Random periodic sequence of globally mean-square exponentially stable discrete-time stochastic genetic regulatory networks with discrete spatial diffusions

1.   Introduction

Genetic regulatory networks (GRNs) refer to the network containing complicated molecular interactions involving messenger ribonucleic acid (mRNAs) and protein molecules that represent the control of gene performance within an organization. Since gene performance regulates cellular functions, molecular/cellular biologists have to learn about and forecast the motions of GRNs. In addition, precise forecasting of gene control procedures can accelerate biotechnology programs. Consequently, mathematical modeling of GRNs and their behaviors are the major study trends in systems biology. Recently, GRNs have been increasingly gained the awareness of researchers because of their easy-to-understand features. These are highly useful and effective methods for describing complex and dynamical transactional relationships. Hence, GRNs are not only the foundation for the investigation of a variety of processes in biological bodies, but also have promising potential applications to systematic biology. Over the recent twenty years, a number of GRNs were already published in the literatures ^{[1,2,3,4,5,6,7,8]}. Remarkably, global exponential stability and (almost) periodicity are important and essential dynamical behaviours in GRNs, which have been extensively studied by numerous scholars (see ^{[5,7,9,10,11]}) in recent twenty years. In particular, in the case of stochastic models, the concept of random periodicity was introduced in lietrature ^[12] based on the theories of semi-flow and metric dynamical system, and the existence of random periodicity to several continuous time stochastic models had been discussed in literatures ^{[12,13,14,15]}. Yet, the problems of global mean-square exponential stability and random periodicity of stochastic GRNs have not been deeply addressed.

Stochastic perturbations not only separate the models from deterministic cases, but also can bring about substantial modifications in dynamic actions of GRNs, see reports ^[16,17]. In general, the behaviors of stochastic systems are highly reliant on time and spatial dependence. Consequently, reaction diffusion is necessary to be taken into account and this induces the investigations of stochastic reaction diffusion systems. Besides, in biologically based network systems, the concentration of constituents is not uniform, resulting in diffusion of cytoplasm from higher to lower concentrations. As it was indicated in report ^[18], the concentration of cellular modules in any region has usually been related to spatial heterogeneity, i.e., mRNA and protein concentrations change according to time and space. Consequently, it is significant to consider GRNs involving the effect of spatial diffusions and more attentions (see ^{[19,20,21,22,23]}) had been paid to GRNs with spatial diffusions in recent years. Such as, the authors in paper ^[19] considered the problem of the stability of GRNs involved with spatial diffusions. With the help of the theory of stable differential equations, a novel generation requirement was created to ensure global exponential stability of reaction diffusions GRNs. Sun et al. ^[20] discussed $H_\infty$ state estimation to GRNs with spatial diffusions and stochastic gain fluctuations, in which the $H_\infty$ performance index is induced to evaluate the ability of the system to resist disturbances and a fuzzy model-based approach is employed to evaluate the immunity of the system. Song et al. ^[21] investigated delayed GRNs with spatial diffusions. By using Lyapunov functional, Wirtinger's and Halanay's inequalities, etc, a novel standard to the networks is developed to guarantee that the estimation error converges to zero. For more researches about GRNs with spatial diffusions, please refer to literatures ^[22,23].

For both computational simulation and analysis, engineers frequently discretize time continuous models to evaluate their structural behaviors. The received and operated signals in digital networks are predicated on discrete time rather than continuous time. Accordingly, discrete-time GRNs have been discussed by numerous scholars, e.g., ^{[24,25,26,27]}. It is remarkable that 1) the difference methods used in papers ^{[24,25,26,27]} were Euler difference; 2) the majority of reported findings on GRNs only concerned time discrete GRNs ^{[24,25,26,27]}, while the corresponding results about space discrete GRNs have not been appreciated adequately in the existing researches, probably due to the partial invalidity of the traditional methods in space-time continuous networks and the difficulty of computing the difference of Lyapunov-Krasovskii functional in discrete space and time networks. These situations trigger this discussion to discuss the space-time discrete GRNs by using exponential Euler difference for time variable and central finite difference for space variable. Notably, the exponential Euler difference is a more effective method than Euler difference (see references ^{[28,29,30,31,32,33]}).

Introduced by the above motivations, the main purpose of this article is to newly formulate a discrete-space and discrete-time stochastic GRNs by employing the methods of exponential Euler difference and finite difference. Subsequently, the constant variation formula, global existence, mean square finiteness and boundedness to the discrete-space and discrete-time GRNs are addressed. Additionally, based on the discrete-space and discrete-time constant variation formula, global mean square exponential stability and the optimal exponential convergence rate are discussed by solving nonlinear constrained optimization problem. At last, random periodicity of discrete-space and discrete-time stochastic GRNs is explored. In the following, the leading researching content and novelties are summarized as follows. 1) Discrete-space and discrete-time stochastic GRNs are newly established to expand the discrete-time GRNs ^{[24,25,26,27]}. 2) Global boundedness and exponential stability in mean square sense are discussed based on discrete-space and discrete-time constant variation formula. 3) The optimal exponential convergence rate can be gained by solving a constrained optimization problem. 4) Random periodicity of discrete-space and discrete-time stochastic GRNs is explored by the theories of semi-flow and metric dynamical systems. 5) This discussion starts the studies of global exponential stability and random periodicity of discrete-space and discrete-time stochastic GRNs.

