Neuroethics: what the study of brain disorders can tell about moral behavior

1.   Introduction and preliminaries

First of all, it is important to keep in mind that $\mathbb{N}_{0}$, $\mathbb{Z}$, $\mathbb{R}$, $\mathbb{R}^{+}$ and $\mathbb{R}^{+}\cup\left\lbrace 0\right\rbrace$, refer to the non-negative integer set, integer set, real set, positive real set and non-negative real set, respectively. If $\Phi, \Psi\in \mathbb{Z}$ and $\Phi\leq \Psi$, the notation $k = \overline {\Phi, \Psi}$ stands for $\left\lbrace k\in \mathbb{Z}: \Phi\leq k \leq \Psi\right\rbrace$.

It is of utmost importance to use different branches of science in tandem. Mathematics can also be deployed in research fields of other scientific areas. Especially, mathematical models of many sorts have been developed for many areas of study, such as ecology, medicine, population biology, biostatistics and molecular biology in ^[1]. In order to apply mathematical modeling in biology, both mathematical and biological information are required. As such, multidisciplinary or interdisciplinary research studies are very popular these days (see ^{[2,3,4,5,6,7,8,9,10,11,12,13,14,15]}).

Difference equations, which represent a topic of applied mathematics, can also be used in mathematical modeling. In addition, difference equations are used to define real discrete models in different areas of advanced science, such as control theory, biology, physics, economics and psychology. One of the applications of difference equations in biology is the population model. Some population models involve exponential-form difference equations. Difference equations of the mentioned type attract the attention of mathematicians. But, their stability analysis process can be complex. Therefore, studying these difference equations and systems is worthwhile. To date, a significant number of papers concerning difference equations have been published (see ^{[16,17,18,19,20,21,22,23,24,25,26,27]}).

As a model in the field of mathematical biology, several difference equations of the exponential-type were studied. For example, the following single-species population model

where $\kappa, \epsilon \in \mathbb{R}^{+}$ and $U_{-1}, U_{0}\in\mathbb{R}^{+}\cup\left\lbrace 0\right\rbrace$, was investigated by El-Metwally et al. in ^[28]. $\kappa$ was used as the migration rate and $\epsilon$ was taken as the population growth rate in Eq (1.1).

There is another example, in which the authors of ^[29] changed Eq (1.1) to a newer version as;

where $U_{-1}, U_{0}\in \mathbb{R}^{+}\cup\left\lbrace 0\right\rbrace$ and $\kappa, \epsilon \in \mathbb{R}^{+}$. As a model in the field of mathematical biology, Eq (1.2) was investigated. In Eq (1.2), $\kappa$ is the migration rate and $\epsilon$ is the population growth rate.

The authors of ^[30] researched the dynamical properties of the next exponential-type difference equation:

where $U_{-1}, U_{0}\in\mathbb{R}^{+}\cup\left\lbrace 0\right\rbrace$ and $\kappa, \epsilon, \zeta\in\mathbb{R}^{+}$. Equation (1.3) represent a mathematical biology-purposed model, where $\kappa$ is the migration rate, $\epsilon$ is the population growth proportion and $\zeta$ is the carrying capacity.

Comert et al. in ^[31], explored the global behavior of the next exponential-type difference equation:

where $k$ is an even number, the initial values $U_{-k}, U_{-k+1}, \dots, U_{0}\in\mathbb{R}^{+}\cup\left\lbrace 0\right\rbrace$ and the parameters $\delta, \epsilon, \zeta\in\mathbb{R}^{+}$.

Moreover, there are numerous studies that can be viewed as a model for difference equations of the exponential-type in the literature ^{[32,33,34,35]}.

Some authors extended Eq (1.3) to the two-dimensional exponential-type difference equations systems. For instance, Papaschinopoulos et al. examined the following systems of two-dimensional exponential-form difference equations:

for $n\in \mathbb{N}_{0}$, where $T_{-1}, T_{0}, R_{-1}, R_{0}\in\mathbb{R}^{+}$ and $\kappa, \epsilon, \zeta, \delta, \lambda, \mu\in\mathbb{R}^{+}$ in ^[36]. The systems given by in (1.5) are two-species population models.

