γ5+zn-1α5+β5e-zn+zn+1-zn1-z-γ6+xn-1α6+β6e-xn,≤U4-L4α4+β4e-L5-x-γ4+L5α4+β4e-L5+U5-L5α5+β5e-L6-y-γ5+L6α5+β5e-L6+U6-L6α6+β6e-L4-z-γ6+L4α6+β6e-L4≤0,for all n ≥0. It follows that Ωn≥0. Hence, we obtain that limn→∞Ωn+1-Ωn=0. Therefore, (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of (7) is globally asymptotically stable.In the following we will investigate the rate of convergence that converges to(x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7) by employing the method [16, 17].


### 2.5. Rate of Convergence

Theorem 6.
If{(xn,yn,zn)}n=-1∞ is a positive solution of (7) such that(51)limn→∞xn=x-,limn→∞yn=y-,limn→∞zn=z-,where(52)x-∈L4,U4,y-∈L5,U5,z-∈L6,U6,then the error vector en=(en4,en-14,en5,en-15,en6,en-16)T of every solution of (7) satisfies both the following asymptotic relations: (53)limn→∞en1/n=λ1,2,3,4,5,6FJx-,y-,z-,limn→∞en+1en=λ1,2,3,4,5,6FJx-,y-,z-,where λ1,2,3,4,5,6FJx-,y-,z- are the characteristic root of FJx-,y-,z-.Proof.
If{(xn,yn,zn)}n=-1∞ is a positive solution of system (7) such that (51) along with (52) holds. In order to find the error terms, one has from (7)(54)xn+1-x-=α4+β4e-ynγ4+yn-1-α4+β4e-y-γ4+y-,=β4e-yn-e-y-γ4+yn-1yn-y-yn-y--α4+β4e-y-γ4+yn-1γ4+y-yn-1-y-,yn+1-y-=α5+β5e-znγ5+zn-1-α5+β5e-z-γ5+z-,=β5e-zn-e-z-γ5+zn-1zn-z-zn-z--α5+β5e-z-γ5+zn-1γ5+z-zn-1-z-,zn+1-z-=α6+β6e-xnγ6+xn-1-α6+β6e-x-γ6+x-,=β6e-xn-e-x-γ6+xn-1xn-x-xn-x--α6+β6e-x-γ6+xn-1γ6+x-xn-1-x-;that is(55)xn+1-x-=β4e-yn-e-y-γ4+yn-1yn-y-yn-y--α4+β4e-y-γ4+yn-1γ4+y-yn-1-y-,yn+1-y-=β5e-zn-e-z-γ5+zn-1zn-z-zn-z--α5+β5e-z-γ5+zn-1γ5+z-zn-1-z-,zn+1-z-=β6e-xn-e-x-γ6+xn-1xn-x-xn-x--α6+β6e-x-γ6+xn-1γ6+x-xn-1-x-.Set (56)en4=xn-x-,en5=yn-y-,en6=zn-z-.In view of (56), (55) takes the following form:(57)en+14=anen5+bnen-15,en+15=cnen6+dnen-16,en+16=enen4+fnen-14,where(58)an=β4e-yn-e-y-γ4+yn-1yn-y-,bn=-α4+β4e-y-γ4+yn-1γ4+y-,cn=β5e-zn-e-z-γ5+zn-1zn-z-,dn=-α5+β5e-z-γ5+zn-1γ5+z-,en=β6e-xn-e-x-γ6+xn-1xn-x-,fn=-α6+β6e-x-γ6+xn-1γ6+x-.Taking the limits of an,bn,cn,dn,en,andfn, we obtain(59)limn→∞an=-β4e-y-γ4+y-,limn→∞bn=-α4+β4e-y-γ4+y-2,limn→∞cn=-β5e-z-γ5+z-,limn→∞dn=-α5+β5e-z-γ5+z-2,limn→∞en=-β6e-x-γ6+x-,limn→∞fn=-α6+β6e-x-γ6+x-2;that is(60)an=-β4e-y-γ4+y-+α5n,bn=-α4+β4e-y-γ4+y-2+α5n-1,cn=-β5e-z-γ5+z-+α6n,dn=-α5+β5e-z-γ5+z-2+α6n-1,en=-β6e-x-γ6+x-+α4n,fn=-α6+β6e-x-γ6+x-2+α4n-1,where α4n,α4n-1,α5n,α5n-1,α6n,α6n-1→0 as n→∞. Now we have Poincare´ difference system (1.10) of [18], where(61)A=00-β4e-y-γ4+y--α4+β4e-y-γ4+y-2001000000000-β5e-z-γ5+z--α5+β5e-z-γ5+z-2001000-β6e-x-γ6+x--α6+β6e-x-γ6+x-20000000010,and (62)Bn=00α5nα5n-1001000000000α6nα6n-1001000α4nα4n-10000000010,and Bn→0, n→∞. Thus the limiting system of error terms about (x-,y-,z-) can be written as(63)en+14en4en+15en5en+16en6=00-β4e-y-γ4+y--α4+β4e-y-γ4+y-2001000000000-β5e-z-γ5+z--α5+β5e-z-γ5+z-2001000-β6e-x-γ6+x--α6+β6e-x-γ6+x-20000000010en4en-14en5en-15en6en-16,which is similar to linearized system (7) about equilibrium (x-,y-,z-).


## 2.1. Boundedness and Persistence

The following theorem shows that every positive solution{(xn,yn,zn)}n=-1∞ of system (7) is bounded and persists.Theorem 1.
Every positive solution{(xn,yn,zn)}n=-1∞ of system (7) is bounded and persists.Proof.
If{(xn,yn,zn)}n=-1∞is an arbitrary solution of (7), then(12)xn≤α4+β4γ4=U4,yn≤α5+β5γ5=U5,zn≤α6+β6γ6=U6,n=0,1,⋯.In addition from (7) and (12), one gets(13)xn≥α4+β4e-α5+β5/γ5γ4+α5+β5/γ5=L4,yn≥α5+β5e-α6+β6/γ6γ5+α6+β6/γ6=L5,zn≥α6+β6e-α4+β4/γ4γ6+α4+β4/γ4=L6,n=2,3,⋯.Hence from (12) and (13), one gets (14)L4≤xn≤U4,L5≤yn≤U5,L6≤zn≤U6,n=3,4,⋯.


## 2.2. Existence of Invariant Set for Solutions

Theorem 2.
Ifxn,yn,znn=-1∞ is a positive solution of system (7), then [L4,U4]×[L5,U5]×[L6,U6] is invariant set for system (7), where L4=α4+β4e-α5+β5/γ5/γ4+α5+β5/γ5, U4=α4+β4/γ4, L5=α5+β5e-α6+β6/γ6/γ5+α6+β6/γ6, U5=α5+β5/γ5, L6=α6+β6e-α4+β4/γ4/γ6+α4+β4/γ4 and U6=α6+β6/γ6.Proof.
For any positive solution{(xn,yn,zn)}n=-1∞ of system (7) with initial conditions x0,x-1∈[L4,U4],y0,y-1∈[L5,U5] andz0,z-1∈L6,U6, one has from (7)(15)x1=α4+β4e-y0γ4+y-1≤α4+β4γ4,and (16)x1=α4+β4e-y0γ4+y-1≥α4+β4e-α5+β5/γ5γ4+α5+β5/γ5.Also(17)y1=α5+β5e-z0γ5+z-1≤α5+β5γ5,and(18)y1=α5+β5e-z0γ5+z-1≥α5+β5e-α6+β6/γ6γ5+α6+β6/γ6.Finally, (19)z1=α6+β6e-x0γ6+x-1≤α6+β6γ6,and(20)z1=α6+β6e-x0γ6+x-1≥α6+β6e-α4+β4/γ4γ6+α4+β4/γ4.Hence x1∈[L4,U4],y1∈[L5,U5] and z1∈L6,U6. Similarly one can show that if xk∈[L4,U4],yk∈[L5,U5] and zk∈L6,U6, then xk+1∈[L4,U4],yk+1∈[L5,U5] and zk+1∈L6,U6.


## 2.3. Existence and Uniqueness of Positive Equilibrium Point

In the following we will study the existence and uniqueness of positive equilibrium point of system (7).Theorem 3.
If(21)-α5+β5e-L61+γ5+U6/γ5+L62α4+β4e-α5+β5e-U6/γ5+U61+α5+β5e-L6/γ5+L6+γ4β6eΓ1+U6γ4+α5+β5e-L6/γ5+L6α4+β4e-α5+β5e-L6/γ5+L6+γ6α5+β5e-U6/γ5+U6+γ4<1,where (22)Γ1=-α4+β4e-α5+β5e-L6/γ5+L6γ4+α5+β5e-L6/γ5+L6,then system (7) has a unique positive equilibrium point (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6].Proof.
Consider the following system of equations:(23)x=α4+β4e-yγ4+y,y=α5+β5e-zγ5+z,z=α6+β6e-xγ6+x.From (23) set(24)y=gz=α5+β5e-zγ5+z, and (25)x=fy=α4+β4e-yγ4+y.Denote (26)Fz=α6+β6e-kzγ6+kz-z,where(27)kz=x=fgz,=fα5+β5e-zγ5+z,=α4+β4e-α5+β5e-z/γ5+zγ4+α5+β5e-z/γ5+z,and z∈L6,U6. We claim that Fz=0 has a unique solution in [L6,U6]. From (26) and (27), one gets(28)F′z=-k′zβ6e-kzγ6+kz+α6+β6e-kzγ6+kz2-1.(29)k′z=β5γ5e-z+β5ze-z+α5+β5e-z/γ5+z2α4+β4e-α5+β5e-z/γ5+z1+α5+β5e-z/γ5+z+γ4α5+β5e-z/γ5+z+γ42.If z-∈[L6,U6] is a solution of Fz=0 then from (26) and (27), one gets (30)α6+β6e-kz-=z-γ6+kz-,where (31)kz-=α4+β4e-α5+β5e-z-/γ5+z-γ4+α5+β5e-z-/γ5+z-.In view of (29), (30), and (31), (28) becomes(32)F′z=-β5γ5e-z-+β5ze-z-+α5+β5e-z-/γ5+z-2α4+β4e-α5+β5e-z-/γ5+z-1+α5+β5e-z-/γ5+z-+γ4α5+β5e-z-/γ5+z-+γ42β6e-α4+β4e-α5+β5e-z-/γ5+z-/γ4+α5+β5e-z-/γ5+z-+z-α4+β4e-α5+β5e-z-/γ5+z-/γ4+α5+β5e-z-/γ5+z-+γ6-1.After some manipulation, equation (32) reduces into the following form:(33)F′z=-β5γ5e-z-+β5ze-z-+α5+β5e-z-/γ5+z-2α4+β4e-α5+β5e-z-/γ5+z-1+α5+β5e-z-/γ5+z-+γ4β6e-α4+β4e-α5+β5e-z-/γ5+z-/γ4+α5+β5e-z-/γ5+z-+z-α5+β5e-z-/γ5+z-+γ4α4+β4e-α5+β5e-z-/γ5+z-+γ6γ4+α5+β5e-z-/γ5+z--1,≤-α5+β5e-L61+γ5+U6/γ5+L62α4+β4e-α5+β5e-U6/γ5+U61+α5+β5e-L6/γ5+L6+γ4β6eΓ1+U6γ4+α5+β5e-L6/γ5+L6α4+β4e-α5+β5e-L6/γ5+L6+γ6α5+β5e-U6/γ5+U6+γ4-1,where Γ1=-α4+β4e-α5+β5e-L6/γ5+L6/γ4+α5+β5e-L6/γ5+L6. Now assuming that (21) along with (22) hold, then from (33) we get F′z<0. Hence F(z)=0 has a unique positive solution in z-∈[L6,U6].


