Global Dynamics of Some Exponential Type Systems

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1. Introduction

Recently, global dynamical properties of $(2,2)$ 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: $\begin{matrix} (1) & x_{n + 1} = \frac{α_{10} + α_{11} e^{- x_{n}}}{α_{12} + x_{n - 1}}, \end{matrix}$ where $α_{s} (s = 10,11,12)$ and $x_{s} (s = - 1,0)$ are the positive real numbers. Equation (1) may be viewed as a model in mathematical biology where $α_{10}$ is the immigration rate, $α_{11}$ is the population growth rate, and $α_{12}$ is the carrying capacity. Bozkurt [2] has explored the global dynamical properties of the following $(2,3)$ -type exponential difference equation: $\begin{matrix} (2) & x_{n + 1} = \frac{α_{10} e^{- x_{n}} + α_{11} e^{- x_{n - 1}}}{α_{12} + α_{10} x_{n} + α_{11} x_{n - 1}}, \end{matrix}$ where $α_{s} (s = 10,11,12)$ and $x_{s} (s = - 1,0)$ 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]: $\begin{matrix} (3) & x_{n + 1} = \frac{α_{10} e^{- y_{n}} + α_{11} e^{- y_{n - 1}}}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}}, \\ y_{n + 1} = \frac{α_{13} e^{- z_{n}} + α_{14} e^{- z_{n - 1}}}{α_{15} + α_{13} z_{n} + α_{14} z_{n - 1}}, \\ z_{n + 1} = \frac{α_{16} e^{- x_{n}} + α_{17} e^{- x_{n - 1}}}{α_{18} + α_{16} x_{n} + α_{17} x_{n - 1}}, \\ (4) & x_{n + 1} = \frac{α_{19} e^{- z_{n}} + α_{20} e^{- z_{n - 1}}}{α_{21} + α_{19} z_{n} + α_{20} z_{n - 1}}, \\ y_{n + 1} = \frac{α_{22} e^{- x_{n}} + α_{23} e^{- x_{n - 1}}}{α_{24} + α_{22} x_{n} + α_{23} x_{n - 1}}, \\ z_{n + 1} = \frac{α_{25} e^{- y_{n}} + α_{26} e^{- y_{n - 1}}}{α_{27} + α_{25} y_{n} + α_{26} y_{n - 1}}, \\ (5) & x_{n + 1} = \frac{α_{28} e^{- x_{n}} + α_{29} e^{- x_{n - 1}}}{α_{30} + α_{28} x_{n} + α_{29} x_{n - 1}}, \\ y_{n + 1} = \frac{α_{31} e^{- y_{n}} + α_{32} e^{- y_{n - 1}}}{α_{33} + α_{31} z_{n} + α_{32} z_{n - 1}}, \\ z_{n + 1} = \frac{α_{34} e^{- z_{n}} + α_{35} e^{- z_{n - 1}}}{α_{36} + α_{34} y_{n} + α_{35} y_{n - 1}}, \\ (6) & x_{n + 1} = \frac{α_{37} e^{- x_{n}} + α_{38} e^{- x_{n - 1}}}{α_{39} + α_{37} y_{n} + α_{38} y_{n - 1}}, \\ y_{n + 1} = \frac{α_{40} e^{- y_{n}} + α_{41} e^{- y_{n - 1}}}{α_{42} + α_{40} x_{n} + α_{41} x_{n - 1}}, \\ z_{n + 1} = \frac{α_{43} e^{- z_{n}} + α_{44} e^{- z_{n - 1}}}{α_{45} + α_{43} z_{n} + α_{44} z_{n - 1}}, \\ (7) & x_{n + 1} = \frac{α_{46} e^{- x_{n}} + α_{47} e^{- x_{n - 1}}}{α_{48} + α_{46} z_{n} + α_{47} z_{n - 1}}, \\ y_{n + 1} = \frac{α_{49} e^{- y_{n}} + α_{50} e^{- y_{n - 1}}}{α_{51} + α_{49} y_{n} + α_{50} y_{n - 1}}, \\ z_{n + 1} = \frac{α_{52} e^{- z_{n}} + α_{53} e^{- z_{n - 1}}}{α_{54} + α_{52} x_{n} + α_{53} x_{n - 1}}, \\ (8) & x_{n + 1} = \frac{α_{55} e^{- z_{n}} + α_{56} e^{- z_{n - 1}}}{α_{57} + α_{55} x_{n} + α_{56} x_{n - 1}}, \\ y_{n + 1} = \frac{α_{58} e^{- x_{n}} + α_{59} e^{- x_{n - 1}}}{α_{60} + α_{58} y_{n} + α_{59} y_{n - 1}}, \\ z_{n + 1} = \frac{α_{61} e^{- y_{n}} + α_{62} e^{- y_{n - 1}}}{α_{63} + α_{61} z_{n} + α_{62} z_{n - 1}}, \end{matrix}$ where $α_{s} (s = 10, \dots, 63)$ and $x_{s}, y_{s}, z_{s} (s = - 1,0)$ are the positive real numbers.

The rest of the paper is organized as follows: in Section 2, 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 solution ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ of systems (3)–(8) is bounded and persists.

Proof.

(i)

If ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ is a positive solution of (3), then $\begin{matrix} (9) & x_{n} \leq \frac{α_{10} + α_{11}}{α_{12}} = U_{1}, \\ y_{n} \leq \frac{α_{13} + α_{14}}{α_{15}} = U_{2}, \\ z_{n} \leq \frac{α_{16} + α_{17}}{α_{18}} = U_{3}, \\ n = 1,2, \dots . \end{matrix}$

From (3) and (9), one gets $\begin{matrix} (10) & x_{n} \geq \frac{(α_{10} + α_{11}) e^{- ((α_{13} + α_{14}) / α_{15})}}{α_{10} + (α_{11} + α_{12}) ((α_{13} + α_{14}) / α_{15})} = L_{1}, \\ y_{n} \geq \frac{(α_{13} + α_{14}) e^{- ((α_{16} + α_{17}) / α_{18})}}{α_{15} + (α_{13} + α_{14}) ((α_{16} + α_{17}) / α_{18})} = L_{2}, \\ z_{n} \geq \frac{(α_{16} + α_{17}) e^{- ((α_{10} + α_{11}) / α_{12})}}{α_{18} + (α_{16} + α_{17}) ((α_{10} + α_{11}) / α_{12})} = L_{3}, \\ n = 2,3, \dots . \end{matrix}$

So, from (9) and (10), one has $\begin{matrix} (11) & L_{1} \leq x_{n} \leq U_{1}, \\ L_{2} \leq y_{n} \leq U_{2}, \\ L_{3} \leq z_{n} \leq U_{3}, \\ n = 3,4, \dots . \end{matrix}$

(ii)

If ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ is a positive solution of (4), then $\begin{matrix} (12) & x_{n} \leq \frac{α_{19} + α_{20}}{α_{21}} = U_{4}, \\ y_{n} \leq \frac{α_{22} + α_{23}}{α_{24}} = U_{5}, \\ z_{n} \leq \frac{α_{25} + α_{26}}{α_{27}} = U_{6}, \\ n = 1,2, \dots . \end{matrix}$

From (4) and (12), one gets $\begin{matrix} (13) & x_{n} \geq \frac{(α_{19} + α_{20}) e^{- ((α_{25} + α_{26}) / α_{27})}}{α_{21} + (α_{19} + α_{20}) ((α_{25} + α_{26}) / α_{27})} = L_{4}, \\ y_{n} \geq \frac{(α_{22} + α_{23}) e^{- ((α_{19} + α_{20}) / α_{21})}}{α_{24} + (α_{22} + α_{23}) ((α_{19} + α_{20}) / α_{21})} = L_{5}, \\ z_{n} \geq \frac{(α_{25} + α_{26}) e^{- ((α_{22} + α_{23}) / α_{24})}}{α_{27} + (α_{25} + α_{26}) ((α_{22} + α_{23}) / α_{24})} = L_{6}, \\ n = 2,3, \dots . \end{matrix}$

So, from (12) and (13), one gets $\begin{matrix} (14) & L_{4} \leq x_{n} \leq U_{4}, \\ L_{5} \leq y_{n} \leq U_{5}, \\ L_{6} \leq z_{n} \leq U_{6}, \\ n = 3,4, \dots . \end{matrix}$

(iii)

If ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ is a positive solution of (5), then $\begin{matrix} (15) & x_{n} \leq \frac{α_{28} + α_{29}}{α_{30}} = U_{7}, \\ y_{n} \leq \frac{α_{31} + α_{32}}{α_{33}} = U_{8}, \\ z_{n} \leq \frac{α_{34} + α_{35}}{α_{36}} = U_{9}, \\ n = 1,2, \dots . \end{matrix}$

From (5) and (15), one gets $\begin{matrix} (16) & x_{n} \geq \frac{(α_{28} + α_{29}) e^{- ((α_{28} + α_{29}) / α_{30})}}{α_{30} + (α_{28} + α_{29}) ((α_{28} + α_{29}) / α_{30})} = L_{7}, \\ y_{n} \geq \frac{(α_{31} + α_{32}) e^{- ((α_{31} + α_{32}) / α_{33})}}{α_{33} + (α_{31} + α_{32}) ((α_{34} + α_{35}) / α_{36})} = L_{8}, \\ z_{n} \geq \frac{(α_{34} + α_{35}) e^{- ((α_{34} + α_{35}) / α_{35})}}{α_{36} + (α_{34} + α_{35}) ((α_{31} + α_{32}) / α_{33})} = L_{9}, \\ n = 2,3, \dots . \end{matrix}$

So, from (15) and (16), one gets $\begin{matrix} (17) & L_{7} \leq x_{n} \leq U_{7}, \\ L_{8} \leq y_{n} \leq U_{8}, \\ L_{9} \leq z_{n} \leq U_{9}, \\ n = 3,4, \dots . \end{matrix}$

(iv)

If ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ is a positive solution of (6), then $\begin{matrix} (18) & x_{n} \leq \frac{α_{37} + α_{38}}{α_{39}} = U_{10}, \\ y_{n} \leq \frac{α_{40} + α_{41}}{α_{42}} = U_{11}, \\ z_{n} \leq \frac{α_{43} + α_{44}}{α_{45}} = U_{12}, \\ n = 1,2, \dots . \end{matrix}$

From (6) and (18), one gets $\begin{matrix} (19) & x_{n} \geq \frac{(α_{37} + α_{38}) e^{- ((α_{37} + α_{38}) / α_{39})}}{α_{39} + (α_{37} + α_{38}) ((α_{40} + α_{41}) / α_{42})} = L_{10}, \\ y_{n} \geq \frac{(α_{40} + α_{41}) e^{- ((α_{40} + α_{41}) / α_{42})}}{α_{42} + (α_{40} + α_{41}) ((α_{37} + α_{38}) / α_{39})} = L_{11}, \\ z_{n} \geq \frac{(α_{43} + α_{44}) e^{- ((α_{43} + α_{44}) / α_{45})}}{α_{45} + (α_{43} + α_{44}) ((α_{43} + α_{44}) / α_{45})} = L_{12}, \\ n = 2,3, \dots . \end{matrix}$

So, from (18) and (19), one gets $\begin{matrix} (20) & L_{10} \leq x_{n} \leq U_{10}, \\ L_{11} \leq y_{n} \leq U_{11}, \\ L_{12} \leq z_{n} \leq U_{12}, \\ n = 3,4, \dots . \end{matrix}$

(v)

If ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ is a positive solution of (7), then $\begin{matrix} (21) & x_{n} \leq \frac{α_{46} + α_{47}}{α_{48}} = U_{13}, \\ y_{n} \leq \frac{α_{49} + α_{50}}{α_{51}} = U_{14}, \\ z_{n} \leq \frac{α_{52} + α_{53}}{α_{54}} = U_{15}, \\ n = 1,2, \dots . \end{matrix}$

