# Global Dynamics of Some Exponential Type Systems

Authors: A. Q. Khan; H. M. Arshad; B. A. Younis; KH. I. Osman; Tarek F. Ibrahim; I. Ahmed; A. Khaliq
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

We explore the boundedness and persistence, existence of an invariant rectangle, local dynamical properties about the unique positive fixed point, global dynamics by the discrete-time Lyapunov function, and the rate of convergence of some2,3-type exponential systems of difference equations. Finally, theoretical results are numerically verified.


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## Body

## 1. Introduction

Recently, global dynamical properties of2,2 as well as 2,3-type exponential difference equations or systems of exponential difference equations have widely been explored. In this regard, Ozturk et al. [1] have explored the global dynamical properties of the following 2,2-type exponential difference equation:(1)xn+1=α10+α11e−xnα12+xn−1,where αss=10,11,12 and xss=−1,0 are the positive real numbers. Equation (1) may be viewed as a model in mathematical biology where α10 is the immigration rate, α11 is the population growth rate, and α12 is the carrying capacity. Bozkurt [2] has explored the global dynamical properties of the following 2,3-type exponential difference equation:(2)xn+1=α10e−xn+α11e−xn−1α12+α10xn+α11xn−1,where αss=10,11,12 and xss=−1,0 are the positive real numbers. More precisely, .Bozkurt [2] has explored the local asymptotic stability of the equilibrium point by linearized stability theorem, gasymptotic stability behavior by Lyapunov function, and semicycle analysis of positive solutions of the exponential difference equation, which is depicted in (2). Finally, theoretical results are verified numerically. Motivated from the aforementioned studies, here our purpose is to explore the global dynamical properties of the following 2,3-type exponential systems, that is the extension of the work of Bozkurt [2]:(3)xn+1=α10e−yn+α11e−yn−1α12+α10yn+α11yn−1,yn+1=α13e−zn+α14e−zn−1α15+α13zn+α14zn−1,zn+1=α16e−xn+α17e−xn−1α18+α16xn+α17xn−1,(4)xn+1=α19e−zn+α20e−zn−1α21+α19zn+α20zn−1,yn+1=α22e−xn+α23e−xn−1α24+α22xn+α23xn−1,zn+1=α25e−yn+α26e−yn−1α27+α25yn+α26yn−1,(5)xn+1=α28e−xn+α29e−xn−1α30+α28xn+α29xn−1,yn+1=α31e−yn+α32e−yn−1α33+α31zn+α32zn−1,zn+1=α34e−zn+α35e−zn−1α36+α34yn+α35yn−1,(6)xn+1=α37e−xn+α38e−xn−1α39+α37yn+α38yn−1,yn+1=α40e−yn+α41e−yn−1α42+α40xn+α41xn−1,zn+1=α43e−zn+α44e−zn−1α45+α43zn+α44zn−1,(7)xn+1=α46e−xn+α47e−xn−1α48+α46zn+α47zn−1,yn+1=α49e−yn+α50e−yn−1α51+α49yn+α50yn−1,zn+1=α52e−zn+α53e−zn−1α54+α52xn+α53xn−1,(8)xn+1=α55e−zn+α56e−zn−1α57+α55xn+α56xn−1,yn+1=α58e−xn+α59e−xn−1α60+α58yn+α59yn−1,zn+1=α61e−yn+α62e−yn−1α63+α61zn+α62zn−1,where αss=10,…,63 and xs,ys,zss=−1,0 are the positive real numbers.The rest of the paper is organized as follows: in Section2, we explore that every positive solution of systems (3)–(8) is bounded and persists, whereas construction of an invariant rectangle is explored in Section 3. In Section 4, we explore the existence as well as uniqueness of the positive equilibrium point of systems (3)–(8). In Section 5, we explore the local dynamical properties about the unique positive equilibrium point of systems (3)–(8). In Section 6, we explore global dynamics about the positive equilibrium by the discrete-time Lyapunov function. We study the rate of convergence in Section 7, whereas discussion along with numerical simulations is presented in Section 8.


## 2. Boundedness and Persistence of Systems (3)–(8)

Theorem 1.
Every positive solutionΩnn=−1∞ of systems (3)–(8) is bounded and persists.Proof.
(i)
IfΩnn=−1∞ is a positive solution of (3), then(9)xn≤α10+α11α12=U1,yn≤α13+α14α15=U2,zn≤α16+α17α18=U3,n=1,2,….From (3) and (9), one gets(10)xn≥α10+α11e−α13+α14/α15α10+α11+α12α13+α14/α15=L1,yn≥α13+α14e−α16+α17/α18α15+α13+α14α16+α17/α18=L2,zn≥α16+α17e−α10+α11/α12α18+α16+α17α10+α11/α12=L3,n=2,3,….So, from (9) and (10), one has(11)L1≤xn≤U1,L2≤yn≤U2,L3≤zn≤U3,n=3,4,….(ii)
IfΩnn=−1∞ is a positive solution of (4), then(12)xn≤α19+α20α21=U4,yn≤α22+α23α24=U5,zn≤α25+α26α27=U6,n=1,2,….From (4) and (12), one gets(13)xn≥α19+α20e−α25+α26/α27α21+α19+α20α25+α26/α27=L4,yn≥α22+α23e−α19+α20/α21α24+α22+α23α19+α20/α21=L5,zn≥α25+α26e−α22+α23/α24α27+α25+α26α22+α23/α24=L6,n=2,3,….So, from (12) and (13), one gets(14)L4≤xn≤U4,L5≤yn≤U5,L6≤zn≤U6,n=3,4,….(iii)
IfΩnn=−1∞ is a positive solution of (5), then(15)xn≤α28+α29α30=U7,yn≤α31+α32α33=U8,zn≤α34+α35α36=U9,n=1,2,….From (5) and (15), one gets(16)xn≥α28+α29e−α28+α29/α30α30+α28+α29α28+α29/α30=L7,yn≥α31+α32e−α31+α32/α33α33+α31+α32α34+α35/α36=L8,zn≥α34+α35e−α34+α35/α35α36+α34+α35α31+α32/α33=L9,n=2,3,….So, from (15) and (16), one gets(17)L7≤xn≤U7,L8≤yn≤U8,L9≤zn≤U9,n=3,4,….(iv)
IfΩnn=−1∞ is a positive solution of (6), then(18)xn≤α37+α38α39=U10,yn≤α40+α41α42=U11,zn≤α43+α44α45=U12,n=1,2,….From (6) and (18), one gets(19)xn≥α37+α38e−α37+α38/α39α39+α37+α38α40+α41/α42=L10,yn≥α40+α41e−α40+α41/α42α42+α40+α41α37+α38/α39=L11,zn≥α43+α44e−α43+α44/α45α45+α43+α44α43+α44/α45=L12,n=2,3,….So, from (18) and (19), one gets(20)L10≤xn≤U10,L11≤yn≤U11,L12≤zn≤U12,n=3,4,….(v)
IfΩnn=−1∞ is a positive solution of (7), then(21)xn≤α46+α47α48=U13,yn≤α49+α50α51=U14,zn≤α52+α53α54=U15,n=1,2,….From (7) and (21), one gets(22)xn≥α46+α47e−α46+α47/α48α48+α46+α47α52+α53/α54=L13,yn≥α49+α50e−α49+α50/α51α51+α49+α50α49+α50/α51=L14,zn≥α52+α53e−α52+α53/α54α54+α52+α53α46+α47/α48=L15,n=2,3,….So, from (21) and (22), one gets(23)L13≤xn≤U13,L14≤yn≤U14,L15≤zn≤U15,n=3,4,….(vi)
IfΩnn=−1∞ is a positive solution of (8), then(24)xn≤α55+α56α57=U16,yn≤α58+α59α60=U17,zn≤α61+α62α63=U18,n=1,2,….From (8) and (24), one gets(25)xn≥α55+α56e−α61+α62/α63α57+α55+α56α55+α56/α57=L16,yn≥α58+α59e−α55+α56/α57α60+α58+α59α58+α59/α60=L17,zn≥α61+α62e−α58+α59/α60α63+α61+α62α61+α62/α63=L18,n=2,3,….So, from (24) and (25), one gets(26)L16≤xn≤U16,L17≤yn≤U17,L18≤zn≤U18,n=3,4,….