Plan of this paper: In Section 2, discrete-time stochastic GRNs with discrete spatial diffusions are formulated and some important lemmas are introduced. Section 3 discusses global existence, mean square finiteness and boundedness to GRNs based on discrete constant variation formula. In Section 4, global exponential stability is addressed and the optimal exponential convergence rate can be solved by a constrained optimization problem. In addition, random periodicity of discrete space and time stochastic GRNs is displayed in Section 5. A numerical example of realizing global exponential stability with optimal convergence rate and random periodicity for discrete stochastic GRNs is given in Section 6. Section 7 states the conclusions and perspectives of this article.

Symbols: $\mathbb{R}^m$ denotes the space of $m$-dimensional real vectors; $\mathbb{Z}$ denotes the set of integers; $\mathbb{Z}_0 = \{0, 1, 2, \ldots\}$; $\mathbb{N} = \mathbb{Z}_0\setminus\{0\}$; $I_{J} = I\cap J$, $\forall I, J\subseteq\mathbb{R}$. Let

Define $\partial\mho_{\varsigma}: = \bar{\mho}_{\varsigma}\backslash\mho_{\varsigma}$, where

For any function $u: \bar{\mho}_\varsigma\times\mathbb{Z}\rightarrow \mathbb{R}^m$ with $u: = u_{k}^{\langle\varsigma\rangle} = (u_{1, k}^{\langle\varsigma\rangle}, \ldots, u_{m, k}^{\langle\varsigma\rangle})^T$, denote

where $(\varsigma, k)\in\bar{\mho}_\varsigma\times\mathbb{Z}$, $\hbar > 0$, $q = 1, 2, \ldots, n$. Hereon, $\Delta$ and $\Delta^2_{\hbar_{\cdot}}$ denote the first order difference and second order central finite difference with respect to time and space variables (i.e., $k$ and $\varsigma$), respectively.

2.   Space-time discrete stochastic GRNs

This section firstly gives discrete-time stochastic GRNs with discrete spatial diffusions, which can be regarded as a full dicretization scheme of continuous time stochastic GRNs with reaction diffusions. Next, based on the theory of difference equations, constant variation formula for such discrete networks is addressed. In the end, some important inequalities such as Minkowski inequality in Lemma 2.2, etc. are recalled.

This article considers the following discrete space and time stochastic GRNs in the shape of

where ${\bf {m}}_i$ and ${\bf {p}}_i$ represent the concentrations of the $i$th mRNA and $i$th protein, respectively, $i = 1, 2, \ldots, m$; $\hbar$ and $h$ of less than 1 denote the space and time steps' length in order;

$a_i^* > 0$ and $c_i^* > 0$ are the decay rates of the $i$th mRNA and $i$th protein, respectively; $d_i > 0$ is the translation rate; $\mu_{iq}$ and $\nu_{iq}$ stand for the transmission diffusion matrixes; $I_i = \sum_{j\in {\bf I}_i} w_{ij}$, $w_{ij}\geq0$ is bounded and ${\bf I}_i$ is the set of all the $j$ which is a repressor of gene $i$; $B = (b_{ij})\in \mathbb{R}^{m\times m}$ with

$\gamma_{ij}$ and $\varpi_{ij}$ denote noise intensities, $i, j = 1, 2, \ldots, m$; $f_j, \sigma_j, \eta_j$ take the Hill function, i.e.,

$H_{pj}$ is the Hill coefficient and $\alpha_{pj}$ is a positive constant, $p = 1, 2, 3$, $j = 1, 2, \ldots, m$;

$j = 1, 2, \ldots, m$; $\mathbb{B}_{11}, \ldots, \mathbb{B}_{1m}, \mathbb{B}_{21}, \ldots, \mathbb{B}_{2m}$ are scalar mutually independent two sides standard Brown motions on complete probability space $(\Omega, \mathcal{F}, \mathscr{F}_\cdot, {\bf P})$ with filtration

The Dirichlet boundary conditions of GRNs (2.1) are described as

Hereon, $\mho_{\varsigma}$ can be regarded as discrete form of rectangle area $\mho$ in $\mathbb{R}^{m}$, which is described by

Let ${\bf {m}}_{i, k}^{\langle\varsigma\rangle} = M_i(\varsigma\hbar, kh)$ and ${\bf {p}}_{i, k}^{\langle\varsigma\rangle} = P_i(\varsigma\hbar, kh)$ for $(\varsigma, k)\in\mho_{\varsigma}\times\mathbb{Z}$, $i = 1, 2, \ldots, m$. Then GRNs (2.1) is full discretization scheme to the following stochastic GRNs with reaction diffusions

where $x = (x_1, \ldots, x_n)^T\in\mho\subseteq\mathbb{R}^{n}$ refers to space variable, $i = 1, 2, \ldots, m.$ Hereby, the discrete techniques in GRNs (2.1) are exponential Euler difference (EED in short) for time variable and finite difference method (FDM in short) for space variable. More information about EED and FDM, please refer to papers ^{[28,29,30,31,32,33]} and ^[34,35,36], respectively.

According to the technique of FDM in papers ^[34,35,36], it follows from GRNs (2.1) that

where

and

for $q = 1, 2, \ldots, n$, $i = 1, 2, \ldots, m$.

Define ${\bf {m}}_{i, k}^{\langle\varsigma\rangle} = M_i(\varsigma\hbar, kh)$, ${\bf {p}}_{i, k}^{\langle\varsigma\rangle} = P_i(\varsigma\hbar, kh)$, $a_{i, k}: = a_i(kh)$, $b_{i, k}: = b_i(kh)$, $c_{i, k}: = c_i(kh)$, $d_{i, k}: = d_i(kh)$, $\gamma_{ij, k}: = \gamma_{ij}(kh)$, $\varpi_{ij, k}: = \varpi_{ij}(kh)$ and $I_{i, k}: = I_{i}(kh)$ for $(\varsigma, k)\in\mho_\varsigma\times\mathbb{Z}$, $i, j = 1, 2, \ldots, m$. By applying EED in literatures ^{[28,29,30,31,32,33]} into (2.3), it yields GRNs (2.1).