From (7) and (21), one gets $\begin{matrix} (22) & x_{n} \geq \frac{(α_{46} + α_{47}) e^{- ((α_{46} + α_{47}) / α_{48})}}{α_{48} + (α_{46} + α_{47}) ((α_{52} + α_{53}) / α_{54})} = L_{13}, \\ y_{n} \geq \frac{(α_{49} + α_{50}) e^{- ((α_{49} + α_{50}) / α_{51})}}{α_{51} + (α_{49} + α_{50}) ((α_{49} + α_{50}) / α_{51})} = L_{14}, \\ z_{n} \geq \frac{(α_{52} + α_{53}) e^{- ((α_{52} + α_{53}) / α_{54})}}{α_{54} + (α_{52} + α_{53}) ((α_{46} + α_{47}) / α_{48})} = L_{15}, \\ n = 2,3, \dots . \end{matrix}$

So, from (21) and (22), one gets $\begin{matrix} (23) & L_{13} \leq x_{n} \leq U_{13}, \\ L_{14} \leq y_{n} \leq U_{14}, \\ L_{15} \leq z_{n} \leq U_{15}, \\ n = 3,4, \dots . \end{matrix}$

(vi)

If ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ is a positive solution of (8), then $\begin{matrix} (24) & x_{n} \leq \frac{α_{55} + α_{56}}{α_{57}} = U_{16}, \\ y_{n} \leq \frac{α_{58} + α_{59}}{α_{60}} = U_{17}, \\ z_{n} \leq \frac{α_{61} + α_{62}}{α_{63}} = U_{18}, \\ n = 1,2, \dots . \end{matrix}$

From (8) and (24), one gets $\begin{matrix} (25) & x_{n} \geq \frac{(α_{55} + α_{56}) e^{- ((α_{61} + α_{62}) / α_{63})}}{α_{57} + (α_{55} + α_{56}) ((α_{55} + α_{56}) / α_{57})} = L_{16}, \\ y_{n} \geq \frac{(α_{58} + α_{59}) e^{- ((α_{55} + α_{56}) / α_{57})}}{α_{60} + (α_{58} + α_{59}) ((α_{58} + α_{59}) / α_{60})} = L_{17}, \\ z_{n} \geq \frac{(α_{61} + α_{62}) e^{- ((α_{58} + α_{59}) / α_{60})}}{α_{63} + (α_{61} + α_{62}) ((α_{61} + α_{62}) / α_{63})} = L_{18}, \\ n = 2,3, \dots . \end{matrix}$

So, from (24) and (25), one gets $\begin{matrix} (26) & L_{16} \leq x_{n} \leq U_{16}, \\ L_{17} \leq y_{n} \leq U_{17}, \\ L_{18} \leq z_{n} \leq U_{18}, \\ n = 3,4, \dots . \end{matrix}$

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

Theorem 2.

If ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ is a positive solution of systems (3)–(8), then their corresponding invariant rectangles, respectively, are $[L_{1}, U_{1}] \times [L_{2}, U_{2}] \times [L_{3}, U_{3}], [L_{4}, U_{4}] \times [L_{5}, U_{5}] \times [L_{6}, U_{6}]$ , $[L_{7}, U_{7}] \times [L_{8}, U_{8}] \times [L_{9}, U_{9}]$ , $[L_{10}, U_{10}] \times [L_{11}, U_{11}] \times [L_{12}, U_{12}]$ , $[L_{13}, U_{13}] \times [L_{14}, U_{14}] \times [L_{15}, U_{15}]$ , and $[L_{16}, U_{16}] \times [L_{17}, U_{17}] \times [L_{18}, U_{18}]$ .

Proof.

If ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ is a positive solution with $x_{0}, x_{- 1} \in [L_{1}, U_{1}], y_{0}, y_{- 1} \in [L_{2}, U_{2}]$ , and $z_{0}, z_{- 1} \in [L_{3}, U_{3}]$ , then from (3), one has $\begin{matrix} (27) & x_{1} = \frac{α_{10} e^{- y_{0}} + α_{11} e^{- y_{- 1}}}{α_{12} + α_{10} y_{0} + α_{11} y_{- 1}} \leq \frac{α_{10} + α_{11}}{α_{12}}, \\ x_{1} = \frac{α_{10} e^{- y_{0}} + α_{11} e^{- y_{- 1}}}{α_{12} + α_{10} y_{0} + α_{11} y_{- 1}} \geq \frac{(α_{10} + α_{11}) e^{- ((α_{13} + α_{14}) / α_{15})}}{α_{12} + (α_{10} + α_{11}) ((α_{13} + α_{14}) / α_{15})}, \\ y_{1} = \frac{α_{13} e^{- z_{0}} + α_{14} e^{- z_{- 1}}}{α_{15} + α_{13} z_{0} + α_{14} z_{- 1}} \leq \frac{α_{13} + α_{14}}{α_{15}}, \\ y_{1} = \frac{α_{13} e^{- z_{0}} + α_{14} e^{- z_{- 1}}}{α_{15} + α_{13} z_{0} + α_{14} z_{- 1}} \geq \frac{(α_{13} + α_{14}) e^{- ((α_{16} + α_{17}) / α_{18})}}{α_{15} + (α_{13} + α_{14}) ((α_{16} + α_{17}) / α_{18})}, \\ z_{1} = \frac{α_{16} e^{- x_{0}} + α_{17} e^{- x_{- 1}}}{α_{18} + α_{16} x_{0} + α_{17} x_{- 1}} \leq \frac{α_{16} + α_{17}}{α_{18}}, \\ z_{1} = \frac{α_{16} e^{- x_{0}} + α_{17} e^{- x_{- 1}}}{α_{18} + α_{16} x_{0} + α_{17} x_{- 1}} \geq \frac{(α_{16} + α_{17}) e^{- ((α_{10} + α_{11}) / α_{12})}}{α_{18} + (α_{16} + α_{17}) ((α_{10} + α_{11}) / α_{12})} . \end{matrix}$

Hence (27) then implies that $x_{1} \in [L_{1}, U_{1}], y_{1} \in [L_{2}, U_{2}], and z_{1} \in [L_{3}, U_{3}]$ . Finally from (3), it is easy to establish that $x_{k + 1} \in [L_{1}, U_{1}] (resp ., y_{k + 1} \in [L_{2}, U_{2}] and z_{k + 1} \in [L_{3}, U_{3}])$ if $x_{k} \in [L_{1}, U_{1}] (resp ., y_{k} \in [L_{2}, U_{2}] and z_{k} \in [L_{3}, U_{3}])$ .

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: $ϒ_{1} = (\bar{x}, \bar{y}, \bar{z}) \in [L_{1}, U_{1}] \times [L_{2}, U_{2}] \times [L_{3}, U_{3}]$ if $\begin{matrix} (28) & \frac{(α_{13} + α_{14}) e^{- L_{1}} ((U_{1} + 1) (α_{13} + α_{14}) + α_{15}) (α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- U_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1}))}}{{(α_{15} + (α_{13} + α_{14}) z)}^{2} {(α_{18} + (α_{16} + α_{17}))}^{2}} \\ \times \frac{((α_{10} + α_{11}) ((((α_{13} + α_{14}) e^{- L_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1})) + 1) + α_{12})}{{(α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- L_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1})))}^{2}} (α_{16} + α_{17}) e^{- ((α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- L_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1}))} / (α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- L_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1}))))} \\ \times ((α_{16} + α_{17}) (Λ_{1} + 1) + α_{18}) < Λ_{1}^{2}, \end{matrix}$

where $\begin{matrix} (29) & Λ_{1} = \frac{(α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- U_{3}}) / (α_{15} + (α_{13} + α_{14}) L_{3}))}}{α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- L_{3}}) / (α_{15} + (α_{13} + α_{14}) L_{3}))}; \end{matrix}$

(ii)

System (4) has a unique positive fixed point: $ϒ_{2} = (\bar{x}, \bar{y}, \bar{z}) \in [L_{4}, U_{4}] \times [L_{5}, U_{5}] \times [L_{6}, U_{6}]$ if $\begin{matrix} (30) & \frac{(α_{19} + α_{20}) e^{- L_{6}} ((U_{6} + 1) (α_{19} + α_{20}) + α_{21}) (α_{22} + α_{23}) e^{- (((α_{19} + α_{20}) e^{- U_{6}}) / (α_{21} + (α_{19} + α_{20}) L_{6}))}}{{(α_{21} + (α_{19} + α_{20}) U_{6})}^{2}} \\ \times \frac{((α_{22} + α_{23}) ((((α_{19} + α_{20}) e^{- L_{6}}) / (α_{21} + (α_{19} + α_{20}) L_{6})) + 1) + α_{24}) (α_{25} + α_{26}) e^{- (((α_{22} + α_{23}) e^{- (((α_{19} + α_{20}) e^{- L_{6}}) / (α_{21} + (α_{19} + α_{20}) L_{6}))}) / ((α_{24} + (α_{22} + α_{23})) (((α_{19} + α_{20}) e^{- L_{6}}) / (α_{21} + (α_{19} + α_{20}) L_{6}))))}}{{(α_{24} + (α_{22} + α_{23}) (((α_{19} + α_{20}) e^{- L_{6}}) / (α_{21} + (α_{19} + α_{20}) L_{6})))}^{2}} \\ \times \frac{((α_{25} + α_{26}) (Λ_{2} + 1) + α_{27})}{{(α_{27} + (α_{25} + α_{26}))}^{2}} < Λ_{2}^{2}, \end{matrix}$

where $\begin{matrix} (31) & Λ_{2} = \frac{(α_{22} + α_{23}) e^{- (((α_{19} + α_{20}) e^{- U_{6}}) / (α_{21} + (α_{19} + α_{20}) L_{6}))}}{α_{24} + (α_{22} + α_{23}) (((α_{19} + α_{20}) e^{- L_{6}}) / (α_{21} + (α_{19} + α_{20}) L_{6}))}; \end{matrix}$

(iii)

System (5) has a unique positive fixed point: $ϒ_{3} = (\bar{x}, \bar{y}, \bar{z}) \in [L_{7}, U_{7}] \times [L_{8}, U_{8}] \times [L_{9}, U_{9}]$ if $\begin{matrix} (32) & \frac{e^{- ((e^{- U_{8}} / L_{8}) - (α_{33} / (α_{31} + α_{32})))} e^{- L_{8}} (U_{8} + 1) ((e^{- L_{8}} / L_{8}) - (α_{33} / (α_{31} + α_{32})) + 1)}{{((e^{- L_{8}} / L_{8}) - (α_{33} / (α_{31} + α_{32})))}^{2} U_{8}^{2}} e^{- ((e^{- ((e^{- L_{8}} / U_{8}) - (α_{33} / (α_{31} + α_{32})))} / (e^{- L_{8}} / L_{8}) - (α_{33} / (α_{31} + α_{32}))) - (α_{36} / (α_{34} + α_{35})))} < Λ_{3}^{2}, \end{matrix}$

where $\begin{matrix} (33) & Λ_{3} = \frac{e^{- ((e^{- U_{8}} / L_{8}) - (α_{33} / (α_{31} + α_{32})))}}{((e^{- L_{8}} / L_{8}) - (α_{33} / (α_{31} + α_{32})))} - \frac{α_{36}}{α_{34} + α_{35}}; \end{matrix}$

(iv)

System (6) has a unique positive fixed point: $ϒ_{4} = (\bar{x}, \bar{y}, \bar{z}) \in [L_{10}, U_{10}] \times [L_{11}, U_{11}] \times [L_{12}, U_{12}]$ if $\begin{matrix} (34) & \frac{e^{- ((e^{- U_{10}} / L_{10}) - (α_{39} / (α_{37} + α_{38})))} e^{- L_{10}} (U_{10} + 1) ((e^{- L_{10}} / L_{10}) - (α_{39} / (α_{37} + α_{38})) + 1)}{{((e^{- L_{10}} / L_{10}) - (α_{39} / (α_{37} + α_{38})))}^{2} U_{10}^{2}} e^{- ((e^{- ((e^{- L_{10}} / U_{10}) - (α_{39} / (α_{37} + α_{38})))} / (e^{- L_{10}} / L_{10}) - (α_{39} / (α_{37} + α_{38}))) - (α_{42} / (α_{40} + α_{41})))} < Λ_{4}^{2}, \end{matrix}$

where $\begin{matrix} (35) & Λ_{4} = \frac{e^{- ((e^{- U_{10}} / L_{10}) - (α_{39} / (α_{37} + α_{38})))}}{((e^{- L_{10}} / L_{10}) - (α_{39} / (α_{37} + α_{38})))} - \frac{α_{42}}{α_{40} + α_{41}}; \end{matrix}$

