Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion

1.   Introduction

Since shunting inhibitory cellular neural networks were proposed by Bouzerdoum and Pinter ^[1] as a new type of neural networks, they have received more and more attention and have been widely applied in optimisation, psychophysics, speech and other fields. At the same time, since time delays are ubiquitous, many research results have been obtained on the dynamics of shunting inhibitory cellular neural networks with time delays ^[2,3,4,5,6].

On the one hand, the quaternion is a generalization of real and complex numbers ^[7]. The skew field of quaternions is defined by

where $q^{R}, q^{I}, q^{J}, q^{K}\in\mathbb{R}$ and the elements $i, j$ and $k$ obey the Hamilton's multiplication rules:

For $q = q^{R}+iq^{I}+jq^{J}+kq^{K}$, we denote $\check{q} = iq^{I}+jq^{J}+kq^{K}$ and $q^R = q-\check{q}$. The norm of $q$ is defined by $\|q\|_{\mathbb{H}} = \sqrt{(q^{R})^2+(q^{I})^2+(q^{J})^2+(q^{K})^2}$. For $y = (y_1, y_2, \cdots, y_n)^T\in\mathbb{H}^n$, we define $\|y\|_{\mathbb{H}^n} = \max_{1\leq p\leq n}\{\|y\|_{\mathbb{H}}\}$, then $(\mathbb{H}^n, \|\cdot\|_{\mathbb{H}^n})$ is a Banach space. As we all know, quaternion-valued neural networks include real-valued neural networks and complex-valued neural networks as their special cases. Compared with complex-valued neural networks, quaternion-valued neural networks only needs half of the connection weight parameters of complex-valued neural networks when dealing with multi-level information ^[8]. In recent years, quaternion-valued neural networks have attracted the attention of many researchers, and their various dynamic behaviors, including fractional-order and stochastic quaternion-valued neural networks, have been extensively studied ^{[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]}.

On the other hand, because periodic and almost periodic oscillations are important dynamics of neural networks, the periodic and almost periodic oscillations of neural networks have been studied a lot in the past few decades ^{[25,26,27,28,29,30,31,32,33]}. Weyl almost periodicity is a generalization of Bohr almost periodicity and Stepanov almost periodicity ^{[34,35,36,37]}. It is a more complex recurrent oscillation. Because the spaces composed of Bohr almost periodic functions and Stepanov almost periodic functions are Banach spaces, it brings some convenience to study the existence of almost periodic solutions in these two senses of differential equations. Therefore, many results have been obtained on the Bohr almost periodic oscillation and Stepanov almost periodic oscillation of neural networks. However, the space composed of Weyl almost periodic functions is incomplete ^[38]. Therefore, the results of Weyl almost periodic solutions of neural networks are still very rare. Therefore, it is a meaningful and challenging work to study the existence of Weyl almost periodic solutions of neural networks.

Motivated by the above, in this paper, we consider the following shunting inhibitory cellular neural networks with time-varying delays:

where $ij\in\{11, 12, \dots, 1n, \dots, m1, m2, \dots, mn\}: = \Lambda$, $C_{ij}$ denotes the cell at the $(i, j)$ position of the lattice. The $r$-neighborhood $N_{r}(i, j)$ of $C_{ij}$ is given as

and $N_{s}(i, j)$ is similarly specified; $x_{ij}(t)\in\mathbb{H}$ denotes the activity of the cell of $C_{ij}$, $I_{ij}(t)\in\mathbb{H}$ is the external input to $C_{ij}$, $a_{ij}(t)\in\mathbb{H}$ is the coefficient of the leakage term, which represents the passive decay rate of the activity of the cell $C_{ij}$, $B_{ij}^{kl}(t)\geq0$ and $C_{ij}^{kl}(t)\geq0$ represent the connection or coupling strength of postsynaptic of activity of the cell transmitted to the cell $C_{ij}$, the activity functions $f_{ij}, g_{ij}:\mathbb{H}\rightarrow \mathbb{H}$ are continuous functions representing the output or firing rate of the cell $C_{ij}$, $\tau_{kl}(t)$ corresponds to the transmission delay and satisfies $0 \leq \tau_{kl}(t) \leq \tau$.