Remark 2.1. Recently, in literatures ^{[28,29,30,31,32,33]}, EED has been widely employed to study discrete-time systems arisen from many fields. Initially, the authors ^[29] discussed the long time behaviours of time discrete neural networks by utilizing EED. Next, EED in papers ^[30,31,32] had been used to study the time discrete stochastic models. In the wake of high level of attention on fractional calculus, papers ^[28,33] established the frame of EED to multi-delay Caputo-Fabrizio fractional-order differential equations and BAM neural networks. From the viewpoints of both theories and numerical examples in literatures ^{[28,29,30,31,32,33]}, it is a powerful demonstration that EED is a more precise portrayal of time-continuous systems than Euler difference. To date, nevertheless, the concerns of discrete-time GRNs with discrete spatial diffusions have been addressed by very few scholars. Thus, GRNs (2.1) has a excellent research value.

Remark 2.2. As is well-known, FDM is an important way to solve partial differential equations in the area of numerical computations. Numerous literatures paid their attentions on the researches of space discrete models arisen from many fields of science and engineering and these models were called the lattice models (see ^[34,35,36]). Up to now, it exists several reports focusing on GRNs with reaction diffusions ^{[19,20,21,22,23]}. To the present knowledge of the authors, there are almost no papers dealing with the study of discrete space-time GRNs. As a result, the work at hand is expected to address such a void.

In line with the theory of difference equations, a discrete-space and discrete-time constant variation formula to GRNs (2.1) will be established as follows.

Lemma 2.1. GRNs (2.1) can be given expression to

where $(\varsigma, k)\in\mho_\varsigma\times[k_0, \infty)_{\mathbb{Z}}$ with some initial point $k_0\in\mathbb{Z}$, $i = 1, 2, \ldots, m$. Besides, it holds

Remark 2.3. Let $\mu_{iq} = \nu_{iq} = 0$ for $i = 1, 2, \ldots, m, q = 1, 2, \ldots, n,$ and get rid of the space variable in GRNs (2.1), then Eq (2.4) is turned into

for all $i, j = 1, 2, \ldots, m.$ Based on Eq 2.3, we can study the existence of various solutions for discrete-time GRNs, e.g., almost periodicity ^[10], almost automorphism ^[28], etc.

Let $L^2(\Omega, \mathbb{R}^{m})$ denote the family of all square integrable $\mathbb{R}^{m}$-valued random variables with the norm

in which ${\bf E}$ denotes the expectation operator with respect to probability space $(\Omega, \mathcal{F}, {\bf P})$. Let $\mathbb{B}(\bar{\mho}_\varsigma\times \mathbb{Z}, L^2(\Omega, \mathbb{R}^{2m}))$ denote the whole bounded mappings from $\bar{\mho}_\varsigma\times \mathbb{Z}$ to $L^2(\Omega, \mathbb{R}^{2m})$ and define

endowed with the norm

Definition 2.1. A discrete-space and discrete-time stochastic process ${\bf {w}} = ({\bf {m}}_{1}, \ldots, {\bf {m}}_{m}, {\bf {p}}_{1}, \ldots, {\bf {p}}_{m})\in\mathbb{X}$ is said to be the solution of GRNs (2.1) on $[k_0, \infty)_{\mathbb{Z}}$ if it is $\mathscr{F}_k$-adaptive and meets constant variation Eq (2.4), $\forall k\in[k_0, \infty)_{\mathbb{Z}}$.

Lemma 2.2. (^[37]) ${\rm(Minkowski\ inequality)}$ If $X, Y\in L^2(\Omega, \mathbb{R})$, then

Lemma 2.3. ${\bf E}|w_{j, k}|^2 = \frac{1}{h}$ for $k\in\mathbb{Z}$, $j = 1, 2, \ldots, n$.

Proof. By the definition of $w_{j, k}$ and Itô isometric property, it derives

This completes the proof.

3.   Global existence and mean square boundedness

This section mainly focuses on global existence, global finiteness and global mean square boundedness to GRNs (2.1). In the first place, via the constant variation formula in Lemma 2.1 and the theory of stochastic calculus, global existence and finiteness of the solutions to GRNs (2.1) have been studied. Additionally, global mean square boundedness to GRNs (2.1) is addressed with the helps of constant variation Eq (2.4) and Minkowski inequality in Lemma 2.2, etc.

Set

where $i, j = 1, 2, \ldots, m$.

The initial values of GRNs (2.1) are described by

where $\delta_{\cdot}$ and $\rho_{\cdot}$ are $\mathscr{F}_{k_0}$-adaptive and $\mathscr{F}_{k_0+1}$-adaptive, respectively.

Assume that ${\bf {w}} = ({\bf {m}}, {\bf {p}})^T$ with ${\bf {m}} = ({\bf {m}}_{1}, \ldots, {\bf {m}}_{m})^{T}$ and ${\bf {p}} = ({\bf {p}}_{1}, \ldots, {\bf {p}}_{m})^{T}$ is a solution of GRNs (2.1) with initial values as those in (3.1). Define $\delta = (\delta_{1}, \ldots, \delta_{m})^T$ and $\rho = (\rho_{1}, \ldots, \rho_{m})^T$.

Theorem 3.1. Suppose that $\|\delta\|_{\mathbb{X}} < \infty$, $\|\rho\|_{\mathbb{X}} < \infty$ and the assumptions below hold.