## 2.4. Local and Global Asymptotic Stability

Theorem 4.
For local stability about(x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7), the following statements hold:
(i) Equilibrium(x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7) is locally asymptotically stable if(34)
β
4
e
-
L
4
-
L
5
γ
4
+
L
5
γ
6
+
L
4
γ
5
+
L
6
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, (ii) Equilibrium (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7) is an unstable if(35)β4e-U4-U5γ4+U5γ6+U4γ5+U6β5α6+β6e-U4γ6+U4+β6α5+β5e-U6γ5+U6+β5e-U6α4+β4e-U5γ5+U6γ6+U4γ4+U5β6e-U4γ4+U5+α6+β6e-U4γ6+U4γ4+U5+α5+β5e-U6γ6+U4γ4+U5γ5+U62β4e-U5α6+β6e-U4γ6+U4+β6e-U4α4+β4e-U5γ4+U5+1γ4+U5γ6+U4γ5+U6β4β5β6e-U4-U5-U6+α4+β4e-U5α5+β5e-U6α6+β6e-U4γ4+U5γ6+U4γ5+U6>1.Proof.
If (x-,y-,z-) is equilibrium point of system (7), then(36)x=α4+β4e-y-γ4+y-,y=α5+β5e-z-γ5+z-,z=α6+β6e-x-γ6+x-.In order to construct the corresponding linearized form of system (7), we consider the following transformation:(37)xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1,where(38)f=α4+β4e-ynγ4+yn-1,f1=xn,g=α5+β5e-znγ5+zn-1,g1=yn,h=α6+β6e-xnγ6+xn-1,h1=zn.The Jacobian matrix FJx-,y-,z- about (x-,y-,z-) under the transformation (37) is given by(39)FJx-,y-,z-=00A1A2001000000000B1B2001000C1C20000000010,where(40)A1=-β4e-y-γ4+y-,A2=-α4+β4e-y-γ4+y-2,B1=-β5e-z-γ5+z-,B2=-α5+β5e-z-γ5+z-2,C1=-β6e-x-γ6+x-,C2=-α6+β6e-x-γ6+x-2.The characteristic equation of FJx-,y-,z- about (x-,y-,z-) is given by(41)λ6+μ1λ3+μ2λ2+μ3λ+μ4=0,where(42)μ1=-A1B1C1,μ2=-A1B1C2+A1B2C1+A2B1C1,μ3=-A1B2C2+A2B1C2+A2B2C1,μ4=-A2B2C2.Now(43)∑i=14μi=β4β5β6e-x--y--z-γ4+y-γ6+x-γ5+z-+β4β5e-x--y-α6+β6e-x-γ4+y-γ5+z-γ6+x-2+β4β6e-x--y-α5+β5e-z-γ4+y-γ6+x-γ5+z-2+β5β6e-x--z-α4+β4e-y-γ5+z-γ6+x-γ4+y-2+β4e-y-α6+β6e-x-α5+β5e-z-γ4+y-γ6+x-2γ5+z-2+β5e-z-α4+β4e-y-α6+β6e-x-γ5+z-γ6+x-2γ4+y-2+β6e-x-α4+β4e-y-α5+β5e-z-γ6+x-γ5+z-2γ4+y-2+α4+β4e-y-α6+β6e-x-α5+β5e-z-γ5+z-2γ4+y-2γ6+x-2,≤β4β5β6e-L4-L5-L6γ4+L5γ6+L4γ5+L6+β4β5e-L4-L5α6+β6e-L4γ4+L5γ5+L6γ6+L42+β4β6e-L4-L5α5+β5e-L6γ4+L5γ6+L4γ5+L62+β5β6e-L4-L6α4+β4e-L5γ5+L6γ6+L4γ4+L52+β4e-L5α6+β6e-L4α5+β5e-L6γ4+L5γ5+L62γ6+L42β5e-L6α4+β4e-L5α6+β6e-L4γ5+L6γ4+L52γ6+L42+β6e-L4α4+β4e-L5α5+β5e-L6γ6+L4γ4+L52γ5+L62+α4+β4e-L5α6+β6e-L6α5+β5e-L4γ5+L62γ4+L52γ6+L42,=β4e-L4-L5γ4+L5γ6+L4γ5+L6β5α6+β6e-L4γ6+L4+β6α5+β5e-L6γ5+L6+β5e-L6α4+β4e-L5γ5+L6γ6+L4γ4+L5β6e-L4γ4+L5+α6+β6e-L4γ6+L4γ4+L5+α5+β5e-L6γ6+L4γ4+L5γ5+L62β4e-L5α6+β6e-L4γ6+L4+β6e-L4α4+β4e-L5γ4+L5+1γ4+L5γ6+L4γ5+L6β4β5β6e-L4-L5-L6+α4+β4e-L5α5+β5e-L6α6+β6e-L4γ4+L5γ6+L4γ5+L6<1.Assuming condition (34) holds, then from (43) one gets ∑i=14μi<1 and hence Theorem 1.5 of [15] implies that equilibrium (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7) is locally asymptotically stable.
Proof (ii). Using same manipulations as in the proof of (i), we have(44)∑i=14μi=β4β5β6e-x--y--z-γ4+y-γ6+x-γ5+z-+β4β5e-x--y-α6+β6e-x-γ4+y-γ5+z-γ6+x-2+β4β6e-x--y-α5+β5e-z-γ4+y-γ6+x-γ5+z-2+β5β6e-x--z-α4+β4e-y-γ5+z-γ6+x-γ4+y-2+β4e-y-α6+β6e-x-α5+β5e-z-γ4+y-γ6+x-2γ5+z-2+β5e-z-α4+β4e-y-α6+β6e-x-γ5+z-γ6+x-2γ4+y-2+β6e-x-α4+β4e-y-α5+β5e-z-γ6+x-γ5+z-2γ4+y-2+α4+β4e-y-α6+β6e-x-α5+β5e-z-γ5+z-2γ4+y-2γ6+x-2,≤β4β5β6e-U4-U5-U6γ4+U5γ6+U4γ5+U6+β4β5e-U4-U5α6+β6e-U4γ4+U5γ5+U6γ6+U42+β4β6e-U4-U5α5+β5e-U6γ4+U5γ6+U4γ5+U62+β5β6e-U4-U6α4+β4e-U5γ5+U6γ6+U4γ4+U52+β4e-U5α6+β6e-U4α5+β5e-U6γ4+U5γ5+U62γ6+U42+β5e-U6α4+β4e-U5α6+β6e-U4γ5+U6γ4+U52γ6+U42+β6e-U4α4+β4e-U5α5+β5e-U6γ6+U4γ4+U52γ5+U62+α4+β4e-U5α6+β6e-U6α5+β5e-U4γ5+U62γ4+U52γ6+U42,=β4e-U4-U5γ4+U5γ6+U4γ5+U6β5α6+β6e-U4γ6+U4+β6α5+β5e-U6γ5+U6+β5e-U6α4+β4e-U5γ5+U6γ6+U4γ4+U5β6e-U4γ4+U5+α6+β6e-U4γ6+U4γ4+U5+α5+β5e-U6γ6+U4γ4+U5γ5+U62β4e-U5α6+β6e-U4γ6+U4+β6e-U4α4+β4e-U5γ4+U5+1γ4+U5γ6+U4γ5+U6β4β5β6e-U4-U5-U6+α4+β4e-U5α5+β5e-U6α6+β6e-U4γ4+U5γ6+U4γ5+U6>1.Assume that (35) hold, then from (44) we get ∑i=14μi>1. Hence by Theorem 1.5 of [15], (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of (7) is unstable.Hereafter we will study the global dynamics of system (7) about the unique positive equilibrium point (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6]. This is main component in the theory of dynamical systems based on the knowledge of its present state. In recent years it is challenging task to determine the global dynamics for higher-order difference equations or systems of difference equations. So, in this paper we study the global dynamics of (7) about the (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] by Lyapunov stability theory because semiconjugacy and weak contraction cannot be used to analyze global dynamics of system (7). Here we construct the discrete-time Lyapunov function: Ωn=x-xn/x--1-ln⁡xn/x-+y-yn/y--1-ln⁡yn/y-+z-zn/z--1-ln⁡zn/z- whose nonnegativity follows from inequality: x-1-ln⁡x≥0,∀x>0 (see [12–14]).Theorem 5.
Equilibrium(x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of (7) is globally asymptotically stable if(45)α4+β4e-L5<x-γ4+L5,α5+β5e-L6<y-γ5+L6,α6+β6e-L4<z-γ6+L4.Proof.
Consider the following discrete-time Lyapunov function:(46)Ωn=x-xnx--1-ln⁡xnx-+y-yny--1-ln⁡yny-+z-znz--1-ln⁡znz-,where the nonnegativity of Ωn follows from the following inequality:(47)x-1-ln⁡x≥0,∀x>0.Moreover, we have (48)-ln⁡xn+1xn=ln⁡xnxn+1=ln⁡1-1-xnxn+1,≤1-xnxn+1≤-xn+1-xnxn+1,-ln⁡yn+1yn=ln⁡ynyn+1=ln⁡1-1-ynyn+1,≤1-ynyn+1≤-yn+1-ynyn+1,-ln⁡zn+1zn=ln⁡znzn+1=ln⁡1-1-znzn+1,≤1-znzn+1≤-zn+1-znzn+1.Now,(49)Ωn+1-Ωn=x-xn+1x--1-ln⁡xn+1x-+y-yn+1y--1-ln⁡yn+1y-+z-zn+1z--1-ln⁡zn+1z--x-xnx--1-ln⁡xnx--y-yny--1-ln⁡yny--z-znz--1-ln⁡znz-,=xn+1-xn+yn+1-yn+zn+1-zn-x-ln⁡xn+1xn-y-ln⁡yn+1yn-z-ln⁡zn+1zn,Ωn+1-Ωn≤xn+1-xn+yn+1-yn+zn+1-zn-x-ln⁡xn+1xn-y-ln⁡yn+1yn-z-ln⁡zn+1zn.Using (48) in (49) and assume that (45) hold then one gets(50)Ωn+1-Ωn≤xn+1-xn+yn+1-yn+zn+1-zn-x-xn+1-xnxn+1-y-yn+1-ynyn+1-z-zn+1-znzn+1,=xn+1-xn1-x-xn+1+yn+1-yn1-y-yn+1+zn+1-zn1-z-zn+1,=xn+1-xn1-x-γ4+yn-1α4+β4e-yn+yn+1-yn1-y-γ5+zn-1α5+β5e-zn+zn+1-zn1-z-γ6+xn-1α6+β6e-xn,≤U4-L4α4+β4e-L5-x-γ4+L5α4+β4e-L5+U5-L5α5+β5e-L6-y-γ5+L6α5+β5e-L6+U6-L6α6+β6e-L4-z-γ6+L4α6+β6e-L4≤0,for all n ≥0. It follows that Ωn≥0. Hence, we obtain that limn→∞Ωn+1-Ωn=0. Therefore, (x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of (7) is globally asymptotically stable.In the following we will investigate the rate of convergence that converges to(x-,y-,z-)∈[L4,U4]×[L5,U5]×[L6,U6] of system (7) by employing the method [16, 17].


## 2.5. Rate of Convergence

Theorem 6.
If{(xn,yn,zn)}n=-1∞ is a positive solution of (7) such that(51)limn→∞xn=x-,limn→∞yn=y-,limn→∞zn=z-,where(52)x-∈L4,U4,y-∈L5,U5,z-∈L6,U6,then the error vector en=(en4,en-14,en5,en-15,en6,en-16)T of every solution of (7) satisfies both the following asymptotic relations: (53)limn→∞en1/n=λ1,2,3,4,5,6FJx-,y-,z-,limn→∞en+1en=λ1,2,3,4,5,6FJx-,y-,z-,where λ1,2,3,4,5,6FJx-,y-,z- are the characteristic root of FJx-,y-,z-.Proof.
If{(xn,yn,zn)}n=-1∞ is a positive solution of system (7) such that (51) along with (52) holds. In order to find the error terms, one has from (7)(54)xn+1-x-=α4+β4e-ynγ4+yn-1-α4+β4e-y-γ4+y-,=β4e-yn-e-y-γ4+yn-1yn-y-yn-y--α4+β4e-y-γ4+yn-1γ4+y-yn-1-y-,yn+1-y-=α5+β5e-znγ5+zn-1-α5+β5e-z-γ5+z-,=β5e-zn-e-z-γ5+zn-1zn-z-zn-z--α5+β5e-z-γ5+zn-1γ5+z-zn-1-z-,zn+1-z-=α6+β6e-xnγ6+xn-1-α6+β6e-x-γ6+x-,=β6e-xn-e-x-γ6+xn-1xn-x-xn-x--α6+β6e-x-γ6+xn-1γ6+x-xn-1-x-;that is(55)xn+1-x-=β4e-yn-e-y-γ4+yn-1yn-y-yn-y--α4+β4e-y-γ4+yn-1γ4+y-yn-1-y-,yn+1-y-=β5e-zn-e-z-γ5+zn-1zn-z-zn-z--α5+β5e-z-γ5+zn-1γ5+z-zn-1-z-,zn+1-z-=β6e-xn-e-x-γ6+xn-1xn-x-xn-x--α6+β6e-x-γ6+xn-1γ6+x-xn-1-x-.Set (56)en4=xn-x-,en5=yn-y-,en6=zn-z-.In view of (56), (55) takes the following form:(57)en+14=anen5+bnen-15,en+15=cnen6+dnen-16,en+16=enen4+fnen-14,where(58)an=β4e-yn-e-y-γ4+yn-1yn-y-,bn=-α4+β4e-y-γ4+yn-1γ4+y-,cn=β5e-zn-e-z-γ5+zn-1zn-z-,dn=-α5+β5e-z-γ5+zn-1γ5+z-,en=β6e-xn-e-x-γ6+xn-1xn-x-,fn=-α6+β6e-x-γ6+xn-1γ6+x-.Taking the limits of an,bn,cn,dn,en,andfn, we obtain(59)limn→∞an=-β4e-y-γ4+y-,limn→∞bn=-α4+β4e-y-γ4+y-2,limn→∞cn=-β5e-z-γ5+z-,limn→∞dn=-α5+β5e-z-γ5+z-2,limn→∞en=-β6e-x-γ6+x-,limn→∞fn=-α6+β6e-x-γ6+x-2;that is(60)an=-β4e-y-γ4+y-+α5n,bn=-α4+β4e-y-γ4+y-2+α5n-1,cn=-β5e-z-γ5+z-+α6n,dn=-α5+β5e-z-γ5+z-2+α6n-1,en=-β6e-x-γ6+x-+α4n,fn=-α6+β6e-x-γ6+x-2+α4n-1,where α4n,α4n-1,α5n,α5n-1,α6n,α6n-1→0 as n→∞. Now we have Poincare´ difference system (1.10) of [18], where(61)A=00-β4e-y-γ4+y--α4+β4e-y-γ4+y-2001000000000-β5e-z-γ5+z--α5+β5e-z-γ5+z-2001000-β6e-x-γ6+x--α6+β6e-x-γ6+x-20000000010,and (62)Bn=00α5nα5n-1001000000000α6nα6n-1001000α4nα4n-10000000010,and Bn→0, n→∞. Thus the limiting system of error terms about (x-,y-,z-) can be written as(63)en+14en4en+15en5en+16en6=00-β4e-y-γ4+y--α4+β4e-y-γ4+y-2001000000000-β5e-z-γ5+z--α5+β5e-z-γ5+z-2001000-β6e-x-γ6+x--α6+β6e-x-γ6+x-20000000010en4en-14en5en-15en6en-16,which is similar to linearized system (7) about equilibrium (x-,y-,z-).