The purpose of this paper is to use the fixed point theorem and a variant of Gronwall inequality to establish the existence and global exponential stability of Weyl almost periodic solutions for a class of quaternion-valued shunting inhibitory cellular neural networks whose coefficients of the leakage terms are quaternions. This is the first paper to study the existence and global exponential stability of Weyl almost periodic solutions of system (1.1) by using the fixed point theorem and a variant of Gronwall inequality. Our result of this paper is new, and our method can be used to study other types of quaternion-valued neural networks.

For convenience, we introduce the following notations:

The initial condition of system (1.1) is given by

where $\varphi_{ij}\in C(\mathbb{R}, \mathbb{H}), ij\in \Lambda$.

Throughout this paper, we assume that:

$(H_{1})$ For $ij, kl\in\Lambda$, functions $a^R_{ij}, B_{ij}^{kl}, C_{ij}^{kl}\in AP(\mathbb{R}, \mathbb{R}^+)$, $\check{a}_{ij}\in AP(\mathbb{R}, \mathbb{H})$, $I_{ij} \in APW^p(\mathbb{R}, \mathbb{H})$, $\tau_{kl}\in AP(\mathbb{R}, \mathbb{R}^{+})\cap C^1(\mathbb{R}, \mathbb{R})$ and $\tau' < 1$.

$(H_{2})$ For $ij\in\Lambda$, there exist positive constants $L_{ij}^{f}$ and $L_{ij}^{g}$ such that for all $x, y\in\mathbb{H}$,

and $f_{ij}(\mathbf{0}) = g_{ij}(\mathbf{0}) = 0$.

The rest of this paper is arranged as follows. In Section 2, we introduce some definitions and lemmas. In Section 3, we study the existence and global exponential stability of Weyl almost periodic solutions of (1.1). In Section 4, an example is given to verify the theoretical results. This paper ends with a brief conclusion in Section 5.

2.   Preliminaries

Let $(\mathbb{X}, \|\cdot\|_{\mathbb{X}})$ be a Banach space and $BC(\mathbb{R}, \mathbb{X})$ be the set of all bounded continuous functions from $\mathbb{R}$ to $\mathbb{X}$.

Definition 2.1. ^[38] A function $f\in BC(\mathbb{R}, \mathbb{X})$ is said to be almost periodic, if for every $\epsilon > 0$, there exists a constant $l = l(\epsilon) > 0$ such that in every interval of length $l(\epsilon)$ contains at least one $\sigma$ such that

Denote by $AP(\mathbb{R}, \mathbb{X})$ the set of all such functions.

For $p\in[1, \infty)$, we denote by $L^p_{loc}(\mathbb{R}, \mathbb{X})$ the space of all functions from $\mathbb{R}$ into $\mathbb{X}$ which are locally $p$-integrable. For $f\in L^p_{loc}(\mathbb{R}, \mathbb{X})$, we define the following seminorm:

Definition 2.2. ^[38] A function $f\in L^p_{loc}(\mathbb{R}, \mathbb{X})$ is said to be $p$-th Weyl almost periodic ($W^p$-almost periodic for short), if for every $\epsilon > 0$, there exists a constant $l = l(\epsilon) > 0$ such that in every interval of length $l(\epsilon)$ contains at least one $\sigma$ such that

This $\sigma$ is called on $\epsilon$-translation number of $f$. The set of all such functions will be denoted by $APW^p(\mathbb{R}, \mathbb{X})$.

Remark 2.1. By Definitions 2.1 and 2.2, it is easy to see that if $f\in AP(\mathbb{R}, \mathbb{X})$, then $f\in APW^p(\mathbb{R}, \mathbb{X})$.

Similar to the proofs of the lemma on page 83 and the lemma on page 84 of ^[39], it is not difficult to prove the following two lemmas.