$(E_1)$ $a_{i, k}$, $c_{i, k}$, $\mu_{iq}$, $\nu_{iq}$, $b_{ij, k}$, $d_{i, k}$, $\gamma_{ij, k}$, $\varpi_{ij, k}$ and $I_{i, k}$ are bounded constants or sequences, $k\in\mathbb{Z}$, $q = 1, 2, \ldots, n$, $i, j = 1, 2, \ldots, m$. Further, $\min_{1\leq i\leq m}\{\underline{a}, \underline{c}\} > 0$.

$(E_2)$ $\max_{j = 1, 2, \ldots, m}\{|f_j(s)|, |\sigma_j(s)|, |\eta_j(s)|\}\leq \varsigma_1|s|+\varsigma_2$, where $\varsigma_1, \varsigma_2$ are two positive constants.

Then GRNs (2.1) with initial values (3.1) possesses a solution ${\bf {w}} = ({\bf {m}}, {\bf {p}})$ with ${\bf {m}} = ({\bf {m}}_{1}, \ldots, {\bf {m}}_{m})^{T}$ and ${\bf {p}} = ({\bf {p}}_{1}, \ldots, {\bf {p}}_{m})^{T}$ on $[k_0, +\infty)_{\mathbb{Z}}$.

Proof. According to initial values in (3.1), it fulfills that

${\bf{(a)}}\ {\bf{(Existence)} }\ {\bf {m}}_{k}^{\langle\varsigma\rangle}$ and ${\bf {p}}_{k}^{\langle\varsigma\rangle}$ exist for $(\varsigma, k)\in\bar{\mho}_\varsigma\times[k_0, k_0+1]_{\mathbb{Z}}$.

${\bf(b)} \ {\bf(Adaptability)}\ {\bf {m}}_{k}^{\langle\varsigma\rangle}$ and ${\bf {p}}_{k}^{\langle\varsigma\rangle}$ are $\mathscr{F}_{k}$-adaptive for $(\varsigma, k)\in\bar{\mho}_\varsigma\times[k_0, k_0+1]_{\mathbb{Z}}$.

${\bf(c)}\ {\bf(Mean\ square\ finiteness)} \ \big\| {\bf {m}}_{k}^{\langle\varsigma\rangle}\big\|_{2} < \infty$ and $\big\| {\bf {p}}_{k}^{\langle\varsigma\rangle}\big\|_{2} < \infty$ for $(\varsigma, k)\in\bar{\mho}_\varsigma\times[k_0, k_0+1]_{\mathbb{Z}}$.

Next, a method of mathematical induction will be employed to prove ${\bf {m}}_{k}^{\langle\cdot\rangle}$ and ${\bf {p}}_{k}^{\langle\cdot\rangle}$ satisfying ${\bf(a)}–{\bf(c)}$ for $k\in[k_0, K]_{\mathbb{Z}}$, where $K\geq k_0+2$ is an arbitrary integer. Assume that ${\bf {m}}_{k'}^{\langle\cdot\rangle}$ and ${\bf {p}}_{k'}^{\langle\cdot\rangle}$ meet ${\bf(a)}–{\bf(c)}$ for $k'\in[k_0+1, K]_{\mathbb{Z}}$.

In accordance to constant variation Eq (2.4), ${\bf {m}}_{i, k'+1}^{\langle\varsigma\rangle}$ and ${\bf {m}}_{i, k'+1}^{\langle\varsigma\rangle}$ are $\mathscr{F}_{k'+1}$-adaptive and exist for $\varsigma\in\bar{\mho}_\varsigma$ by the adaptability and existences of ${\bf {m}}_{i, s}^{\langle\varsigma\rangle}$ and ${\bf {p}}_{i, s}^{\langle\varsigma\rangle}$ in $(\varsigma, s)\in\bar{\mho}_\varsigma\times[k_0, k']_{\mathbb{Z}}$, $i = 1, 2, \ldots, m$. Further, by Minkowski inequality in Lemmas (2.2) and (2.3), it derives

Similarly, it deduces

where $\varsigma\in\mho_\varsigma.$

By the mean square finiteness of ${\bf {u}}_{\cdot, s}^{\langle\varsigma\rangle}$ and ${\bf {p}}_{\cdot, s}^{\langle\varsigma\rangle}$ in $(\varsigma, s)\in\bar{\mho}_\varsigma\times[k_0, k']_{\mathbb{Z}}$, it concludes

for all $(\varsigma, s)\in\bar{\mho}_\varsigma\times[k_0, k']_{\mathbb{Z}}$, $q = 1, 2, \ldots, n.$ Consequently, ${\bf {m}}_{k'+1}^{\langle\cdot\rangle}$ and ${\bf {p}}_{k'+1}^{\langle\cdot\rangle}$ satisfy mean square finiteness in ${\bf(c)}$. By the arbitrariness of $K$, ${\bf {m}}_{k}^{\langle\cdot\rangle}$ and ${\bf {p}}_{k}^{\langle\cdot\rangle}$ meet $${\bf(a)}–{\bf(c)}$$ for $k\in[k_0, \infty)_{\mathbb{Z}}$, which induces global existence of the solution of GRNs (2.1) with initial values (3.1). This completes the proof.

Theorem 3.1 only gives global finiteness in mean square sense, the application value is relatively small. In the following, global boundedness in mean square sense of GRNs (2.1) with initial values (3.1), which has great potential for application in practical processes, will be addressed on the basis of constant variation Eq (2.4) and Minkowski inequality.

Theorem 3.2. Supposing that the following conditions are valid.

$(G_1)$ $f_j(0) = \sigma_{j}(0) = \eta_{j}(0) = 0$ and it exists positive numbers $L_{j}^{f}$, $L_{j}^{\sigma}$ and $L_{j}^{\eta}$ such that

for any $u, v\in\mathbb{R}$, $j = 1, 2, \ldots, m$.