## 3. Existence of Invariant Rectangle of Systems (3)–(8)

Theorem 2.
IfΩnn=−1∞ is a positive solution of systems (3)–(8), then their corresponding invariant rectangles, respectively, are L1,U1×L2,U2×L3,U3,L4,U4×L5,U5×L6,U6, L7,U7×L8,U8×L9,U9, L10,U10×L11,U11×L12,U12, L13,U13×L14,U14×L15,U15, and L16,U16×L17,U17×L18,U18.Proof.
IfΩnn=−1∞ is a positive solution with x0,x−1∈L1,U1,y0,y−1∈L2,U2, and z0,z−1∈L3,U3, then from (3), one has(27)x1=α10e−y0+α11e−y−1α12+α10y0+α11y−1≤α10+α11α12,x1=α10e−y0+α11e−y−1α12+α10y0+α11y−1≥α10+α11e−α13+α14/α15α12+α10+α11α13+α14/α15,y1=α13e−z0+α14e−z−1α15+α13z0+α14z−1≤α13+α14α15,y1=α13e−z0+α14e−z−1α15+α13z0+α14z−1≥α13+α14e−α16+α17/α18α15+α13+α14α16+α17/α18,z1=α16e−x0+α17e−x−1α18+α16x0+α17x−1≤α16+α17α18,z1=α16e−x0+α17e−x−1α18+α16x0+α17x−1≥α16+α17e−α10+α11/α12α18+α16+α17α10+α11/α12.
Hence (27) then implies that x1∈L1,U1,y1∈L2,U2,andz1∈L3,U3. Finally from (3), it is easy to establish that xk+1∈L1,U1resp.,yk+1∈L2,U2andzk+1∈L3,U3 if xk∈L1,U1resp.,yk∈L2,U2andzk∈L3,U3.Remark 1.
In a similar way, one can prove the invariant rectangle for systems (4)–(8).