Lemma 2.1. If $f\in APW^p(\mathbb{R}, \mathbb{X})$, then $f$ is bounded and uniformly continuous on $\mathbb{R}$ withrespect to the seminorn $\|\cdot\|_{W^p}$.

Using the argumentation contained in the proof of Proposition 3.21 in ^[38], one can easily prove the following.

Lemma 2.2. If $f_{k}\in APW^p(\mathbb{R}, \mathbb{X})$, $k = 1, 2, \ldots, n$. Then, for every $\epsilon > 0$, there exist common $\epsilon$-translation numbers for these functions.

Lemma 2.3. ^[40]Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that, for every $t\in\mathbb{R}$,

for some locally integrable function $\rho:\mathbb{R}\rightarrow \mathbb{R}$, and for some constants$\gamma_1, \ldots, \gamma_n\geq0$, and some constants $\eta_1, \ldots, \eta_n\geq\gamma$, where$\gamma = \sum\limits_{p = 1}^n\gamma_p$. We assume that the integrals in the right hand side of (2.1)are convergent. Let $\eta = \min\{\eta_1, \ldots, \eta_n\}$. Then, for every $\xi\in(0, \eta-\gamma]$such that $\int_{-\infty}^0e^{\xi s}\rho(s)ds$ converges, we have, for every $t\in\mathbb{R}$,

In particular, if $\rho(t)$ is constant, we have

3.   Main results

Let $BUC(\mathbb{R}, \mathbb{H}^{m\times n})$ be a collection of bounded and uniformly continuous functions from $\mathbb{R}$ to $\mathbb{H}^{m\times n}$, then, the space $BUC(\mathbb{R}, \mathbb{H}^{m\times n})$ with the norm $\|x\|_{\infty} = \sup\limits_{t\in\mathbb{R}}\|x(t)\|_{\mathbb{H}^{m\times n}}$ is a Banach space, where $x\in BUC(\mathbb{R}, \mathbb{H}^{m\times n})$.

Denote $\phi^0 = (\phi_{11}^0, \cdots, \phi_{1n}^0, \phi_{21}^0, \cdots, \phi_{2n}^0, \cdots, \phi_{m1}^0, \cdots, \phi_{mn}^0)^T$, where

We will show that $\phi^0$ is well defined under assumption $(H_1)$. In fact, by $I_{ij}\in APW^p(\mathbb{R}, \mathbb{H})$ and Lemma 2.1, there exists a constant $M > 0$ such that $\|I_{ij}\|_{W^p}\leq M$ for all $ij\in \Lambda$. According to the Hölder inequality, one has

where $\frac{1}{p}+\frac{1}{q} = 1$, which means that $\phi^0$ is well defined.

Take a positive constant $\alpha\geq\|\phi^0\|_{\infty}$. Let

Then, for every $\phi\in\Omega$, one has

Theorem 3.1. Assume $(H_{1})$–$(H_{2})$ hold. Furthermore, suppose that

$(H_{3})$

$(H_{4})$for $p > 2$,

and, for $p = 2$,

then system (1.1) has a unique $W^p$-almost periodic solution in $\Omega$.

Proof. It is easy to check that if $x = (x_{11}, \cdots, x_{1n}, x_{21}, \cdots, x_{2n}, \cdots, x_{m1}, \cdots, x_{mn})^T\in \Omega$ is a solution of the integral equation

then $x$ is a solution of system (1.1).

Define an operator $T:\Omega\rightarrow \mathbb{H}^{m\times n}$ by

where

Now, we will prove that $T\phi$ is well defined. Actually, by $(H_1)$–$(H_3)$ and (3.1), for $ij\in \Lambda$, one deduces that

That is, $T\phi$ is well defined.

We will divide the rest of the proof into four steps.

$Step$ 1, we will prove that $T\phi\in BUC(\mathbb{R}, \mathbb{H}^{m\times n})$, for every $\phi\in \Omega$.