## 3. On the System:xn+1=α7+β7e-zn/γ7+zn-1, yn+1=α8+β8e-xn/γ8+xn-1, zn+1=α9+β9e-yn/γ9+yn-1

In this section our focus is to investigate the behavior of system (8).


### 3.1. Boundedness and Persistence

Theorem 7.
Every positive solution{(xn,yn,zn)}n=-1∞ of system (8) is bounded and persists.Proof.
If{(xn,yn,zn)}n=-1∞is an arbitrary solution of (8), then(64)xn≤α7+β7γ7=U7,yn≤α8+β8γ8=U8,zn≤α9+β9γ9=U9,n=0,1,⋯.Additionally from (8) and (64), one gets(65)xn≥α7+β7e-α9+β9/γ9γ7+α9+β9/γ9=L7,yn≥α8+β8e-α7+β7/γ7γ8+α7+β7/γ7=L8,zn≥α9+β9e-α8+β8/γ8γ9+α8+β8/γ8=L9,n=2,3,⋯.Finally from (64) and (65), one gets(66)L7≤xn≤U7,L8≤yn≤U8,L9≤zn≤U9,n=3,4,⋯.


### 3.2. Existence of Invariant Set for Solutions

Theorem 8.
If{(xn,yn,zn)}n=-1∞ is a positive solution of system (8), then [L7,U7]×[L8,U8]×[L9,U9] is invariant set for system (8), where L7=α7+β7e-α9+β9/γ9/γ7+α9+β9/γ9, U7=α7+β7/γ7, L8=α8+β8e-α7+β7/γ7/γ8+α7+β7/γ7, U8=α8+β8/γ8, L9=α9+β9e-α8+β8/γ8/γ9+α8+β8/γ8 and U9=α9+β9/γ9.Proof.
For any positive solution{(xn,yn,zn)}n=-1∞ of (8) with initial conditions x0,x-1∈[L7,U7],y0,y-1∈[L8,U8]andz0,z-1∈L9,U9. We have from system (8)(67)x1=α7+β7e-z0γ7+z-1≤α7+β7γ7,and(68)x1=α7+β7e-z0γ7+z-1≥α7+β7e-α9+β9/γ9γ7+α9+β9/γ9.Also(69)y1=α8+β8e-x0γ8+x-1≤α8+β8γ8,and (70)y1=α8+β8e-x0γ8+x-1≥α8+β8e-α7+β7/γ7γ8+α7+β7/γ7.Finally(71)z1=α9+β9e-y0γ9+y-1≤α9+β9γ9,and(72)z1=α9+β9e-y0γ9+y-1≥α9+β9e-α8+β8/γ8γ9+α8+β8/γ8.Hence x1∈[L7,U7],y1∈[L8,U8]andz1∈L9,U9. Similarly one can show that if xk∈[L7,U7],yk∈[L8,U8]andzk∈L9,U9, then xk+1∈[L7,U7],yk+1∈[L8,U8]andzk+1∈L9,U9.


### 3.3. Existence and Uniqueness of Positive Equilibrium Point

Theorem 9.
If(73)-α7+β7e-L91+γ7+U9/γ7+L92α8+β8e-α7+β7e-U9/γ7+U91+α7+β7e-L9/γ7+L9+γ8β9eΓ2+U9γ8+α7+β7e-L9/γ7+L9α8+β8e-α7+β7e-L9/γ7+L9+γ8α7+β7e-U9/γ7+U9+γ8<1,where (74)Γ2=-α8+β8e-α7+β7e-L9/γ7+L9γ8+α7+β7e-L9/γ7+L9,then system (8) has a unique positive equilibrium point (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9].Proof.
Consider(75)x=α7+β7e-zγ7+z,y=α8+β8e-xγ8+x,z=α9+β9e-yγ9+y.From (75) set (76)y=gx=α8+β8e-xγ8+x,and(77)x=fz=α7+β7e-zγ7+z.Denote (78)Fz=α9+β9e-kzγ9+kz-z,where(79)kz=y=gfz,=gα7+β7e-zγ7+z,=α8+β8e-α7+β7e-z/γ7+zγ8+α7+β7e-z/γ7+z,and z∈L9,U9. We claim that F(z)=0 has a unique solution in [L9,U9]. From (78) and (79) one gets(80)F′z=-k′zβ9e-kzγ9+kz+α9+β9e-kzγ9+kz2-1.And(81)k′z=β7γ7e-z+β7ze-z+α7+β7e-z/γ7+z2α8+β8e-α7+β7e-z/γ7+z1+α7+β7e-z/γ7+z+γ8α7+β7e-z/γ7+z+γ82.Now if z-∈[L9,U9] is a solution of Fz=0, then from (78) and (79) one gets (82)α9+β9e-kz-=z-γ9+kz-,where (83)kz-=α8+β8e-α7+β7e-z-/γ7+z-γ8+α7+β7e-z-/γ7+z-.In view of equation (81), (82), and (83), (80) becomes (84)F′z=-β7γ7e-z-+β7ze-z-+α7+β7e-z-/γ7+z-2α8+β8e-α7+β7e-z-/γ7+z-1+α7+β7e-z-/γ7+z-+γ8α7+β7e-z-/γ7+z-+γ82β9e-α8+β8e-α7+β7e-z-/γ7+z-/γ8+α7+β7e-z-/γ7+z-+z-α8+β8e-α7+β7e-z-/γ7+z-/γ8+α7+β7e-z-/γ7+z-+γ9-1.After some manipulation, (84) reduces into the following form:(85)F′z=-β7γ7e-z-+β7ze-z-+α7+β7e-z-/γ7+z-2α8+β8e-α7+β7e-z-/γ7+z-1+α7+β7e-z-/γ7+z-+γ8β9e-α8+β8e-α7+β7e-z-/γ7+z-/γ8+α7+β7e-z-/γ7+z-+z-α7+β7e-z-/γ7+z-+γ8α8+β8e-α7+β7e-z-/γ7+z-+γ9γ8+α7+β7e-z-/γ7+z--1,≤-α7+β7e-L91+γ7+U9/γ7+L92α8+β8e-α7+β7e-U9/γ7+U91+α7+β7e-L9/γ7+L9+γ8β9eΓ2+U9γ8+α7+β7e-L9/γ7+L9α8+β8e-α7+β7e-L9/γ7+L9+γ9α7+β7e-U9/γ7+U9+γ8-1,where Γ2=-α8+β8e-α7+β7e-L9/γ7+L9/γ8+α7+β7e-L9/γ7+L9. Now assuming that (73) along with (74) hold then from (85) one gets F′z<0. Hence F(z)=0 has a unique positive solution in z-∈[L9,U9].


### 3.4. Local and Global Asymptotic Stability

Theorem 10.
For local stability about(x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8), the following statements hold:
(i) Equilibrium(x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is locally asymptotically stable if(86)
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2
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<
1
,(ii) Equilibrium (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of (8) is unstable if(87)
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+
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+
1
γ
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γ
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+
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7
γ
9
+
U
8
β
7
β
8
β
9
e
-
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7
-
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8
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9
+
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7
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>
1
.Proof.
If (x-,y-,z-) is equilibrium point (8), then(88)x-=α7+β7e-z-γ7+z-,y-=α8+β8e-x-γ8+x-,z=α6+β6e-y-γ6+y-.To construct the corresponding linearized form (8), we consider the following transformation:(89)xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1,where(90)f=α7+β7e-znγ7+zn-1,f1=xn,g=α8+β8e-xnγ8+xn-1,g1=yn,h=α9+β9e-ynγ9+yn-1,h1=zn.The Jacobian matrix about (x-,y-,z-) under the transformation (89) is given by (91)FJx-,y-,z-=0000A1A2100000B1B2000000100000C1C200000010,where (92)A1=-β7e-z-γ7+z-,A2=-α7+β7e-z-γ7+z-2,B1=-β8e-x-γ8+x-,B2=-α8+β8e-x-γ8+x-2,C1=-β9e-y-γ9+y-,C2=-α9+β9e-y-γ9+y-2.The characteristic equation of FJx-,y-,z- about (x-,y-,z-) is given by(93)λ6+μ1λ3+μ2λ2+μ3λ+μ4=0,where(94)μ1=-A1B1C1,μ2=-A1B1C2+A1B2C1+A2B1C1,μ3=-A1B2C2+A2B1C2+A2B2C1,μ4=-A2B2C2.Now compute ∑i=14μi as follows:(95)
∑
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<
1
.Assuming that condition (86) holds, then from (95) one gets ∑i=14μi<1 and hence by Theorem 1.5 of [15] equilibrium (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is locally asymptotically stable.
Proof (ii). Using similar calculation one can prove that (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is unstable.Hereafter we will prove that(x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is globally asymptotically stable.Theorem 11.
Equilibrium(x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is globally asymptotically stable if(96)α7+β7e-L9<x-γ7+L9,α8+β8e-L7<y-γ8+L7,α9+β9e-L8<z-γ9+L8.Proof.
Using similar arrangements as for the proof of Theorem5 and from (48) we obtain(97)Ωn+1-Ωn=x-xn+1x--1-ln⁡xn+1x-+y-yn+1y--1-ln⁡yn+1y-+z-zn+1z--1-ln⁡zn+1z--x-xnx--1-ln⁡xnx--y-yny--1-ln⁡yny--z-znz--1-ln⁡znz-,Ωn+1-Ωn≤xn+1-xn+yn+1-yn+zn+1-zn-x-ln⁡xn+1xn-y-ln⁡yn+1yn-z-ln⁡zn+1zn,Ωn+1-Ωn≤xn+1-xn+yn+1-yn+zn+1-zn-x-xn+1-xnxn+1-y-yn+1-ynyn+1-z-zn+1-znzn+1,=xn+1-xn1-x-xn+1+yn+1-yn1-y-yn+1+zn+1-zn1-z-zn+1,=xn+1-xn1-x-γ7+zn-1α7+β7e-zn+yn+1-yn1-y-γ8+xn-1α8+β8e-xn+zn+1-zn1-z-γ9+yn-1α9+β9e-yn,≤U7-L7α7+β7e-L9-x-γ7+L9α7+β7e-L9+U8-L8α8+β8e-L7-y-γ8+L7α8+β8e-L7+U9-L9α9+β9e-L8-z-γ9+L8α9+β9e-L8≤0,for all n ≥0. In view of (96) it follows that Ωn≥0. Thus we obtain that limn→∞Ωn+1-Ωn=0 and hence (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is globally asymptotically stable.