## 4. Existence as well as Uniqueness of Positive Fixed Point of Systems (3)–(8)

Existence as well as uniqueness of a positive fixed point of systems (3)–(8) is explored in this section, as follows.Theorem 3.
(i)
System (3) has a unique positive fixed point: ϒ1=x¯,y¯,z¯∈L1,U1×L2,U2×L3,U3 if(28)α13+α14e−L1U1+1α13+α14+α15α10+α11e−α13+α14e−U1/α15+α13+α14L1α15+α13+α14z2α18+α16+α172×α10+α11α13+α14e−L1/α15+α13+α14L1+1+α12α12+α10+α11α13+α14e−L1/α15+α13+α14L12α16+α17e−α10+α11e−α13+α14e−L1/α15+α13+α14L1/α12+α10+α11α13+α14e−L1/α15+α13+α14L1×α16+α17Λ1+1+α18<Λ12,where(29)Λ1=α10+α11e−α13+α14e−U3/α15+α13+α14L3α12+α10+α11α13+α14e−L3/α15+α13+α14L3;(ii)
System (4) has a unique positive fixed point: ϒ2=x¯,y¯,z¯∈L4,U4×L5,U5×L6,U6 if(30)α19+α20e−L6U6+1α19+α20+α21α22+α23e−α19+α20e−U6/α21+α19+α20L6α21+α19+α20U62×α22+α23α19+α20e−L6/α21+α19+α20L6+1+α24α25+α26e−α22+α23e−α19+α20e−L6/α21+α19+α20L6/α24+α22+α23α19+α20e−L6/α21+α19+α20L6α24+α22+α23α19+α20e−L6/α21+α19+α20L62×α25+α26Λ2+1+α27α27+α25+α262<Λ22,where(31)Λ2=α22+α23e−α19+α20e−U6/α21+α19+α20L6α24+α22+α23α19+α20e−L6/α21+α19+α20L6;(iii)
System (5) has a unique positive fixed point: ϒ3=x¯,y¯,z¯∈L7,U7×L8,U8×L9,U9 if(32)e−e−U8/L8−α33/α31+α32e−L8U8+1e−L8/L8−α33/α31+α32+1e−L8/L8−α33/α31+α322U82e−e−e−L8/U8−α33/α31+α32/e−L8/L8−α33/α31+α32−α36/α34+α35<Λ32,where(33)Λ3=e−e−U8/L8−α33/α31+α32e−L8/L8−α33/α31+α32−α36α34+α35;(iv)
System (6) has a unique positive fixed point: ϒ4=x¯,y¯,z¯∈L10,U10×L11,U11×L12,U12 if(34)e−e−U10/L10−α39/α37+α38e−L10U10+1e−L10/L10−α39/α37+α38+1e−L10/L10−α39/α37+α382U102e−e−e−L10/U10−α39/α37+α38/e−L10/L10−α39/α37+α38−α42/α40+α41<Λ42,where(35)Λ4=e−e−U10/L10−α39/α37+α38e−L10/L10−α39/α37+α38−α42α40+α41;(v)
System (7) has a unique positive fixed point: ϒ5=x¯,y¯,z¯∈L13,U13×L14,U14×L15,U15 if(36)e−e−U13/L13−α48/α46+α47e−L13U13+1e−L13/L13−α48/α46+α47+1e−L13/L13−α48/α46+α472U132e−e−e−L13/U13−α48/α46+α47/e−L13/L13−α48/α46+α47−α54/α52+α53<Λ52,where(37)Λ5=e−e−U13/L13−α48/α46+α47e−L13/L13−α48/α46+α47−α54α52+α53;(vi)
System (8) has a unique positive fixed point: ϒ6=x¯,y¯,z¯∈L16,U16×L17,U17×L18,U18 if(38)α57+2U16α55+α56α63+2U16α61+α62lnα55+α56/L16α57+α55+α56L16α63lnα55+α56/L16α57+α55+α56L16+α61+α62lnα55+α56/L16α57+α55+α56L162×α57+2U16α55+α56α63+2α61+α62lnα55+α56/L16α57+α55+α56L16α63lnα55+α56/L16α57+α55+α56L16+Λ6×α60α58+α59L16α57+α55+α56L16α60+α58+α59<Λ6,where(39)Λ6=lnα61+α62lnα55+α56/L16α57+α55+α56L16α63+α61+α62lnα55+α56/L16α57+α55+α56L16.Proof.
(i)
From (3), one has(40)x=α10+α11e−yα12+α10+α11y,y=α13+α14e−zα15+α13+α14z,z=α16+α17e−xα18+α16+α17x.
From (40), one gets(41)z=α16+α17e−xα18+α16+α17x,x=α10+α11e−yα12+α10+α11y,y=α13+α14e−zα15+α13+α14z.
From (41), setting(42)y=gz=α13+α14e−zα15+α13+α14z,x=fy=α10+α11e−yα12+α10+α11y.
Denote(43)Fz≔α16+α17e−kzα18+α16+α17kz−z,where(44)kz≔x=fgz,=fα13+α14e−zα15+α13+α14z,=α10+α11e−α13+α14e−z/α15+α13+α14zα12+α10+α11α13+α14e−z/α15+α13+α14z,and z∈L3,U3. Here, our finding is that Fz=0 has a unique solution, where z∈L3,U3. From (43) and (44), one gets(45)F′z=−k′zα16+α17e−kzkz+1α16+α17+α18α18+α16+α17kz2−1,where(46)k′z=−α13+α14e−zz+1α13+α14+α15α10+α11e−α13+α14e−z/α15+α13+α14zα15+α13+α14z2α12+α10+α11α13+α14e−z/α15+α13+α14z2×α10+α11α13+α14e−zα15+α13+α14z+1+α12.
Using (44) and (46) in (45), we obtain(47)F′z=α13+α14e−zz+1α13+α14+α15α10+α11e−α13+α14e−z/α15+α13+α14zα15+α13+α14z2α18+α16+α17α10+α11e−α13+α14e−z/α15+α13+α14z/α12+α10+α11α13+α14e−z/α15+α13+α14z2×α10+α11α13+α14e−z/α15+α13+α14z+1+α12α12+α10+α11α13+α14e−z/α15+α13+α14z2α16+α17e−α10+α11e−α13+α14e−z/α15+α13+α14z/α12+α10+α11α13+α14e−z/α15+α13+α14z×α16+α17α10+α11e−α13+α14e−z/α15+α13+α14zα12+α10+α11α13+α14e−z/α15+α13+α14z+1+α18−1≤α13+α14e−L1U1+1α13+α14+α15α10+α11e−α13+α14e−U1/α15+α13+α14L1α15+α13+α14L12α18+α16+α17Λ12×α10+α11α13+α14e−L1/α15+α13+α14L1+1+α12α12+α10+α11α13+α14e−L1/α15+α13+α14L12α16+α17e−α10+α11e−α13+α14e−L1/α15+α13+α14L1/α12+α10+α11α13+α14e−L1/α15+α13+α14L1×α16+α17Λ1+1+α18−1,where Λ1 is depicted in (29). Now, assuming that if (28) along with (29) holds then from (47), one gets F′z<0.Remark 2.
The proof of (ii)–(vi) is same as the proof of (i). So, it is omitted.