$(G_2)$ $\min\{a_{\diamond}, c_{\diamond}\} > 0$ and

Then the solution of GRNs (2.1) is global mean square boundedness, i.e.,

where $\lambda_0: = \max\limits_{1\leq i\leq m}\left\{\|\delta\|_{\mathbb{X}} +\frac{1}{\underline{a}_i}\bar{I}_i, \|\rho\|_{\mathbb{X}}\right\}.$

Proof. For any constant $K\in[k_0, \infty)_{\mathbb{Z}}$, define

In view of Theorem 3.1, $\mathcal{M}_K$, $\mathcal{P}_K$ and $\mathcal{Z}_K$ are finite. With the helps of Eq (2.4), Minkowski inequality in Lemmas (2.2) and (2.3), it computes

Similarly,

Combining (3.2) and (3.3), it follows from $(G_2)$ that

which induces from the arbitrariness of $K$ that

This completes the proof.

Remark 3.1. The assumption $\min\{a_{\diamond}, c_{\diamond}\} > 0$ in $(G_2)$, in which $a_{i, \cdot}: = a_{i, \cdot}^*+2\sum_{q = 1}^{n}\frac{\mu_{iq}}{\hbar^2}$ and $c_{i, \cdot}: = c_{i, \cdot}^*+2\sum_{q = 1}^{n}\frac{\nu_{iq}}{\hbar^2}$ for $i = 1, 2, \ldots, m$, implies that the sum of the diffusions' intensities of the $i$th mRNA or $i$th protein is greater than -$\frac{1}{2}\hbar^2$ times the decay rates of the $i$th mRNA or $i$th protein, respectively, $i = 1, 2, \ldots, m$. Furthermore, if $\mu_{i\cdot}$ and $\nu_{i\cdot}$ are nonnegative constants, then $\lambda_1 < 1$ in $(G_2)$ is equal to

This indicates that the spatial diffusions with nonnegative intensive coefficients have no influence on global mean square boundedness of GRNs (2.1).

Remark 3.2. To assure the validity of condition $(G_2)$, the following aspects should be paid attention in the application.

${\bf (i)}$ The coefficients of GRNs (2.1), except for $a_{i, k}$ and $c_{i, k}$, should be better to choose lesser constants, instead, $a_{i, k}$ and $c_{i, k}$ should be selected biggish positive constants for any $i = 1, 2, \ldots, m$, $k\in\mathbb{Z}$.

${\bf (ii)}$ Generally, small positive constants are selected for the time and space steps' length $h$ and $\hbar$.

${\bf (iii)}$ The activation functions $f_{j}$, $\sigma_{j}$ and $\eta_{j}$ of GRNs (2.1) are best to select some small enough positive constants $L_{j}^{f}$, $L_{j}^{\sigma}$ and $L_{j}^{\eta}$ for any $i = 1, 2, \ldots, m$.

Remark 3.3. Remarkably, the assumptions $f_j(0) = \sigma_{j}(0) = \eta_{j}(0) = 0$ in $(G_1)$ are not obligatory. We can remove this assumption, but it will increase the computational difficulties of this discussion.

Remark 3.4. If $h, \hbar\rightarrow0$ in assumption $(G_2)$, then $\lambda_1 < 1$ is hard to be valid. So it is a disadvantage of the proposed method in this discussion. We hope it can be improved in the future works.

When $\gamma_{ij} = \varpi_{ij} = 0$ in GRNs (2.1) for $i, j = 1, 2, \ldots, m$, then it is turned into a determined networks as noted below

where $i = 1, 2, \ldots, m$.

Further, let $\mu_{i\cdot} = \nu_{i\cdot} = 0$ in GRNs (3.4), it is changed to the following discrete-time networks

where $i = 1, 2, \ldots, m$.

Corollary 3.1. Supposing that $(G_1)$ and the following condition hold.

$(G_3)$ $\min\{a_{\diamond}, c_{\diamond}\} > 0$ and

Then the solution of GRNs (3.4) is global mean square boundedness, i.e.,

Corollary 3.2. Supposing that $(G_1)$ and the following condition hold.

$(G_4)$ $\min\{a_{\diamond}, c_{\diamond}\} > 0$ and $\lambda_3: = \max_{1\leq i\leq m}\Bigg\{ \frac{1}{\underline{a}_i}\bigg[\sum_{j = 1}^m\bar{b}_{ij}L_j^f\bigg], \frac{1}{\underline{c}_i}\bar{d}_i \bigg\} < 1.$

Then the solution of GRNs (3.5) is global mean square boundedness, i.e.,

Remark 3.5. The existing literatures ^{[24,25,26,27]} had reported various dynamical explorations of discrete-time GRNs (3.5). However, for discrete spatial diffusions networks (2.1), almost no paper involves.

4.   Global exponential stability and optimal convergent rate

This section is chiefly concerned with global exponential stability in mean square sense to GRNs (2.1) via the theory of stochastic calculus and some inequalities' skills. Firstly, the definition of global exponential stability in mean square sense with the exponential convergent rate to GRNs (2.1) is described as a mathematical expression, which displays the asymptotic relationships of two arbitrary solutions of GRNs (2.1) with different initial values. Subsequently, global exponential stability in mean square sense to GRNs (2.1) is achieved and one optimization problem under nonlinear constraints is created to gain the optimal convergent rate. At last, to acquire the optimal solution of convergent rate in a feasible region, an enforceable algorithm is proposed to solve this regard.