(v)

System (7) has a unique positive fixed point: $ϒ_{5} = (\bar{x}, \bar{y}, \bar{z}) \in [L_{13}, U_{13}] \times [L_{14}, U_{14}] \times [L_{15}, U_{15}]$ if $\begin{matrix} (36) & \frac{e^{- ((e^{- U_{13}} / L_{13}) - (α_{48} / (α_{46} + α_{47})))} e^{- L_{13}} (U_{13} + 1) ((e^{- L_{13}} / L_{13}) - (α_{48} / (α_{46} + α_{47})) + 1)}{{((e^{- L_{13}} / L_{13}) - (α_{48} / (α_{46} + α_{47})))}^{2} U_{13}^{2}} e^{- ((e^{- ((e^{- L_{13}} / U_{13}) - (α_{48} / (α_{46} + α_{47})))} / (e^{- L_{13}} / L_{13}) - (α_{48} / (α_{46} + α_{47}))) - (α_{54} / (α_{52} + α_{53})))} < Λ_{5}^{2}, \end{matrix}$

where $\begin{matrix} (37) & Λ_{5} = \frac{e^{- ((e^{- U_{13}} / L_{13}) - (α_{48} / (α_{46} + α_{47})))}}{(((e^{- L_{13}} / L_{13}) - (α_{48} / (α_{46} + α_{47}))))} - \frac{α_{54}}{α_{52} + α_{53}}; \end{matrix}$

(vi)

System (8) has a unique positive fixed point: $ϒ_{6} = (\bar{x}, \bar{y}, \bar{z}) \in [L_{16}, U_{16}] \times [L_{17}, U_{17}] \times [L_{18}, U_{18}]$ if $\begin{matrix} (38) & \frac{(α_{57} + 2 U_{16} (α_{55} + α_{56})) (α_{63} + 2 U_{16} (α_{61} + α_{62}) \ln ((α_{55} + α_{56}) / (L_{16} (α_{57} + (α_{55} + α_{56}) L_{16}))))}{(α_{63} \ln (α_{55} + α_{56}) / L_{16} (α_{57} + (α_{55} + α_{56}) L_{16}) + (α_{61} + α_{62}) {(\ln ((α_{55} + α_{56}) / (L_{16} (α_{57} + (α_{55} + α_{56}) L_{16}))))}^{2})} \\ \times (\frac{α_{57} + 2 U_{16} (α_{55} + α_{56}) (α_{63} + 2 (α_{61} + α_{62})) \ln ((α_{55} + α_{56}) / (L_{16} (α_{57} + (α_{55} + α_{56}) L_{16})))}{α_{63} \ln ((α_{55} + α_{56}) / (L_{16} (α_{57} + (α_{55} + α_{56}) L_{16})))} + Λ_{6}) \\ \times \frac{α_{60} (α_{58} + α_{59})}{L_{16} (α_{57} + (α_{55} + α_{56}) L_{16}) (α_{60} + (α_{58} + α_{59}))} < Λ_{6}, \end{matrix}$

where $\begin{matrix} (39) & Λ_{6} = ln (\frac{α_{61} + α_{62}}{ln ((α_{55} + α_{56}) / (L_{16} (α_{57} + (α_{55} + α_{56}) L_{16}))) (α_{63} + (α_{61} + α_{62}) ln ((α_{55} + α_{56}) / (L_{16} (α_{57} + (α_{55} + α_{56}) L_{16})))}) . \end{matrix}$

Proof.

(i)

From (3), one has

\begin{matrix} (40) & x = \frac{(α_{10} + α_{11}) e^{- y}}{α_{12} + (α_{10} + α_{11}) y}, \\ y = \frac{(α_{13} + α_{14}) e^{- z}}{α_{15} + (α_{13} + α_{14}) z}, \\ z = \frac{(α_{16} + α_{17}) e^{- x}}{α_{18} + (α_{16} + α_{17}) x} . \end{matrix}

From (40), one gets $\begin{matrix} (41) & z = \frac{(α_{16} + α_{17}) e^{- x}}{α_{18} + (α_{16} + α_{17}) x}, \\ x = \frac{(α_{10} + α_{11}) e^{- y}}{α_{12} + (α_{10} + α_{11}) y}, \\ y = \frac{(α_{13} + α_{14}) e^{- z}}{α_{15} + (α_{13} + α_{14}) z} . \end{matrix}$

From (41), setting $\begin{matrix} (42) & y = g (z) = \frac{(α_{13} + α_{14}) e^{- z}}{α_{15} + (α_{13} + α_{14}) z}, \\ x = f (y) = \frac{(α_{10} + α_{11}) e^{- y}}{α_{12} + (α_{10} + α_{11}) y} . \end{matrix}$

Denote $\begin{matrix} (43) & F (z) ≔ \frac{(α_{16} + α_{17}) e^{- k (z)}}{α_{18} + (α_{16} + α_{17}) k (z)} - z, \end{matrix}$ where $\begin{matrix} (44) & k (z) ≔ x = f (g (z)), \\ = f (\frac{(α_{13} + α_{14}) e^{- z}}{α_{15} + (α_{13} + α_{14}) z}), \\ = \frac{(α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z))}}{α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z))}, \end{matrix}$ and $z \in [L_{3}, U_{3}]$ . Here, our finding is that $F (z) = 0$ has a unique solution, where $z \in [L_{3}, U_{3}]$ . From (43) and (44), one gets $\begin{matrix} (45) & F^{'} (z) = - \frac{k^{'} (z) (α_{16} + α_{17}) e^{- k (z)} ((k (z) + 1) (α_{16} + α_{17}) + α_{18})}{{(α_{18} + (α_{16} + α_{17}) k (z))}^{2}} - 1, \end{matrix}$ where $\begin{matrix} (46) & k^{'} (z) = - \frac{(α_{13} + α_{14}) e^{- z} ((z + 1) (α_{13} + α_{14}) + α_{15}) (α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z))}}{{(α_{15} + (α_{13} + α_{14}) z)}^{2} {(α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z)))}^{2}} \\ \times ((α_{10} + α_{11}) (\frac{(α_{13} + α_{14}) e^{- z}}{α_{15} + (α_{13} + α_{14}) z} + 1) + α_{12}) . \end{matrix}$

Using (44) and (46) in (45), we obtain $\begin{matrix} (47) & F^{'} (z) = \frac{(α_{13} + α_{14}) e^{- z} ((z + 1) (α_{13} + α_{14}) + α_{15}) (α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z))}}{{(α_{15} + (α_{13} + α_{14}) z)}^{2} {(α_{18} + (α_{16} + α_{17}) (((α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z))}) / (α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z)))))}^{2}} \times \\ \frac{((α_{10} + α_{11}) ((((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z)) + 1) + α_{12})}{{(α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z)))}^{2}} (α_{16} + α_{17}) e^{- (((α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z))}) / (α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z))))} \times \\ ((α_{16} + α_{17}) (\frac{(α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z))}}{α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- z}) / (α_{15} + (α_{13} + α_{14}) z))} + 1) + α_{18}) - 1 \\ \leq \frac{(α_{13} + α_{14}) e^{- L_{1}} ((U_{1} + 1) (α_{13} + α_{14}) + α_{15}) (α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- U_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1}))}}{{(α_{15} + (α_{13} + α_{14}) L_{1})}^{2} {(α_{18} + (α_{16} + α_{17}) Λ_{1})}^{2}} \times \\ \frac{((α_{10} + α_{11}) ((((α_{13} + α_{14}) e^{- L_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1})) + 1) + α_{12})}{{(α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- L_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1})))}^{2}} (α_{16} + α_{17}) e^{- ((α_{10} + α_{11}) e^{- (((α_{13} + α_{14}) e^{- L_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1}))} / α_{12} + (α_{10} + α_{11}) (((α_{13} + α_{14}) e^{- L_{1}}) / (α_{15} + (α_{13} + α_{14}) L_{1})))} \times \\ ((α_{16} + α_{17}) (Λ_{1} + 1) + α_{18}) - 1, \end{matrix}$ where $Λ_{1}$ 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 about $ϒ_{i} (i = 1, \dots, 6)$ , respectively, of systems (3) to (8) is explored in this section, as follows.

Theorem 4.

For $ϒ_{1}$ of (3), the following holds:

(i)

$ϒ_{1}$ is a sink if

\begin{matrix} (48) & \frac{α_{10} α_{13} (α_{14} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} \\ + \frac{α_{10} α_{14} (α_{13} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} \\ + \frac{(α_{11} α_{13} + α_{10} α_{14}) (α_{16} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} < 1; \end{matrix}

(ii)

$ϒ_{1}$ is a source if

\begin{matrix} (49) & \frac{α_{10} α_{13} (α_{14} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} \\ + \frac{α_{10} α_{14} (α_{13} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} \\ + \frac{(α_{11} α_{13} + α_{10} α_{14}) (α_{16} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} > 1. \end{matrix}

Proof.

(i)

If $ϒ_{1}$ is a fixed point of (3), then $\begin{matrix} (50) & \bar{x} = \frac{(α_{10} + α_{11}) e^{- \bar{y}}}{α_{12} + (α_{10} + α_{11}) \bar{y}}, \\ \bar{y} = \frac{(α_{13} + α_{14}) e^{- \bar{z}}}{α_{15} + (α_{13} + α_{14}) \bar{z}}, \\ \bar{z} = \frac{(α_{16} + α_{17}) e^{- \bar{x}}}{α_{18} + (α_{16} + α_{17}) \bar{x}} . \end{matrix}$

Moreover, we have the following map for constructing the corresponding linearized form of (3): $\begin{matrix} (51) & (x_{n + 1}, x_{n}, y_{n + 1}, y_{n}, z_{n + 1}, z_{n}) ⟶ (f, f_{1}, g, g_{1}, h, h_{1}), \end{matrix}$

where $\begin{matrix} (52) & f = \frac{α_{10} e^{- y_{n}} + α_{11} e^{- y_{n - 1}}}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}}, \\ f_{1} = x_{n}, \\ g = \frac{α_{13} e^{- z_{n}} + α_{14} e^{- z_{n - 1}}}{α_{15} + α_{13} z_{n} + α_{14} z_{n - 1}}, \\ g_{1} = y_{n}, \\ h = \frac{α_{16} e^{- x_{n}} + α_{17} e^{- x_{n - 1}}}{α_{18} + α_{16} x_{n} + α_{17} x_{n - 1}}, \\ h_{1} = z_{n} . \end{matrix}$

The ${J|}_{ϒ_{1}}$ about $ϒ_{1}$ subject to the map (51) is $\begin{matrix} (53) & {J|}_{Υ_{1}} = (\begin{matrix} 0 & 0 & b_{13} & b_{14} & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & b_{35} & b_{36} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ b_{51} & b_{52} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}), \end{matrix}$

where $\begin{matrix} (54) & b_{13} = - \frac{α_{10} (e^{- \bar{y}} + \bar{x})}{α_{12} + (α_{10} + α_{11}) \bar{y}}, \\ b_{14} = - \frac{α_{11} (e^{- \bar{y}} + \bar{x})}{α_{12} + (α_{10} + α_{11}) \bar{y}}, \\ b_{35} = - \frac{α_{13} (e^{- \bar{z}} + \bar{y})}{α_{15} + (α_{13} + α_{14}) \bar{z}}, \\ b_{36} = - \frac{α_{14} (e^{- \bar{z}} + \bar{y})}{α_{15} + (α_{13} + α_{14}) \bar{z}}, \\ b_{51} = - \frac{α_{16} (e^{- \bar{x}} + \bar{z})}{α_{18} + (α_{16} + α_{17}) \bar{x}}, \\ b_{52} = - \frac{α_{17} (e^{- \bar{x}} + \bar{z})}{α_{18} + (α_{16} + α_{17}) \bar{x}} . \end{matrix}$