## 5. Local Dynamics about Unique Positive Fixed Point of Systems (3)–(8)

The local dynamics aboutϒii=1,…,6, respectively, of systems (3) to (8) is explored in this section, as follows.Theorem 4.
Forϒ1 of (3), the following holds:(i)
ϒ1 is a sink if(48)α10α13α14+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1+α10α14α13+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1+α11α13+α10α14α16+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1<1;(ii)
ϒ1 is a source if(49)α10α13α14+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1+α10α14α13+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1+α11α13+α10α14α16+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1>1.Proof.
(i)
Ifϒ1 is a fixed point of (3), then(50)x¯=α10+α11e−y¯α12+α10+α11y¯,y¯=α13+α14e−z¯α15+α13+α14z¯,z¯=α16+α17e−x¯α18+α16+α17x¯.Moreover, we have the following map for constructing the corresponding linearized form of (3):(51)xn+1,xn,yn+1,yn,zn+1,zn⟶f,f1,g,g1,h,h1,where(52)f=α10e−yn+α11e−yn−1α12+α10yn+α11yn−1,f1=xn,g=α13e−zn+α14e−zn−1α15+α13zn+α14zn−1,g1=yn,h=α16e−xn+α17e−xn−1α18+α16xn+α17xn−1,h1=zn.TheJϒ1 about ϒ1 subject to the map (51) is(53)JΥ1=00b13b14001000000000b35b36001000b51b520000000010,where(54)b13=−α10e−y¯+x¯α12+α10+α11y¯,b14=−α11e−y¯+x¯α12+α10+α11y¯,b35=−α13e−z¯+y¯α15+α13+α14z¯,b36=−α14e−z¯+y¯α15+α13+α14z¯,b51=−α16e−x¯+z¯α18+α16+α17x¯,b52=−α17e−x¯+z¯α18+α16+α17x¯.The auxiliary equation ofJϒ1 about ϒ1 is(55)Pλ=λ6+L1λ3+L2λ2+L3λ+L4=0,where(56)L1=−b13b35b51,L2=−b13b35b52−b13b36b51−b14b35b51,L3=−b13b36b52−b14b35b52−b14b36b51,L4=−b14b36b52.Now,(57)∑i=14Li=α10α13α14+α17e−x¯+z¯e−y¯+x¯e−z¯+y¯α12+α10+α11y¯α15+α13+α14z¯α18+α16+α17x¯+α10α14α13+α17e−x¯+z¯e−y¯+x¯e−z¯+y¯α12+α10+α11y¯α15+α13+α14z¯α18+α16+α17x¯+α11α13α16+α17e−x¯+z¯e−y¯+x¯e−z¯+y¯α12+α10+α11y¯α15+α13+α14z¯α18+α16+α17x¯+α10α14α16+α17e−x¯+z¯e−y¯+x¯e−z¯+y¯α12+α10+α11y¯α15+α13+α14z¯α18+α16+α17x¯≤α10α13α14+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1+α10α14α13+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1+α11α13α16+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1+α10α14α16+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1=α10α13α14+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1+α10α14α13+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1+α11α13+α10α14α16+α17e−L1+L3e−L2+L1e−L3+L2α12+α10+α11L2α15+α13+α14L3α18+α16+α17L1.Assuming that (48) holds, and then from (57), one gets ∑i=14Li<1. Hence, ϒ1 of (3) is a sink.(ii)
Using similar manipulation as in the proof of (i), one has(58)∑i=14Li≥α10α13α14+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1+α10α14α13+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1+α11α13α16+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1+α10α14α16+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1=α10α13α14+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1+α10α14α13+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1+α11α13+α10α14α16+α17e−U1+U3e−U2+U1e−U3+U2α12+α10+α11U2α15+α13+α14U3α18+α16+α17U1.
Assuming that (49) holds, and then from (58), one gets ∑i=14Li>1. Hence, ϒ1 of (3) is a source.In similar manner, one can explore the local dynamics aboutYii=2,…,6, respectively, of systems (4)–(8), as follows.Theorem 5.
(i)
Forϒ2 of (4), the following holds:(i.1)ϒ2 is a sink if(59)α19α25α22+α23e−L4+L5e−L6+L4e−L4+L6α27+α25+α26L5α21+α19+α20L6α24+α22+α23L4+α19α26α22+α23e−L4+L5e−L5+L6e−L6+L4α27+α25+α26L5α21+α19+α20L6α24+α22+α23L4+α20α22+α20α23α26+α25e−L4+L5e−L5+L6e−L6+L4α27+α25+α26L5α21+α19+α20L6α24+α22+α23L4<1.(i.2)ϒ2 is a source if(60)α19α25α22+α23e−U4+U5e−U6+U4e−U4+U6α27+α25+α26U5α21+α19+α20U6α24+α22+α23U4+α19α26α22+α23e−U4+U5e−U5+U6e−U6+U4α27+α25+α26U5α21+α19+α20U6α24+α22+α23U4+α20α22+α20α23α26+α25e−U4+U5e−U5+U6e−U6+U4α27+α25+α26U5α21+α19+α20U6α24+α22+α23U4>1,with(61)Jϒ2=0000b15b16100000b31b3200000100000b53b5400000010,where(62)b15=−α19e−x¯+y¯α21+α19+α20x¯,b16=−α20e−x¯+y¯α21+α19+α20x¯,b31=−α22e−z¯+x¯α24+α22+α23z¯,b32=−α23e−z¯+x¯α24+α22+α23z¯,b53=−α25e−y¯+z¯α27+α25+α26y¯,b54=−α26e−y¯+z¯α27+α25+α26y¯;(ii)