Let $({\bf {m}}, {\bf {p}})^T$ and $(\tilde{ {\bf {m}}}, \tilde{ {\bf {p}}})^T$ with ${\bf {m}} = ({\bf {m}}_1, \ldots, {\bf {m}}_{m})^T$, $\tilde{ {\bf {m}}} = (\tilde{ {\bf {m}}}_1, \ldots, \tilde{ {\bf {m}}}_{m})^T$, ${\bf {p}} = ({\bf {p}}_1, \ldots, {\bf {p}}_{m})^T$ and $\tilde{ {\bf {p}}} = (\tilde{ {\bf {p}}}_1, \ldots, \tilde{ {\bf {p}}}_{m})^T$ be two arbitrary solutions of GRNs (2.1) with initial values

where $\tilde{\delta}^{\langle\varsigma\rangle}$ and $\tilde{\rho}^{\langle\varsigma\rangle}$ are $\mathscr{F}_{k_0}$-adaptive and $\mathscr{F}_{k_0+1}$-adaptive, respectively; $\varsigma\in\mho_\varsigma$.

Let ${\bf {e}} = ({\bf {e}}_{1}, \ldots, {\bf {e}}_{m})^T$, $\tilde{ {\bf {e}}} = (\tilde{ {\bf {e}}}_{1}, \ldots, \tilde{ {\bf {e}}}_{m})^T$, $\hat{\delta} = (\hat{\delta}_1, \ldots, \hat{\delta}_m)^T$ and $\hat{\rho} = (\hat{\rho}_1, \ldots, \hat{\rho}_m)^T$, where ${\bf {e}}_{i} = {\bf {m}}_{i}-\tilde{ {\bf {m}}}_{i}$, $\tilde{ {\bf {e}}}_{i} = {\bf {p}}_{i}-\tilde{ {\bf {p}}}_{i}$, $\hat{\delta}_i = \delta_i-\tilde{\delta}_i$ and $\hat{\rho}_i = \rho_i-\tilde{\rho}_i$, $i = 1, 2, \ldots, m$. From (2.4), it gets

where $(\varsigma, k)\in\mho_\varsigma\times[k_0, \infty)_{\mathbb{Z}}$ with some initial point $k_0\in\mathbb{Z}$,

$i, j = 1, 2, \ldots, m$. Besides, it holds ${\bf {e}}_{k}^{\langle\varsigma\rangle}\Big|_{\varsigma\in \partial \mho_\varsigma} = 0$ and $\tilde{ {\bf {e}}}_{k}^{\langle\varsigma\rangle}\Big|_{\varsigma\in \partial \mho_\varsigma} = 0$ for all $k\in[k_0, \infty)_{\mathbb{Z}}$.

Definition 4.1. GRNs (2.1) is said to be globally mean-square $\kappa$-exponential convergent if it exists $\mathscr{L} > 0$ and $0 < \kappa < 1$ such that

Hereon, $\kappa$ is called the convergent rate of GRNs (2.1).

Theorem 4.1. Let $(G_1)$–$(G_2)$ hold. Then GRNs (2.1) is globally mean-square exponential convergent with the best convergent rate $\kappa$ with respect to a two-tuples $(k_0, \mathscr{L})$, which can be addressed by solving the following optimization problem under nonlinear constraints

where $\mathscr{L}': = \frac{1}{\mathscr{L}}$.

Proof. Owing to $(G_2)$, it has $\mathscr{L} > 1$ and $0 < \kappa < \min_{1\leq i\leq m}\{\underline{a}_i, \underline{c}_i\}$ ensuring

In the light of the error system (4.1) it yields

where $(\varsigma, k)\in\mho_\varsigma\times[k_0, \infty)_{\mathbb{Z}}$, $i = 1, 2, \ldots, n$. By Minkowski, $C_p$ inequalities and Lemma 2.3, one has

where $(\varsigma, k)\in\mho_\varsigma\times[k_0, \infty)_{\mathbb{Z}}$, $i = 1, 2, \ldots, n$.

Similarly,

where $(\varsigma, k)\in\mho_\varsigma\times[k_0, \infty)_{\mathbb{Z}}$, $i = 1, 2, \ldots, n$.

Subsequently, we will use the proof by contradiction. Supposing that

If not, it must exist $k'\in(k_0, \infty)_{\mathbb{Z}}$ ensuring that

and

In line with (4.4), (4.7) and (4.3) in turn, it induces

This induces a confliction with (4.8) and (4.6) is valid.

In the light of (4.5), (4.7) and (4.3) in turn, it results in

This induces a confliction with (4.8) and (4.6) is valid. As a consequence, GRNs (2.1) is globally mean-square exponential convergent. The proof is finished.

In view of assumption $(G_2)$, the feasible region of optimization problem (4.2) is not empty. However, the optimal solution to optimization problem (4.2) may not exist. To ensure the existence of optimal solution, we can simplify optimization problem (4.2) according to the following algorithm.

Similar to the arguments as those in Corollaries 3.1 and 3.2, it directly obtains the following two corollaries according to Theorem 4.1.

Corollary 4.1. Supposing that $(G_1)$ and $(G_3)$ hold. Then GRNs (3.4) is globally mean-square exponential convergent with the best convergent rate $\kappa$ with respect to a two-tuples $(k_0, \mathscr{L})$, which can be addressed by solving the following optimization problem under nonlinear constraints

Corollary 4.2. Supposing that $(G_1)$ and $(G_4)$ hold. Then GRNs (3.5) is globally mean-square exponential convergent with the best convergent rate $\kappa$ with respect to a two-tuples $(k_0, \mathscr{L})$, which can be addressed by solving the following optimization problem under nonlinear constraints

Remark 4.1. Under the assumptions in Theorem 4.1 and as analysed in Remark 3.1, the spatial diffusions with nonnegative intensive coefficients have no influence on global exponential convergence of GRNs (2.1) as well.