The auxiliary equation of ${J|}_{ϒ_{1}}$ about $ϒ_{1}$ is $\begin{matrix} (55) & P (λ) = λ^{6} + L_{1} λ^{3} + L_{2} λ^{2} + L_{3} λ + L_{4} = 0, \end{matrix}$

where $\begin{matrix} (56) & L_{1} = - b_{13} b_{35} b_{51}, \\ L_{2} = - b_{13} b_{35} b_{52} - b_{13} b_{36} b_{51} - b_{14} b_{35} b_{51}, \\ L_{3} = - b_{13} b_{36} b_{52} - b_{14} b_{35} b_{52} - b_{14} b_{36} b_{51}, \\ L_{4} = - b_{14} b_{36} b_{52} . \end{matrix}$

Now,

\begin{matrix} (57) & \sum_{i = 1}^{4} |L_{i}| = \frac{α_{10} α_{13} (α_{14} + α_{17}) (e^{- \bar{x}} + \bar{z}) (e^{- \bar{y}} + \bar{x}) (e^{- \bar{z}} + \bar{y})}{(α_{12} + (α_{10} + α_{11}) \bar{y}) (α_{15} + (α_{13} + α_{14}) \bar{z}) (α_{18} + (α_{16} + α_{17}) \bar{x})} \\ + \frac{α_{10} α_{14} (α_{13} + α_{17}) (e^{- \bar{x}} + \bar{z}) (e^{- \bar{y}} + \bar{x}) (e^{- \bar{z}} + \bar{y})}{(α_{12} + (α_{10} + α_{11}) \bar{y}) (α_{15} + (α_{13} + α_{14}) \bar{z}) (α_{18} + (α_{16} + α_{17}) \bar{x})} \\ + \frac{α_{11} α_{13} (α_{16} + α_{17}) (e^{- \bar{x}} + \bar{z}) (e^{- \bar{y}} + \bar{x}) (e^{- \bar{z}} + \bar{y})}{(α_{12} + (α_{10} + α_{11}) \bar{y}) (α_{15} + (α_{13} + α_{14}) \bar{z}) (α_{18} + (α_{16} + α_{17}) \bar{x})} \\ + \frac{α_{10} α_{14} (α_{16} + α_{17}) (e^{- \bar{x}} + \bar{z}) (e^{- \bar{y}} + \bar{x}) (e^{- \bar{z}} + \bar{y})}{(α_{12} + (α_{10} + α_{11}) \bar{y}) (α_{15} + (α_{13} + α_{14}) \bar{z}) (α_{18} + (α_{16} + α_{17}) \bar{x})} \\ \leq \frac{α_{10} α_{13} (α_{14} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} \\ + \frac{α_{10} α_{14} (α_{13} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} \\ + \frac{α_{11} α_{13} (α_{16} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} \\ + \frac{α_{10} α_{14} (α_{16} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} \\ = \frac{α_{10} α_{13} (α_{14} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} \\ + \frac{α_{10} α_{14} (α_{13} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} \\ + \frac{(α_{11} α_{13} + α_{10} α_{14}) (α_{16} + α_{17}) (e^{- L_{1}} + L_{3}) (e^{- L_{2}} + L_{1}) (e^{- L_{3}} + L_{2})}{(α_{12} + (α_{10} + α_{11}) L_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})} . \end{matrix}

Assuming that (48) holds, and then from (57), one gets $\sum_{i = 1}^{4} |L_{i}| < 1$ . Hence, $ϒ_{1}$ of (3) is a sink.

(ii)

Using similar manipulation as in the proof of (i), one has

\begin{matrix} (58) & \sum_{i = 1}^{4} |L_{i}| \geq \frac{α_{10} α_{13} (α_{14} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} \\ + \frac{α_{10} α_{14} (α_{13} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} \\ + \frac{α_{11} α_{13} (α_{16} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} \\ + \frac{α_{10} α_{14} (α_{16} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} \\ = \frac{α_{10} α_{13} (α_{14} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} \\ + \frac{α_{10} α_{14} (α_{13} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} \\ + \frac{(α_{11} α_{13} + α_{10} α_{14}) (α_{16} + α_{17}) (e^{- U_{1}} + U_{3}) (e^{- U_{2}} + U_{1}) (e^{- U_{3}} + U_{2})}{(α_{12} + (α_{10} + α_{11}) U_{2}) (α_{15} + (α_{13} + α_{14}) U_{3}) (α_{18} + (α_{16} + α_{17}) U_{1})} . \end{matrix}

Assuming that (49) holds, and then from (58), one gets $\sum_{i = 1}^{4} |L_{i}| > 1$ . Hence, $ϒ_{1}$ of (3) is a source.

In similar manner, one can explore the local dynamics about $Y_{i} (i = 2, \dots, 6)$ , respectively, of systems (4)–(8), as follows.

Theorem 5.

(i)

For $ϒ_{2}$ of (4), the following holds:

(i.1) $ϒ_{2}$ is a sink if

\begin{matrix} (59) & \frac{α_{19} α_{25} (α_{22} + α_{23}) (e^{- L_{4}} + L_{5}) (e^{- L_{6}} + L_{4}) (e^{- L_{4}} + L_{6})}{(α_{27} + (α_{25} + α_{26}) L_{5}) (α_{21} + (α_{19} + α_{20}) L_{6}) (α_{24} + (α_{22} + α_{23}) L_{4})} \\ + \frac{α_{19} α_{26} (α_{22} + α_{23}) (e^{- L_{4}} + L_{5}) (e^{- L_{5}} + L_{6}) (e^{- L_{6}} + L_{4})}{(α_{27} + (α_{25} + α_{26}) L_{5}) (α_{21} + (α_{19} + α_{20}) L_{6}) (α_{24} + (α_{22} + α_{23}) L_{4})} \\ + \frac{(α_{20} α_{22} + α_{20} α_{23}) (α_{26} + α_{25}) (e^{- L_{4}} + L_{5}) (e^{- L_{5}} + L_{6}) (e^{- L_{6}} + L_{4})}{(α_{27} + (α_{25} + α_{26}) L_{5}) (α_{21} + (α_{19} + α_{20}) L_{6}) (α_{24} + (α_{22} + α_{23}) L_{4})} < 1. \end{matrix}

(i.2) $ϒ_{2}$ is a source if

\begin{matrix} (60) & \frac{α_{19} α_{25} (α_{22} + α_{23}) (e^{- U_{4}} + U_{5}) (e^{- U_{6}} + U_{4}) (e^{- U_{4}} + U_{6})}{(α_{27} + (α_{25} + α_{26}) U_{5}) (α_{21} + (α_{19} + α_{20}) U_{6}) (α_{24} + (α_{22} + α_{23}) U_{4})} \\ + \frac{α_{19} α_{26} (α_{22} + α_{23}) (e^{- U_{4}} + U_{5}) (e^{- U_{5}} + U_{6}) (e^{- U_{6}} + U_{4})}{(α_{27} + (α_{25} + α_{26}) U_{5}) (α_{21} + (α_{19} + α_{20}) U_{6}) (α_{24} + (α_{22} + α_{23}) U_{4})} \\ + \frac{(α_{20} α_{22} + α_{20} α_{23}) (α_{26} + α_{25}) (e^{- U_{4}} + U_{5}) (e^{- U_{5}} + U_{6}) (e^{- U_{6}} + U_{4})}{(α_{27} + (α_{25} + α_{26}) U_{5}) (α_{21} + (α_{19} + α_{20}) U_{6}) (α_{24} + (α_{22} + α_{23}) U_{4})} > 1, \end{matrix}

with $\begin{matrix} (61) & {J|}_{ϒ_{2}} = (\begin{matrix} 0 & 0 & 0 & 0 & b_{15} & b_{16} \\ 1 & 0 & 0 & 0 & 0 & 0 \\ b_{31} & b_{32} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & b_{53} & b_{54} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}), \end{matrix}$

where $\begin{matrix} (62) & b_{15} = - \frac{α_{19} (e^{- \bar{x}} + \bar{y})}{α_{21} + (α_{19} + α_{20}) \bar{x}}, \\ b_{16} = - \frac{α_{20} (e^{- \bar{x}} + \bar{y})}{α_{21} + (α_{19} + α_{20}) \bar{x}}, \\ b_{31} = - \frac{α_{22} (e^{- \bar{z}} + \bar{x})}{α_{24} + (α_{22} + α_{23}) \bar{z}}, \\ b_{32} = - \frac{α_{23} (e^{- \bar{z}} + \bar{x})}{α_{24} + (α_{22} + α_{23}) \bar{z}}, \\ b_{53} = - \frac{α_{25} (e^{- \bar{y}} + \bar{z})}{α_{27} + (α_{25} + α_{26}) \bar{y}}, \\ b_{54} = - \frac{α_{26} (e^{- \bar{y}} + \bar{z})}{α_{27} + (α_{25} + α_{26}) \bar{y}}; \end{matrix}$

(ii)

For $ϒ_{3}$ of (5), the following holds:

(ii.1) $ϒ_{3}$ is a sink if

\begin{matrix} (63) & \frac{(α_{28} + α_{29}) (e^{- L_{7}} + L_{7})}{α_{30} + (α_{28} + α_{29}) L_{7}} + \frac{(α_{31} + α_{32}) e^{- L_{8}}}{α_{33} + (α_{31} + α_{32}) L_{9}} + \frac{(α_{34} + α_{35}) e^{- L_{9}}}{α_{36} + (α_{34} + α_{35}) L_{8}} \\ + \frac{(α_{34} α_{28} + α_{29} α_{34} + α_{35} α_{28} + α_{29} α_{35} + α_{29} α_{32}) e^{- L_{7} - L_{9}}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{31} α_{34} + α_{35} α_{31} + α_{32} α_{35}) e^{- L_{8} - L_{9}}}{(α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{28} α_{34} + α_{29} α_{34} + α_{28} α_{35} + α_{29} α_{32} + α_{29} α_{35}) L_{7} e^{- L_{9}}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{31} α_{34} + α_{32} α_{34} + α_{31} α_{35} + α_{32} α_{35}) L_{7} L_{8}}{(α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{28} α_{31} α_{34} + α_{29} α_{31} α_{34} + α_{28} α_{32} α_{34} + α_{28} α_{35} α_{31}) e^{- L_{7} - L_{8} - L_{9}}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{29} α_{32} α_{34} + α_{29} α_{31} α_{35} + α_{28} α_{32} α_{35} + α_{29} α_{32} α_{35}) e^{- L_{7} - L_{8} - L_{9}}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{28} α_{31} α_{34} + α_{29} α_{31} α_{34} + α_{28} α_{32} α_{34} + α_{28} α_{35} α_{31}) L_{7} L_{8} L_{9}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{29} α_{32} α_{34} + α_{29} α_{31} α_{35} + α_{28} α_{32} α_{35} + α_{29} α_{32} α_{35}) L_{7} L_{8} L_{9}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{28} α_{31} α_{34} + α_{29} α_{31} α_{34} + α_{28} α_{32} α_{34} + α_{28} α_{35} α_{31}) e^{- L_{7}} L_{8} L_{9}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{29} α_{32} α_{34} + α_{29} α_{31} α_{35} + α_{28} α_{32} α_{35} + α_{29} α_{32} α_{35}) e^{- L_{7}} L_{8} L_{9}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{28} α_{31} α_{34} + α_{29} α_{31} α_{34} + α_{28} α_{32} α_{34} + α_{28} α_{35} α_{31}) L_{7} e^{- L_{8} - L_{9}}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} \\ + \frac{(α_{29} α_{32} α_{34} + α_{29} α_{31} α_{35} + α_{28} α_{32} α_{35} + α_{29} α_{32} α_{35}) L_{7} e^{- L_{8} - L_{9}}}{(α_{30} + (α_{28} + α_{29}) L_{7}) (α_{33} + (α_{31} + α_{32}) L_{9}) (α_{36} + (α_{34} + α_{35}) L_{8})} < 1. \end{matrix}