In fact, by (3.3), we see that $T\phi$ is bounded on $\mathbb{R}$. So, we only need to show that $T\phi$ is uniformly continuous on $\mathbb{R}$. Based on the Hölder inequality for $0\leq h\leq 1$ and $q\geq 1$ with $\frac{1}{p}+\frac{1}{q} = 1$, one has

where $ij\in\Lambda$. Hence, letting $h\rightarrow0^+$, by (3.1), we have

which means that $(T_{ij}\phi)$ is uniformly continuous on $\mathbb{R}$, $ij\in\Lambda$. Therefore, $T\phi\in BUC(\mathbb{R}, \mathbb{H}^{m\times n})$.

$Step$ 2, we will prove that $T$ is a self-mapping from $\Omega$ to $\Omega$.

Actually, for arbitrary $\phi\in\Omega$, from $(H_2)$–$(H_3)$, we have

which implies that $T\phi\in\Omega$. Consequently, $T$ is a self-mapping from $\Omega$ to $\Omega$.

$Step$ 3, we will prove $T$ is a contraction mapping.

As a matter of fact, in view of $(H_{1})$–$(H_{2})$, for any $\phi, \nu\in \Omega$, we can get

From this and $(H_3)$, one has

Noticing that $\kappa < 1$, $T$ is a contraction mapping. Consequently, system $(1.1)$ has a unique solution $x$ in $\Omega$.

$Step$ 4, we will prove that the unique solution $x\in\Omega$ is $W^p$-almost periodic.

Indeed, since $x = (x_{11}, \cdots, x_{1n}, x_{21}, \cdots, x_{2n}, \cdots, x_{m1}, \cdots, x_{mn})^T\in \Omega$, $x$ is bounded and uniformly continuous. Hence, for every $\epsilon > 0$, there exists a $\delta\in (0, \epsilon)$ such that for any $t_1, t_2\in \mathbb{R}$ with $|t_1-t_2| < \delta$ and $ij\in \Lambda$, we have

Also, for this $\delta$, in view of $(H_1)$ and Lemma 2.2, we see that there exists a common $\delta$-translation number $\sigma$ such that

and

where $ij, kl\in\Lambda$. Consequently, from (3.4) and (3.9), we get

Since $x$ is a solution of system (1.1), by (3.2), for $ij\in\Lambda$, we have

When $p > 2$, it follows from Hölder's inequality $\left(\frac{2}{p}, \frac{p-2}{p}\right)$, Hölder's inequality $\left(\frac{1}{2}, \frac{1}{2}\right)$ and $(H_2)$ that

for $ij\in\Lambda$. Similarly, we have

and

for $ij\in\Lambda$.

Besides, combining with Hölder's inequality $\left(\frac{2}{p}, \frac{p-2}{p}\right)$, Hölder's inequality $\left(\frac{1}{2}, \frac{1}{2}\right)$ and $(H_2)$ that

for $ij\in\Lambda$. In a similar way, one can get that

and

for $ij\in\Lambda$. Hence, together with a change of variables, Fubini's theorem, Hölder's inequality, (3.8) and (3.13), we derive that

where

and

and, together with a change of variables, Fubini's theorem, Hölder's inequality, (3.6) and (3.12), we derive that

where

Moreover, based on a change of variables, Fubini's theorem, Hölder's inequality, (3.7), (3.10) and (3.14), we deduce that

where

In view of (3.5), (3.8) and (3.15)–(3.19), we can easily obtain that

and

for $ij\in\Lambda$. Consequently, combining with (3.11) and (3.20)–(3.27), we obtain

where $\eta = \frac{p}{4}a^m$,

and

By $(H_4)$, we have $\gamma < \eta$. Thus, it follows from Lemma 2.3 that

Hence, $x\in APW^p(\mathbb{R}, \mathbb{H}^{m\times n})$.

When $p = 2$, similar to the proof of the case of $p > 2$, one can obtain

where $\tilde{\eta} = \frac{a^m}{2}$,

and

By $(H_4)$, we have $\tilde{\gamma} < \tilde{\eta}$. Thus, it follows from Lemma 2.3 that

which means that $x\in APW^2(\mathbb{R}, \mathbb{H}^{m\times n})$. The proof is complete.