Remark 4.2. Global exponential stability of GRNs is an important researching subject in the discussions of the past decades, because it not only depends on the scale of the initial values, but also tends to the equilibrium state with a constant convergence rate, and literatures ^{[5,7,9,10,11]} had been made extensive research on this subject. However, it exists few papers focusing on the optimal convergence rate to global exponential stability of GRNs, and Theorem 4.1, Corollaries 4.1 and 4.2 in this article accomplishes this work with respect to discrete space and time GRNs (2.1), (3.4) and (3.5), respectively.

5.   Random $\tau$-periodic solution

Define $\blacktriangle: = \{(k, s)\in\mathbb{Z}\times \mathbb{Z}, s\leq k\}$, coordinate function $\mathbb{B}_{pj, k}(\omega): = \mathbb{B}_{pj}(kh, \omega) = \omega_{pj, k}$ and $\vartheta:\mathbb{Z}\times \Omega\rightarrow \Omega\subseteq\mathbb{R}^{2m}$ by

where $\omega = (\omega_{11}, \ldots, \omega_{1m}, \omega_{21}, \ldots, \omega_{2m})^T\in\Omega$, $k, s\in\mathbb{Z}$, $p = 1, 2$, $j = 1, 2, \ldots, m$. In view of reference ^[38], $(\Omega, \mathcal{F}, {\bf P}, (\vartheta_k)_{k\in\mathbb{Z}})$ is a metric dynamical system. Considering a stochastic periodic semi-flow $X: \blacktriangle\times\Omega \times\mathbb{R}^{2m}\rightarrow \mathbb{R}^{2m}$ of period $\tau\in\mathbb{Z}$, which fulfils the semi-flow relationship

and the periodic property

Definition 5.1. (^[12]) Random periodic path of period $\tau\in\mathbb{Z}$ of the semi-flow $X: \blacktriangle\times\Omega\times\mathbb{R}^{2m}\rightarrow \mathbb{R}^{2m}$ is an $\mathcal{F}$-measurable sequence $Y: \mathbb{Z}\times\Omega\rightarrow \mathbb{R}^n$ ensuring

In this section, we use ${\bf {w}}_{\cdot}^{\langle\cdot\rangle}(k_0, \varphi) = ({\bf {m}}_{\cdot}^{\langle\cdot\rangle}(k_0, \varphi), {\bf {p}}_{\cdot}^{\langle\cdot\rangle} (k_0, \varphi))^T$ to denote the solution of GRNs (2.1) with initial value $\varphi = (\delta, \rho)^T$ starting from time $k_0$, where $\delta$ and $\rho$ are defined as that in Section 3. Consequently, $X^{\langle\cdot\rangle}(k, s): \Omega \times \mathbb{R}^{2m} \rightarrow \mathbb{R}^{2m}$ defined by $X^{\langle\cdot\rangle}(k, s)\varphi = {\bf {w}}_{k}^{\langle\cdot\rangle}(s, \varphi)$ becomes a semi-flow, $\forall k\in[s, \infty)_{\mathbb{Z}}$, where $s\in \mathbb{Z}$ is the starting point here.

Theorem 5.1. Let assumptions $(G_1)$–$(G_2)$ and the following condition hold.

$(G_5)$ $a_{i, k}$, $b_{i, k}$, $c_{i, k}$, $d_{i, k}$, $\gamma_{ij, k}$, $\varpi_{ij, k}$ and $I_{i, k}$ are $\tau$-periodic sequences with respect to (w.r.t.) time variable $k\in\mathbb{Z}$, $i, j = 1, 2, \ldots, m$.

Then a random $\tau$-periodic process ${\bf {w}}_{*, k}^{\langle\cdot\rangle} = \left({\bf {m}}_{*, k}^{\langle\cdot\rangle}, {\bf {p}}_{*, k}^{\langle\cdot\rangle}\right)^T\in L^2(\Omega, \mathbb{R}^{2m})\, (k\in\mathbb{N})$ solves GRNs (2.1).

Proof. According to Theorem 3.2, ${\bf {w}}_\cdot^{\langle\iota\rangle}(k_0, \cdot):$ $L^2(\Omega, \mathbb{R}^{2m})\rightarrow L^2(\Omega, \mathbb{R}^{2m})$. Let $\varphi = (\delta, \rho)^T\in L^2(\Omega, \mathbb{R}^{2m})$. By the property of semi-flow, for any integers $k, k', k''\geq 0$,

In accordance with Theorem 4.1, for any $\epsilon > 0$, it has $k_* > 0$ such that

So it exists $k_{**} > 0$ such that for any $k', k'''\geq k_{**}$,

i.e., $\left\{ {\bf {w}}_{k}^{\langle\cdot\rangle}(-k'\tau, \varphi):k'\in\mathbb{N}\right\}$ is a Cauchy sequence and

Define a semi-flow $X^{\langle\cdot\rangle}(k, s, \varphi): = {\bf {w}}_{k}^{\langle\cdot\rangle}(k, s, \varphi)$ for all $s, k\in\mathbb{Z}$. It attains

which is addressed by the continuity of ${\bf {w}}_{k}^{\langle\cdot\rangle}(s, \cdot):L^2(\Omega, \mathbb{R}^{2m})\rightarrow L^2(\Omega, \mathbb{R}^{2m})$, $\forall s, k\in\mathbb{Z}$. Besides,

As a result,

Let $\tilde{\varphi} = (\tilde{\delta}, \tilde{\rho})^T\in L^2(\Omega, \mathbb{R}^{2m})$ be another initial values of GRNs (2.1). From Theorem 4.1, it has

tends to 0 as $k'\rightarrow \infty$ for $k\in\mathbb{N}$. So the convergence is not related to the initial value.