(ii.2) $ϒ_{3}$ is a source if

\begin{matrix} (64) & \frac{(α_{28} + α_{29}) (e^{- U_{7}} + U_{7})}{α_{30} + (α_{28} + α_{29}) U_{7}} + \frac{(α_{31} + α_{32}) e^{- U_{8}}}{α_{33} + (α_{31} + α_{32}) U_{9}} + \frac{(α_{34} + α_{35}) e^{- U_{9}}}{α_{36} + (α_{34} + α_{35}) U_{8}} \\ + \frac{(α_{34} α_{28} + α_{29} α_{34} + α_{35} α_{28} + α_{29} α_{35} + α_{29} α_{32}) e^{- U_{7} - U_{9}}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{31} α_{34} + α_{35} α_{31} + α_{32} α_{35}) e^{- U_{8} - U_{9}}}{(α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{28} α_{34} + α_{29} α_{34} + α_{28} α_{35} + α_{29} α_{32} + α_{29} α_{35}) U_{7} e^{- U_{9}}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{31} α_{34} + α_{32} α_{34} + α_{31} α_{35} + α_{32} α_{35}) U_{7} U_{8}}{(α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{28} α_{31} α_{34} + α_{29} α_{31} α_{34} + α_{28} α_{32} α_{34} + α_{28} α_{35} α_{31}) e^{- U_{7} - U_{8} - U_{9}}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{29} α_{32} α_{34} + α_{29} α_{31} α_{35} + α_{28} α_{32} α_{35} + α_{29} α_{32} α_{35}) e^{- U_{7} - U_{8} - U_{9}}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{28} α_{31} α_{34} + α_{29} α_{31} α_{34} + α_{28} α_{32} α_{34} + α_{28} α_{35} α_{31}) U_{7} U_{8} U_{9}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{29} α_{32} α_{34} + α_{29} α_{31} α_{35} + α_{28} α_{32} α_{35} + α_{29} α_{32} α_{35}) U_{7} U_{8} U_{9}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{28} α_{31} α_{34} + α_{29} α_{31} α_{34} + α_{28} α_{32} α_{34} + α_{28} α_{35} α_{31}) e^{- U_{7}} U_{8} U_{9}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{29} α_{32} α_{34} + α_{29} α_{31} α_{35} + α_{28} α_{32} α_{35} + α_{29} α_{32} α_{35}) e^{- U_{7}} U_{8} U_{9}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{28} α_{31} α_{34} + α_{29} α_{31} α_{34} + α_{28} α_{32} α_{34} + α_{28} α_{35} α_{31}) U_{7} e^{- U_{8} - U_{9}}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} \\ + \frac{(α_{29} α_{32} α_{34} + α_{29} α_{31} α_{35} + α_{28} α_{32} α_{35} + α_{29} α_{32} α_{35}) U_{7} e^{- U_{8} - U_{9}}}{(α_{30} + (α_{28} + α_{29}) U_{7}) (α_{33} + (α_{31} + α_{32}) U_{9}) (α_{36} + (α_{34} + α_{35}) U_{8})} > 1, \end{matrix}

with

\begin{matrix} (65) & {J|}_{ϒ_{3}} = (\begin{matrix} b_{11} & b_{12} & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & b_{33} & b_{34} & b_{35} & b_{36} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & b_{53} & b_{54} & b_{55} & b_{56} \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}), \end{matrix}

where $\begin{matrix} (66) & b_{11} = - \frac{α_{28} (e^{- \bar{x}} + \bar{x})}{α_{30} + (α_{28} + α_{29}) \bar{x}}, \\ b_{12} = - \frac{α_{29} (e^{- \bar{x}} + \bar{x})}{α_{30} + (α_{28} + α_{29}) \bar{x}}, \\ b_{33} = - \frac{α_{31} e^{- \bar{y}}}{α_{33} + (α_{31} + α_{32}) \bar{z}}, \\ b_{34} = - \frac{α_{32} e^{- \bar{y}}}{α_{33} + (α_{31} + α_{32}) \bar{z}}, \\ b_{35} = - \frac{α_{31} \bar{y}}{α_{33} + (α_{31} + α_{32}) \bar{z}}, \\ b_{36} = - \frac{α_{32} \bar{y}}{α_{33} + (α_{31} + α_{32}) \bar{z}}, \\ b_{53} = - \frac{α_{34} \bar{z}}{α_{36} + (α_{34} + α_{35}) \bar{y}}, \\ b_{54} = - \frac{α_{35} \bar{z}}{α_{36} + (α_{34} + α_{35}) \bar{y}}, \\ b_{55} = - \frac{α_{34} e^{- \bar{z}}}{α_{36} + (α_{34} + α_{35}) \bar{y}}, \\ b_{56} = - \frac{α_{35} e^{- \bar{z}}}{α_{36} + (α_{34} + α_{35}) \bar{y}}; \end{matrix}$

(iii)

For $ϒ_{4}$ of (6), the following holds:

(iii.1) $ϒ_{4}$ is a sink if

\begin{matrix} (67) & \frac{(α_{37} + α_{38}) e^{- L_{10}}}{α_{39} + (α_{37} + α_{38}) L_{11}} + \frac{(α_{40} + α_{41}) e^{- L_{11}}}{α_{42} + (α_{40} + α_{41}) L_{10}} \\ + \frac{(α_{43} + α_{44}) (e^{- L_{12}} + L_{12})}{α_{45} + (α_{43} + α_{44}) L_{12}} \\ + \frac{(α_{37} α_{40} + α_{41} α_{37} + α_{38} α_{40} + α_{37} α_{41} + α_{38} α_{41}) e^{- L_{10} - L_{11}}}{(α_{39} + (α_{37} + α_{38}) L_{11}) (α_{42} + (α_{40} + α_{41}) L_{10})} \\ + \frac{(α_{37} α_{40} + α_{38} α_{40} + α_{37} α_{41} + α_{38} α_{41}) L_{10} L_{11}}{(α_{39} + (α_{37} + α_{38}) L_{11}) (α_{42} + (α_{40} + α_{41}) L_{10})} \\ + \frac{(α_{40} α_{43} + α_{41} α_{44} + α_{41} α_{43} + α_{4} α_{44}) e^{- L_{11}} (e^{- L_{12}} + L_{12})}{(α_{42} + (α_{40} + α_{41}) L_{10}) (α_{45} + (α_{43} + α_{44}) L_{12})} \\ + \frac{(α_{37} α_{43} + α_{38} α_{43} + α_{38} α_{44} + α_{37} α_{44}) e^{- L_{10}} (e^{- L_{12}} + L_{10})}{(α_{39} + (α_{37} + α_{38}) L_{11}) (α_{45} + (α_{43} + α_{44}) L_{12})} \\ + [(α_{37} α_{40} α_{43} + α_{38} α_{40} α_{43} + α_{37} α_{41} α_{43} + α_{37} α_{40} α_{44} \\ + α_{38} α_{41} α_{43} + α_{38} α_{40} α_{44} + α_{37} α_{41} α_{44} + α_{38} α_{41} α_{44}) e^{- L_{10} - L_{11} - L_{12}} \\ + (α_{37} α_{40} α_{43} + α_{38} α_{40} α_{43} + α_{37} α_{41} α_{43} + α_{37} α_{40} α_{44} \\ + α_{38} α_{41} α_{43} + α_{38} α_{40} α_{44} + α_{37} α_{41} α_{44} + α_{38} α_{41} α_{44}) L_{10} L_{11} L_{12} \\ + (α_{37} α_{40} α_{43} + α_{38} α_{40} α_{43} + α_{37} α_{41} α_{43} + α_{37} α_{40} α_{44} \\ + α_{38} α_{41} α_{43} + α_{38} α_{40} α_{44} + α_{37} α_{41} α_{44} + α_{38} α_{41} α_{44}) e^{- L_{12}} L_{10} L_{11} \\ + (α_{37} α_{40} α_{43} + α_{38} α_{40} α_{43} + α_{37} α_{41} α_{43} + α_{37} α_{40} α_{44} + α_{38} α_{41} α_{43} \\ + α_{38} α_{40} α_{44} + α_{37} α_{41} α_{44} + α_{38} α_{41} α_{44}) L_{12} e^{- L_{10} - L_{11}}] \\ \frac{1}{(α_{39} + (α_{37} + α_{38}) L_{11}) (α_{42} + (α_{40} + α_{41}) L_{10}) (α_{45} + (α_{43} + α_{44}) L_{12})} < 1. \end{matrix}

(iii.2) $ϒ_{4}$ is a source if

\begin{matrix} (68) & \frac{(α_{37} + α_{38}) e^{- U_{10}}}{α_{39} + (α_{37} + α_{38}) U_{11}} + \frac{(α_{40} + α_{41}) e^{- U_{11}}}{α_{42} + (α_{40} + α_{41}) U_{10}} + \frac{(α_{43} + α_{44}) (e^{- U_{12}} + L_{12})}{α_{45} + (α_{43} + α_{44}) U_{12}} \\ + \frac{(α_{37} α_{40} + α_{41} α_{37} + α_{38} α_{40} + α_{37} α_{41} + α_{38} α_{41}) e^{- U_{10} - U_{11}}}{(α_{39} + (α_{37} + α_{38}) U_{11}) (α_{42} + (α_{40} + α_{41}) U_{10})} \\ + \frac{(α_{37} α_{40} + α_{38} α_{40} + α_{37} α_{41} + α_{38} α_{41}) U_{10} U_{11}}{(α_{39} + (α_{37} + α_{38}) U_{11}) (α_{42} + (α_{40} + α_{41}) U_{10})} \\ + \frac{(α_{40} α_{43} + α_{41} α_{44} + α_{41} α_{43} + α_{4} α_{44}) e^{- U_{11}} (e^{- U_{12}} + U_{12})}{(α_{42} + (α_{40} + α_{41}) U_{10}) (α_{45} + (α_{43} + α_{44}) U_{12})} \\ + \frac{(α_{37} α_{43} + α_{38} α_{43} + α_{38} α_{44} + α_{37} α_{44}) e^{- U_{10}} (e^{- U_{12}} + U_{10})}{(α_{39} + (α_{37} + α_{38}) U_{11}) (α_{45} + (α_{43} + α_{44}) U_{12})} \\ + [(α_{37} α_{40} α_{43} + α_{38} α_{40} α_{43} + α_{37} α_{41} α_{43} + α_{37} α_{40} α_{44} + \\ α_{38} α_{41} α_{43} + α_{38} α_{40} α_{44} + α_{37} α_{41} α_{44} + α_{38} α_{41} α_{44}) e^{- U_{10} - U_{11} - U_{12}} \\ + (α_{37} α_{40} α_{43} + α_{38} α_{40} α_{43} + α_{37} α_{41} α_{43} + α_{37} α_{40} α_{44} \\ + α_{38} α_{41} α_{43} + α_{38} α_{40} α_{44} + α_{37} α_{41} α_{44} + α_{38} α_{41} α_{44}) U_{10} U_{11} U_{12} \\ + (α_{37} α_{40} α_{43} + α_{38} α_{40} α_{43} + α_{37} α_{41} α_{43} + α_{37} α_{40} α_{44} \\ + α_{38} α_{41} α_{43} + α_{38} α_{40} α_{44} + α_{37} α_{41} α_{44} + α_{38} α_{41} α_{44}) e^{- U_{12}} U_{10} U_{11} \\ + (α_{37} α_{40} α_{43} + α_{38} α_{40} α_{43} + α_{37} α_{41} α_{43} + α_{37} α_{40} α_{44} + α_{38} α_{41} α_{43} \\ + α_{38} α_{40} α_{44} + α_{37} α_{41} α_{44} + α_{38} α_{41} α_{44}) U_{12} e^{- U_{10} - U_{11}}] \\ \times \frac{1}{(α_{39} + (α_{37} + α_{38}) U_{11}) (α_{42} + (α_{40} + α_{41}) U_{10}) (α_{45} + (α_{43} + α_{44}) U_{12})} > 1, \end{matrix}

with

\begin{matrix} (69) & {J|}_{ϒ_{4}} = (\begin{matrix} b_{11} & b_{12} & b_{13} & b_{14} & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ b_{31} & b_{32} & b_{33} & b_{34} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & b_{55} & b_{56} \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}), \end{matrix}

where $\begin{matrix} (70) & b_{11} = - \frac{α_{37} e^{- \bar{x}}}{α_{39} + (α_{37} + α_{38}) \bar{y}}, \\ b_{12} = - \frac{α_{38} e^{- \bar{x}}}{α_{39} + (α_{37} + α_{38}) \bar{y}}, \\ b_{13} = - \frac{α_{37} \bar{x}}{α_{39} + (α_{37} + α_{38}) \bar{y}}, \\ b_{14} = - \frac{α_{38} \bar{x}}{α_{39} + (α_{37} + α_{38}) \bar{y}}, \\ b_{31} = - \frac{α_{40} \bar{y}}{α_{42} + (α_{40} + α_{41}) \bar{x}}, \\ b_{32} = - \frac{α_{41} \bar{y}}{α_{42} + (α_{40} + α_{41}) \bar{x}}, \\ b_{33} = - \frac{α_{40} e^{- \bar{y}}}{α_{42} + (α_{40} + α_{41}) \bar{x}}, \\ b_{34} = - \frac{α_{41} e^{- \bar{y}}}{α_{42} + (α_{40} + α_{41}) \bar{x}}, \\ b_{55} = - \frac{α_{43} (e^{- \bar{z}} + \bar{z})}{α_{45} + (α_{43} + α_{44}) \bar{z}}, \\ b_{56} = - \frac{α_{44} (e^{- \bar{z}} + \bar{z})}{α_{45} + (α_{43} + α_{44}) \bar{z}}; \end{matrix}$