Definition 3.1. ^[14] Let $x$ be a solution of system (1.1) with the initial value $\varphi$ and $y$ be an arbitrary solution of system (1.1) with the initial value $\psi$, respectively. If there exist positive constants $\lambda$ and $M$ such that

where $\|\varphi-\psi\|_{\tau} = \sup\limits_{t\in[-\tau, 0]}\|\varphi(t)-\psi(t)\|_{\mathbb{H}^{m\times n}}.$ Then the solution $x$ of system (1.1) is said to be globally exponentially stable.

Theorem 3.2. Assume that $(H_{1})$–$(H_{3})$ hold, then system (1.1) has a unique $W^p$-almost periodic solution that is globallyexponentially stable.

Proof. Let $x(t)$ be the $W^p$-almost periodic solution with the initial value $\varphi(t)$ and $y(t)$ be an arbitrary solution with the initial value $\psi(t)$. Taking

we have

For $ij\in\Lambda$, we define the following functions:

From $(H_3)$, we get

Since $\Pi_{ij}(u)$ is continuous on $[0, +\infty)$ and $\Pi_{ij}(u)\rightarrow -\infty$ as $u\rightarrow \infty$, there exists $\zeta_{ij} > 0$ such that $\Pi_{ij}(\zeta_{ij}) = 0$ and $\Pi_{ij}(u) > 0$, for $u\in(0, \zeta_{ij})$, $ij\in\Lambda$. Let $\varsigma = \min\limits_{ij\in\Lambda}\{\zeta_{ij}\}$, then we have $\Pi_{ij}(\varsigma)\geq0, ij\in\Lambda$. Hence, we can choose a positive constant $\lambda$ such that $0 < \lambda < \min\{\varsigma, a^m\}$ and $\Pi_{ij}(\lambda) > 0$. Thus, one has

where $ij\in\Lambda$. Take a constant $M = \max\limits_{i\in\Lambda}\bigg\{\frac{a^m}{\check{a}_{ij}^M+\sum\limits_{C_{kl}\in N_r(i, j)}4B_{ij}^{kl^M}L_{ij}^f\alpha+\sum\limits_{C_{kl}\in N_s(i, j)}4C_{ij}^{kl^M} L_{ij}^g\alpha}\bigg\}$, then by $(H_3)$, we have $M > 1$. Thus,

From (3.28), we have

Multiplying both sides of (3.31) by $e^{\int_{0}^{t}a_{ij}^R(v)dv}$ and integrating over $[0, t]$, we have

Hence, for any $\epsilon > 0$, it is easy to see that

We claim that

Otherwise, there exists $t^* > 0$ such that

and

From (3.29), (3.30) and (3.34), we get

which contradicts the Eq (3.33). Hence, (3.32) holds. Letting $\epsilon\rightarrow0^+$, from (3.32), we have

Therefore, the $W^p$-almost periodic solution of system (1.1) is globally exponentially stable. This completes the proof.

4.   A numerical example

Example 4.1. In system (1.1), let $i, j = 1, 2$, $r = s = 1$ and take

By computing, we obtain

Choose a constant $\alpha = \frac{1}{16}\geq \|\phi^0\|_{\infty}$, for $i, j = 1, 2$, we obtain that

For $i, j = 1, 2$, take $p = 3$, it is easy to obtain that

Take $p = 2$, we have

Thus, conditions $(H_1)$–$(H_4)$ of Theorems 3.1 and Theorems 3.2 are satisfied. Hence, system (1.1) has a unique $W^p$-almost periodic solution that is globally exponentially stable (see Figures 1, 2).

Remark 4.1. No known results are available to give the results of Example 4.1.

5.   Conclusions

In this paper, the existence and global exponential stability of Weyl almost periodic solutions for a class of quaternion-valued neural networks with time-varying delays are established. Even when the system we consider is a real-valued system, our results are brand-new. In addition, the method in this paper can be used to study the existence of Weyl almost periodic solutions for other types of neural networks.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 11861072.

Conflict of interest

The authors declare no conflict of interest in this paper.