### 3.5. Rate of Convergence

Theorem 12.
Ifxn,yn,znn=-1∞is a positive solution of system (8) such that (51) along with the following holds:(98)x-∈L7,U7,y-∈L8,U8,z-∈L9,U9,then the error vector en=(en7,en-17,en8,en-18,en9,en-19)Tof every solution of (8) satisfies asymptotic relations (53), where λ1,2,3,4,5,6FJx-,y-,z- are the characteristic root of Jacobian matrix FJx-,y-,z-.Proof.
Ifxn,yn,znn=-1∞ is a positive solution of (8) such that (51) along with (98) holds, then in order to find the error terms, one has from (8)(99)xn+1-x-=α7+β7e-znγ7+zn-1-α7+β7e-z-γ7+z-,=β7e-zn-e-z-γ7+zn-1zn-z-zn-z--α7+β7e-z-γ7+zn-1γ7+z-zn-1-z-,yn+1-y-=α8+β8e-xnγ8+xn-1-α8+β8e-x-γ8+x-,=β8e-xn-e-x-γ8+xn-1xn-x-xn-x--α8+β8e-x-γ8+xn-1γ8+x-xn-1-x-,zn+1-z-=α9+β9e-ynγ9+yn-1-α9+β9e-y-γ9+y-,=β9e-yn-e-y-γ9+yn-1yn-y-yn-y--α9+β9e-y-γ9+yn-1γ9+y-yn-1-y-; that is,(100)xn+1-x-=β7e-zn-e-z-γ7+zn-1zn-z-zn-z--α7+β7e-z-γ7+zn-1γ7+z-zn-1-z-,yn+1-y-=β8e-xn-e-x-γ8+xn-1xn-x-xn-x--α8+β8e-x-γ8+xn-1γ8+x-xn-1-x-,zn+1-z-=β9e-yn-e-y-γ9+yn-1yn-y-yn-y--α9+β9e-y-γ9+yn-1γ9+y-yn-1-y-.Set(101)en7=xn-x-,en8=yn-y-,en9=zn-z-.In view of (101), (100) take the following form:(102)en+17=anen9+bnen-19,en+18=cnen7+dnen-17,en+19=enen8+fnen-18,where(103)an=β7e-zn-e-z-γ7+zn-1zn-z-,bn=-α7+β7e-z-γ7+zn-1γ7+z-,cn=β8e-xn-e-x-γ8+xn-1xn-x-,dn=-α8+β8e-x-γ8+xn-1γ8+x-,en=β9e-yn-e-y-γ9+yn-1yn-y-,fn=-α9+β9e-y-γ9+yn-1γ9+y-.Taking the limits one gets(104)limn→∞an=-β7e-z-γ7+z-,limn→∞bn=-α7+β7e-z-γ7+z-2,limn→∞cn=-β8e-x-γ8+x-,limn→∞dn=-α8+β8e-x-γ8+x-2,limn→∞en=-β9e-y-γ9+y-,limn→∞fn=-α9+β9e-y-γ9+y-2.That is,(105)an=-β7e-z-γ7+z-+α9n,bn=-α7+β7e-z-γ7+z-2+α9n-1,cn=-β8e-x-γ8+x-+α7n,dn=-α8+β8e-x-γ8+x-2+α7n-1,en=-β9e-y-γ9+y-+α8n,fn=-α9+β9e-y-γ9+y-2+α8n-1.where α7n,α7n-1,α8n,α8n-1,α9n,α9n-1→0 as n→∞. Now we have system 1.10 of [18], where (106)A=0000-β7e-z-γ7+z--α7+β7e-z-γ7+z-2100000-β8e-x-γ8+x--α8+β8e-x-γ8+x-2000000100000-β9e-y-γ9+y--α9+β9e-y-γ9+y-200000010,Bn=0000α9nα9n-1100000α7nα7n-1000000100000α8nα8n-100000010,and Bn→0,n→∞. Thus the limiting system of error terms about (x-,y-,z-) can be written as(107)en+17en7en+18en8en+19en9=0000-β7e-z-γ7+z--α7+β7e-z-γ7+z-2100000-β8e-x-γ8+x--α8+β8e-x-γ8+x-2000000100000-β9e-y-γ9+y--α9+β9e-y-γ9+y-200000010en7en-17en8en-18en9en-19,which is similar to linearized system (8) about equilibrium (x-,y-,z-).


## 3.1. Boundedness and Persistence

Theorem 7.
Every positive solution{(xn,yn,zn)}n=-1∞ of system (8) is bounded and persists.Proof.
If{(xn,yn,zn)}n=-1∞is an arbitrary solution of (8), then(64)xn≤α7+β7γ7=U7,yn≤α8+β8γ8=U8,zn≤α9+β9γ9=U9,n=0,1,⋯.Additionally from (8) and (64), one gets(65)xn≥α7+β7e-α9+β9/γ9γ7+α9+β9/γ9=L7,yn≥α8+β8e-α7+β7/γ7γ8+α7+β7/γ7=L8,zn≥α9+β9e-α8+β8/γ8γ9+α8+β8/γ8=L9,n=2,3,⋯.Finally from (64) and (65), one gets(66)L7≤xn≤U7,L8≤yn≤U8,L9≤zn≤U9,n=3,4,⋯.


## 3.2. Existence of Invariant Set for Solutions

Theorem 8.
If{(xn,yn,zn)}n=-1∞ is a positive solution of system (8), then [L7,U7]×[L8,U8]×[L9,U9] is invariant set for system (8), where L7=α7+β7e-α9+β9/γ9/γ7+α9+β9/γ9, U7=α7+β7/γ7, L8=α8+β8e-α7+β7/γ7/γ8+α7+β7/γ7, U8=α8+β8/γ8, L9=α9+β9e-α8+β8/γ8/γ9+α8+β8/γ8 and U9=α9+β9/γ9.Proof.
For any positive solution{(xn,yn,zn)}n=-1∞ of (8) with initial conditions x0,x-1∈[L7,U7],y0,y-1∈[L8,U8]andz0,z-1∈L9,U9. We have from system (8)(67)x1=α7+β7e-z0γ7+z-1≤α7+β7γ7,and(68)x1=α7+β7e-z0γ7+z-1≥α7+β7e-α9+β9/γ9γ7+α9+β9/γ9.Also(69)y1=α8+β8e-x0γ8+x-1≤α8+β8γ8,and (70)y1=α8+β8e-x0γ8+x-1≥α8+β8e-α7+β7/γ7γ8+α7+β7/γ7.Finally(71)z1=α9+β9e-y0γ9+y-1≤α9+β9γ9,and(72)z1=α9+β9e-y0γ9+y-1≥α9+β9e-α8+β8/γ8γ9+α8+β8/γ8.Hence x1∈[L7,U7],y1∈[L8,U8]andz1∈L9,U9. Similarly one can show that if xk∈[L7,U7],yk∈[L8,U8]andzk∈L9,U9, then xk+1∈[L7,U7],yk+1∈[L8,U8]andzk+1∈L9,U9.


## 3.3. Existence and Uniqueness of Positive Equilibrium Point

Theorem 9.
If(73)-α7+β7e-L91+γ7+U9/γ7+L92α8+β8e-α7+β7e-U9/γ7+U91+α7+β7e-L9/γ7+L9+γ8β9eΓ2+U9γ8+α7+β7e-L9/γ7+L9α8+β8e-α7+β7e-L9/γ7+L9+γ8α7+β7e-U9/γ7+U9+γ8<1,where (74)Γ2=-α8+β8e-α7+β7e-L9/γ7+L9γ8+α7+β7e-L9/γ7+L9,then system (8) has a unique positive equilibrium point (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9].Proof.
Consider(75)x=α7+β7e-zγ7+z,y=α8+β8e-xγ8+x,z=α9+β9e-yγ9+y.From (75) set (76)y=gx=α8+β8e-xγ8+x,and(77)x=fz=α7+β7e-zγ7+z.Denote (78)Fz=α9+β9e-kzγ9+kz-z,where(79)kz=y=gfz,=gα7+β7e-zγ7+z,=α8+β8e-α7+β7e-z/γ7+zγ8+α7+β7e-z/γ7+z,and z∈L9,U9. We claim that F(z)=0 has a unique solution in [L9,U9]. From (78) and (79) one gets(80)F′z=-k′zβ9e-kzγ9+kz+α9+β9e-kzγ9+kz2-1.And(81)k′z=β7γ7e-z+β7ze-z+α7+β7e-z/γ7+z2α8+β8e-α7+β7e-z/γ7+z1+α7+β7e-z/γ7+z+γ8α7+β7e-z/γ7+z+γ82.Now if z-∈[L9,U9] is a solution of Fz=0, then from (78) and (79) one gets (82)α9+β9e-kz-=z-γ9+kz-,where (83)kz-=α8+β8e-α7+β7e-z-/γ7+z-γ8+α7+β7e-z-/γ7+z-.In view of equation (81), (82), and (83), (80) becomes (84)F′z=-β7γ7e-z-+β7ze-z-+α7+β7e-z-/γ7+z-2α8+β8e-α7+β7e-z-/γ7+z-1+α7+β7e-z-/γ7+z-+γ8α7+β7e-z-/γ7+z-+γ82β9e-α8+β8e-α7+β7e-z-/γ7+z-/γ8+α7+β7e-z-/γ7+z-+z-α8+β8e-α7+β7e-z-/γ7+z-/γ8+α7+β7e-z-/γ7+z-+γ9-1.After some manipulation, (84) reduces into the following form:(85)F′z=-β7γ7e-z-+β7ze-z-+α7+β7e-z-/γ7+z-2α8+β8e-α7+β7e-z-/γ7+z-1+α7+β7e-z-/γ7+z-+γ8β9e-α8+β8e-α7+β7e-z-/γ7+z-/γ8+α7+β7e-z-/γ7+z-+z-α7+β7e-z-/γ7+z-+γ8α8+β8e-α7+β7e-z-/γ7+z-+γ9γ8+α7+β7e-z-/γ7+z--1,≤-α7+β7e-L91+γ7+U9/γ7+L92α8+β8e-α7+β7e-U9/γ7+U91+α7+β7e-L9/γ7+L9+γ8β9eΓ2+U9γ8+α7+β7e-L9/γ7+L9α8+β8e-α7+β7e-L9/γ7+L9+γ9α7+β7e-U9/γ7+U9+γ8-1,where Γ2=-α8+β8e-α7+β7e-L9/γ7+L9/γ8+α7+β7e-L9/γ7+L9. Now assuming that (73) along with (74) hold then from (85) one gets F′z<0. Hence F(z)=0 has a unique positive solution in z-∈[L9,U9].


## 3.4. Local and Global Asymptotic Stability

Theorem 10.
For local stability about(x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8), the following statements hold:
(i) Equilibrium(x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is locally asymptotically stable if(86)
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,(ii) Equilibrium (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of (8) is unstable if(87)
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.Proof.
If (x-,y-,z-) is equilibrium point (8), then(88)x-=α7+β7e-z-γ7+z-,y-=α8+β8e-x-γ8+x-,z=α6+β6e-y-γ6+y-.To construct the corresponding linearized form (8), we consider the following transformation:(89)xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1,where(90)f=α7+β7e-znγ7+zn-1,f1=xn,g=α8+β8e-xnγ8+xn-1,g1=yn,h=α9+β9e-ynγ9+yn-1,h1=zn.The Jacobian matrix about (x-,y-,z-) under the transformation (89) is given by (91)FJx-,y-,z-=0000A1A2100000B1B2000000100000C1C200000010,where (92)A1=-β7e-z-γ7+z-,A2=-α7+β7e-z-γ7+z-2,B1=-β8e-x-γ8+x-,B2=-α8+β8e-x-γ8+x-2,C1=-β9e-y-γ9+y-,C2=-α9+β9e-y-γ9+y-2.The characteristic equation of FJx-,y-,z- about (x-,y-,z-) is given by(93)λ6+μ1λ3+μ2λ2+μ3λ+μ4=0,where(94)μ1=-A1B1C1,μ2=-A1B1C2+A1B2C1+A2B1C1,μ3=-A1B2C2+A2B1C2+A2B2C1,μ4=-A2B2C2.Now compute ∑i=14μi as follows:(95)
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.Assuming that condition (86) holds, then from (95) one gets ∑i=14μi<1 and hence by Theorem 1.5 of [15] equilibrium (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is locally asymptotically stable.
Proof (ii). Using similar calculation one can prove that (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is unstable.Hereafter we will prove that(x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is globally asymptotically stable.Theorem 11.
Equilibrium(x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is globally asymptotically stable if(96)α7+β7e-L9<x-γ7+L9,α8+β8e-L7<y-γ8+L7,α9+β9e-L8<z-γ9+L8.Proof.
Using similar arrangements as for the proof of Theorem5 and from (48) we obtain(97)Ωn+1-Ωn=x-xn+1x--1-ln⁡xn+1x-+y-yn+1y--1-ln⁡yn+1y-+z-zn+1z--1-ln⁡zn+1z--x-xnx--1-ln⁡xnx--y-yny--1-ln⁡yny--z-znz--1-ln⁡znz-,Ωn+1-Ωn≤xn+1-xn+yn+1-yn+zn+1-zn-x-ln⁡xn+1xn-y-ln⁡yn+1yn-z-ln⁡zn+1zn,Ωn+1-Ωn≤xn+1-xn+yn+1-yn+zn+1-zn-x-xn+1-xnxn+1-y-yn+1-ynyn+1-z-zn+1-znzn+1,=xn+1-xn1-x-xn+1+yn+1-yn1-y-yn+1+zn+1-zn1-z-zn+1,=xn+1-xn1-x-γ7+zn-1α7+β7e-zn+yn+1-yn1-y-γ8+xn-1α8+β8e-xn+zn+1-zn1-z-γ9+yn-1α9+β9e-yn,≤U7-L7α7+β7e-L9-x-γ7+L9α7+β7e-L9+U8-L8α8+β8e-L7-y-γ8+L7α8+β8e-L7+U9-L9α9+β9e-L8-z-γ9+L8α9+β9e-L8≤0,for all n ≥0. In view of (96) it follows that Ωn≥0. Thus we obtain that limn→∞Ωn+1-Ωn=0 and hence (x-,y-,z-)∈[L7,U7]×[L8,U8]×[L9,U9] of system (8) is globally asymptotically stable.