(iv)

For $ϒ_{5}$ of (7), the following holds:

(iv.1) $ϒ_{5}$ is a sink if

\begin{matrix} (71) & \frac{(α_{46} + α_{47}) e^{- L_{13}}}{α_{48} + (α_{46} + α_{47}) L_{15}} + \frac{(α_{52} + α_{53}) e^{- L_{15}}}{α_{54} + (α_{52} + α_{53}) L_{13}} + \frac{(α_{49} + α_{50}) (e^{- L_{14}} + L_{14})}{α_{51} + (α_{49} + α_{50}) L_{14}} \\ + \frac{(α_{46} α_{52} + α_{47} α_{52} + α_{46} α_{53} + α_{47} α_{53}) e^{- L_{13} - L_{15}}}{(α_{48} + (α_{46} + α_{47}) L_{15}) (α_{54} + (α_{52} + α_{53}) L_{13})} \\ + \frac{(α_{46} α_{52} + α_{47} α_{52} + α_{46} α_{53} + α_{47} α_{53}) L_{13} L_{15}}{(α_{48} + (α_{46} + α_{47}) L_{15}) (α_{54} + (α_{52} + α_{53}) L_{13})} \\ + \frac{(α_{α_{46}} α_{49} + α_{47} α_{47} + α_{46} α_{50} + α_{47} α_{50}) e^{- L_{13}} (e^{- L_{14}} + L_{14})}{(α_{48} + (α_{46} + α_{47}) L_{15}) (α_{51} + (α_{49} + α_{50}) L_{14})} \\ + \frac{(α_{49} α_{52} + α_{50} α_{52} + α_{49} α_{53} + α_{50} α_{53}) e^{- L_{15}} (e^{- L_{14}} + L_{14})}{(α_{54} + (α_{52} + α_{53}) L_{13}) (α_{51} + (α_{49} + α_{50}) L_{14})} \\ + [(α_{46} α_{49} α_{52} + α_{47} α_{49} α_{52} + α_{46} α_{50} α_{52} + α_{46} α_{49} α_{53} \\ + α_{47} α_{50} α_{52} + α_{47} α_{49} α_{53} + α_{46} α_{50} α_{53} + α_{47} α_{50} α_{53}) e^{- L_{13} - L_{14} - L_{15}} \\ + (α_{46} α_{49} α_{52} + α_{47} α_{49} α_{52} + α_{46} α_{50} α_{52} + α_{46} α_{49} α_{53} \\ + α_{47} α_{50} α_{52} + α_{47} α_{49} α_{53} + α_{46} α_{50} α_{53} + α_{47} α_{50} α_{53}) L_{13} L_{14} L_{15} \\ + (α_{46} α_{49} α_{52} + α_{47} α_{49} α_{52} + α_{46} α_{50} α_{52} + α_{46} α_{49} α_{53} \\ + α_{47} α_{50} α_{52} + α_{47} α_{49} α_{53} + α_{46} α_{50} α_{53} + α_{47} α_{50} α_{53}) e^{- L_{14}} L_{13} L_{15} \\ + (α_{46} α_{49} α_{52} + α_{47} α_{49} α_{52} + α_{46} α_{50} α_{52} + α_{46} α_{49} α_{53} \\ + α_{47} α_{50} α_{52} + α_{47} α_{49} α_{53} + α_{46} α_{50} α_{53} + α_{47} α_{50} α_{53}) L_{14} e^{- L_{13} - L_{15}}] \\ \times \frac{1}{(α_{48} + (α_{46} + α_{47}) L_{15}) (α_{51} + (α_{49} + α_{50}) L_{14}) (α_{54} + (α_{52} + α_{53}) L_{13})} < 1. \end{matrix}

(iv.2) $ϒ_{5}$ is a source if

\begin{matrix} (72) & \frac{(α_{46} + α_{47}) e^{- U_{13}}}{α_{48} + (α_{46} + α_{47}) U_{15}} + \frac{(α_{52} + α_{53}) e^{- U_{15}}}{α_{54} + (α_{52} + α_{53}) U_{13}} + \frac{(α_{49} + α_{50}) (e^{- U_{14}} + U_{14})}{α_{51} + (α_{49} + α_{50}) U_{14}} + \\ \frac{(α_{46} α_{52} + α_{47} α_{52} + α_{46} α_{53} + α_{47} α_{53}) e^{- U_{13} - U_{15}}}{(α_{54} + (α_{52} + α_{53}) U_{13}) (α_{48} + (α_{46} + α_{47}) U_{15})} \\ + \frac{(α_{46} α_{52} + α_{47} α_{52} + α_{46} α_{53} + α_{47} α_{53}) U_{13} U_{15}}{(α_{54} + (α_{52} + α_{53}) U_{13}) (α_{48} + (α_{46} + α_{47}) U_{15})} \\ + \frac{(α_{α_{46}} α_{49} + α_{47} α_{47} + α_{46} α_{50} + α_{47} α_{50}) e^{- U_{13}} (e^{- U_{14}} + U_{14})}{(α_{54} + (α_{52} + α_{53}) U_{15}) (α_{51} + (α_{49} + α_{48}) U_{14})} \\ + \frac{(α_{49} α_{52} + α_{50} α_{52} + α_{49} α_{53} + α_{50} α_{53}) e^{- U_{15}} (e^{- U_{14}} + U_{14})}{(α_{54} + (α_{52} + α_{53}) U_{15}) (α_{51} + (α_{49} + α_{48}) U_{14})} \\ + [(α_{46} α_{49} α_{52} + α_{47} α_{49} α_{52} + α_{46} α_{50} α_{52} + α_{46} α_{49} α_{53} + \\ α_{47} α_{50} α_{52} + α_{47} α_{49} α_{53} + α_{46} α_{50} α_{53} + α_{47} α_{50} α_{53}) e^{- U_{13} - U_{14} - U_{15}} \\ + (α_{46} α_{49} α_{52} + α_{47} α_{49} α_{52} + α_{46} α_{50} α_{52} + α_{46} α_{49} α_{53} \\ + α_{47} α_{50} α_{52} + α_{47} α_{49} α_{53} + α_{46} α_{50} α_{53} + α_{47} α_{50} α_{53}) U_{13} U_{14} U_{15} \\ + (α_{46} α_{49} α_{52} + α_{47} α_{49} α_{52} + α_{46} α_{50} α_{52} + α_{46} α_{49} α_{53} \\ + α_{47} α_{50} α_{52} + α_{47} α_{49} α_{53} + α_{46} α_{50} α_{53} + α_{47} α_{50} α_{53}) e^{- U_{14}} U_{13} U_{15} \\ + (α_{46} α_{49} α_{52} + α_{47} α_{49} α_{52} + α_{46} α_{50} α_{52} + α_{46} α_{49} α_{53} + \\ α_{47} α_{50} α_{52} + α_{47} α_{49} α_{53} + α_{46} α_{50} α_{53} + α_{47} α_{50} α_{53}) U_{14} e^{- U_{13} - U_{15}}] \\ \times \frac{1}{(α_{48} + (α_{46} + α_{47}) U_{15}) (α_{51} + (α_{49} + α_{50}) U_{14}) (α_{54} (α_{52} + α_{53}) U_{13})} > 1, \end{matrix}

with

\begin{matrix} (73) & {J|}_{ϒ_{5}} = (\begin{matrix} b_{11} & b_{12} & 0 & 0 & b_{15} & b_{16} \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & b_{33} & b_{34} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ b_{51} & b_{52} & 0 & 0 & b_{55} & b_{56} \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}), \end{matrix}

where $\begin{matrix} (74) & b_{11} = - \frac{α_{46} e^{- \bar{x}}}{α_{48} + (α_{46} + α_{47}) \bar{z}}, \\ b_{12} = - \frac{α_{47} e^{- \bar{x}}}{α_{48} + (α_{46} + α_{47}) \bar{z}}, \\ b_{15} = - \frac{α_{46} \bar{x}}{α_{48} + (α_{46} + α_{47}) \bar{z}}, \\ b_{16} = - \frac{α_{47} \bar{x}}{α_{48} + (α_{46} + α_{47}) \bar{z}}, \\ b_{33} = - \frac{α_{49} (e^{- \bar{y}} + \bar{y})}{α_{51} + (α_{49} + α_{50}) \bar{y}}, \\ b_{34} = - \frac{α_{50} (e^{- \bar{y}} + \bar{y})}{α_{51} + (α_{49} + α_{50}) \bar{y}}, \\ b_{51} = - \frac{α_{52} \bar{z}}{α_{54} + (α_{52} + α_{53}) \bar{x}}, \\ b_{52} = - \frac{α_{53} \bar{z}}{α_{54} + (α_{52} + α_{53}) \bar{x}}, \\ b_{55} = - \frac{α_{52} e^{- \bar{z}}}{α_{54} + (α_{52} + α_{53}) \bar{x}}, \\ b_{56} = - \frac{α_{53} e^{- \bar{z}}}{α_{54} + (α_{52} + α_{53}) \bar{x}}; \end{matrix}$

(v)

For $ϒ_{6}$ of (8), the following holds:

(v.1) $ϒ_{6}$ is a sink if

\begin{matrix} (75) & \frac{(α_{55} α_{58} + α_{56} α_{58} + α_{55} α_{59} + α_{56} α_{59}) L_{16} L_{17}}{(α_{57} + (α_{55} + α_{56}) L_{16}) (α_{60} + (α_{58} + α_{59}) L_{17})} + \frac{(α_{55} + α_{56}) L_{16}}{α_{57} + (α_{55} + α_{56}) L_{16}} \\ + \frac{(α_{55} α_{61} + α_{55} α_{62} + α_{56} α_{61} + α_{56} α_{62}) L_{16} L_{18}}{(α_{57} + (α_{55} + α_{56}) L_{16}) (α_{63} + (α_{61} + α_{62}) L_{18})} + \frac{(α_{58} + α_{59}) L_{17}}{α_{60} + (α_{58} + α_{59}) L_{17}} \\ + \frac{(α_{58} α_{62} + α_{59} α_{61} + α_{59} α_{62} + α_{58} α_{61}) L_{16} L_{18}}{(α_{57} + (α_{55} + α_{56}) L_{16}) (α_{63} + (α_{61} + α_{62}) L_{18})} + \frac{(α_{61} + α_{62}) L_{18}}{α_{63} + (α_{61} + α_{62}) L_{18}} \\ + [(α_{55} α_{58} α_{61} + α_{56} α_{58} α_{61} + α_{55} α_{58} α_{61} + α_{55} α_{58} α_{62} \\ + α_{56} α_{58} α_{61} + α_{56} α_{58} α_{62} + α_{55} α_{58} α_{62} + α_{56} α_{58} α_{62}) e^{- L_{16} - L_{17} - L_{18}} \\ + (α_{55} α_{58} α_{61} + α_{55} α_{58} α_{62} + α_{56} α_{58} α_{61} + α_{55} α_{59} α_{61} \\ + α_{56} α_{58} α_{62} + α_{55} α_{59} α_{62} + α_{56} α_{59} α_{61} + α_{56} α_{59} α_{62}) L_{16} L_{17} L_{18}] \\ \times \frac{1}{(α_{57} + (α_{55} + α_{56}) L_{16}) (α_{60} + (α_{58} + α_{59}) L_{17}) (α_{63} + (α_{61} + α_{62}) L_{18})} < 1. \end{matrix}