Forϒ3 of (5), the following holds:(ii.1)ϒ3 is a sink if(63)α28+α29e−L7+L7α30+α28+α29L7+α31+α32e−L8α33+α31+α32L9+α34+α35e−L9α36+α34+α35L8+α34α28+α29α34+α35α28+α29α35+α29α32e−L7−L9α30+α28+α29L7α36+α34+α35L8+α31α34+α35α31+α32α35e−L8−L9α33+α31+α32L9α36+α34+α35L8+α28α34+α29α34+α28α35+α29α32+α29α35L7e−L9α30+α28+α29L7α36+α34+α35L8+α31α34+α32α34+α31α35+α32α35L7L8α33+α31+α32L9α36+α34+α35L8+α28α31α34+α29α31α34+α28α32α34+α28α35α31e−L7−L8−L9α30+α28+α29L7α33+α31+α32L9α36+α34+α35L8+α29α32α34+α29α31α35+α28α32α35+α29α32α35e−L7−L8−L9α30+α28+α29L7α33+α31+α32L9α36+α34+α35L8+α28α31α34+α29α31α34+α28α32α34+α28α35α31L7L8L9α30+α28+α29L7α33+α31+α32L9α36+α34+α35L8+α29α32α34+α29α31α35+α28α32α35+α29α32α35L7L8L9α30+α28+α29L7α33+α31+α32L9α36+α34+α35L8+α28α31α34+α29α31α34+α28α32α34+α28α35α31e−L7L8L9α30+α28+α29L7α33+α31+α32L9α36+α34+α35L8+α29α32α34+α29α31α35+α28α32α35+α29α32α35e−L7L8L9α30+α28+α29L7α33+α31+α32L9α36+α34+α35L8+α28α31α34+α29α31α34+α28α32α34+α28α35α31L7e−L8−L9α30+α28+α29L7α33+α31+α32L9α36+α34+α35L8+α29α32α34+α29α31α35+α28α32α35+α29α32α35L7e−L8−L9α30+α28+α29L7α33+α31+α32L9α36+α34+α35L8<1.(ii.2)ϒ3 is a source if(64)α28+α29e−U7+U7α30+α28+α29U7+α31+α32e−U8α33+α31+α32U9+α34+α35e−U9α36+α34+α35U8+α34α28+α29α34+α35α28+α29α35+α29α32e−U7−U9α30+α28+α29U7α36+α34+α35U8+α31α34+α35α31+α32α35e−U8−U9α33+α31+α32U9α36+α34+α35U8+α28α34+α29α34+α28α35+α29α32+α29α35U7e−U9α30+α28+α29U7α36+α34+α35U8+α31α34+α32α34+α31α35+α32α35U7U8α33+α31+α32U9α36+α34+α35U8+α28α31α34+α29α31α34+α28α32α34+α28α35α31e−U7−U8−U9α30+α28+α29U7α33+α31+α32U9α36+α34+α35U8+α29α32α34+α29α31α35+α28α32α35+α29α32α35e−U7−U8−U9α30+α28+α29U7α33+α31+α32U9α36+α34+α35U8+α28α31α34+α29α31α34+α28α32α34+α28α35α31U7U8U9α30+α28+α29U7α33+α31+α32U9α36+α34+α35U8+α29α32α34+α29α31α35+α28α32α35+α29α32α35U7U8U9α30+α28+α29U7α33+α31+α32U9α36+α34+α35U8+α28α31α34+α29α31α34+α28α32α34+α28α35α31e−U7U8U9α30+α28+α29U7α33+α31+α32U9α36+α34+α35U8+α29α32α34+α29α31α35+α28α32α35+α29α32α35e−U7U8U9α30+α28+α29U7α33+α31+α32U9α36+α34+α35U8+α28α31α34+α29α31α34+α28α32α34+α28α35α31U7e−U8−U9α30+α28+α29U7α33+α31+α32U9α36+α34+α35U8+α29α32α34+α29α31α35+α28α32α35+α29α32α35U7e−U8−U9α30+α28+α29U7α33+α31+α32U9α36+α34+α35U8>1,with(65)Jϒ3=b11b12000010000000b33b34b35b3600100000b53b54b55b56000010,where(66)b11=−α28e−x¯+x¯α30+α28+α29x¯,b12=−α29e−x¯+x¯α30+α28+α29x¯,b33=−α31e−y¯α33+α31+α32z¯,b34=−α32e−y¯α33+α31+α32z¯,b35=−α31y¯α33+α31+α32z¯,b36=−α32y¯α33+α31+α32z¯,b53=−α34z¯α36+α34+α35y¯,b54=−α35z¯α36+α34+α35y¯,b55=−α34e−z¯α36+α34+α35y¯,b56=−α35e−z¯α36+α34+α35y¯;(iii)
Forϒ4 of (6), the following holds:(iii.1)ϒ4 is a sink if(67)α37+α38e−L10α39+α37+α38L11+α40+α41e−L11α42+α40+α41L10+α43+α44e−L12+L12α45+α43+α44L12+α37α40+α41α37+α38α40+α37α41+α38α41e−L10−L11α39+α37+α38L11α42+α40+α41L10+α37α40+α38α40+α37α41+α38α41L10L11α39+α37+α38L11α42+α40+α41L10+α40α43+α41α44+α41α43+α4α44e−L11e−L12+L12α42+α40+α41L10α45+α43+α44L12+α37α43+α38α43+α38α44+α37α44e−L10e−L12+L10α39+α37+α38L11α45+α43+α44L12+α37α40α43+α38α40α43+α37α41α43+α37α40α44+α38α41α43+α38α40α44+α37α41α44+α38α41α44e−L10−L11−L12+α37α40α43+α38α40α43+α37α41α43+α37α40α44+α38α41α43+α38α40α44+α37α41α44+α38α41α44L10L11L12+α37α40α43+α38α40α43+α37α41α43+α37α40α44+α38α41α43+α38α40α44+α37α41α44+α38α41α44e−L12L10L11+α37α40α43+α38α40α43+α37α41α43+α37α40α44+α38α41α43+α38α40α44+α37α41α44+α38α41α44L12e−L10−L111α39+α37+α38L11α42+α40+α41L10α45+α43+α44L12<1.(iii.2)ϒ4 is a source if(68)α37+α38e−U10α39+α37+α38U11+α40+α41e−U11α42+α40+α41U10+α43+α44e−U12+L12α45+α43+α44U12+α37α40+α41α37+α38α40+α37α41+α38α41e−U10−U11α39+α37+α38U11α42+α40+α41U10+α37α40+α38α40+α37α41+α38α41U10U11α39+α37+α38U11α42+α40+α41U10+α40α43+α41α44+α41α43+α4α44e−U11e−U12+U12α42+α40+α41U10α45+α43+α44U12+α37α43+α38α43+α38α44+α37α44e−U10e−U12+U10α39+α37+α38U11α45+α43+α44U12+α37α40α43+α38α40α43+α37α41α43+α37α40α44+α38α41α43+α38α40α44+α37α41α44+α38α41α44e−U10−U11−U12+α37α40α43+α38α40α43+α37α41α43+α37α40α44+α38α41α43+α38α40α44+α37α41α44+α38α41α44U10U11U12+α37α40α43+α38α40α43+α37α41α43+α37α40α44+α38α41α43+α38α40α44+α37α41α44+α38α41α44e−U12U10U11+α37α40α43+α38α40α43+α37α41α43+α37α40α44+α38α41α43+α38α40α44+α37α41α44+α38α41α44U12e−U10−U11×1α39+α37+α38U11α42+α40+α41U10α45+α43+α44U12>1,with(69)Jϒ4=b11b12b13b1400100000b31b32b33b34000010000000b55b56000010,where(70)b11=−α37e−x¯α39+α37+α38y¯,b12=−α38e−x¯α39+α37+α38y¯,b13=−α37x¯α39+α37+α38y¯,b14=−α38x¯α39+α37+α38y¯,b31=−α40y¯α42+α40+α41x¯,b32=−α41y¯α42+α40+α41x¯,b33=−α40e−y¯α42+α40+α41x¯,b34=−α41e−y¯α42+α40+α41x¯,b55=−α43e−z¯+z¯α45+α43+α44z¯,b56=−α44e−z¯+z¯α45+α43+α44z¯;(iv)