## 3.5. Rate of Convergence

Theorem 12.
Ifxn,yn,znn=-1∞is a positive solution of system (8) such that (51) along with the following holds:(98)x-∈L7,U7,y-∈L8,U8,z-∈L9,U9,then the error vector en=(en7,en-17,en8,en-18,en9,en-19)Tof every solution of (8) satisfies asymptotic relations (53), where λ1,2,3,4,5,6FJx-,y-,z- are the characteristic root of Jacobian matrix FJx-,y-,z-.Proof.
Ifxn,yn,znn=-1∞ is a positive solution of (8) such that (51) along with (98) holds, then in order to find the error terms, one has from (8)(99)xn+1-x-=α7+β7e-znγ7+zn-1-α7+β7e-z-γ7+z-,=β7e-zn-e-z-γ7+zn-1zn-z-zn-z--α7+β7e-z-γ7+zn-1γ7+z-zn-1-z-,yn+1-y-=α8+β8e-xnγ8+xn-1-α8+β8e-x-γ8+x-,=β8e-xn-e-x-γ8+xn-1xn-x-xn-x--α8+β8e-x-γ8+xn-1γ8+x-xn-1-x-,zn+1-z-=α9+β9e-ynγ9+yn-1-α9+β9e-y-γ9+y-,=β9e-yn-e-y-γ9+yn-1yn-y-yn-y--α9+β9e-y-γ9+yn-1γ9+y-yn-1-y-; that is,(100)xn+1-x-=β7e-zn-e-z-γ7+zn-1zn-z-zn-z--α7+β7e-z-γ7+zn-1γ7+z-zn-1-z-,yn+1-y-=β8e-xn-e-x-γ8+xn-1xn-x-xn-x--α8+β8e-x-γ8+xn-1γ8+x-xn-1-x-,zn+1-z-=β9e-yn-e-y-γ9+yn-1yn-y-yn-y--α9+β9e-y-γ9+yn-1γ9+y-yn-1-y-.Set(101)en7=xn-x-,en8=yn-y-,en9=zn-z-.In view of (101), (100) take the following form:(102)en+17=anen9+bnen-19,en+18=cnen7+dnen-17,en+19=enen8+fnen-18,where(103)an=β7e-zn-e-z-γ7+zn-1zn-z-,bn=-α7+β7e-z-γ7+zn-1γ7+z-,cn=β8e-xn-e-x-γ8+xn-1xn-x-,dn=-α8+β8e-x-γ8+xn-1γ8+x-,en=β9e-yn-e-y-γ9+yn-1yn-y-,fn=-α9+β9e-y-γ9+yn-1γ9+y-.Taking the limits one gets(104)limn→∞an=-β7e-z-γ7+z-,limn→∞bn=-α7+β7e-z-γ7+z-2,limn→∞cn=-β8e-x-γ8+x-,limn→∞dn=-α8+β8e-x-γ8+x-2,limn→∞en=-β9e-y-γ9+y-,limn→∞fn=-α9+β9e-y-γ9+y-2.That is,(105)an=-β7e-z-γ7+z-+α9n,bn=-α7+β7e-z-γ7+z-2+α9n-1,cn=-β8e-x-γ8+x-+α7n,dn=-α8+β8e-x-γ8+x-2+α7n-1,en=-β9e-y-γ9+y-+α8n,fn=-α9+β9e-y-γ9+y-2+α8n-1.where α7n,α7n-1,α8n,α8n-1,α9n,α9n-1→0 as n→∞. Now we have system 1.10 of [18], where (106)A=0000-β7e-z-γ7+z--α7+β7e-z-γ7+z-2100000-β8e-x-γ8+x--α8+β8e-x-γ8+x-2000000100000-β9e-y-γ9+y--α9+β9e-y-γ9+y-200000010,Bn=0000α9nα9n-1100000α7nα7n-1000000100000α8nα8n-100000010,and Bn→0,n→∞. Thus the limiting system of error terms about (x-,y-,z-) can be written as(107)en+17en7en+18en8en+19en9=0000-β7e-z-γ7+z--α7+β7e-z-γ7+z-2100000-β8e-x-γ8+x--α8+β8e-x-γ8+x-2000000100000-β9e-y-γ9+y--α9+β9e-y-γ9+y-200000010en7en-17en8en-18en9en-19,which is similar to linearized system (8) about equilibrium (x-,y-,z-).


## 4. On the System:xn+1=α10+β10e-xn/γ10+xn-1, yn+1=α11+β11e-yn/γ11+zn-1, zn+1=α12+β12e-zn/γ12+yn-1

In this we will investigate the dynamics of system (9). If (x-,y-,z-) is equilibrium point of system (9), then(108)x-=α10+β10e-x-γ10+x-,y-=α11+β11e-y-γ11+z-,z-=α12+β12e-z-γ12+y-.To construct the corresponding linearized form of system (9), we consider the following transformation:(109)xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1,where(110)f=α10+β10e-xnγ10+xn-1,f1=xn,g=α11+β11e-ynγ11+zn-1,g1=yn,h=α12+β12e-znγ12+yn-1,h1=zn.The Jacobian matrixFJx-,y-,z- about (x-,y-,z-) under the transformation xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1 is given by(111)FJx-,y-,z-=A1A2000010000000B100B2001000000C1C20000010,where(112)A1=-β10e-x-γ10+x-,A2=-α10+β10e-x-γ10+x-2,B1=-β11e-y-γ11+z-,B2=-α11+β11e-y-γ11+z-2,C1=-α12+β12e-z-γ12+y-2,C2=-β12e-z-γ12+y-.Theorem 13.
Every positive solutionxn,yn,znn=-1∞of system (9) is bounded and persists.Proof.
Ifxn,yn,znn=-1∞is an arbitrary solution of (9), then(113)xn≤α10+β10γ10=U10,yn≤α11+β11γ11=U11,zn≤α12+β12γ12=U12,n=0,1,⋯.Now from (9) and (113) we have (114)xn≥α10+β10e-α10+β10/γ10γ10+α10+β10/γ10=L10,yn≥α11+β11e-α11+β11/γ11γ11+α12+β12/γ12=L11,zn≥α12+β12e-α12+β12/γ12γ12+α11+β11/γ11=L12,n=2,3⋯.Hence from (113) and (114) one gets (115)L10≤xn≤U10,L11≤yn≤U11,L12≤zn≤U12,n=3,4,⋯.Theorem 14.
Ifxn,yn,znn=-1∞ is a positive solution of system (9) then [L10,U10]×[L11,U11]×[L12,U12] is invariant set for system (9) where L10=α10+β10e-α10+β10/γ10/γ10+α10+β10/γ10, U10=α10+β10/γ10, L11=α11+β11e-α11+β11/γ11/γ11+α12+β12/γ12, U11=α11+β11/γ11, L12=α12+β12e-α12+β12/γ12/γ12+α11+β11/γ11 and U12=α12+β12/γ12.Proof.
Follows by induction.Note. In remaining section, we will only state other results for system (9) and their proofs are similar as we have done in Sections 2 and 3.Theorem 15.
If(116)α11+β11e-L111+U11/L112α12+β12e-α11+β11e-U11/U11-γ111+α11+β11e-L11/L11-γ11β11eΓ3+γ11+U12γ11-α11+β11e-L11/L11α12+β12e-α11+β11e-L11/L11-γ11-γ12α11+β11e-U11/U11-γ11<1,where(117)Γ3=-α12+β12e-α11+β11e-L11/L11-γ11α11+β11e-L11/γ11-γ11-γ12,then system (9) has a unique positive equilibrium point (x-,y-,z-)∈[L10,U10]×[L11,U11]×[L12,U12].Theorem 16.
For local stability about(x-,y-,z-)∈[L10,U10]×[L11,U11]×[L12,U12] of system (9), the following statements hold:
(i) Equilibrium(x-,y-,z-)∈[L10,U10]×[L11,U11]×[L12,U12] of system (9) is locally asymptotically stable if(118)β10e-L10γ10+L101+β11e-L11γ11+L12+β12e-L12γ12+L11+β11e-L11γ11+L121+α10+β10e-L10γ10+L102+β12e-L12γ12+L11+β12e-L12γ12+L111+α10+β10e-L10γ10+L102+α10+β10e-L10γ10+L1021+β11β12e-L11-L12γ12+L12γ12+L11+α11+β11e-L11α12+β12e-L12γ11+L122γ12+L1121+β10e-L10γ10+L10+1γ10+L11γ11+L12γ12+L11β10β11β12e-L10-L11-L12+α10+β10e-L10α11+β11e-L11α12+β12e-L12γ10+L10γ11+L12γ12+L11<1,(ii) Equilibrium (x-,y-,z-)∈[L10,U10]×[L11,U11]×[L12,U12] of system (9) is unstable if (119)β10e-U10γ10+U101+β11e-U11γ11+U12+β12e-U12γ12+U11+β11e-U11γ11+U121+α10+β10e-U10γ10+U102+β12e-U12γ12+U11+β12e-U12γ12+U111+α10+β10e-U10γ10+U102+α10+β10e-U10γ10+U1021+β11β12e-U11-U12γ12+U12γ12+U11+α11+β11e-U11α12+β12e-U12γ11+U122γ12+U1121+β10e-U10γ10+U10+1γ10+U11γ11+U12γ12+U11β10β11β12e-U10-U11-U12+α10+β10e-U10α11+β11e-U11α12+β12e-U12γ10+U10γ11+U12γ12+U11>1.Theorem 17.
Equilibrium(x-,y-,z-)∈[L10,U10]×[L11,U11]×[L12,U12] of system (9) is globally asymptotically stable if(120)α10+β10e-L10<x-γ10+L10,α11+β11e-L11<y-γ11+L12,α12+β12e-L12<z-γ12+L11.Theorem 18.
Ifxn,yn,znn=-1∞is a positive solution of system (9) such that (51) along with the following holds(121)x-∈L10,U10,y-∈L11,U11,z-∈L12,U12,then the error vector en=en10,en-110,en11,en-111,en12,en-112T of every solution of (9) satisfies asymptotic relations (53), where λ1,2,3,4,5,6FJx-,y-,z- are the characteristic root of FJx-,y-,z-.Proof.
Ifxn,yn,znn=-1∞is a positive solution of system (9) such that (51) along with (121) holds then in order to find the error terms, one has from (9).(122)xn+1-x-=α10+β10e-xnγ10+xn-1-α10+β10e-x-γ10+x-,=β10e-xn-e-x-γ10+xn-1xn-x-xn-x--α10+β10e-x-γ10+xn-1γ10+x-xn-1-x-,yn+1-y-=α11+β11e-ynγ11+zn-1-α11+β11e-y-γ11+z-,=β11e-yn-e-y-γ11+zn-1yn-y-yn-y--α11+β11e-y-γ11+zn-1γ11+z-zn-1-z-,zn+1-z-=α12+β12e-znγ12+yn-1-α12+β12e-z-γ12+y-,=β12e-zn-e-z-γ12+yn-1zn-z-zn-z--α12+β12e-z-γ12+yn-1γ12+y-yn-1-y-;that is,(123)xn+1-x-=β10e-xn-e-x-γ10+xn-1xn-x-xn-x--α10+β10e-x-γ10+xn-1γ10+x-xn-1-x-,yn+1-y-=β11e-yn-e-y-γ11+zn-1yn-y-yn-y--α11+β11e-y-γ11+zn-1γ11+z-zn-1-z-,zn+1-z-=β12e-zn-e-z-γ12+yn-1zn-z-zn-z--α12+β12e-z-γ12+yn-1γ12+y-yn-1-y-.Set (124)en10=xn-x-,en11=yn-y-,en12=zn-z-.Using (124), (123) become (125)en+110=anen10+bnen-110,en+111=cnen11+dnen-112,en+112=enen-111+fnen12,where(126)an=β10e-xn-e-x-γ10+xn-1xn-x-,bn=-α10+β10e-x-γ10+xn-1γ10+x-,cn=β11e-yn-e-y-γ11+zn-1yn-y-,dn=-α11+β11e-y-γ11+zn-1γ11+z-,en=-α12+β12e-z-γ12+yn-1γ12+y-,fn=β12e-zn-e-z-γ12+yn-1zn-z-.Taking the limits one gets(127)limn→∞an=-β10e-x-γ10+x-,limn→∞bn=-α10+β10e-x-γ10+x-2,limn→∞cn=-β11e-y-γ11+z-,limn→∞dn=-α11+β11e-y-γ11+z-2,limn→∞en=-α12+β12e-z-γ12+y-2,limn→∞fn=-β12e-z-γ12+y-;that is,(128)an=-β10e-x-γ10+x-+α10n,bn=-α10+β10e-x-γ10+x-2+α10n-1,cn=-β11e-y-γ11+z-+α11n,dn=-α11+β11e-y-γ11+z-2+α12n-1,en=-α12+β12e-z-γ12+y-2+α11n-1,fn=-β12e-z-γ12+y-+α12n,where α10n,α10n-1,α11n,α11n-1,α12n,α12n-1→0 as n→∞. Now we have system 1.10 of [18], where(129)A=-β10e-x-γ10+x--α10+β10e-x-γ10+x-2000010000000-β11e-y-γ11+z-00-α11+β11e-y-γ11+z-2001000000-α12+β12e-z-γ12+y-2-β12e-z-γ12+y-0000010,Bn=α10nα10n-1000010000000α11n00α12n-1001000000α11n-1α12n0000010,and Bn→0,n→∞. Thus the limiting system of error terms about (x-,y-,z-) is(130)en+110en10en+111en11en+112en12=-β10e-x-γ10+x--α10+β10e-x-γ10+x-2000010000000-β11e-y-γ11+z-00-α11+β11e-y-γ11+z-2001000000-α12+β12e-z-γ12+y-2-β12e-z-γ12+y-0000010en10en-110en11en-111en12en-112,which is similar to linearized system (9) about equilibrium (x-,y-,z-).