(v.2) $ϒ_{6}$ is a source if $\begin{matrix} (76) & \frac{(α_{55} α_{58} + α_{56} α_{58} + α_{55} α_{59} + α_{56} α_{59}) U_{16} U_{17}}{(α_{57} + (α_{55} + α_{56}) U_{16}) (α_{60} + (α_{58} + α_{59}) U_{17})} + \frac{(α_{55} + α_{56}) U_{16}}{α_{57} + (α_{55} + α_{56}) U_{16}} \\ + \frac{(α_{55} α_{61} + α_{55} α_{62} + α_{56} α_{61} + α_{56} α_{62}) U_{16} U_{18}}{(α_{57} + (α_{55} + α_{56}) U_{16}) (α_{63} + (α_{61} + α_{62}) U_{18})} + \frac{(α_{58} + α_{59}) U_{17}}{α_{60} + (α_{58} + α_{59}) U_{17}} \\ + \frac{(α_{58} α_{62} + α_{59} α_{61} + α_{59} α_{62} + α_{58} α_{61}) U_{16} U_{18}}{(α_{57} + (α_{55} + α_{56}) U_{16}) (α_{63} + (α_{61} + α_{62}) U_{18})} + \frac{(α_{61} + α_{62}) U_{18}}{α_{63} + (α_{61} + α_{62}) U_{18}} \\ + [(α_{55} α_{58} α_{61} + α_{56} α_{58} α_{61} + α_{55} α_{58} α_{61} + α_{55} α_{58} α_{62} \\ + α_{56} α_{58} α_{61} + α_{56} α_{58} α_{62} + α_{55} α_{58} α_{62} + α_{56} α_{58} α_{62}) e^{- U_{16} - U_{17} - U_{18}} \\ + (α_{55} α_{58} α_{61} + α_{55} α_{58} α_{62} + α_{56} α_{58} α_{61} + α_{55} α_{59} α_{61} \\ + α_{56} α_{58} α_{62} + α_{55} α_{59} α_{62} + α_{56} α_{59} α_{61} + α_{56} α_{59} α_{62}) U_{16} U_{17} U_{18}] \\ \times \frac{1}{(α_{57} + (α_{55} + α_{56}) U_{16}) (α_{60} + (α_{58} + α_{59}) U_{17}) (α_{63} + (α_{61} + α_{62}) U_{18})} > 1, \end{matrix}$

with

\begin{matrix} (77) & {J|}_{ϒ_{6}} = (\begin{matrix} b_{11} & b_{12} & 0 & 0 & b_{15} & b_{16} \\ 1 & 0 & 0 & 0 & 0 & 0 \\ b_{31} & b_{32} & b_{33} & b_{34} & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & b_{53} & b_{54} & b_{55} & b_{56} \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}), \end{matrix}

where

\begin{matrix} (78) & b_{11} = - \frac{α_{55} \bar{x}}{α_{57} + (α_{55} + α_{56}) \bar{x}}, \\ b_{12} = - \frac{α_{56} \bar{x}}{α_{57} + (α_{55} + α_{56}) \bar{x}}, \\ b_{15} = - \frac{α_{55} e^{- \bar{z}}}{α_{57} + (α_{55} + α_{56}) \bar{x}}, \\ b_{16} = - \frac{α_{56} e^{- \bar{z}}}{α_{57} + (α_{55} + α_{56}) \bar{x}}, \\ b_{31} = - \frac{α_{58} e^{- \bar{x}}}{α_{60} + (α_{58} + α_{59}) \bar{y}}, \\ b_{32} = - \frac{α_{59} e^{- \bar{x}}}{α_{60} + (α_{58} + α_{59}) \bar{y}}, \\ b_{33} = - \frac{α_{58} \bar{y}}{α_{60} + (α_{58} + α_{59}) \bar{y}}, \\ b_{34} = - \frac{α_{59} \bar{y}}{α_{60} + (α_{58} + α_{59}) \bar{y}}, \\ b_{53} = - \frac{α_{61} e^{- \bar{y}}}{α_{63} + (α_{61} + α_{62}) \bar{z}}, \\ b_{54} = - \frac{α_{62} e^{- \bar{y}}}{α_{63} + (α_{61} + α_{62}) \bar{z}}, \\ b_{55} = - \frac{α_{61} \bar{z}}{α_{63} + (α_{61} + α_{62}) \bar{z}}, \\ b_{56} = - \frac{α_{62} \bar{z}}{α_{63} + (α_{61} + α_{62}) \bar{z}} . \end{matrix}

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 $Y_{i} (i = 1, \dots, 6)$ , respectively, of systems (3)–(8) is explored.

6. Global Dynamics of Systems (3)–(8)

Theorem 6.

For global dynamics about $Y_{i} (i = 1, \dots, 6)$ , respectively, of systems (3)–(8), following statements hold:

(i)

$ϒ_{1}$ of (3) is global asymptotically stable if

\begin{matrix} (79) & (α_{10} + α_{11}) e^{- L_{2}} < (2 \bar{x} - U_{1}) (α_{12} + (α_{10} + α_{11}) L_{2}), \\ (α_{13} + α_{14}) e^{- L_{3}} < (2 \bar{y} - U_{2}) (α_{15} + (α_{13} + α_{14}) L_{3}), \\ (α_{16} + α_{17}) e^{- L_{1}} < (2 \bar{z} - U_{3}) (α_{18} + (α_{16} + α_{17}) L_{1}); \end{matrix}

(ii)

$ϒ_{2}$ of (4) is global asymptotically stable if

\begin{matrix} (80) & (α_{19} + α_{20}) e^{- L_{6}} < (2 \bar{x} - U_{4}) (α_{21} + (α_{19} + α_{20}) L_{6}), \\ (α_{22} + α_{23}) e^{- L_{4}} < (2 \bar{y} - U_{5}) (α_{24} + (α_{22} + α_{23}) L_{4}), \\ (α_{25} + α_{26}) e^{- L_{5}} < (2 \bar{z} - U_{6}) (α_{27} + (α_{25} + α_{26}) L_{5}); \end{matrix}

(iii)

$ϒ_{3}$ of (5) is global asymptotically stable if

\begin{matrix} (81) & (α_{28} + α_{29}) e^{- L_{7}} < (2 \bar{x} - U_{7}) (α_{30} + (α_{28} + α_{29}) L_{7}), \\ (α_{31} + α_{32}) e^{- L_{8}} < (2 \bar{y} - U_{8}) (α_{33} + (α_{31} + α_{32}) L_{9}), \\ (α_{34} + α_{35}) e^{- L_{9}} < (2 \bar{z} - U_{9}) (α_{36} + (α_{34} + α_{35}) L_{8}); \end{matrix}

(iv)

$ϒ_{4}$ of (6) is global asymptotically stable if

\begin{matrix} (82) & (α_{37} + α_{38}) e^{- L_{10}} < (2 \bar{x} - U_{10}) (α_{39} + (α_{37} + α_{38}) L_{11}), \\ (α_{40} + α_{41}) e^{- L_{11}} < (2 \bar{y} - U_{11}) (α_{42} + (α_{40} + α_{41}) L_{10}), \\ (α_{43} + α_{44}) e^{- L_{12}} < (2 \bar{z} - U_{12}) (α_{45} + (α_{43} + α_{44}) L_{12}); \end{matrix}

(v)

$ϒ_{5}$ of (7) is global asymptotically stable if

\begin{matrix} (83) & (α_{46} + α_{47}) e^{- L_{13}} < (2 \bar{x} - U_{13}) (α_{48} + (α_{46} + α_{47}) L_{15}), \\ (α_{49} + α_{50}) e^{- L_{14}} < (2 \bar{y} - U_{14}) (α_{51} + (α_{49} + α_{50}) L_{14}), \\ (α_{52} + α_{53}) e^{- L_{15}} < (2 \bar{z} - U_{15}) (α_{54} + (α_{52} + α_{53}) L_{13}); \end{matrix}

(vi)

$ϒ_{6}$ of (8) is global asymptotically stable if

\begin{matrix} (84) & (α_{55} + α_{56}) e^{- L_{18}} < (2 \bar{x} - U_{16}) (α_{57} + (α_{56} + α_{57}) L_{16}), \\ (α_{58} + α_{59}) e^{- L_{16}} < (2 \bar{y} - U_{17}) (α_{60} + (α_{58} + α_{59}) L_{17}), \\ (α_{61} + α_{62}) e^{- L_{17}} < (2 \bar{z} - U_{18}) (α_{63} + (α_{61} + α_{62}) L_{18}) . \end{matrix}

Proof.

(i)

Consider the following discrete-time Lyapunov function:

\begin{matrix} (85) & Γ_{n} = {(x_{n} - \bar{x})}^{2} + {(y_{n} - \bar{y})}^{2} + {(z_{n} - \bar{z})}^{2} . \end{matrix}

Now $\begin{matrix} (86) & Δ Γ_{n} = Γ_{n + 1} - Γ_{n} \\ = (x_{n + 1} - x_{n}) (x_{n + 1} + x_{n} - 2 \bar{x}) + (y_{n + 1} - y_{n}) (y_{n + 1} + y_{n} - 2 \bar{y}) + (z_{n + 1} - z_{n}) (z_{n + 1} + z_{n} - 2 \bar{z}), \\ = (x_{n + 1} - x_{n}) (\frac{α_{10} e^{- y_{n}} + α_{11} e^{- y_{n - 1}}}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} + x_{n} - 2 \bar{x}) \\ + (y_{n + 1} - y_{n}) (\frac{α_{13} e^{- z_{n}} + α_{14} e^{- z_{n - 1}}}{α_{15} + α_{13} z_{n} + α_{14} z_{n - 1}} + y_{n} - 2 \bar{y}) \\ + (z_{n + 1} - z_{n}) (\frac{α_{16} e^{- x_{n}} + α_{17} e^{- x_{n - 1}}}{α_{18} + α_{16} x_{n} + α_{17} x_{n - 1}} + z_{n} - 2 \bar{z}) \\ \leq (U_{1} - L_{1}) (\frac{(α_{10} + α_{11}) e^{- L_{2}}}{α_{12} + (α_{10} + α_{11}) L_{2}} + U_{1} - 2 \bar{x}) \\ + (U_{2} - L_{2}) (\frac{(α_{13} + α_{14}) e^{- L_{3}}}{α_{15} + (α_{13} + α_{14}) L_{3}} + U_{2} - 2 \bar{y}) \\ + (U_{3} - L_{3}) (\frac{(α_{16} + α_{17}) e^{- L_{1}}}{α_{18} + (α_{16} + α_{17}) L_{1}} + U_{3} - 2 \bar{z}) \\ = (U_{1} - L_{1}) (\frac{(α_{10} + α_{11}) e^{- L_{2}} - (2 \bar{x} - U_{1}) (α_{12} + (α_{10} + α_{11}) L_{2})}{α_{12} + (α_{10} + α_{11}) L_{2}}) \\ + (U_{2} - L_{2}) (\frac{(α_{13} + α_{14}) e^{- L_{3}} - (2 \bar{y} - U_{2}) (α_{15} + (α_{13} + α_{14}) L_{3})}{α_{15} + (α_{13} + α_{14}) L_{3}}) \\ + (U_{3} - L_{3}) (\frac{(α_{16} + α_{17}) e^{- L_{1}} - (2 \bar{z} - U_{3}) (α_{18} + (α_{16} + α_{17}) L_{1})}{α_{18} + (α_{16} + α_{17}) L_{1}}) . \end{matrix}$

From (80) and (87), one gets $Δ Γ_{n} < 0 \forall n \geq 0$ . Hence, we obtain that ${lim}_{n ⟶ \infty} (Γ_{n + 1} - Γ_{n}) = 0$ , and thus $ϒ_{1}$ 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 ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ , $s . t$ . $\begin{matrix} (87) & lim_{n ⟶ \infty} x_{n} = \bar{x}, \\ lim_{n ⟶ \infty} y_{n} = \bar{y}, \\ lim_{n ⟶ \infty} z_{n} = \bar{z}, \end{matrix}$ where $\begin{matrix} (88) & \bar{x} \in [L_{1}, U_{1}], \\ \bar{y} \in [L_{2}, U_{2}], \\ \bar{z} \in [L_{3}, U_{3}], \end{matrix}$ then the error vector, $i . e$ ., $\begin{matrix} (89) & θ_{n} = (\begin{matrix} θ_{n}^{1} \\ θ_{n - 1}^{1} \\ θ_{n}^{2} \\ θ_{n - 1}^{2} \\ θ_{n}^{3} \\ θ_{n - 1}^{3} \end{matrix}), \end{matrix}$ satisfies the following relation: $\begin{matrix} (90) & \lim_{n ⟶ \infty} {(‖θ_{n}‖)}^{1 / n} = |λ_{1, \dots, 6} J_{|ϒ_{1}|}, \\ \lim_{n ⟶ \infty} \frac{‖θ_{n + 1}‖}{‖θ_{n}‖} = |λ_{1, \dots, 6} J_{|ϒ_{1}|}, \end{matrix}$ where ${|λ_{1, \dots, 6} J|}_{ϒ_{1}|}$ are the roots of ${J|}_{ϒ_{1}}$ .

