Strategies to investigate protein turnover with fluorescent protein reporters in eukaryotic organisms

Jonathan Trauth; Johannes Scheffer; Sophia Hasenjäger; Christof Taxis; Jonathan Trauth; Johannes Scheffer; Sophia Hasenjäger; Christof Taxis

doi:10.3934/biophy.2020008

AIMS Biophysics

2020, Volume 7, Issue 2: 90-118. doi: 10.3934/biophy.2020008

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Strategies to investigate protein turnover with fluorescent protein reporters in eukaryotic organisms

Molecular Genetics, Department of Biology, Philipps-University Marburg, Germany

Received: 26 February 2020 Accepted: 16 April 2020 Published: 23 April 2020

In higher eukaryotes, defects in regulated protein turnover are intimately linked to development of diseases and aging. Systematic investigation of proteostasis and protein degradation pathways is of high importance for understanding basic cellular events as well as developmental processes in higher organisms. Recently, novel fluorescent protein-based tools for monitoring protein degradation and mapping degradation pathways were described that facilitate this task. Here we give an overview of these tools and relate them to biophysical properties of fluorescent proteins. We focus on methods for the identification of degradation pathways, the discovery of novel degradation sequences, the investigation of proteome dynamics, and the characterization of protein stability. One can expect systematic application of these tools in the near future by systems biology approaches enhancing understanding of the ubiquitin-proteasome system from single protein degradation pathways to its influence on developmental processes.

Keywords:

Citation: Jonathan Trauth, Johannes Scheffer, Sophia Hasenjäger, Christof Taxis. Strategies to investigate protein turnover with fluorescent protein reporters in eukaryotic organisms[J]. AIMS Biophysics, 2020, 7(2): 90-118. doi: 10.3934/biophy.2020008

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Abstract

1. Introduction

Due to the difficulty in measuring the biomass of plankton, mathematical models of plankton populations are important methods for understanding the physical and biological processes of plankton. Various phytoplankton-zooplankton models (PZMs) have been studied: An NPZD model ^[1], a stochastic PZM ^[2], a plankton-nutrient system ^[3], a marine phytoplankton-zooplankton ecological model (PZEM) ^[4], Trophic phytoplankton-zooplankton models (PZM) ^[5], a bioeconomic PZM with time delays ^[6], a two-zooplankton one-phytoplankton PZM ^[7], a phytoplankton and two zooplankton PZM with toxin-producing delay ^[8], planktonic animal and plant PZM ^[9], and other PZMs ^{[10, 11, 12]}. In ^[12], Wang et al. investigated the following PZEM:

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{dP}{dt} = \overset{plant\ growth}{\overbrace{rP\left( 1-\frac{P}{K} \right) }}-\overset{plant\ loss}{\overbrace{\frac{e_1PZ}{1+e_1h_1P+e_2h_2A}}}, \\ \frac{dZ}{dt} = \underset{animal\ gain}{\underbrace{\frac{n_1e_1PZ+n_2e_2AZ}{1+e_1h_1P+e_2h_2A}}}-\underset{natural\ death}{\underbrace{\tau Z}}-\underset{animal\ death}{\underbrace{\sigma \frac{PZ}{h+P}}}. \end{array} \right. \end{eqnarray}$

(1.1)

Fractional-order reaction-diffusion equation can describe the dynamic behavior of real systems more accurately. In recent years, fractional-order models have made remarkable progress in fields such as biomedicine, ecology, and public health. In biomedicine, fractional-order models have been used to study tumor growth ^[13] and hepatitis B virus (HBV) treatment ^[14], revealing their advantages in describing the complex dynamic behaviors of biological systems. In ecology, fractional-order models have demonstrated their potential in the study of ecosystem complexity by analyzing the dynamic behaviors of plant-herbivore interactions ^[15]. In public health, the fractional model has been applied to analyze the transmission of smoking behavior ^[16] and the correlation between human papilloma virus (HPV) and cervical cancer ^[17], highlighting their utility in public health research. These studies not only enrich the theoretical application of fractional models but also have made significant progress in numerical analysis, providing new tools and methods for solving complex biomedical and ecological problems ^{[13, 17]}. For example, in ecology, fractional PZEMs can more accurately describe the complex dynamics of ecosystems, including population fluctuations and interactions. Li et al. ^[18] studied an established fractional-order delayed zooplankton-phytoplankton model. Javidi and Ahmad ^[19] studied dynamic analysis of time fractional-order phytoplankton-toxic PZM. Kumar et al. ^[20] studied a fractional plankton-oxygen modeling. The proposed fractional-order PZEM extends traditional integer-order models by incorporating memory effects through fractional derivatives, which better capture long-term ecological interactions. This model accounts for toxic substances and additional food transmission, factors critical to understanding algal blooms, and population collapse. Models often neglect cross-diffusion and fractional dynamics, limiting their ability to simulate complex spatiotemporal patterns. Our work bridges this gap, offering insights into chaotic attractors and pattern formation under realistic ecological constraints. In this paper, we investigate the following fractional-order PZEM:

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{\partial^{\alpha_1}P}{\partial t^{\alpha_1}} = \overset{plant\ growth}{\overbrace{rP\left( 1-\frac{P}{K} \right) }}-\overset{plant\ loss}{\overbrace{\frac{e_1PZ}{1+e_1h_1P+e_2h_2A}}}+ \overset{plant\ diffusion}{\overbrace{d_{11} \nabla^{2} P}}, \\ \frac{\partial^{\alpha_2}Z}{\partial t^{\alpha_2}} = \underset{animal\ gain}{\underbrace{\frac{n_1e_1PZ+n_2e_2AZ}{1+e_1h_1P+e_2h_2A}}}-\underset{natural\ death}{\underbrace{\tau Z}}-\underset{animal\ death}{\underbrace{\sigma \frac{PZ}{h+P}}}+\underset{plant\ and \ animal \ diffusion}{\underbrace{ d_{21} \nabla^{2} P+d_{31} \nabla^{2} Z}}, \end{array} \right. \end{eqnarray}$

(1.2)

where $P$ and $Z$ represent the population densities of phytoplankton and zooplankton, respectively, at time t. $\nabla ^2$ stands for the Laplacian operator of the two-dimensional plane, $\varOmega$ is a bounded connected region with smooth boundaries $\partial\varOmega$ , and $d_{11}$ and $d_{31}$ represent the diffusion coefficients of phytoplankton and zooplankton, respectively. These parameters characterize the stochastic movement of individuals within a population from areas of high concentration to those of lower density. $d_{21}$ is a cross-diffusion coefficient that indicates the effect of the presence of phytoplankton on the movement of zooplankton populations. $\alpha_i (i = 1, 2)$ is the fractional-order orders, $0 < \alpha_i < 1$ and $\frac{\partial^{\alpha_1}P}{\partial t^{\alpha_1}}, \frac{\partial^{\alpha_2}Z}{\partial t^{\alpha_2}}$ are the Grünwald-Letnikov (GL) fractional derivative. $t$ is the time variable. The meanings of the parameters in the model are shown in Table 1.

Table 1. Parameter and biological significance in the model.

Parameter	Ecological Significance
$r$	The intrinsic growth rate of phytoplankton
$K$	The carrying capacity of the environment for phytoplankton
$h_{1}$	The time zooplankton need to process one unit of phytoplankton
$h_{2}$	The time zooplankton need to process one unit of supplementary food
$e_{1}$	The capacity of zooplankton to prey on phytoplankton
$e_{2}$	The capacity of zooplankton to forage for supplementary food
$n_{1}$	The nutritional value of phytoplankton
$n_{2}$	The nutritional value of supplementary food
$A$	The biomass of supplementary food
$\tau$	The mortality rate of zooplankton
$\sigma$	The proportion of toxins released by phytoplankton
$h$	The half-saturation constant

| Show Table

DownLoad: CSV

Several numerical schemes have been used to solve the Fractional-order reaction-diffusion equation, the Galerkin method ^{[21, 22]}, the Fourier spectral method ^{[23, 24]}, the finite difference method ^{[25, 26]}, the operational matrix method ^{[27, 28]}, and so on ^{[29, 30, 31, 32, 33, 34]}. In this paper, we investigate dynamic properties and numerical solutions of the fractional-order PZEM (1.2). The major contributions of this paper are as follows:

(a) The concept of the phytoplankton-zooplankton ecological model is first extended to fractional-order PZEM that incorporates the effects of toxic substances and additional food transmission in the environment. This model leverages the memory effect of fractional-order derivatives to more effectively capture biological significance compared to traditional integer-order models.

(b) Stability, Turing instability, Hopf bifurcation, and weakly nonlinear property are analyzed for the PZEM. a new high-precision numerical method is developed for the fractional PZEM without diffusion term, and a discretization method is established for the PZEM with a diffusion term.

The paper is organized as follows: In Section 2, we give the stability and Hopf bifurcation analysis. In Section 3, we detail weakly nonlinear analysis. In Section 4, we give the numerical algorithm and the numerical simulation results of the system. Finally, in Section 5, we the summarize paper.

2. Stability analysis

In this section, we investigate the stability of the equilibrium point and the emergence of Hopf bifurcation. Let

$\begin{eqnarray*} \begin{aligned} &\bar{u} = e_1 h_1 P, \quad \bar{v} = \frac{e_1}{r} Z, \quad t = r T, \quad \kappa = e_1 h_1 K, \quad \alpha = \frac{n_1 h_2}{n_2 h_1}, \\ & \tau = \frac{n_1}{r h_1}, \quad \zeta = \frac{n_2 e_2 h_1}{n_1} A, \quad m = \frac{\tau}{r}, \quad \mu = \frac{\tau}{r}, \quad G = e_1 h h_1. \end{aligned} \end{eqnarray*}$

Equation (1.2) is converted to the following form:

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{\partial^{\alpha_1}}{\partial t^{\alpha_1}} \bar{u} = \bar{u}\left(1 - \frac{\bar{u}}{\kappa}\right) - \frac{\bar{u}\bar{v}}{1 + \alpha\zeta + \bar{u}} + d_{1} \nabla^{2} \bar{u}, \\ \frac{\partial^{\alpha_2}}{\partial t^{\alpha_2}} \bar{v} = \frac{\tau(\bar{u} + \zeta) \bar{v}}{1 + \alpha\zeta + \bar{u}} - m \bar{v} - \mu \frac{\bar{u}\bar{v}}{G + \bar{u}} + d_{2} \nabla^{2} \bar{u} + d_{3} \nabla^{2} \bar{v}, \end{array} \right. \end{eqnarray}$

(2.1)

where, $d_1 = e_1h_1d_{11}, d_2 = e_1h_1d_{21}, d_3 = \frac{e_1}{r}d_{31}$ .