## 5. On the System:xn+1=α13+β13e-xn/γ13+zn-1, yn+1=α14+β14e-yn/γ14+yn-1, zn+1=α15+β15e-zn/γ15+xn-1

This section deals with the study of main results about system (10). If (x-,y-,z-) is equilibrium point of system (10), then(131)x=α13+β13e-x-γ13+z-,y=α14+β14e-y-γ14+y-,z=α15+β15e-z-γ15+x-.To construct the corresponding linearized form of system (10), we consider the following transformation: (132)xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1,where(133)f=α13+β13e-xnγ13+zn-1,f1=xn,g=α14+β14e-ynγ14+yn-1,g1=yn,h=α15+β15e-znγ15+xn-1,h1=zn.The Jacobian matrix FJx-,y-,z- about (x-,y-,z-) under the transformation xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1 is given by(134)FJx-,y-,z-=A10000A210000000B1B2000010000C100C20000010,where(135)A1=-β13e-x-γ13+z-,A2=-α13+β13e-x-γ13+z-2,B1=-β14e-y-γ14+y-,B2=-α14+β14e-y-γ14+y-2,C1=-α15+β15e-z-γ15+x-2,C2=-β15e-z-γ15+x-.Theorem 19.
Every positive solutionxn,yn,znn=-1∞ of system (10) is bounded and persists.Proof.
Ifxn,yn,znn=-1∞ is an arbitrary solution of (10) then(136)xn≤α13+β13γ13=U13,yn≤α14+β14γ14=U14,zn≤α15+β15γ15=U15,n=0,1,⋯.From (10) and (136) one gets(137)xn≥α13+β13e-α13+β13/γ13γ13+α15+β15/γ15=L13,yn≥α14+β14e-α14+β14/γ14γ14+α14+β14/γ14=L14,zn≥α15+β15e-α15+β15/γ15γ15+α13+β13/γ13=L15,n=2,3,⋯.Lastly from (136) and (137) one gets(138)L13≤xn≤U13,L14≤yn≤U14,L15≤zn≤U15,n=3,4,⋯.Theorem 20.
Ifxn,yn,zn is a positive solution of system (10), then [L13,U13]×[L14,U14]×[L15,U15] is invariant set for system (10), where L13=α13+β13e-α13+β13/γ13/γ13+α15+β15/γ15, U13=α13+β13/γ13, L14=α14+β14e-α14+β14/γ14/γ14+α14+β14/γ14,U14=α14+β14/γ14, L15=α15+β15e-α15+β15/γ15/γ15+α13+β13/γ13 and U15=α15+β15/γ15.Remark 21.
In the following we will state the results for the existence and uniqueness of positive equilibrium, condition under which the unique positive equilibrium is locally asymptotically stable, unstable, and globally asymptotically stable for the system (10).Theorem 22.
If(139)α13+β13e-L131+U13/L132α15+β15e-α13+β13e-U13/U13-γ131+α13+β13e-L13/L13-γ13β13eΓ4+γ13+U15γ13-α13+β13e-L13/L13α15+β15e-α13+β13e-L13/L13-γ13-γ15α13+β13e-U13/U13-γ13<1,where(140)Γ4=-α15+β15e-α13+β13e-L13/L13-γ13α13+β13e-L13/L13-γ13-γ15,then system (10) has a unique positive equilibrium point (x-,y-,z-)∈L13,U13]×[L14,U14]×[L15,U15].Theorem 23.
For local stability about(x-,y-,z-)∈[L13,U13]×[L14,U14]×[L15,U15] of system (10), the following statements hold:
(i) Equilibrium(x-,y-,z-)∈[L13,U13]×[L14,U14]×[L15,U15] of system (10) is locally asymptotically stable if(141)β13e-L13γ13+L131+β14e-L14γ14+L14+β15e-L15γ15+L15+α14+β14e-L14γ14+L142+β14e-L14γ14+L141+β15e-L15γ15+L13+β15e-L15γ15+L131+α14+β14e-L14γ14+L142+α14+β14e-L14γ14+L1421+β13β15e-L13-L15γ15+L13γ13+L15+α13+β13e-L13α15+β15e-L15γ13+L152γ15+L1321+β14e-L14γ14+L14+1γ13+L15γ14+L14γ15+L13β13β14β15e-L13-L14-L15+α13+β13e-L13α14+β14e-L14α15+β15e-L15γ13+L15γ14+L14γ15+L13<1, (ii) Equilibrium (x-,y-,z-)∈[L13,U13]×[L14,U14]×[L15,U15] of system (10) is unstable if(142)β13e-U13γ13+U131+β14e-U14γ14+U14+β15e-U15γ15+U15+α14+β14e-U14γ14+U142+β14e-U14γ14+U141+β15e-U15γ15+U13+β15e-U15γ15+U131+α14+β14e-U14γ14+U142+α14+β14e-U14γ14+U1421+β13β15e-U13-U15γ15+U13γ13+U15+α13+β13e-U13α15+β15e-U15γ13+U152γ15+U1321+β14e-U14γ14+U14+1γ13+U15γ14+U14γ15+U13β13β14β15e-U13-U14-U15+α13+β13e-U13α14+β14e-U14α15+β15e-U15γ13+U15γ14+U14γ15+U13>1.Theorem 24.
Equilibrium(x-,y-,z-)∈[L13,U13]×[L14,U14]×[L15,U15] of system (10) is globally asymptotically stable if(143)α13+β13e-L13<x-γ13+L15,α14+β14e-L14<y-γ14+L14,α15+β15e-L15<z-γ15+L13.Theorem 25.
Ifxn,yn,znn=-1∞ is a positive solution of system (10) such that (51) along with the following relation holds:(144)x-∈L13,U13,y-∈L14,U14,z-∈L15,U15,then the error vector en=(en13,en-113,en14,en-114,en15,en-115)T of every solution of (10) satisfies both asymptotic relations defined in (53), where λ1,2,3,4,5,6FJx-,y-,z- are the characteristic root of FJx-,y-,z-.


## 6. On the Systemxn+1=α16+β16e-xn/γ16+yn-1, yn+1=α17+β17e-yn/γ17+xn-1, zn+1=α18+β18e-zn/γ18+zn-1

In this section, we will study the behavior of system (11). If (x-,y-,z-) is equilibrium point of system (11), then(145)x=α16+β16e-x-γ16+y-,y=α17+β17e-y-γ17+x-,z=α18+β18e-z-γ18+z-.To construct the corresponding linearized form of system (11), we consider the following transformation:(146)xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1,where(147)f=α16+β16e-xnγ16+y,f1=xn,g=α17+β17e-ynγ17+xn-1,g1=yn,h=α18+β18e-znγ18+zn-1,h1=zn.The Jacobian matrix FJx-,y-,z- about (x-,y-,z-) under xn,xn-1,yn,yn-1,zn,zn-1→f,f1,g,g1,h,h1 is given by (148)FJx-,y-,z-=A100A2001000000B1B20000010000000C1C2000010.where (149)A1=-β16e-x-γ16+y-,A2=-α16+β16e-x-γ16+y-2,B1=-α17+β17e-y-γ17+x-2,B2=-β17e-y-γ17+x-,C1=-β18e-z-γ18+z-,C2=-α18+β18e-z-γ18+z-2.Theorem 26.
Every positive solutionxn,yn,znn=-1∞ of system (11) is bounded and persists.Proof.
Ifxn,yn,znn=-1∞is an arbitrary solution of (11), then(150)xn≤α16+β16γ16=U16,yn≤α17+β17γ17=U17,zn≤α18+β18γ18=U18.Additionally from (11) and (150), one gets(151)xn≥α16+β16e-α16+β16/γ16γ16+α17+β17/γ17=L16,yn≥α17+β17e-α17+β17/γ17γ17+α16+β16/γ16=L17,zn≥α18+β18e-α18+β18/γ18γ18+α18+β18/γ18=L18,n=0,1,2,⋯.Therefore from (150) and (151), one gets(152)L16≤xn≤U16,L17≤yn≤U17,L18≤zn≤U18,n=3,4,⋯.Hereafter we will state the main results for system (11) and their proof is similar as we have done in Sections 2 and 3.Theorem 27.
Ifxn,yn,znn=-1∞ is a positive solution of system (11), then [L16,U16]×[L17,U17]×[L18,U18] is invariant set for system (11) where L16=α16+β16e-α16+β16/γ16/γ16+α17+β17/γ17, U16=α16+β16/γ16, L17=α17+β17e-α17+β17/γ17/γ17+α16+β16/γ16, U17=α17+β17/γ17, L18=α18+β18e-α18+β18/γ18/γ18+α18+β18/γ18 and U18=α18+β18/γ18.Theorem 28.
If(153)α17+β17e-L171+U17/L172α16+β16e-α17+β17e-U17/U17-γ171+α17+β17e-L17/L17-γ17β17eΓ5+γ17+U16γ17-α17+β17e-L17/L17α16+β16e-α17+β17e-L17/L17-γ17-γ16α17+β17e-U17/U17-γ17<1,where (154)Γ5=-α16+β16e-α17+β17e-L17/L17-γ17α17+β17e-L17/L17-γ17-γ16,then system (11) has a unique positive equilibrium point (x-,y-,z-)∈[L16,U16]×[L17,U17]×[L18,U18].Theorem 29.
For equilibrium(x-,y-,z-)∈[L16,U16]×[L17,U17]×[L18,U18] of system (11), the following statements hold:
(i) Equilibrium(x-,y-,z-)∈[L16,U16]×[L17,U17]×[L18,U18] of system (11) is locally asymptotically stable if(155)β16e-L16γ16+L171+β18e-L18γ18+L18+β17e-L17γ17+L16+α18+β18e-L18γ18+L182+β18e-L18γ18+L181+β17e-L17γ17+L16+β17e-L17γ17+L161+α18+β18e-L18γ18+L182+α18+β18e-L18γ18+L1821+β16β17e-L16-L17γ16+L17γ17+L16+α16+β16e-L16α17+β17e-L17γ17+L162γ16+L1721+β18e-L18γ18+L18+1γ16+L17γ17+L16γ18+L18β16β17β18e-L16-L17-L18+α16+β16e-L16α17+β17e-L17α18+β18e-L18γ16+L17γ17+L16γ18+L18<1, (ii) Equilibrium (x-,y-,z-)∈[L16,U16]×[L17,U17]×[L18,U18] of system (11) is unstable if (156)β16e-U16γ16+U171+β18e-U18γ18+U18+β17e-U17γ17+U16+α18+β18e-U18γ18+U182+β18e-U18γ18+U181+β17e-U17γ17+U16+β17e-U17γ17+U161+α18+β18e-U18γ18+U182+α18+β18e-U18γ18+U1821+β16β17e-U16-U17γ16+U17γ17+U16+α16+β16e-U16α17+β17e-U17γ17+U162γ16+U1721+β18e-U18γ18+U18+1γ16+U17γ17+U16γ18+U18β16β17β18e-U16-U17-U18+α16+β16e-U16α17+β17e-U17α18+β18e-U18γ16+U17γ17+U16γ18+U18>1.Theorem 30.
Equilibrium (x-,y-,z-)∈[L16,U16]×[L17,U17]×L18,U18 of system (11) is globally asymptotically stable if(157)α16+β16e-L16<x-γ16+L17,α17+β17e-L17<y-γ17+L16,α18+β18e-L18<z-γ18+L18.Theorem 31.
Ifxn,yn,znn=-1∞ is a positive solution of system (11) such that (51) holds along with the relations (158)x-∈L16,U16,y-∈L17,U17,z-∈L18,U18,then the error vector en=(en16,en-116,en17,en-117,en18,en-118)Tof every solution of (11) satisfies (53), where λ1,2,3,4,5,6FJx-,y-,z- are the characteristic root of FJx-,y-,z-.