In addition, Thai et al. in ^[37], studied the dynamical properties of the following systems:

for $n\in \mathbb{N}_{0}$, where the parameters $\alpha_{k}, \beta_{k}, \gamma_{k}\in\mathbb{R}^{+}$, for $k\in\{1, 2\}$ and $T_{-1}, T_{0}, R_{-1}, R_{0}\in\mathbb{R}^{+}$. Two-dimensional exponential-form difference equations systems have been studied by many authors in recent years (see ^{[38,39,40,41,42,43,44]}).

Our aim in this paper is to generalize the systems given by (1.6) to the following three-dimensional system of difference equations of exponential-form:

for $n\in \mathbb{N}_{0}$, where the parameters $\Gamma_{k}$, $\delta_{k}$, $\Theta_{k}$ for $k = \overline{1, 3}$ and the initial values $\Upsilon_{-\nu}$, $\Psi_{-\nu}$, $\Omega_{-\nu}$, for $\nu\in\{0, 1\}$ are positive constants. Another goal of this study is to contribute to the literature, since there are few studies on systems of three-dimensional exponential-form difference equations in the literature (see ^[45,46]).

In mathematical biology, the systems given by (1.7) and (1.8) can be viewed as three-species population models. Also, the biological parameters of systems (1.7) and (1.8) are shown in Table 1.

Our paper is organized as follows: In the following section, we study the boundedness, local and global stability of the unique positive equilibrium point and rate of convergence of system (1.7). In the third section, we study the boundedness, local and global stability of the unique positive equilibrium point and rate of convergence of system (1.8). The conclusion is given in the last section.

Before we start our analysis, recall some lemmas and definitions which are used throughout this work. For more particulars, one can refer to the references [47,48,49,50].

where $f:U^{2}\times V^{2}\times W^{2} \to U$, $g:U^{2}\times V^{2}\times W^{2} \to V$, $h:U^{2}\times V^{2}\times W^{2} \to W$ are continuous differentiable functions and $U$, $V$, $W$ are some intervals of real numbers. Also, a solution $\left\lbrace \mho_{n}, \rho_{n}, \sigma_{n}\right\rbrace_{n = -1} ^{\infty}$ of system (1.9) is uniquely defined by the initial values $\left(\mho_{-\mu}, \rho_{-\mu}, \sigma_{-\mu}\right) \in U \times V \times W$ for $\mu\in\{0, 1\}$. Along with system (1.9), we take into account the suitable vector map $\mathbb{F} = \left(f, \mho_{n}, g, \rho_{n}, h, \sigma_{n}\right)$. An equilibrium point of system (1.9) is a point $\left(\overline \mho, \overline \rho, \overline \sigma\right)$ that supplies

The point $\left(\overline \mho, \overline \mho, \overline \rho, \overline \rho, \overline \sigma, \overline \sigma\right)$ is named a fixed point of the vector map $\mathbb{F}$.

Definition 1. Assume that $\left(\overline \mho, \overline \mho, \overline \rho, \overline \rho, \overline \sigma, \overline \sigma\right)$ is a fixed point of the vector map $\mathbb{F} = \left(f, \mho_{n}, g, \rho_{n}, h, \sigma_{n}\right)$, where $f$, $g$ and $h$ are continuous differentiable functions at $\left(\overline \mho, \overline \rho, \overline \sigma\right)$. The linearized system given by system (1.9) about the equilibrium point $\left(\overline \mho, \overline \rho, \overline \sigma\right)$ is

where $K_{n} = \begin{pmatrix} \mho_{n} \\ \mho_{n-1} \\ \rho_{n} \\ \rho_{n-1} \\ \sigma_{n} \\ \sigma_{n-1} \end{pmatrix}$ and $J_{\mathbb{F}}$ is the Jacobian matrix of system (1.9) about the equilibrium point $\left(\overline \mho, \overline \rho, \overline \sigma\right)$.

Lemma 1. Let $K_{n+1} = \mathbb{F}\left(K_{n}\right)$, $n\in \mathbb{N}_{0}$, be a system of difference equations where $\overline{K}$ is a fixed point of $\mathbb{F}$. If all eigenvalues of the Jacobian matrix $J_{\mathbb{F}}$ about $\overline{K}$ lie inside the open unit disk $\left| \lambda\right| < 1$, then $\overline{K}$ is locally asymptotically stable. If one of them has a modulus greater than one, then $\overline{K}$ is unstable.

Definition 2. If there exist positive constants $t$ and $T$ and an integer $N\geq -1$, the positive solution $\left\lbrace \mho_{n}, \rho_{n}, \sigma_{n}\right\rbrace_{n = -1} ^{\infty}$ of system (1.9) is bounded and persists such that

The following lemma gives the rate of convergence of solutions of the systems of difference equations.