2.1. Stability analysis

We analyze the stability of Eq (2.1) without the diffusion term.

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{\partial^{\alpha_1}}{\partial t^{\alpha_1}} \bar{u} = \bar{u}\left(1 - \frac{\bar{u}}{\kappa}\right) - \frac{\bar{u}\bar{v}}{1 + \alpha\zeta + \bar{u}}, \\ \frac{\partial^{\alpha_2}}{\partial t^{\alpha_2}} \bar{v} = \frac{\tau(\bar{u} + \zeta) \bar{v}}{1 + \alpha\zeta + \bar{u}} - m \bar{v} - \mu \frac{\bar{u}\bar{v}}{G + \bar{u}}. \end{array} \right. \end{eqnarray}$

(2.2)

Solving the following equation:

$\begin{eqnarray} \left\{ \begin{array}{l} \bar{u}\left(1 - \frac{\bar{u}}{\kappa}\right) - \frac{\bar{u} \bar{v}}{1 + \alpha \zeta + \bar{u}} = 0 , \\ \frac{\tau (\bar{u} + \zeta) \bar{v}}{1 + \alpha \zeta + \bar{u}} - m \bar{v} - \mu \frac{\bar{u} \bar{v}}{G + \bar{u}} = 0. \end{array} \right. \end{eqnarray}$

(2.3)

We can get the following equilibrium points of Eq (2.2):

$\begin{eqnarray} E_{0} = (0, 0), \quad E_{1} = (\kappa, 0), \quad E^{*} = \left(\bar{u}^{*}, \bar{v}^{*}\right), \end{eqnarray}$

(2.4)

where

$\begin{eqnarray*} \bar{v}^* = \left(1 + \alpha \zeta + \bar{u}^*\right) \left(1 - \frac{\bar{u}^*}{\kappa}\right). \end{eqnarray*}$

$u^{*}$ represents the positive solution to the quadratic equation (2.5)

$\begin{eqnarray} \bar{D}_{1} \bar{u}^{2} + \bar{D}_{2} \bar{u} + \bar{D}_{3} = 0, \end{eqnarray}$

(2.5)

where

$\begin{eqnarray*} \bar{D}_{1} = \tau - m - \mu, \bar{D}_{3} = \tau \zeta G - m G - m \alpha \zeta G, \bar{D}_{2} = \tau \zeta + \tau G - m \alpha \zeta - m G - \mu \alpha \zeta - \mu - m. \end{eqnarray*}$

Solving Eq (2.5), we can get:

$\begin{eqnarray} \left\{ \begin{array}{l} \bar{u}_{1, 2}^{*} = \frac{-(\tau \zeta + \tau H - m \alpha \zeta - mH - \mu \alpha \zeta - \mu - m)}{2(\tau - m - \mu)} \pm \frac{\sqrt{(\tau \zeta + \tau H - m \alpha \zeta - mH - \mu \alpha \zeta - \mu - m)^{2} - 4(\tau - m - \mu)(\tau \zeta H - mH - m \alpha \zeta H)}}{2(\tau - m - \mu)}, \\ \\ \bar{v}^{*} = \left(1 + \alpha \zeta + \bar{u}^{*}\right) \left(1 - \frac{\bar{u}^{*}}{\kappa}\right). \end{array} \right. \end{eqnarray}$

(2.6)

The Jacobian matrix associated with $E^{*} = \left(\bar{u}^{*}, \bar{v}^{*}\right)$ is formulated as follows

$\begin{eqnarray} J = \left[ \begin{matrix} a_{11}& a_{12}\\ a_{21}& a_{22}\\ \end{matrix} \right] , \end{eqnarray}$

(2.7)

where

$\begin{gather*} a_{11} = 1 - \frac{2 \bar{u}^*}{\kappa} - \frac{\bar{v}^* (1 + \alpha \zeta)}{(1 + \alpha \zeta + \bar{u}^*)^2}, \ a_{12} = -\frac{\bar{u}^*}{1 + \alpha \zeta + \bar{u}^*}, \\ a_{21} = \frac{\tau \bar{v}^* (1 + \alpha \zeta - \zeta)}{(1 + \alpha \zeta + \bar{u}^*)^2} - \frac{\mu \bar{v}^* G}{(G + \bar{u}^*)^2}, \ a_{22} = \frac{\tau (\bar{u}^* + \zeta)}{1 + \alpha \zeta + \bar{u}^*} - m - \frac{\mu \bar{u}^*}{G + \bar{u}^*}. \end{gather*}$

When $(\alpha_{1}, \alpha_{2}) = (1, 1)$ , the characteristic equation at the equilibrium point is as follows:

$\begin{eqnarray} \lambda ^2+Tr_{0}\lambda +\bigtriangleup_{0} = 0, \end{eqnarray}$

(2.8)

where,

$\begin{eqnarray*} Tr_{0} = a_{11}+a_{22}, \bigtriangleup_{0} = a_{11}a_{22}-a_{21}a_{12}. \end{eqnarray*}$

Let $\alpha = 0.55, \tau = 4.1, \zeta = 0.72, G = 8.43, \mu = 0.79, \kappa = 2.38$ , and $m = 2.45$ . The system's equilibrium point and its associated eigenvalues are presented in Table 2.

Table 2. The calculation results of the equilibrium point, eigenvalue, and argument of the system.