Proof.

If the positive solution of (3) is ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ , $s . t$ . (88) along with (89) holds. To find the error terms, one has $\begin{matrix} (91) & x_{n + 1} - \bar{x} = \frac{α_{10} e^{- y_{n}} + α_{11} e^{- y_{n - 1}}}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} - \frac{(α_{10} + α_{11}) e^{- \bar{y}}}{α_{12} + (α_{10} + α_{11}) \bar{y}} \\ = - \frac{α_{10} e^{- y_{n}} (e^{y_{n} - \bar{y}} - 1)}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} - \frac{α_{11} e^{- y_{n - 1}} (e^{y_{n - 1} - \bar{y}} - 1)}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} - \frac{α_{10} \bar{x} (y_{n} - \bar{y})}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} \\ - \frac{α_{11} \bar{x} (y_{n - 1} - \bar{y})}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} \\ = - \frac{α_{10} (e^{- y_{n}} (y_{n} - \bar{y} + O_{1} ({(y_{n} - \bar{y})}^{2})) + \bar{x} (y_{n} - \bar{y}))}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} (y_{n} - \bar{y}) \\ - \frac{α_{11} (e^{- y_{n - 1}} (y_{n - 1} - \bar{y} + O_{2} ({(y_{n - 1} - \bar{y})}^{2}) + \bar{x} (y_{n - 1} - \bar{y}))}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} (y_{n - 1} - \bar{y}) . \end{matrix}$

So, $\begin{matrix} (92) & x_{n + 1} - \bar{x} = - \frac{α_{10} (e^{- y_{n}} + \bar{x})}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} (y_{n} - \bar{y}) - \frac{α_{11} (e^{- y_{n - 1}} + \bar{x})}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} (y_{n - 1} - \bar{y}) \\ + O_{1} ({(y_{n} - \bar{y})}^{2}) + O_{2} ({(y_{n - 1} - \bar{y})}^{2}) . \end{matrix}$

Similarly, $\begin{matrix} (93) & y_{n + 1} - \bar{y} = - \frac{α_{13} (e^{- z_{n}} + \bar{y})}{α_{15} + α_{13} z_{n} + α_{14} z_{n - 1}} (z_{n} - \bar{z}) - \frac{α_{14} (e^{- z_{n - 1}} + \bar{y})}{α_{15} + α_{13} z_{n} + α_{14} z_{n - 1}} (z_{n - 1} - \bar{z}) \\ + O_{3} ({(z_{n} - \bar{z})}^{2}) + O_{4} ({(z_{n - 1} - \bar{z})}^{2}), \\ z_{n + 1} - \bar{z} = - \frac{α_{16} (e^{- x_{n}} + \bar{z})}{α_{18} + α_{16} x_{n} + α_{17} x_{n - 1}} (x_{n} - \bar{x}) - \frac{α_{17} (e^{- x_{n - 1}} + \bar{z})}{α_{18} + α_{16} x_{n} + α_{17} x_{n - 1}} (x_{n - 1} - \bar{x}) + \\ O_{5} ({(x_{n} - \bar{x})}^{2}) + O_{6} ({(x_{n - 1} - \bar{x})}^{2}) . \end{matrix}$

From (92) and (93), one gets $\begin{matrix} (94) & x_{n + 1} - \bar{x} \approx - \frac{α_{10} (e^{- y_{n}} + \bar{x})}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} (y_{n} - \bar{y}) - \frac{α_{11} (e^{- y_{n - 1}} + \bar{x})}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}} (y_{n - 1} - \bar{y}), \\ y_{n + 1} - \bar{y} \approx - \frac{α_{13} (e^{- z_{n}} + \bar{y})}{α_{15} + α_{13} z_{n} + α_{14} z_{n - 1}} (z_{n} - \bar{z}) - \frac{α_{14} (e^{- z_{n - 1}} + \bar{y})}{α_{15} + α_{13} z_{n} + α_{14} z_{n - 1}} (z_{n - 1} - \bar{z}), \\ z_{n + 1} - \bar{z} \approx - \frac{α_{16} (e^{- x_{n}} + \bar{z})}{α_{18} + α_{16} x_{n} + α_{17} x_{n - 1}} (x_{n} - \bar{x}) - \frac{α_{17} (e^{- x_{n - 1}} + \bar{z})}{α_{18} + α_{16} x_{n} + α_{17} x_{n - 1}} (x_{n - 1} - \bar{x}) . \end{matrix}$

Denote $\begin{matrix} (95) & θ_{n}^{1} = x_{n} - \bar{x}, \\ θ_{n}^{2} = y_{n} - \bar{y}, \\ θ_{n}^{3} = z_{n} - \bar{z} . \end{matrix}$

In view of (95), from (94), one gets $\begin{matrix} (96) & θ_{n + 1}^{1} = ω_{1 n} θ_{n}^{2} + ω_{2 n} θ_{n - 1}^{2}, \\ θ_{n + 1}^{2} = ω_{3 n} θ_{n}^{3} + ω_{4 n} θ_{n - 1}^{3}, \\ θ_{n + 1}^{3} = ω_{5 n} θ_{n}^{1} + ω_{6 n} θ_{n - 1}^{1}, \end{matrix}$ where $\begin{matrix} (97) & ω_{1 n} = - \frac{α_{10} (e^{- y_{n}} + \bar{x})}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}}, \\ ω_{2 n} = - \frac{α_{11} (e^{- y_{n - 1}} + \bar{x})}{α_{12} + α_{10} y_{n} + α_{11} y_{n - 1}}, \\ ω_{3 n} = - \frac{α_{13} (e^{- z_{n}} + \bar{y})}{α_{15} + α_{13} z_{n} + α_{14} z_{n - 1}}, \\ ω_{4 n} = - \frac{α_{14} (e^{- z_{n - 1}} + \bar{y})}{α_{15} + α_{13} z_{n} + α_{14} z_{n - 1}}, \\ ω_{5 n} = - \frac{α_{16} (e^{- x_{n}} + \bar{z})}{α_{18} + α_{16} x_{n} + α_{17} x_{n - 1}}, \\ ω_{6 n} = - \frac{α_{17} (e^{- x_{n - 1}} + \bar{z})}{α_{18} + α_{16} x_{n} + α_{17} x_{n - 1}} . \end{matrix}$

From (97), one gets $\begin{matrix} (98) & lim_{n ⟶ \infty} ω_{1 n} = - \frac{α_{10} (e^{- \bar{y}} + \bar{x})}{α_{12} + (α_{10} + α_{11}) \bar{y}}, \\ lim_{n ⟶ \infty} ω_{2 n} = - \frac{α_{11} (e^{- \bar{y}} + \bar{x})}{α_{12} + (α_{10} + α_{11}) \bar{y}}, \\ lim_{n ⟶ \infty} ω_{3 n} = - \frac{α_{13} (e^{- \bar{z}} + \bar{y})}{α_{15} + (α_{13} + α_{14}) \bar{z}}, \\ lim_{n ⟶ \infty} ω_{4 n} = - \frac{α_{14} (e^{- \bar{z}} + \bar{y})}{α_{15} + (α_{13} + α_{14}) \bar{z}}, \\ lim_{n ⟶ \infty} ω_{5 n} = - \frac{α_{16} (e^{- \bar{x}} + \bar{z})}{α_{18} + (α_{16} + α_{17}) \bar{x}}, \\ lim_{n ⟶ \infty} ω_{6 n} = - \frac{α_{17} (e^{- \bar{x}} + \bar{z})}{α_{18} + (α_{16} + α_{17}) \bar{x}}, \end{matrix}$ that is, $\begin{matrix} (99) & ω_{1 n} = - \frac{α_{10} (e^{- \bar{y}} + \bar{x})}{α_{12} + (α_{10} + α_{11}) \bar{y}} + b 1_{n}, \\ ω_{2 n} = - \frac{α_{11} (e^{- \bar{y}} + \bar{x})}{α_{12} + (α_{10} + α_{11}) \bar{y}} + b 1_{n - 1}, \\ ω_{3 n} = - \frac{α_{13} (e^{- \bar{z}} + \bar{y})}{α_{15} + (α_{13} + α_{14}) \bar{z}} + b 2_{n}, \\ ω_{4 n} = - \frac{α_{14} (e^{- \bar{z}} + \bar{y})}{α_{15} + (α_{13} + α_{14}) \bar{z}} + b 2_{n - 1}, \\ ω_{5 n} = - \frac{α_{16} (e^{- \bar{x}} + \bar{z})}{α_{18} + (α_{16} + α_{17}) \bar{x}} + b 3_{n}, \\ ω_{6 n} = - \frac{α_{17} (e^{- \bar{x}} + \bar{z})}{α_{18} + (α_{16} + α_{17}) \bar{x}} + b 3_{n - 1}, \end{matrix}$ where $b 1_{n}, b 1_{n - 1}, b 2_{n}, b 2_{n - 1}, b 3_{n}, and b 3_{n - 1} ⟶ 0$ as $n ⟶ \infty$ . Now, we have system 1.10 of [6], where $A = {J|}_{ϒ_{1}}$ and $\begin{matrix} (100) & B (n) = (\begin{matrix} 0 & 0 & b 1_{n} & b 1_{n - 1} & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & b 2_{n} & b 2_{n - 1} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ b 3_{n} & b 3_{n - 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}), \end{matrix}$ such that $‖B (n)‖ ⟶ \infty, n ⟶ \infty$ . So, about $ϒ_{1}$ of (3) the limiting system becomes $\begin{matrix} (101) & (\begin{matrix} θ_{n + 1}^{1} \\ θ_{n}^{1} \\ θ_{n + 1}^{2} \\ θ_{n}^{2} \\ θ_{n + 1}^{3} \\ θ_{n}^{3} \end{matrix}) = {J|}_{ϒ_{1}} (\begin{matrix} θ_{n}^{1} \\ θ_{n - 1}^{1} \\ θ_{n}^{2} \\ θ_{n - 1}^{2} \\ θ_{n}^{3} \\ θ_{n - 1}^{3} \end{matrix}), \end{matrix}$ which is as ${J|}_{ϒ_{1}}$ about $ϒ_{1}$ .

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 ${\{Ω_{n}\}}_{n = - 1}^{\infty}$ , $s . t$ . (87) along with the following relation holds:

\begin{matrix} (102) & \bar{x} \in [L_{4}, U_{4}], \\ \bar{y} \in [L_{5}, U_{5}], \\ \bar{z} \in [L_{6}, U_{6}], \end{matrix}