Lemma 2. (^[37])

where $X_{n}$ is a $k$-dimensional vector, $\alpha \in \mathbb{C}^{k\times k}$ is a constant matrix and $\beta: \mathbb{Z}^{+} \to \mathbb{C}^{k\times k}$ is a matrix function satisfying

where $\left| \left|. \right| \right|$ denotes any matrix norm which is associated with the vector norm

2.   Global behavior of system (1.7)

Lemma 3. Every positive solution of system (1.7) is bounded and persists.

Proof. Suppose that $\left\lbrace \left(\Upsilon_{n}, \Psi_{n}, \Omega_{n}\right) \right\rbrace$ is an arbitrary solution of system (1.7). We get

Then, from system (1.7) and Eq (2.1), we have

for $n\in \mathbb{N}$.

Therefore, from Eqs (2.1) and (2.2), the proof of the lemma is complete. □

We can indicate the following lemma, which is useful for our study of system (1.7). We consider the following general system of difference equations:

where $f$, $g$ and $h$ are continuous functions and the initial conditions $x_{-1}$, $x_{0}$, $y_{-1}$, $y_{0}$, $z_{-1}$ and $z_{0}$ are positive numbers.

Lemma 4. Let $f$, $g$, $h$, $f:\mathbb{R}^+ \to \mathbb{R}^+$, $g:\mathbb{R}^+ \to \mathbb{R}^+$ and $h:\mathbb{R}^+ \to \mathbb{R}^+$ be continuous functions. Let $a_{1}$, $b_{1}$, $a_{2}$, $b_{2}$, $a_{3}$ and $b_{3}$ be positive numbers such that $a_{1} < b_{1}$, $a_{2} < b_{2}$, $a_{3} < b_{3}$ and

Suppose that the function $f\left(u, v\right)$ is a decreasing function with respect to $u$ (resp. $v$) for every $v$ (resp. $u$), $g\left(w, t\right)$ is a decreasing function with respect to $w$ (resp. $t$) for all $t$ (resp. $w$) and $h\left(p, s\right)$ is a decreasing function with respect to $p$ (resp. $s$) for every $s$ (resp. $p$). Finally, suppose that, if $m$, $M$, $r$, $R$, $t$ and $T$ are real numbers such that, if

then $m = M$, $r = R$ and $t = T$. Then, system (2.3) has a unique positive equilibrium $\left(\overline{x}, \overline{y}, \overline{z}\right)$ and every positive solution of system (2.3) that satisfies

tends to the unique positive equilibrium of system (2.3).

Proof. System (2.3) is equivalent to the following system of separated equations:

We consider the equation

From the conditions of $f$, $g$, $h$, we have that $F:\left[ a_{1}, b_{1}\right] \times \left[ a_{1}, b_{1}\right] \times \left[ a_{1}, b_{1}\right] \to \left[ a_{1}, b_{1}\right]$ and $F\left(\alpha, \beta, \gamma, \delta \right)$ is increasing in $\alpha$ for all $\beta, \gamma, \delta$; increasing in $\beta$ for all $\alpha, \gamma, \delta$; increasing in $\gamma$ for all $\alpha, \beta, \delta$; increasing in $\delta$ for all $\alpha, \beta, \gamma$. Now, let us take $m$ and $M$ as positive numbers so that

By setting $h\left(m, m\right) = T$ and $h\left(M, M\right) = t$, we get

By setting $g\left(t, t\right) = R$ and $g\left(T, T\right) = r$, we have that the relations given by Eq (2.4) are satisfied. Then, from a hypothesis of Lemma 4, we have that $m = M$. Equation (2.5) has a unique positive equilibrium $\overline{x}$, and every positive solution of Eq (2.5) tends to the unique positive equilibrium $\overline{x}$. Similarly, we can prove that the equation

has a unique positive equilibrium $\overline{y}$ and every positive solution of Eq (2.6) tends to the unique positive equilibrium $\overline{y}$. Similarly, we can prove that the equation

has a unique positive equilibrium $\overline{z}$ and every positive solution of Eq (2.7) tends to the unique positive equilibrium $\overline{z}$. This completes the proof of the lemma. □

2.1.   Local and global asymptotic stability

Theorem 3. For the local stability about $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right) \in \left[ \Lambda_{1}, \Pi_{1} \right] \times \left[ \Lambda_{2}, \Pi_{2} \right] \times \left[ \Lambda_{3}, \Pi_{3} \right]$, i.e., the equilibrium point of system (1.7), the next declarations are valid:

(i) $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is locally asymptotically stable if $L_{1} < 1$,

(ii) $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is unstable if $U_{1} > 1$,

where

Proof. (i) From system (1.7), we obtain

In order to construct the corresponding linearized form of system (1.7), we consider the following transformation:

where

By using the transformation given by Eq (2.11), we obtain

where

The characteristic equation of $F_{J}\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is below:

where

Because $xe^{-x} < e^{-1}$, for $x > 0$, we have

By supposing that $L_{1} < 1$, from Eq (2.17), we have that $\sum_{i = 1}^{4}\left| m_{i}\right| < 1$. According to the Rouche theorem, $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is locally asymptotically stable.

(ii) We get

By supposing that $U_{1} > 1$, from Eq (2.18), we have that $\sum_{i = 1}^{4}\left| m_{i}\right| > 1$. According to the Rouche theorem, $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is unstable.

□

Theorem 4. System (1.7) has a unique positive equilibrium $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$, and every positive solution of system (1.7) tends to the unique positive equilibrium of system (1.7) as $n\to\infty$ if

Proof. We consider the functions

where

From Eqs (2.20) and (2.21) we get the following relations for $k, l \in \left[ \Lambda_{1}, \Pi_{1} \right]$, $u, v \in \left[ \Lambda_{2}, \Pi_{2} \right]$ and $w, t \in \left[ \Lambda_{3}, \Pi_{3} \right]$:

thus, $f:\left[ \Lambda_{2}, \Pi_{2} \right]\times \left[ \Lambda_{2}, \Pi_{2} \right] \to \left[ \Lambda_{1}, \Pi_{1} \right]$, $g:\left[ \Lambda_{3}, \Pi_{3} \right]\times \left[ \Lambda_{3}, \Pi_{3} \right] \to \left[ \Lambda_{2}, \Pi_{2} \right]$ and $h:\left[ \Lambda_{1}, \Pi_{1} \right]\times \left[ \Lambda_{1}, \Pi_{1} \right] \to \left[ \Lambda_{3}, \Pi_{3} \right]$. Let $\left\lbrace \left(\Upsilon_{n}, \Psi_{n}, \Omega_{n}\right) \right\rbrace$ be an arbitrary solution of system (1.7). Therefore, from Lemma 3, for $n\in \mathbb{N}$, we get

Now, let $m$, $M$, $r$, $R$, $t$ and $T$ be positive numbers such that

Then, we consider the functions

Note that $F$ maps the interval $\left[ \Lambda_{1}, \Pi_{1} \right]$ into itself. We claim that the equation $F\left(x\right) = 0$ has a unique solution in $\left[ \Lambda_{1}, \Pi_{1} \right]$. From Eq (2.23), we get

Let $\overline{x}$, $\overline{x}\in \left[ \Lambda_{1}, \Pi_{1} \right]$ be a solution of the equation $F\left(x\right) = 0$. Then, from Eq (2.23), we get

Now, observe that Eqs (2.26) and (2.27) imply that

Then, from Eqs (2.19), (2.26) and (2.28), we have

Therefore, from Eq (2.29), we see that the equation $F\left(x\right) = 0$ has a unique solution in $\left[ \Lambda_{1}, \Pi_{1} \right]$.

Note that $G$ maps the interval $\left[ \Lambda_{2}, \Pi_{2} \right]$ into itself. We claim that the equation $G\left(y\right) = 0$ has a unique solution in $\left[ \Lambda_{2}, \Pi_{2} \right]$. From Eq (2.24), we get

Let $\overline{y}$, $\overline{y}\in \left[ \Lambda_{2}, \Pi_{2} \right]$ be a solution of the equation $G\left(y\right) = 0$. Then, from Eq (2.24), we get

Now, observe that Eqs (2.30) and (2.31) imply that

Then, from Eqs (2.19), (2.30) and (2.32), we have

Therefore, from Eq (2.33), we see that the equation $G\left(y\right) = 0$ has a unique solution in $\left[ \Lambda_{2}, \Pi_{2} \right]$.