Equilibrium point		Eigenvalues of Jacobian matrix		Stability
0	0	–0.3354	1	Unstability
2.3800	0	–1	0.7421	Unstability
0.3131	1.4843	0.0138–0.4838i	0.0138+0.4838i	Unstability

| Show Table

DownLoad: CSV

It can be seen from Table 3 that all three equilibrium points are unstable points, and the system may generate chaos. Next, we introduce a high-precision numerical method for numerical simulation.

Table 3. Comparison of methods for problem (2.9) when

$\beta = 0.9$ and

$t \in [0, 1]$ .

Method	$\alpha=0.4$		$\alpha=0.7$		$\alpha=1$
Method	$h=0.01$	$h=0.05$	$h=0.01$	$h=0.05$	$h=0.01$	$h=0.05$
Closed-from solution ^[35]	3.2307e-02	5.4302e-02	4.5734e-02	2.5713e-01	2.0716e-01	4.5136e-01
Present method	8.9463e-05	5.6724e-04	9.1684e-05	7.3524e-04	2.1527e-04	4.5136e-03

| Show Table

DownLoad: CSV

2.2. Numerical comparison

In simulations of this section, let $\alpha = 1.1, \tau = 4.4, \zeta = 0.8, G = 8, n = 1.75, \kappa = 5.3$ , and $m = 2.5$ and initial conditions $u_0 = 1$ and $v_0 = 1$ . We set the time step size $h = 0.01$ and the time as $T = 600$ for simulation. shows numerical results at $(\alpha_1, \alpha_2) = (0.895, 0.994)$ .

Figure 1. Comparative numerical result of the two methods of the model

$\left(\alpha, \tau, \zeta, G, n, \kappa, m\right) = \left(1.1, 4.4, 0.8, 8, 1.75, 5.3, 2.5\right)$ ,

$\alpha_1 = 0.895$ , and

$\alpha_2 = 0.994$ .

DownLoad: Full-Size Img PowerPoint

We consider the following fractional order GL initial value problem

$\begin{equation} \left\{ \begin{aligned} D_t^\alpha y(t) & = t^\beta, \\ y(0) & = 0. \end{aligned} \right. \end{equation}$

(2.9)

Numerical simulations are performed for Eq (2.9). Compared with the closed-from solution, the simulation results of the high-precision numerical method are in good agreement with the simulation results of other methods, and the accuracy is higher. This verifies the effectiveness of the high-precision numerical method.

2.3. Brief description of the numerical methodology

We first consider the following GL differential equation:

$\begin{eqnarray} \frac{d^{\alpha}}{dt^{\alpha}} y\left( t \right) = f\left( t, y\left( t \right) \right). \end{eqnarray}$

(2.10)

According to ^[36], we can get the high-precision numerical method ^{[36, 37, 38, 39]}:

$\begin{eqnarray} y\left( t_k \right) = h^{\alpha}f\left( t_k, y\left( t_{k-1} \right) \right) -\sum\limits_{i = 1}^m{\vartheta_{j}^{\left( \alpha, p \right)} y\left( t_{k-j} \right)}, \end{eqnarray}$

(2.11)

where,

$\begin{eqnarray} \begin{array}{ll} \left\{ \begin{array}{l} \vartheta^{(\alpha, p)}_0 = \mathbf{g}_0, k = 0, \\ \vartheta^{(\alpha, p)}_k = -\frac{1}{\mathbf{g}_0}\sum\limits_{i = 0}^{k-1}(1-i\frac{1+\alpha}{k})\mathbf{g}_i \vartheta^{(\alpha, p)}_{k-i}, k = 1, 2, ...., p-1, \\ \vartheta^{(\alpha, p)}_k = -\frac{1}{\mathbf{g}_0}\sum\limits_{i = 0}^{p}(1-i\frac{1+\alpha}{k})\mathbf{g}_i \vartheta^{(\alpha, p)}_{k-i}, k = p, p+1, p+2, .... \end{array} \right.\end{array} \end{eqnarray}$

(2.12)

$\begin{eqnarray} \begin{array}{ll} \begin{pmatrix} \mathbf{g}_0 \\ \mathbf{g}_1\\ \mathbf{g}_2\\ \vdots\\ \mathbf{g}_p\\ \end{pmatrix} = - \begin{pmatrix} 1& 1 & 1& \cdots & 1 \\ 1 & 2 & 3 & \cdots & p+1\\ 1 & 2^2 & 3^2 & \cdots &(p+1)^2\\ \vdots & \vdots & \vdots & \cdots & \vdots\\ 1 & 2^p& 3^p & \cdots& (p+1)^p\\ \end{pmatrix}^{-1} \begin{pmatrix} 0 \\ 1\\ 2\\ \vdots\\ p\\ \end{pmatrix}. \end{array} \end{eqnarray}$

(2.13)

Where $m = \left[t_k/h \right] +1$ , $h$ is the step size. Therefore, a high-precision numerical method for the system (2.2) is the following