## 7. Numerical Simulations

In this section, we will provide some numerical simulations to not only verify obtained theoretical results in Sections2–6 but also exhibit more interesting behavior of these systems. If α4=195,β4=0.4,γ4=7.0,α5=9,β5=0.5,γ5=0.0008,α6=9,β6=9,γ6=0.08 then system (7) with initial values x0=0.2,x-1=1.7,y0=1.4,y-1=0.9,z0=0.14,z-1=0.09 can be written as (159)xn+1=195+0.4e-yn7.0+yn-1,yn+1=9+0.5e-zn0.0008+zn-1,zn+1=9+9e-xn0.08+xn-1.From computations it is easy to verify that for above chosen values of parameters α4=195,β4=0.4,γ4=7.0,α5=9,β5=0.5,γ5=0.0008,α6=9,β6=9,γ6=0.08 the condition under which the unique positive equilibrium point of (159) is satisfied is as follows:(160)β4e-L4-L5γ4+L5γ6+L4γ5+L6β5α6+β6e-L4γ6+L4+β6α5+β5e-L6γ5+L6+β5e-L6α4+β4e-L5γ5+L6γ6+L4γ4+L5β6e-L4γ4+L5+α6+β6e-L4γ6+L4γ4+L5+α5+β5e-L6γ6+L4γ4+L5γ5+L62β4e-L5α6+β6e-L4γ6+L4+β6e-L4α4+β4e-L5γ4+L5+1γ4+L5γ6+L4γ5+L6β4β5β6e-L4-L5-L6+α4+β4e-L5α5+β5e-L6α6+β6e-L4γ4+L5γ6+L4γ5+L6=0.109087754918294<1.Moreover, in Figure 1, plot of xn is shown in Figure 1(a), plot of yn is shown in Figure 1(b), plot of zn is shown in Figure 1(c) and attractor of system (159) is shown in Figure 1(d). Now if α7=5,β7=414,γ7=7.0,α8=42,β8=295,γ8=0.8,α9=9,β9=29,γ9=0.8 then system (8) with initial values x0=0.2,x-1=0.7,y0=2.4,y-1=52.9,z0=0.4,z-1=0.9 can be written as(161)xn+1=5+414e-xn7.0+zn-1,yn+1=42+295e-xn0.8+xn-1,zn+1=9+29e-yn0.8+yn-1.For chosen values of parameters α7=5,β7=414,γ7=7.0,α8=42,β8=295,γ8=0.8,α9=9,β9=29,γ9=0.8 the condition under which the unique positive equilibrium point of (161) is satisfied, is as follows:(162)β7e-L9α9+β9e-L8γ7+L9γ8+L7γ9+L82β8e-L7+α8+β8e-L7γ8+L7+β9e-L8α8+β8e-L7γ7+L9γ9+L8γ8+L72β7e-L9+α7+β7e-L9γ7+L9+β8e-L7α7+β7e-L9γ8+L7γ9+L8γ7+L92β9e-L8+α9+β9e-L8γ9+L8+1γ7+L9γ8+L7γ9+L8β7β8β9e-L7-L8-L9+α7+β7e-L9α8+β8e-L7α9+β9e-L8γ7+L9γ8+L7γ9+L8=0.2357754309080341<1.Moreover, in Figure 2, plot of xn is shown in Figure 2(a), plot of yn is shown in Figure 2(b), plot of zn is shown in Figure 2(c) and attractor of system (161) is shown in Figure 2(d). For system (9), we choose if α10=15,β10=2,γ10=7.0,α11=2,β11=25,γ11=22,α12=9,β12=4329,γ12=0.8 with initial values x0=9.2,x-1=1.7,y0=7.4,y-1=2.9,z0=4.4,z-1=3.9 then(163)xn+1=15+2e-xn7.0+xn-1,yn+1=2+25e-xn22+zn-1,zn+1=9+4329e-zn0.8+yn-1.For these values α10=15,β10=2,γ10=7.0,α11=2,β11=25,γ11=22,α12=9,β12=4329,γ12=0.8 the condition under which the unique positive equilibrium point of (163) is satisfied is as follows:(164)β10e-L10γ10+L101+β11e-L11γ11+L12+β12e-L12γ12+L11+β11e-L11γ11+L121+α10+β10e-L10γ10+L102+β12e-L12γ12+L11+β12e-L12γ12+L111+α10+β10e-L10γ10+L102+α10+β10e-L10γ10+L1021+β11β12e-L11-L12γ12+L12γ12+L11+α11+β11e-L11α12+β12e-L12γ11+L122γ12+L1121+β10e-L10γ10+L10+1γ10+L11γ11+L12γ12+L11β10β11β12e-L10-L11-L12+α10+β10e-L10α11+β11e-L11α12+β12e-L12γ10+L10γ11+L12γ12+L11=0.03159295074271<1.Moreover, in Figure 3, plot of xn is shown in Figure 3(a), plot of yn is shown in Figure 3(b), plot of zn is shown in Figure 3(c) and attractor of system (163) is shown in Figure 3(d). Again if we choose α13=25,β13=94,γ13=788,α14=42,β14=26,γ14=5,α15=6,β15=59,γ15=37 with initial values x0=5.2,x-1=14.7,y0=23.4,y-1=2.9,z0=7.4,z-1=3.9, then system (10) can be written as(165)xn+1=25+94e-xn788+zn-1,yn+1=42+26e-yn5+yn-1,zn+1=6+59e-zn37+xn-1.For parametric values α13=25,β13=94,γ13=788,α14=42,β14=26,γ14=5,α15=6,β15=59,γ15=37 the condition under which the unique positive equilibrium point of (165) is satisfied is as follows:(166)β13e-L13γ13+L131+β14e-L14γ14+L14+β15e-L15γ15+L15+α14+β14e-L14γ14+L142+β14e-L14γ14+L141+β15e-L15γ15+L13+β15e-L15γ15+L131+α14+β14e-L14γ14+L142+α14+β14e-L14γ14+L1421+β13β15e-L13-L15γ15+L13γ13+L15+α13+β13e-L13α15+β15e-L15γ13+L152γ15+L1321+β14e-L14γ14+L14+1γ13+L15γ14+L14γ15+L13β13β14β15e-L13-L14-L15+α13+β13e-L13α14+β14e-L14α15+β15e-L15γ13+L15γ14+L14γ15+L13=0.60229445538307<1.Moreover, in Figure 4, plot of xn is shown in Figure 4(a), plot of yn is shown in Figure 4(b), plot of zn is shown in Figure 4(c) and attractor of system (165) is shown in Figure 4(d). Finally, if we choose α16=5,β16=4,γ16=8,α17=4695,β17=6,γ17=75,α18=6,β18=459,γ18=379 with initial values x0=5.2,x-1=44.7,y0=23.4,y-1=3.9,z0=7.4,z-1=0.9 then system (11) can be written as(167)xn+1=25+94e-xn788+yn-1,yn+1=42+26e-yn5+xn-1,zn+1=6+59e-zn37+zn-1.For parametric values α16=5,β16=4,γ16=8,α17=4695,β17=6,γ17=75,α18=6,β18=459,γ18=379 the condition under which the unique positive equilibrium point of (167) is satisfied is as follows:(168)β16e-L16γ16+L171+β18e-L18γ18+L18+β17e-L17γ17+L16+α18+β18e-L18γ18+L182+β18e-L18γ18+L181+β17e-L17γ17+L16+β17e-L17γ17+L161+α18+β18e-L18γ18+L182+α18+β18e-L18γ18+L1821+β16β17e-L16-L17γ16+L17γ17+L16+α16+β16e-L16α17+β17e-L17γ17+L162γ16+L1721+β18e-L18γ18+L18+1γ16+L17γ17+L16γ18+L18β16β17β18e-L16-L17-L18+α16+β16e-L16α17+β17e-L17α18+β18e-L18γ16+L17γ17+L16γ18+L18=0.39998577828719<1.Moreover, in Figure 5, plot of xn is shown in Figure 5(a), plot of yn is shown in Figure 5(b), plot of zn is shown in Figure 5(c) and attractor of system (167) is shown in Figure 5(d).Figure 1
Phase portraits for system (159).
(a) (b) (c) (d)Figure 2
Phase portraits for system (161).
(a) (b) (c) (d)Figure 3
Phase portraits for system (163).
(a) (b) (c) (d)Figure 4
Phase portraits for system (165).
(a) (b) (c) (d)Figure 5
Phase portraits for system (167).
(a) (b) (c) (d)