Note that $H$ maps the interval $\left[ \Lambda_{3}, \Pi_{3} \right]$ into itself. We claim that the equation $H\left(z\right) = 0$ has a unique solution in $\left[ \Lambda_{3}, \Pi_{3} \right]$. From Eq (2.25), we get

Let $\overline{z}$, $\overline{z}\in \left[ \Lambda_{3}, \Pi_{3} \right]$ be a solution of the equation $H\left(z\right) = 0$. Then, from Eq (2.25), we get

Now, observe that Eqs (2.34) and (2.35) imply that

Then, from Eqs (2.19), (2.34) and (2.36), we have

Therefore, from, Eq (2.37) we see that the equation $H\left(z\right) = 0$ has a unique solution in $\left[ \Lambda_{3}, \Pi_{3} \right]$.

In addition, Eq (2.22) implies that $m$ and $M$ are roots of $F\left(x\right) = 0$. Hence, we get that $m = M$. Similarly, Eq (2.22) implies that $r$ and $R$ are roots of $G\left(y\right) = 0$. Hence, we get that $r = R$. Finally, Eq (2.22) implies that $t$ and $T$ are roots of $H\left(z\right) = 0$. Hence, we get that $t = T$. From Lemma 4, system (1.7) has a unique positive equilibrium $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ and every positive solution of system (1.7) tends to the unique positive equilibrium as $n \to \infty$. This completes the proof of the theorem.

□

2.2.   Rate of convergence

Theorem 5. If $\left\lbrace \left(\Upsilon_{n}, \Psi_{n}, \Omega_{n}, \right) \right\rbrace _{n = -1}^{\infty}$ is a positive solution of system (1.7) such that

where

then the error vector $\Phi_{n} = \left(\Phi_{n}^{1}, \Phi_{n-1}^{1}, \Phi_{n}^{2}, \Phi_{n-1}^{2}, \Phi_{n}^{3}, \Phi_{n-1}^{3}\right)^{T}$ of every solution of system (1.7) supplies both of the following asymptotic relations:

where $\left| \lambda_{1, 2, 3, 4, 5, 6}F_{J}\left(\overline{\Upsilon}, \overline{\Psi}, \overline{\Omega} \right)\right|$ denotes the characteristic root of $F_{J}\left(\overline{\Upsilon}, \overline{\Psi}, \overline{\Omega}\right)$.

Proof. To find the error terms, from system (1.7), we obtain

that is,

Set

By using Eq (2.43), Eq (2.42) can be written in the following form:

where

By taking the limits of $d_{n}$, $e_{n}$, $f_{n}$, $g_{n}$, $h_{n}$ and $j_{n}$, we respectively obtain

that is,

where $\xi1_{n}\to 0$, $\xi1_{n-1}\to 0$, $\xi2_{n}\to 0$, $\xi2_{n-1}\to 0$, $\xi3_{n}\to 0$ and $\xi3_{n-1} \to 0$ as $n \to \infty$. Then, we get Poincare difference system (1.10) of ^[51], where

and

and $\left| \left| B_{n} \right| \right| \to 0$ as $n \to \infty$. Moreover, the limiting system of error terms turns into

which is similar to the linearized system (1.7) about the equilibrium $\left(\overline{\Upsilon}, \overline{\Psi}, \overline{\Omega}\right)$. □

3.   Global behavior of system (1.8)

Lemma 5. Every positive solution of system (1.8) is bounded and persists.

Proof. Suppose that $\left\lbrace \left(\Upsilon_{n}, \Psi_{n}, \Omega_{n}\right) \right\rbrace$ is an arbitrary solution of system (1.8). From Lemma 3, by using induction and applying $n\in \mathbb{N}$, we obtain

The proof of the lemma is similar to Lemma 3, so it is omitted. □

3.1.   Local and global asymptotic stability

Theorem 6. For the local stability about $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right) \in \left[ \Lambda_{4}, \Pi_{4} \right] \times \left[ \Lambda_{5}, \Pi_{5} \right] \times \left[ \Lambda_{6}, \Pi_{6} \right]$, i.e., the equilibrium point of system (1.8), the next declarations are valid:

(i) $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is locally asymptotically stable if $L_{2} < 1$,

(ii) $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is unstable if $U_{2} > 1$,

where

Proof. (i) From system (1.8), we have

In order to construct the corresponding linearized form of system (1.8), we consider the following transformation:

where

By using the transformation given by Eq (3.5), we have

where

The characteristic equation of $F_{J}\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is below:

where

Now, we can compute $\sum_{i = 1}^{4}\left| \widehat m_{i}\right|$ as follows:

By supposing that $L_{2} < 1$, from Eq (3.11), we have that $\sum_{i = 1}^{4}\left| \widehat m_{i}\right| < 1$. According to the Rouche theorem, $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is locally asymptotically stable.