$\begin{align} \left\{ \begin{array}{l} \bar{u}_{k} = h^{\alpha_1} \left(\bar{u}_{k-1}\left(1 - \frac{\bar{u}_{k-1}}{\kappa}\right) - \frac{\bar{u}_{k-1} \bar{v}_{k-1}}{1 + \alpha \zeta + \bar{u}_{k-1}} \right) + \bar{u}_0 - \sum\limits_{j = 1}^{m} \vartheta_j^{(\alpha_1, p)} \bar{u}_{k-j}, \\ \bar{v}_{k} = h^{\alpha_2} \left( \frac{\tau (\bar{u}_{k} + \zeta) \bar{v}_{k-1}}{1 + \alpha \zeta + \bar{u}_{k}} - m \bar{v}_{k-1} - \mu \frac{\bar{u}_{k} \bar{v}_{k-1}}{G + \bar{u}_{k}} \right) + \bar{v}_0 - \sum\limits_{j = 1}^{m} \vartheta_j^{(\alpha_2, p)} \bar{v}_{k-j}. \end{array} \right. \end{align}$

(2.14)

According to ^{[36, 40]}, the least common order of the system is identified as $0.9819$ . Let $\left(\alpha _1, \alpha _2 \right) = \left(1.05, 0.95 \right), \alpha = 0.55, \tau = 4.1, \zeta = 0.72, G = 8.43, \mu = 0.79, \kappa = 2.38$ , and $m = 2.45$ , and the characteristic equation of the system is given by

$\begin{eqnarray} \lambda ^{40}-\frac{10667}{28823}\lambda ^{21}-\frac{24789}{90071}\lambda ^{19}+\frac{1039692}{4437077} = 0, \end{eqnarray}$

(2.15)

with characteristic roots $\lambda _{1, 2} = 1.0358\pm 0.0439i$ , as $|arg\left(\lambda _{1, 2} \right) | = 0.0423 < \frac{\pi}{40}$ . This indicates that the instability of the system may generate chaos. It can be seen from Figure 2 that the system generates chaos. The chaotic phenomenon of the system gradually disappears over time and tends to a stable state.

Figure 2. Numerical simulation of phase diagrams and time series plot of system (2.2) at

$(\alpha_{1}, \alpha_{2}) = (1.05, 0.95).$ .

DownLoad: Full-Size Img PowerPoint

We set different orders $\left(\alpha _1, \alpha _2 \right) = \left(1.1, 0.95 \right)$ to observe the influence of the order on the dynamic behavior of the system at parameters $\alpha = 0.55, \tau = 4.1, \zeta = 0.72, G = 8.43, n = 0.79, \; \kappa = 2.38$ , and $m = 2.45$ . The characteristic equation of the system is given by

$\begin{eqnarray} \lambda ^{41}-\frac{10667}{28823}\lambda ^{22}-\frac{24789}{90071}\lambda ^{19}+\frac{1039692}{4437077} = 0, \end{eqnarray}$

(2.16)

with characteristic roots $\lambda _{1, 2} = 1.0021\pm 0.2687i$ , as $|arg\left(\lambda _{1, 2} \right) | = 0.2620 < \frac{\pi}{40}$ . It indicates that the instability of the system may generate chaos. It can be seen from Figure 3 that the system is in a quasi-periodic state.

Figure 3. Numerical simulation of phase diagrams and time series plot of system (2.2) at

$(\alpha_{1}, \alpha_{2}) = (1.1, 0.95).$ .

DownLoad: Full-Size Img PowerPoint

To gain a clearer view of the system's dynamic behavior, shows the phase diagram of the system under different $\alpha$ orders.

Figure 4. Comparison of the phytoplankton-zooplankton phase diagram of the system (2.2) at different fractional-order derivatives.

DownLoad: Full-Size Img PowerPoint

2.4. Hopf bifurcation analysis

Next, we explore the potential for a Hopf bifurcation at $E^{*}$ . A Hopf bifurcation is critical as it marks a transition in the model's stability and the emergence of periodic solutions. To achieve this, the characteristic roots of system (2.2) must be purely complex.

The solution to system (2.2) is obtained as follows:

$\begin{eqnarray} \lambda _{1, 2} = \frac{Tr_0\pm \sqrt{Tr_{0}^{2}-4det_0}}{2}. \end{eqnarray}$

(2.17)

When $\alpha = 1$ , system (2.2) undergoes a destabilizing Hopf bifurcation under the conditions $tr_0 = 0$ and $det_0 > 0$ . Given that the stability of system (2.2) is influenced by the fractional derivative, this derivative can be considered a parameter in the Hopf bifurcation. We now determine the conditions for the Hopf bifurcation for system (2.2) around parameter $E_*$ , $\alpha = \alpha_h$ :

(1) At the equilibrium point $E_*$ , the Jacobian matrix possesses a pair of complex conjugate eigenvalues $\lambda _{1, 2} = a_i+ib_i$ , which transition to being purely imaginary at $\alpha = \alpha_h$ ;

(2) $m\left(\alpha _h \right) = 0$ where $m\left(\alpha \right) = \alpha \frac{\pi}{2}-\underset{1\le i\le 2}{\min}|arg\left(\lambda _i \right) |$ ;

(3) $\left. \frac{\partial m\left(\alpha \right)}{\partial \alpha} \right|_{\alpha = \alpha _h}\ne 0$ .

We now demonstrate that $E_*$ experiences a Hopf bifurcation as $\alpha$ crosses the value $\alpha_h$ .