Forϒ5 of (7), the following holds:(iv.1)ϒ5 is a sink if(71)α46+α47e−L13α48+α46+α47L15+α52+α53e−L15α54+α52+α53L13+α49+α50e−L14+L14α51+α49+α50L14+α46α52+α47α52+α46α53+α47α53e−L13−L15α48+α46+α47L15α54+α52+α53L13+α46α52+α47α52+α46α53+α47α53L13L15α48+α46+α47L15α54+α52+α53L13+αα46α49+α47α47+α46α50+α47α50e−L13e−L14+L14α48+α46+α47L15α51+α49+α50L14+α49α52+α50α52+α49α53+α50α53e−L15e−L14+L14α54+α52+α53L13α51+α49+α50L14+α46α49α52+α47α49α52+α46α50α52+α46α49α53+α47α50α52+α47α49α53+α46α50α53+α47α50α53e−L13−L14−L15+α46α49α52+α47α49α52+α46α50α52+α46α49α53+α47α50α52+α47α49α53+α46α50α53+α47α50α53L13L14L15+α46α49α52+α47α49α52+α46α50α52+α46α49α53+α47α50α52+α47α49α53+α46α50α53+α47α50α53e−L14L13L15+α46α49α52+α47α49α52+α46α50α52+α46α49α53+α47α50α52+α47α49α53+α46α50α53+α47α50α53L14e−L13−L15×1α48+α46+α47L15α51+α49+α50L14α54+α52+α53L13<1.(iv.2)ϒ5 is a source if(72)α46+α47e−U13α48+α46+α47U15+α52+α53e−U15α54+α52+α53U13+α49+α50e−U14+U14α51+α49+α50U14+α46α52+α47α52+α46α53+α47α53e−U13−U15α54+α52+α53U13α48+α46+α47U15+α46α52+α47α52+α46α53+α47α53U13U15α54+α52+α53U13α48+α46+α47U15+αα46α49+α47α47+α46α50+α47α50e−U13e−U14+U14α54+α52+α53U15α51+α49+α48U14+α49α52+α50α52+α49α53+α50α53e−U15e−U14+U14α54+α52+α53U15α51+α49+α48U14+α46α49α52+α47α49α52+α46α50α52+α46α49α53+α47α50α52+α47α49α53+α46α50α53+α47α50α53e−U13−U14−U15+α46α49α52+α47α49α52+α46α50α52+α46α49α53+α47α50α52+α47α49α53+α46α50α53+α47α50α53U13U14U15+α46α49α52+α47α49α52+α46α50α52+α46α49α53+α47α50α52+α47α49α53+α46α50α53+α47α50α53e−U14U13U15+α46α49α52+α47α49α52+α46α50α52+α46α49α53+α47α50α52+α47α49α53+α46α50α53+α47α50α53U14e−U13−U15×1α48+α46+α47U15α51+α49+α50U14α54α52+α53U13>1,with(73)Jϒ5=b11b1200b15b1610000000b33b3400001000b51b5200b55b56000010,where(74)b11=−α46e−x¯α48+α46+α47z¯,b12=−α47e−x¯α48+α46+α47z¯,b15=−α46x¯α48+α46+α47z¯,b16=−α47x¯α48+α46+α47z¯,b33=−α49e−y¯+y¯α51+α49+α50y¯,b34=−α50e−y¯+y¯α51+α49+α50y¯,b51=−α52z¯α54+α52+α53x¯,b52=−α53z¯α54+α52+α53x¯,b55=−α52e−z¯α54+α52+α53x¯,b56=−α53e−z¯α54+α52+α53x¯;(v)
Forϒ6 of (8), the following holds:(v.1)ϒ6 is a sink if(75)α55α58+α56α58+α55α59+α56α59L16L17α57+α55+α56L16α60+α58+α59L17+α55+α56L16α57+α55+α56L16+α55α61+α55α62+α56α61+α56α62L16L18α57+α55+α56L16α63+α61+α62L18+α58+α59L17α60+α58+α59L17+α58α62+α59α61+α59α62+α58α61L16L18α57+α55+α56L16α63+α61+α62L18+α61+α62L18α63+α61+α62L18+α55α58α61+α56α58α61+α55α58α61+α55α58α62+α56α58α61+α56α58α62+α55α58α62+α56α58α62e−L16−L17−L18+α55α58α61+α55α58α62+α56α58α61+α55α59α61+α56α58α62+α55α59α62+α56α59α61+α56α59α62L16L17L18×1α57+α55+α56L16α60+α58+α59L17α63+α61+α62L18<1.(v.2)ϒ6 is a source if(76)α55α58+α56α58+α55α59+α56α59U16U17α57+α55+α56U16α60+α58+α59U17+α55+α56U16α57+α55+α56U16+α55α61+α55α62+α56α61+α56α62U16U18α57+α55+α56U16α63+α61+α62U18+α58+α59U17α60+α58+α59U17+α58α62+α59α61+α59α62+α58α61U16U18α57+α55+α56U16α63+α61+α62U18+α61+α62U18α63+α61+α62U18+α55α58α61+α56α58α61+α55α58α61+α55α58α62+α56α58α61+α56α58α62+α55α58α62+α56α58α62e−U16−U17−U18+α55α58α61+α55α58α62+α56α58α61+α55α59α61+α56α58α62+α55α59α62+α56α59α61+α56α59α62U16U17U18×1α57+α55+α56U16α60+α58+α59U17α63+α61+α62U18>1,with(77)Jϒ6=b11b1200b15b16100000b31b32b33b340000100000b53b54b55b56000010,where(78)b11=−α55x¯α57+α55+α56x¯,b12=−α56x¯α57+α55+α56x¯,b15=−α55e−z¯α57+α55+α56x¯,b16=−α56e−z¯α57+α55+α56x¯,b31=−α58e−x¯α60+α58+α59y¯,b32=−α59e−x¯α60+α58+α59y¯,b33=−α58y¯α60+α58+α59y¯,b34=−α59y¯α60+α58+α59y¯,b53=−α61e−y¯α63+α61+α62z¯,b54=−α62e−y¯α63+α61+α62z¯,b55=−α61z¯α63+α61+α62z¯,b56=−α62z¯α63+α61+α62z¯.Proof.
It is similar to the proof of Theorem 4. So its proof is omitted. □
Hereafter, by constructing a Lyapunov function with discrete time motivated from the work of [2], global dynamics about Yii=1,…,6, respectively, of systems (3)–(8) is explored.