(ii) We have

By supposing that $U_{2} > 1$, from Eq (3.12), we have that $\sum_{i = 1}^{4}\left| \widehat m_{i}\right| > 1$. According to the Rouche theorem, $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$ is unstable. □

Theorem 7. Consider system (1.8). Assume that the next relation is true:

Moreover, system (1.8) has a unique positive equilibrium $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$, and every positive solution of system (1.8) tends to the unique positive equilibrium of system (1.8) as $n \to \infty$.

Proof. We consider the functions

where

and $I_{4}, I_{5}, I_{6}$ are defined in Eq (3.1). From Eqs (3.14) and (3.15), we see that, for $\Upsilon \in I_{4}$, $\Psi \in I_{5}$ and $\Omega \in I_{6}$,

thus,

Let $p$, $P$, $l$, $L$, $w$ and $W$ be positive numbers such that

From Eq (3.16), we get

which imply that

Moreover, we get

Then Eqs (3.17) and (3.18) imply that

thus,

In addition, observe that Eqs (3.13) and (3.20) imply that

from which we have that $P = p$, $L = l$ and $W = w$. Thus from Eq (3.1), system (1.8) has a unique positive equilibrium $\left(\overline \Upsilon, \overline \Psi, \overline \Omega \right)$, and every positive solution of system (1.8) tends to the unique positive equilibrium as $n \to \infty$.

□

3.2.   Rate of convergence

Theorem 8. If $\left\lbrace \left(\Upsilon_{n}, \Psi_{n}, \Omega_{n}, \right) \right\rbrace _{n = -1}^{\infty}$ is a positive solution of system (1.8) such that

where

then the error vector $\Phi_{n} = \left(\Phi_{n}^{4}, \Phi_{n-1}^{4}, \Phi_{n}^{5}, \Phi_{n-1}^{5}, \Phi_{n}^{6}, \Phi_{n-1}^{6}\right)^{T}$ of every solution of system (1.8) supplies both of the following asymptotic relations:

where $\left| \lambda_{1, 2, 3, 4, 5, 6}F_{J}\left(\overline{\Upsilon}, \overline{\Psi}, \overline{\Omega} \right)\right|$ denotes the characteristic root of $F_{J}\left(\overline{\Upsilon}, \overline{\Psi}, \overline{\Omega}\right)$.

Proof. To find the error terms, from system (1.8), we apply

that is,

Set

By using Eq (3.26), Eq (3.25) can be written in the following form:

where

By taking the limits of $\widehat d_{n}$, $\widehat e_{n}$, $\widehat f_{n}$, $\widehat g_{n}$, $\widehat h_{n}$ and $\widehat j_{n}$, we respectively obtain

that is,

where $\xi4_{n}\to 0$, $\xi4_{n-1}\to 0$, $\xi5_{n}\to 0$, $\xi5_{n-1}\to 0$, $\xi6_{n}\to 0$ and $\xi6_{n-1} \to 0$ as $n \to \infty$. Then, we obtain Poincare difference system (1.10) of ^[51], where

and

and $\left| \left| \widehat B_{n} \right| \right| \to 0$ as $n \to \infty$. Moreover, the limiting system of error terms turns into

which is similar to the linearized system (1.8) about the equilibrium $\left(\overline{\Upsilon}, \overline{\Psi}, \overline{\Omega}\right)$. □

4.   Conclusions

In this paper, the global dynamics of two exponential-type systems of difference equations are investigated. The main results are as follows:

(i) All positive solutions of systems (1.7) and (1.8) have been shown to persist and to be bounded.

(ii) It has been shown that the equilibrium points of systems (1.7) and (1.8) are locally asymptotically stable or unstable based on the parameters $L_{1}$, $L_{2}$, $U_{1}$ and $U_{2}$.

(iii) It has been explained, by using both increasing and decreasing functions and a well-known mean-value theorem, that the equilibrium points of systems (1.7) and (1.8) are globally asymptotically stable when the conditions given by Eqs (2.19) and (3.13) are valid.

(iv) Information has been given regarding the rates of convergence of systems (1.7) and (1.8).

Use of AI tools declarations

The author declares that she has not used artificial intelligence (AI) tools in the creation of this article.

Conflict of interest

The author declares no conflict of interest that may influence the publication of this paper.