Proof. Given that $tr_{0}^{2}-4\det _0 < 0$ and $tr_0 > 0$ , the eigenvalues form a complex conjugate pair with a positive real component. Thus,

$\begin{eqnarray} 0 < arg\left( \lambda _{12} \right) = \tan ^{-1}\left( \frac{\sqrt{4\det _0-tr_{0}^{2}}}{tr_0} \right) < \frac{\pi}{2}, \end{eqnarray}$

(2.18)

and $\alpha \frac{\pi}{2} > \Big{|}\tan ^{-1}\left(\frac{\sqrt{4\det _0-tr_{0}^{2}}}{tr_0} \right) \Big{|}$ for some $\alpha$ . Let $\alpha_h \frac{\pi}{2} = \Big{|}\tan ^{-1}\left(\frac{\sqrt{4\det _0-tr_{0}^{2}}}{tr_0} \right) \Big{|}$ , get $\alpha _h = \frac{2}{\pi}\tan ^{-1}\left(\frac{\sqrt{4\det _0-tr_{0}^{2}}}{tr_0} \right)$ . Moreover, $\left. \frac{\partial m\left(\alpha \right)}{\partial \alpha} \right|_{\alpha = \alpha _h} = \frac{\pi}{2}\ne 0$ . Therefore, all Hopf conditions satisfy. □

3. Weakly nonlinear analysis

Above, we analyze the dynamical behavior of fractional system (2.2). The expression of the solution of system (2.1) involves more complex special functions, such as the Mittag-Leffler function and other hypergeometric functions. Although the form of the solution exists, its expression is very large. Therefore, in this section, we reveal various spatiotemporal behaviors around the Turing bifurcation threshold $d _2$ and derive the amplitude equations associated with system (2.1) at $\alpha_1 = \alpha_2 = 1$ using multi-scale and weakly nonlinear analysis. The solution of system (2.1) is written in the following form:

$\begin{eqnarray} \left( \begin{array}{l} \bar{u}\\ \bar{v}\\ \end{array} \right) = \left( \begin{array}{c} \bar{u}^*\\ \bar{v}^*\\ \end{array} \right) +\sum\limits_{j = 1}^3{\left( \begin{array}{l} A_{j}^{\bar{u}^*}\\ A_{j}^{\bar{v}^*}\\ \end{array} \right)}e^{iq_j\cdot r}+\, \, \text{c.c}., \end{eqnarray}$

(3.1)

where $\left(A_{j}^{\bar{u}^*}, A_{j}^{\bar{v}^*} \right) ^T$ represents the magnitude of the wave vector $\boldsymbol{q}_j$ , fulfilling the condition that $\left|\boldsymbol{q}_j \right| = q_T$ .

Adding a perturbation to the equilibrium point $E^{*}(u^*, v^*)$ , such that $\bar{u} = \bar{u}^*+u$ , $\bar{v} = \bar{v}^*+v$ , and substituting them into system (2.1), and performing a Taylor expansion, we can get the following form:

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{\partial u}{\partial t} = a_{11}u+a_{12}v+k_{20}u^2+k_{02}v^2+k_{11}uv+k_{30}u^3+k_{03}v^3+k_{21}u^2v+k_{12}uv^2+d_1\nabla ^2u, \\ \frac{\partial v}{\partial t} = a_{11}u+a_{22}v+m_{20}u^2+m_{02}v^2+m_{11}uv+m_{30}u^3+m_{03}v^3+m_{21}u^2v+m_{12}uv^2+d_2\nabla ^2u+d_3\nabla ^2v, \end{array} \right. \end{eqnarray}$

(3.2)

where

$\begin{gather*} k_{02} = 0, \quad k_{03} = 0, \quad k_{12} = 0, \quad m_{02} = 0, \quad m_{03} = 0, \quad m_{12} = 0, \\ k_{11} = -\frac{1 + \alpha \zeta}{(1 + \alpha \zeta + \bar{u}^*)^2}, \quad m_{11} = \frac{(1 + \alpha \zeta - \zeta) \tau}{(1 + \alpha \zeta + \bar{u}^*)^2} - \frac{\mu G}{(G + \bar{u}^*)^2}, \\ k_{21} = -\frac{1 + \alpha \zeta}{(1 + \alpha \zeta + \bar{u}^*)^3}, \quad m_{21} = \frac{-\tau (1 + \alpha \zeta - \zeta)}{(1 + \alpha \zeta + \bar{u}^*)^3} + \frac{\mu G}{(G + \bar{u}^*)^3}, \\ k_{30} = -\frac{(1 + \alpha \zeta) \left(1 - \frac{\bar{u}^*}{\kappa}\right)}{(1 + \alpha \zeta + \bar{u}^*)^3}, \quad m_{30} = \frac{\tau \left(1 - \frac{\bar{u}^*}{\kappa}\right) (1 + \alpha \zeta - \zeta)}{(1 + \alpha \zeta + \bar{u}^*)^3} - \frac{\mu G \bar{v}^*}{(G + \bar{u}^*)^4}, \\ k_{20} = -\frac{1}{\kappa} + \frac{(1 + \alpha \zeta) \left(1 - \frac{\bar{u}^*}{\kappa}\right)}{(1 + \alpha \zeta + \bar{u}^*)^2}, \quad m_{20} = \frac{-\tau (1 + \alpha \zeta - \zeta) \left(1 - \frac{\bar{u}^*}{\kappa}\right)}{(1 + \alpha \zeta + \bar{u}^*)^2} + \frac{\mu G \bar{v}^*}{(G + \bar{u}^*)^3}. \end{gather*}$