## 6. Global Dynamics of Systems (3)–(8)

Theorem 6.
For global dynamics aboutYii=1,…,6, respectively, of systems (3)–(8), following statements hold:(i)
ϒ1 of (3) is global asymptotically stable if(79)α10+α11e−L2<2x¯−U1α12+α10+α11L2,α13+α14e−L3<2y¯−U2α15+α13+α14L3,α16+α17e−L1<2z¯−U3α18+α16+α17L1;(ii)
ϒ2 of (4) is global asymptotically stable if(80)α19+α20e−L6<2x¯−U4α21+α19+α20L6,α22+α23e−L4<2y¯−U5α24+α22+α23L4,α25+α26e−L5<2z¯−U6α27+α25+α26L5;(iii)
ϒ3 of (5) is global asymptotically stable if(81)α28+α29e−L7<2x¯−U7α30+α28+α29L7,α31+α32e−L8<2y¯−U8α33+α31+α32L9,α34+α35e−L9<2z¯−U9α36+α34+α35L8;(iv)
ϒ4 of (6) is global asymptotically stable if(82)α37+α38e−L10<2x¯−U10α39+α37+α38L11,α40+α41e−L11<2y¯−U11α42+α40+α41L10,α43+α44e−L12<2z¯−U12α45+α43+α44L12;(v)
ϒ5 of (7) is global asymptotically stable if(83)α46+α47e−L13<2x¯−U13α48+α46+α47L15,α49+α50e−L14<2y¯−U14α51+α49+α50L14,α52+α53e−L15<2z¯−U15α54+α52+α53L13;(vi)
ϒ6 of (8) is global asymptotically stable if(84)α55+α56e−L18<2x¯−U16α57+α56+α57L16,α58+α59e−L16<2y¯−U17α60+α58+α59L17,α61+α62e−L17<2z¯−U18α63+α61+α62L18.Proof.
(i)
Consider the following discrete-time Lyapunov function:(85)Γn=xn−x¯2+yn−y¯2+zn−z¯2.
Now(86)ΔΓn=Γn+1−Γn=xn+1−xnxn+1+xn−2x¯+yn+1−ynyn+1+yn−2y¯+zn+1−znzn+1+zn−2z¯,=xn+1−xnα10e−yn+α11e−yn−1α12+α10yn+α11yn−1+xn−2x¯+yn+1−ynα13e−zn+α14e−zn−1α15+α13zn+α14zn−1+yn−2y¯+zn+1−znα16e−xn+α17e−xn−1α18+α16xn+α17xn−1+zn−2z¯≤U1−L1α10+α11e−L2α12+α10+α11L2+U1−2x¯+U2−L2α13+α14e−L3α15+α13+α14L3+U2−2y¯+U3−L3α16+α17e−L1α18+α16+α17L1+U3−2z¯=U1−L1α10+α11e−L2−2x¯−U1α12+α10+α11L2α12+α10+α11L2+U2−L2α13+α14e−L3−2y¯−U2α15+α13+α14L3α15+α13+α14L3+U3−L3α16+α17e−L1−2z¯−U3α18+α16+α17L1α18+α16+α17L1.
From (80) and (87), one gets ΔΓn<0∀n≥0. Hence, we obtain that limn⟶∞Γn+1−Γn=0, and thus ϒ1 of (3) is global asymptotically stable. □Remark 3.
The proof of (ii)–(vi) is same as the proof of (i).


## 7. Rate of Convergence

We will explore the convergence result about the equilibrium point of systems (3)–(8) motivated from the existing literature [3–5], in this section.Theorem 7.
If the positive solution of (3) is Ωnn=−1∞, s.t.(87)limn⟶∞xn=x¯,limn⟶∞yn=y¯,limn⟶∞zn=z¯,where(88)x¯∈L1,U1,y¯∈L2,U2,z¯∈L3,U3,then the error vector, i.e.,(89)θn=θn1θn−11θn2θn−12θn3θn−13,satisfies the following relation:(90)limn⟶∞θn1/n=λ1,…,6Jϒ1,limn⟶∞θn+1θn=λ1,…,6Jϒ1,where λ1,…,6Jϒ1 are the roots of Jϒ1.Proof.
If the positive solution of (3) is Ωnn=−1∞, s.t. (88) along with (89) holds. To find the error terms, one has(91)xn+1−x¯=α10e−yn+α11e−yn−1α12+α10yn+α11yn−1−α10+α11e−y¯α12+α10+α11y¯=−α10e−yneyn−y¯−1α12+α10yn+α11yn−1−α11e−yn−1eyn−1−y¯−1α12+α10yn+α11yn−1−α10x¯yn−y¯α12+α10yn+α11yn−1−α11x¯yn−1−y¯α12+α10yn+α11yn−1=−α10e−ynyn−y¯+O1yn−y¯2+x¯yn−y¯α12+α10yn+α11yn−1yn−y¯−α11e−yn−1yn−1−y¯+O2yn−1−y¯2+x¯yn−1−y¯α12+α10yn+α11yn−1yn−1−y¯.
So,(92)xn+1−x¯=−α10e−yn+x¯α12+α10yn+α11yn−1yn−y¯−α11e−yn−1+x¯α12+α10yn+α11yn−1yn−1−y¯+O1yn−y¯2+O2yn−1−y¯2.
Similarly,(93)yn+1−y¯=−α13e−zn+y¯α15+α13zn+α14zn−1zn−z¯−α14e−zn−1+y¯α15+α13zn+α14zn−1zn−1−z¯+O3zn−z¯2+O4zn−1−z¯2,zn+1−z¯=−α16e−xn+z¯α18+α16xn+α17xn−1xn−x¯−α17e−xn−1+z¯α18+α16xn+α17xn−1xn−1−x¯+O5xn−x¯2+O6xn−1−x¯2.
From (92) and (93), one gets(94)xn+1−x¯≈−α10e−yn+x¯α12+α10yn+α11yn−1yn−y¯−α11e−yn−1+x¯α12+α10yn+α11yn−1yn−1−y¯,yn+1−y¯≈−α13e−zn+y¯α15+α13zn+α14zn−1zn−z¯−α14e−zn−1+y¯α15+α13zn+α14zn−1zn−1−z¯,zn+1−z¯≈−α16e−xn+z¯α18+α16xn+α17xn−1xn−x¯−α17e−xn−1+z¯α18+α16xn+α17xn−1xn−1−x¯.
Denote(95)θn1=xn−x¯,θn2=yn−y¯,θn3=zn−z¯.
In view of (95), from (94), one gets(96)θn+11=ω1nθn2+ω2nθn−12,θn+12=ω3nθn3+ω4nθn−13,θn+13=ω5nθn1+ω6nθn−11,where(97)ω1n=−α10e−yn+x¯α12+α10yn+α11yn−1,ω2n=−α11e−yn−1+x¯α12+α10yn+α11yn−1,ω3n=−α13e−zn+y¯α15+α13zn+α14zn−1,ω4n=−α14e−zn−1+y¯α15+α13zn+α14zn−1,ω5n=−α16e−xn+z¯α18+α16xn+α17xn−1,ω6n=−α17e−xn−1+z¯α18+α16xn+α17xn−1.
From (97), one gets(98)limn⟶∞ω1n=−α10e−y¯+x¯α12+α10+α11y¯,limn⟶∞ω2n=−α11e−y¯+x¯α12+α10+α11y¯,limn⟶∞ω3n=−α13e−z¯+y¯α15+α13+α14z¯,limn⟶∞ω4n=−α14e−z¯+y¯α15+α13+α14z¯,limn⟶∞ω5n=−α16e−x¯+z¯α18+α16+α17x¯,limn⟶∞ω6n=−α17e−x¯+z¯α18+α16+α17x¯,that is,(99)ω1n=−α10e−y¯+x¯α12+α10+α11y¯+b1n,ω2n=−α11e−y¯+x¯α12+α10+α11y¯+b1n−1,ω3n=−α13e−z¯+y¯α15+α13+α14z¯+b2n,ω4n=−α14e−z¯+y¯α15+α13+α14z¯+b2n−1,ω5n=−α16e−x¯+z¯α18+α16+α17x¯+b3n,ω6n=−α17e−x¯+z¯α18+α16+α17x¯+b3n−1,where b1n,b1n−1,b2n,b2n−1,b3n,andb3n−1⟶0 as n⟶∞. Now, we have system 1.10 of [6], where A=Jϒ1 and(100)Bn=00b1nb1n−1001000000000b2nb2n−1001000b3nb3n−10000000010,such that Bn⟶∞,n⟶∞. So, about ϒ1 of (3) the limiting system becomes(101)θn+11θn1θn+12θn2θn+13θn3=Jϒ1θn1θn−11θn2θn−12θn3θn−13,which is as Jϒ1 about ϒ1.In the following theorem, we will summarize the convergence results for systems (4) to (8).Theorem 8.
(i)
If the positive solution of (4) is Ωnn=−1∞, s.t. (87) along with the following relation holds:(102)x¯∈L4,U4,y¯∈L5,U5,z¯∈L6,U6,then the error vector, which is depicted in (89), satisfies the following relations:(103)limn⟶∞θn1/n=λ1,…,6Jϒ2,limn⟶∞θn+1θn=λ1,…,6Jϒ2,whereλ1,…,6Jϒ2 are the roots of Jϒ2.(ii)
If the positive solution of (5) is Ωnn=−1∞, s.t. (87) along with the following relation holds:(104)x¯∈L7,U7,y¯∈L8,U8,z¯∈L9,U9,then the error vector, which is depicted in (89), satisfies the following relations:(105)limn⟶∞θn1/n=λ1,…,6Jϒ3,limn⟶∞θn+1θn=λ1,…,6Jϒ3,whereλ1,…,6Jϒ3 are the roots of Jϒ3.(iii)
If the positive solution of (6) is Ωnn=−1∞, s.t. (87) along with the following relation holds:(106)x¯∈L10,U10,y¯∈L11,U11,z¯∈L12,U12,then the error vector, which is depicted in (89), satisfies the following relations:(107)limn⟶∞θn1/n=λ1,…,6Jϒ4,limn⟶∞θn+1θn=λ1,…,6Jϒ4,whereλ1,…,6Jϒ4 are the roots of Jϒ4.(iv)
If the positive solution of (7) is Ωnn=−1∞, s.t. (87) along with the following relation holds:(108)x¯∈L13,U13,y¯∈L14,U14,z¯∈L15,U15,then the error vector, which is depicted in (89), satisfies the following relations:(109)limn⟶∞θn1/n=λ1,…,6Jϒ5,limn⟶∞θn+1θn=λ1,…,6Jϒ5,whereλ1,…,6Jϒ5 are the roots of Jϒ5.(v)
If the positive solution of (8) is Ωnn=−1∞, s.t. (87) along with the following relation holds:(110)x¯∈L16,U16,y¯∈L17,U17,z¯∈L18,U18,then the error vector, which is depicted in (89), satisfies the following relations:(111)limn⟶∞θn1/n=λ1,…,6Jϒ6,limn⟶∞θn+1θn=λ1,…,6Jϒ6,where λ1,…,6Jϒ6 are the roots of Jϒ6.Proof.
It is similar to Theorem7, and hence its proof is omitted.