Finally, random $\tau$-periodicity should be demonstrated. Based on Eq (2.4), it gains

and

where $\tilde{w}_{k} = (\tilde{w}_{1, k}, \tilde{w}_{2, k})^T = \vartheta_\tau \omega(k)$ with $\tilde{w}_{1, k} = (\tilde{w}_{11, k}, \ldots, \tilde{w}_{1m, k})^T$ and $\tilde{w}_{2, k} = (\tilde{w}_{21, k}, \ldots, \tilde{w}_{2m, k})^T$, $k\in\mathbb{Z}$, $i = 1, 2, \ldots, m$. In addition,

and

where $k, k'\in\mathbb{Z}$, $\vartheta\in\mho_\varsigma$, $i = 1, 2, \ldots, m$.

By pathwise uniqueness of the solution to GRNs (2.1), it gets

Consequently,

as $k'\rightarrow \infty$, where $k\in\mathbb{N}$. Thus, ${\bf {w}}_{*, k+\tau}^{\langle\cdot\rangle}(\omega) = {\bf {w}}_{*, k}^{\langle\cdot\rangle}(\vartheta_\tau \omega)$, ${\bf P}$-a.s., $\forall k\in\mathbb{N}$. This completes the proof.

In allusion to determined GRNs (3.4) and (3.5), we have the following results.

Corollary 5.1. Let assumptions $(G_1)$, $(G_3)$ and $(G_5)$ hold. Then GRNs (3.4) possesses a random $\tau$-periodic solution.

Corollary 5.2.. Let assumptions $(G_1)$, $(G_4)$ and $(G_5)$ hold. Then GRNs (3.5) possesses a random $\tau$-periodic solution.

Remark 5.1. According to assumptions in Theorems 4.1 and 5.1, the rules concluded in Remark 3.1 are also applicative for global exponential stability and random periodicity of GRNs (2.1). In the case of stochastic neural networks, periodic dynamics is an important behavior among various dynamical performances. By employing semi-flow relationship of stochastic models, the existence of random periodic solutions of some nonlinear stochastic systems had been studied in the published literatures ^{[12,13,14,15]}. It should be noted that the research of random periodic solutions to stochastic GRNs, let alone stochastic discrete-time GRNs with discrete spatial diffusions, has not been addressed to the authors' knowledge.

6.   Experimental illustrations

This section gives an experimental example to verify the feasibility of main results for discrete space and time stochastic GRNs, which have been addressed in the former sections of this article.

Considering the following discrete-time stochastic GRNs with discrete spatial diffusions

where $(\varsigma, k)\in(0, 10)\times\mathbb{Z}_0$,

Taking $h = 0.1$ and $\hbar = 0.5$. Corresponding to GRNs (2.1),

Obviously, $L_1^f = L_2^f = 0.1$, $L_1^\sigma = L_2^\sigma = 0.05$, $L_1^\eta = L_2^\eta = \frac{1}{15}$. With the help of MATLAB toolbox, it gains $\lambda_1 = 0.2168 < 1$ and all assumptions of Theorem 4.1 are satisfied. Thus, by Theorem 4.1, GRNs (6.1) is globally mean-square exponential convergent. Additionally, taking $\mathscr{L} = 2$, it computes the optimal convergence rate $\kappa = 7.0859$ by solving the optimization problem under nonlinear constraints displayed in Algorithm 1. The trajectories of global exponential stability of GRNs (6.1) with optimal convergence rate $\kappa = 7.0859$ in 3-dimensional and 2-dimensional spaces have been showed in Figures 1–4. Besides, by Theorem 5.1, GRNs (6.1) possesses a random $2$-periodic oscillation, which is displayed in Figures 5 and 6.

Remark 6.1. Literatures ^{[24,25,26,27]} only researched the time discrete GRNs, e.g., state estimation ^[25], global exponential stability ^[26], bifurcations and chaos ^[27]. Compared with literatures ^{[24,25,26,27]}, this discussion has the following advantages: 1) discrete-time GRNs with discrete spatial diffusions is considered; 2) random periodicity is studied. Thus, the current research expands the works in literatures ^{[24,25,26,27]}.

7.   Conclusions and perspectives

A discrete-time and discrete-space stochastic genetic regulatory networks is proposed, which can be regarded as a fully discretized configuration of stochastic genetic regulatory networks with reaction-diffusions. Based on the constant variational formulation in discrete form of Lemma 2.1, global existence, mean-square boundedness, global exponential stability with optimal convergence speed, and random periodic solutions of this discrete-time stochastic genetic regulatory networks are discussed. In addition, several important inequalities at the end of Section 2 are essential for the discussion in this paper, such as Minkowski inequality. It is worth noting that the work in this paper will open up the study of qualitative problems of discrete-time genetic regulatory networks and lay the theoretical and practical foundation for future work in this field.

In the future, several open topics can be considered further as follows.

● Considerations of the effects of time delays ^[5].

● Considerations of Markovian jumps ^[22].

● Considerations of the issues of controls, e.g., synchronization ^[8], $H_\infty$ states' estimations ^[25], etc.

● Researches of almost periodic sequences ^[5] and almost automorphism ^[28], etc.

● Researches of fractional models ^[39,40].

● Discussions of other stabilities, e.g., finite-time and fixed-time stability ^[41], etc.

Acknowledgments

This work is supported by Science and Technology Plan Project of Guizhou Province (Contract No.:Qiankehe LH [2017] No. 7031).

Conflict of interest

The authors declare there is no conflicts of interest.