Let $U = \left(u, v \right) ^T$ and system (3.2) can be simplified as

$\begin{eqnarray} \frac{\partial U}{\partial t} = LU+N\left( U, U \right), \end{eqnarray}$

(3.3)

$\begin{eqnarray*} L = \left( \begin{matrix} a_{11}+d_1\nabla^{2}& a_{12}\\ a_{21}+d_2\nabla^{2}& a_{22}+d_3\nabla^{2}\\ \end{matrix} \right), \end{eqnarray*}$

where

$\begin{eqnarray*} N = \left(\begin{array}{l} k_{20} u^2 + k_{02} v^2 + k_{11} u v + k_{30} u^3 + k_{03} v^3 + k_{21} u^2 v + k_{12} uv^2 \\ m_{20} u^2 + m_{02} v^2 + m_{11} uc + m_{30} u^3 + m_{03} v^3 + m_{21} u^2 v + m_{12} uv^2 \end{array}\right). \end{eqnarray*}$

Here, $L$ denotes a linear operator and $N$ represents a nonlinear operator.

We expand parameter $d _2$ with a sufficiently small parameter $\varepsilon$ , so we can obtain:

$\begin{eqnarray} d^{T}_{2} = \varepsilon d^{(1)} _2+\varepsilon ^2d^{(2)} _2+\varepsilon ^3d^{(3)} _2+\mathcal{O}\left(\varepsilon ^3 \right). \end{eqnarray}$

(3.4)

We develop variable $U$ and nonlinear term $N$ , respectively, the small parameter $\varepsilon$ :

$\begin{eqnarray} U = \left( \begin{array}{c} u\\ v\\ \end{array} \right) = \varepsilon \left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) +\varepsilon ^2\left( \begin{array}{c} u_2\\ v_2\\ \end{array} \right) +\varepsilon ^3\left( \begin{array}{c} u_3\\ v_3\\ \end{array} \right) +\mathcal{O}\left(\varepsilon ^3 \right), \end{eqnarray}$

(3.5)

$\begin{eqnarray} N = \varepsilon ^2N_2+\varepsilon ^3N_3+\mathcal{O}\left(\varepsilon ^4 \right), \end{eqnarray}$

(3.6)

where

$\begin{gather*} N_2 = \left( \begin{array}{c} k_{20}u_{1}^{2}+k_{11}u_1v_1\\ m_{20}u_{1}^{2}+m_{11}u_1v_1\\ \end{array} \right), \\ N_3 = \left( \begin{array}{c} 2k_{20}u_1u_2+k_{11}u_1v_2+k_{11}u_2v_1+k_{30}u_{1}^{3}+k_{21}u_{1}^{2}v_1\\ 2m_{20}u_1u_2+m_{11}u_1v_2+m_{11}u_2v_1+m_{30}u_{1}^{3}+m_{21}u_{1}^{2}v_1\\ \end{array} \right). \end{gather*}$

We decompose operator $L$ into the following form:

$\begin{eqnarray} L = L_c+\left( d_{2} -d ^{T}_2 \right) M, \end{eqnarray}$

(3.7)

where

$\begin{gather*} L_c = \left( \begin{matrix} d_1\nabla ^2+a_{11}& a_{12}\\ d_2^{T}\nabla ^2+a_{21}& d_3\nabla ^2+a_{22}\\ \end{matrix} \right), M = \left( \begin{matrix} b_{11}& b_{12}\\ b_{21}& b_{22}\\ \end{matrix} \right) . \end{gather*}$

Using the multiple-scale approach, we differentiate the system's time scales into $T_1 = \varepsilon t, T _2 = \varepsilon ^2t, T_3 = \varepsilon ^3t$ , which are mutually independent. Consequently, the time derivative can be articulated as:

$\begin{eqnarray} \frac{\partial}{\partial t} = \varepsilon \frac{\partial}{\partial T_1}+\varepsilon ^2\frac{\partial}{\partial T_2}+\varepsilon ^3\frac{\partial}{\partial T_3}+\mathcal{O}\left(\varepsilon ^3 \right). \end{eqnarray}$

(3.8)

Substituting Eqs (3.4)–(3.8) into Eq (3.3), we can obtain the following equation:

$\begin{eqnarray} \varepsilon :L_c\left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) = 0, \end{eqnarray}$

(3.9)

$\begin{eqnarray} \varepsilon ^2:L_c\left( \begin{array}{c} u_2\\ v_2\\ \end{array} \right) = \frac{\partial}{\partial T_1}\left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) -d^{(1)} _2M\left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) -N_2, \end{eqnarray}$

(3.10)

$\begin{eqnarray} \varepsilon ^3:L_c\left( \begin{array}{c} u_3\\ v_3\\ \end{array} \right) = \frac{\partial}{\partial T_1}\left( \begin{array}{c} u_2\\ v_2\\ \end{array} \right) +\frac{\partial}{\partial T_2}\left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) -d^{(1)} _2M\left( \begin{array}{c} u_2\\ v_2\\ \end{array} \right) -d^{(2)} _2M\left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) -N_3. \end{eqnarray}$

(3.11)

The $1^{st}$ order of $(\mathcal{O}\left(\varepsilon \right))$ :

$\begin{eqnarray} L_c\left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) = 0. \end{eqnarray}$

(3.12)

The solution of Eq (3.12) is

$\begin{eqnarray} \left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) = \left( \begin{array}{c} \varphi\\ 1\\ \end{array} \right) \sum\limits_{j = 1}^3{\left( W_je^{i\boldsymbol{q}_j\cdot r} \right)}+c.c., \end{eqnarray}$

(3.13)

where $\varphi = \frac{-a_{12}}{a_{11}-{d}_{1}q_{c}^{2}}, \left| \boldsymbol{q}_{_j} \right| = q_c, q_c = q_T\left(d ^{T}_2 \right)$ . Additionally, $W_j$ denotes the amplitude of $e^{i\boldsymbol{q}_j\cdot r}$ .