## 8. Discussion and Simulations

In the reported work, we have explored the global dynamics of2,3-type exponential systems of difference equations. We have investigated that Ωnn=−1∞ of systems (3) to (8) is bounded and persists, and the corresponding invariant rectangles, respectively, are L1,U1×L2,U2×L3,U3, L4,U4×L5,U5×L6,U6, L7,U7×L8,U8×L9,U9, L10,U10×L11,U11×L112,U12, L13,U13×L14,U14×L15,U15, and L16,U16×L17,U17×L18,U18. Further, we have explored the existence and uniqueness of the positive equilibrium and the global and local dynamics of systems (3)–(8). We have also investigated the rate of convergence of the positive solution of systems (3)–(8). Finally, some numerical examples are provided to support the theoretical results. For instance, if αii=10,…,18, respectively, are 13, 24, 319, 12, 0.1, 0.2, 1.5,0.4, and 0.002, then from Figures 1(a)–1(c), the positive fixed point 0.06304125281075143,0.567941569702224,0.4220224516533974 of (3) is stable and its corresponding attractor is shown in Figure 1(s). Now, if αii=19,…,27, respectively, are 19, 14, 9, 112, 0.1, 0.2, 15,14, and 2, then from Figures 1(d)–1(f), the positive equilibrium point 0.6122355979161732,0.996719892698477,0.8425637558539959 of (4) is stable and its corresponding attractor is shown in Figure 1(t). For (5), if αii=28,…,36, respectively, are 9, 0.4, 9, 12, 0.1, 0.2, 15,2, and 15, then from Figures 1(g)–1(i), its unique positive equilibrium point 0.277465951,0.924713,0.88573 is stable and its corresponding attractor is shown in Figure 1(u). For (6), if αii=37,…,45, respectively, are 29, 14, 9, 12, 0.1, 0.2, 15,14, and 2, then from Figures 1(j)–1(l), its unique positive equilibrium point 0.9391896591799592,0.9495483631730844,0.8598773120562713 is stable and its corresponding attractor is shown in Figure 1(v). For (7), if αii=46,…,54, respectively, are 29, 14, 1.9, 12, 0.1, 1.2, 15,14, and 2, then from Figures 1(m)–1(o), its unique positive equilibrium point 1.22564,0.8167047,1.007332 is stable and its corresponding attractor is shown in Figure 1(v). Finally, if αii=55,…,63, respectively, are 9, 4, 129, 12, 0.1, 25, 14,14, and 2, then from Figures 1(p)–1(r), the unique positive equilibrium point 0.09228515208223682,9.184938317317343,0.0000975782517529335 of system (8) is stable and its corresponding attractor is shown in Figure 1(w). For more results on dynamical properties of difference equations, we refer the reader to [7, 8] and the references cited therein.Figure 1
Trajectories for (3) to (8) with initial values xs,ys,zss=−1,0, respectively, being 0.7,2.4,0.9,0.4,1.9,and0.4.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)(q)(r)(s)(t)(u)(v)(w)



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