## 8. Conclusions

This work is related to the global dynamics of a list of 3×6 systems of exponential difference equations which is natural extension of single-species model studied by Oztruket al. [8] to three-species model. For each system, we studied the dynamics including boundedness and persistence, existence of invariant interval, existence and uniqueness of positive equilibrium point, and local and global dynamics about the unique positive equilibrium point and conclusion is presented in Tables 2–5. Furthermore rate of convergence that converges to unique positive equilibrium point for each system is also demonstrated. Finally some numerical simulations are presented to verify theoretical results.Table 2
Existence of invariant interval corresponding to each system.
Systems Existence of invariant interval (7) [ L 4 , U 4 ] × [ L 5 , U 5 ] × [ L 6 , U 6 ] where L4=α4+β4e-α5+β5/γ5γ4+α5+β5/γ5, U4=α4+β4γ4, L5=α5+β5e-α6+β6/γ6γ5+α6+β6/γ6, U5=α5+β5γ5, L6=α6+β6e-α4+β4/γ4γ6+α4+β4/γ4, U6=α6+β6γ6. (8) [ L 7 , U 7 ] × [ L 8 , U 8 ] × [ L 9 , U 9 ] where L7=α7+β7e-α9+β9/γ9γ7+α9+β9/γ9, U7=α7+β7γ7, L8=α8+β8e-α7+β7/γ7γ8+α7+β7/γ7, U8=α8+β8γ8, L9=α9+β9e-α8+β8/γ8γ9+α8+β8/γ8, U9=α9+β9γ9. (9) [ L 10 , U 10 ] × [ L 11 , U 11 ] × [ L 12 , U 12 ] where L10=α10+β10e-α10+β10/γ10γ10+α10+β10/γ10, U10=α10+β10γ10, L11=α11+β11e-α11+β11/γ11γ11+α12+β12/γ12, U11=α11+β11γ11, L12=α12+β12e-α12+β12/γ12γ12+α11+β11/γ11, U12=α12+β12γ12. (10) [ L 13 , U 13 ] × [ L 14 , U 14 ] × [ L 15 , U 15 ] where L13=α13+β13e-α13+β13/γ13γ13+α15+β15/γ15, U13=α13+β13γ13, L14=α14+β14e-α14+β14/γ14γ14+α14+β14/γ14, U14=α14+β14γ14, L15=α15+β15e-α15+β15/γ15γ15+α13+β13/γ13, U15=α15+β15γ15. (11) [ L 16 , U 16 ] × [ L 17 , U 17 ] × [ L 18 , U 18 ] where L16=α16+β16e-α16+β16/γ16γ16+α17+β17/γ17, U16=α16+β16γ16, L17=α17+β17e-α17+β17/γ17γ17+α16+β16/γ16, U17=α17+β17γ17, L18=α18+β18e-α18+β18/γ18γ18+α18+β18/γ18, U18=α18+β18γ18.Table 3
Existence and uniqueness of positive equilibrium point for each system.
Systems Existence and uniqueness of positive equilibrium point (7) - α 5 + β 5 e - L 6 1 + γ 5 + U 6 / γ 5 + L 6 2 α 4 + β 4 e - α 5 + β 5 e - U 6 / γ 5 + U 6 1 + α 5 + β 5 e - L 6 / γ 5 + L 6 + γ 4 β 6 e Γ 1 + U 6 γ 4 + α 5 + β 5 e - L 6 / γ 5 + L 6 α 4 + β 4 e - α 5 + β 5 e - L 6 / γ 5 + L 6 + γ 6 α 5 + β 5 e - U 6 / γ 5 + U 6 + γ 4 < 1, where Γ1=-α4+β4e-α5+β5e-L6/γ5+L6γ4+α5+β5e-L6/γ5+L6. (8) - α 7 + β 7 e - L 9 1 + γ 7 + U 9 / γ 7 + L 9 2 α 8 + β 8 e - α 7 + β 7 e - U 9 / γ 7 + U 9 1 + α 7 + β 7 e - L 9 / γ 7 + L 9 + γ 8 β 9 e Γ 2 + U 9 γ 8 + α 7 + β 7 e - L 9 / γ 7 + L 9 α 8 + β 8 e - α 7 + β 7 e - L 9 / γ 7 + L 9 + γ 8 α 7 + β 7 e - U 9 / γ 7 + U 9 + γ 8 < 1, where Γ2=-α8+β8e-α7+β7e-L9/γ7+L9γ8+α7+β7e-L9/γ7+L9. (9) α 11 + β 11 e - L 11 1 + U 11 / L 11 2 α 12 + β 12 e - α 11 + β 11 e - U 11 / U 11 - γ 11 1 + α 11 + β 11 e - L 11 / L 11 - γ 11 β 11 e Γ 3 + γ 11 + U 12 γ 11 - α 11 + β 11 e - L 11 / L 11 α 12 + β 12 e - α 11 + β 11 e - L 11 / L 11 - γ 11 - γ 12 α 11 + β 11 e - U 11 / U 11 - γ 11 < 1, where Γ3=-α12+β12e-α11+β11e-L11/L11-γ11α11+β11e-L11/γ11-γ11-γ12. (10) α 13 + β 13 e - L 13 1 + U 13 / L 13 2 α 15 + β 15 e - α 13 + β 13 e - U 13 / U 13 - γ 13 1 + α 13 + β 13 e - L 13 / L 13 - γ 13 β 13 e Γ 4 + γ 13 + U 15 γ 13 - α 13 + β 13 e - L 13 / L 13 α 15 + β 15 e - α 13 + β 13 e - L 13 / L 13 - γ 13 - γ 15 α 13 + β 13 e - U 13 / U 13 - γ 13 < 1, where Γ4=-α15+β15e-α13+β13e-L13/L13-γ13α13+β13e-L13/L13-γ13-γ15. (11) α 17 + β 17 e - L 17 1 + U 17 / L 17 2 α 16 + β 16 e - α 17 + β 17 e - U 17 / U 17 - γ 17 1 + α 17 + β 17 e - L 17 / L 17 - γ 17 β 17 e Γ 5 + γ 17 + U 16 γ 17 - α 17 + β 17 e - L 17 / L 17 α 16 + β 16 e - α 17 + β 17 e - L 17 / L 17 - γ 17 - γ 16 α 17 + β 17 e - U 17 / U 17 - γ 17 < 1, where Γ5=-α16+β16e-α17+β17e-L17/L17-γ17α17+β17e-L17/L17-γ17-γ16.Table 4
Existence of parametric condition under which the positive equilibrium point is stable and unstable for each system.
Systems Existence of parametric condition under which the positive equilibrium point is stable and unstable (7) ( x - , y - , z - ) ∈ [ L 4 , U 4 ] × [ L 5 , U 5 ] × [ L 6 , U 6 ] is locally asymptotically stable if β4e-L4-L5γ4+L5γ6+L4γ5+L6β5(α6+β6e-L4)γ6+L4+β6(α5+β5e-L6)γ5+L6+β5e-L6(α4+β4e-L5)γ5+L6γ6+L4γ4+L5β6e-L4γ4+L5+α6+β6e-L4γ6+L4γ4+L5+α5+β5e-L6γ6+L4γ4+L5γ5+L62β4e-L5(α6+β6e-L4)γ6+L4+β6e-L4(α4+β4e-L5)γ4+L5+1γ4+L5γ6+L4γ5+L6β4β5β6e-L4-L5-L6+(α4+β4e-L5)(α5+β5e-L6)(α6+β6e-L4)γ4+L5γ6+L4γ5+L6<1.And unstable if β4e-U4-U5γ4+U5γ6+U4γ5+U6β5(α6+β6e-U4)γ6+U4+β6(α5+β5e-U6)γ5+U6+β5e-U6(α4+β4e-U5)γ5+U6γ6+U4γ4+U5β6e-U4γ4+U5+α6+β6e-U4γ6+U4γ4+U5+α5+β5e-U6γ6+U4γ4+U5γ5+U62β4e-U5(α6+β6e-U4)γ6+U4+β6e-U4(α4+β4e-U5)γ4+U5+1γ4+U5γ6+U4γ5+U6β4β5β6e-U4-U5-U6+(α4+β4e-U5)(α5+β5e-U6)(α6+β6e-U4)γ4+U5γ6+U4γ5+U6>1, (8) ( x - , y - , z - ) ∈ [ L 7 , U 7 ] × [ L 8 , U 8 ] × [ L 9 , U 9 ] is locally asymptotically stable if  β7e-L9(α9+β9e-L8)γ7+L9γ8+L7γ9+L82β8e-L7+α8+β8e-L7γ8+L7+β9e-L8(α8+β8e-L7)γ7+L9γ9+L8γ8+L72β7e-L9+α7+β7e-L9γ7+L9+β8e-L7(α7+β7e-L9)γ8+L7γ9+L8γ7+L92β9e-L8+α9+β9e-L8γ9+L8+1γ7+L9γ8+L7γ9+L8β7β8β9e-L7-L8-L9+(α7+β7e-L9)(α8+β8e-L7)(α9+β9e-L8)γ7+L9γ8+L7γ9+L8<1.And unstable if β7e-U9(α9+β9e-U8)γ7+U9γ8+U7γ9+U82β8e-U7+α8+β8e-U7γ8+U7+β9e-U8(α8+β8e-U7)γ7+U9γ9+U8γ8+U72β7e-U9+α7+β7e-U9γ7+U9+β8e-U7(α7+β7e-U9)γ8+U7γ9+U8γ7+U92β9e-U8+α9+β9e-U8γ9+U8+1γ7+U9γ8+U7γ9+U8β7β8β9e-U7-U8-U9+(α7+β7e-U9)(α8+β8e-U7)(α9+β9e-U8)γ7+U9γ8+U7γ9+U8>1. (9) ( x - , y - , z - ) ∈ [ L 10 , U 10 ] × [ L 11 , U 11 ] × [ L 12 , U 12 ] is locally asymptotically stable ifβ10e-L10γ10+L101+β11e-L11γ11+L12+β12e-L12γ12+L11+β11e-L11γ11+L121+α10+β10e-L10γ10+L102+β12e-L12γ12+L11+β12e-L12γ12+L111+α10+β10e-L10γ10+L102+α10+β10e-L10γ10+L1021+β11β12e-L11-L12γ12+L12γ12+L11+α11+β11e-L11α12+β12e-L12γ11+L122γ12+L1121+β10e-L10γ10+L10+1γ10+L11γ11+L12γ12+L11β10β11β12e-L10-L11-L12+(α10+β10e-L10)(α11+β11e-L11)(α12+β12e-L12)γ10+L10γ11+L12γ12+L11<1. And unstable if β10e-U10γ10+U101+β11e-U11γ11+U12+β12e-U12γ12+U11+β11e-U11γ11+U121+α10+β10e-U10γ10+U102+β12e-U12γ12+U11+β12e-U12γ12+U111+α10+β10e-U10γ10+U102+α10+β10e-U10γ10+U1021+β11β12e-U11-U12γ12+U12γ12+U11+α11+β11e-U11α12+β12e-U12γ11+U122γ12+U1121+β10e-U10γ10+U10+1γ10+U11γ11+U12γ12+U11β10β11β12e-U10-U11-U12+(α10+β10e-U10)(α11+β11e-U11)(α12+β12e-U12)γ10+U10γ11+U12γ12+U11>1. (10) ( x - , y - , z - ) ∈ [ L 13 , U 13 ] × [ L 14 , U 14 ] × [ L 15 , U 15 ]is locally asymptotically stable if  β13e-L13γ13+L131+β14e-L14γ14+L14+β15e-L15γ15+L15+α14+β14e-L14γ14+L142+β14e-L14γ14+L141+β15e-L15γ15+L13+β15e-L15γ15+L131+α14+β14e-L14γ14+L142+α14+β14e-L14γ14+L1421+β13β15e-L13-L15γ15+L13γ13+L15+α13+β13e-L13α15+β15e-L15γ13+L152γ15+L1321+β14e-L14γ14+L14+1γ13+L15γ14+L14γ15+L13β13β14β15e-L13-L14-L15+(α13+β13e-L13)(α14+β14e-L14)(α15+β15e-L15)γ13+L15γ14+L14γ15+L13<1. And unstable if β13e-U13γ13+U131+β14e-U14γ14+U14+β15e-U15γ15+U15+α14+β14e-U14γ14+U142+β14e-U14γ14+U141+β15e-U15γ15+U13+β15e-U15γ15+U131+α14+β14e-U14(γ14+U14)2+α14+β14e-U14(γ14+U14)21+β13β15e-U13-U15γ15+U13γ13+U15+α13+β13e-U13α15+β15e-U15(γ13+U15)2(γ15+U13)21+β14e-U14γ14+U14+1γ13+U15γ14+U14γ15+U13β13β14β15e-U13-U14-U15+(α13+β13e-U13)(α14+β14e-U14)(α15+β15e-U15)γ13+U15γ14+U14γ15+U13>1. (11) ( x - , y - , z - ) ∈ [ L 16 , U 16 ] × [ L 17 , U 17 ] × [ L 18 , U 18 ] is locally asymptotically stable if  β16e-L16γ16+L171+β18e-L18γ18+L18+β17e-L17γ17+L16+α18+β18e-L18γ18+L182+β18e-L18γ18+L181+β17e-L17γ17+L16+β17e-L17γ17+L161+α18+β18e-L18γ18+L182+α18+β18e-L18γ18+L1821+β16β17e-L16-L17γ16+L17γ17+L16+α16+β16e-L16α17+β17e-L17γ17+L162γ16+L1721+β18e-L18γ18+L18+1γ16+L17γ17+L16γ18+L18β16β17β18e-L16-L17-L18+(α16+β16e-L16)(α17+β17e-L17)(α18+β18e-L18)γ16+L17γ17+L16γ18+L18<1. And unstable if  β16e-U16γ16+U171+β18e-U18γ18+U18+β17e-U17γ17+U16+α18+β18e-U18(γ18+U18)2+β18e-U18γ18+U181+β17e-U17γ17+U16+β17e-U17γ17+U161+α18+β18e-U18(γ18+U18)2+α18+β18e-U18(γ18+U18)21+β16β17e-U16-U17γ16+U17γ17+U16+α16+β16e-U16α17+β17e-U17(γ17+U16)2(γ16+U17)21+β18e-U18γ18+U18+1γ16+U17γ17+U16γ18+U18β16β17β18e-U16-U17-U18+(α16+β16e-U16)(α17+β17e-U17)(α18+β18e-U18)γ16+U17γ17+U16γ18+U18>1.Table 5
Existence of parametric condition under which the positive equilibrium point is globally asymptotically stable for each system.
Systems Existence of parametric condition under which the positive equilibrium point is globally asymptotically stable (7) ( x - , y - , z - ) ∈ [ L 4 , U 4 ] × [ L 5 , U 5 ] × [ L 6 , U 6 ] is globally asymptotically stable if α4+β4e-L5<x-γ4+L5, α5+β5e-L6<y-γ5+L6 and α6+β6e-L4<z-γ6+L4. (8) ( x - , y - , z - ) ∈ [ L 7 , U 7 ] × [ L 8 , U 8 ] × [ L 9 , U 9 ] is globally asymptotically stable if α7+β7e-L9<x-γ7+L9, α8+β8e-L7<y-γ8+L7 and α9+β9e-L8<z-γ9+L8. (9) ( x - , y - , z - ) ∈ [ L 10 , U 10 ] × [ L 11 , U 11 ] × [ L 12 , U 12 ] is globally asymptotically stable if α10+β10e-L10<x-γ10+L10, α11+β11e-L11<y-(γ11+L12) and α12+β12e-L12<z-γ12+L11. (10) ( x - , y - , z - ) ∈ [ L 13 , U 13 ] × [ L 14 , U 14 ] × [ L 15 , U 15 ] is globally asymptotically stable if α13+β13e-L13<x-γ13+L15, α14+β14e-L14<y-(γ14+L14) and α15+β15e-L15<z-γ15+L13. (11) ( x - , y - , z - ) ∈ [ L 16 , U 16 ] × [ L 17 , U 17 ] × L 18 , U 18 is globally asymptotically stable if α16+β16e-L16<x-γ16+L17, α17+β17e-L17<y-(γ17+L16) and α18+β18e-L18<z-γ18+L18.



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*Source: 8362837-2018-09-02.xml*