The $2^{nd}$ order of $(\mathcal{O}\left(\varepsilon ^2 \right))$ :

$\begin{eqnarray} L_c\left( \begin{array}{c} u_2\\ v_2\\ \end{array} \right) = \frac{\partial}{\partial T_1}\left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) -d^{(1)} _2M\left( \begin{array}{c} u_1\\ v_1\\ \end{array} \right) -N_2 = \left( \begin{array}{c} F_u\\ F_v\\ \end{array} \right), \end{eqnarray}$

(3.14)

$F_u^j$ and $F_v^j (j = 1, 2, 3)$ are the coefficients with respect to $e^{i\boldsymbol{q}_j\cdot r}$ in $F_u$ and $F_v$ , respectively.

Using the Fredholm solvability condition for Eq (3.14), the vector function on the right-hand side must be perpendicular to the zero eigenvalue of $L_{c}^{+}$ . The zero eigenvector of $L_{c}$ is:

$\begin{eqnarray} \left( \begin{array}{c} 1\\ \psi\\ \end{array} \right) e^{-iq_jr}+c.c., \ j = 1, 2, 3, \end{eqnarray}$

(3.15)

with $\psi = -\frac{a_{11}-{d}_{1}q_{c}^{2}}{a_{21}-{d}^{T}_{2}q_{c}^{2}}$ . From the orthogonal condition:

$\begin{eqnarray} \left( 1, \psi \right) \left( \begin{array}{c} F_{u}^{j}\\ F_{v}^{j}\\ \end{array} \right) = 0, \, \, j = 1, 2, 3. \end{eqnarray}$

(3.16)

Utilizing the Fredholm solvability condition as stated in Eq (3.16), we arrive at the following:

$\begin{eqnarray} \left\{ \begin{array}{c} \left( \varphi +\psi \right) \frac{\partial W_1}{\partial T_1} = d^{(1)} _2\eta_0W_1+2\left( \eta_1+\psi \eta_2 \right) \overline{W}_2\overline{W}_3, \\ \left( \varphi +\psi \right) \frac{\partial W_2}{\partial T_1} = d^{(1)} _2\eta_0W_2+2\left( \eta_1+\psi \eta_2 \right) \overline{W}_1\overline{W}_3, \\ \left( \varphi +\psi \right) \frac{\partial W_3}{\partial T_1} = d^{(1)} _2\eta_0W_3+2\left( \eta_1+\psi \eta_2 \right) \overline{W}_1\overline{W}_2, \end{array} \right. \end{eqnarray}$

(3.17)

where

$\begin{eqnarray*} \left\{ \begin{array}{l} \eta_0 = \left( \varphi b_{11}+b_{22} \right) +\psi \left( \varphi b_{21}+b_{22} \right) , \\ \eta_1 = \frac{1}{2}k_{20}\varphi ^2+k_{11}\varphi , \\ \eta_2 = \frac{1}{2}m_{20}\varphi ^2+m_{11}\varphi . \end{array} \right. \end{eqnarray*}$

Next, higher-order disturbance terms are introduced:

$\begin{eqnarray} \left( \begin{array}{c} u_2\\ v_2\\ \end{array} \right) = \left( \begin{array}{c} U_0\\ V_0\\ \end{array} \right) +\sum\limits_{j = 1}^3{\left( \begin{array}{c} U_j\\ V_j\\ \end{array} \right)}e^{i\boldsymbol{q}_j\cdot r}+\sum\limits_{j = 1}^3{\left( \begin{array}{c} U_{jj}\\ V_{jj}\\ \end{array} \right)}e^{2i\boldsymbol{q}_j\cdot r}+\\ \left( \begin{array}{c} U_{12}\\ V_{12}\\ \end{array} \right) e^{i\left( \boldsymbol{q}_1-\boldsymbol{q}_2 \right) \cdot r}+\left( \begin{array}{c} U_{23}\\ V_{23}\\ \end{array} \right) e^{i\left( \boldsymbol{q}_2-\boldsymbol{q}_3 \right) \cdot r}+\left( \begin{array}{c} U_{31}\\ V_{31}\\ \end{array} \right) e^{i\left( \boldsymbol{q}_3-\boldsymbol{q}_1 \right) \cdot r}+c.c.. \end{eqnarray}$

(3.18)

We substitute formulas (3.13) and (3.18) into formula (3.10). By solving the equations corresponding to different modes, it is known that the coefficients have the following forms:

$\begin{eqnarray*} \begin{aligned} \left( \begin{array}{l} U_0\\ V_0\\ \end{array} \right) & = \left( \begin{array}{l} u_{00}\\ v_{00}\\ \end{array} \right) \left( |W_1|^2+|W_2|^2+|W_3|^2 \right) , \ U_j = \varphi Y_j, \\ \left( \begin{array}{l} U_{jj}\\ V_{jj}\\ \end{array} \right) & = \left( \begin{array}{l} u_{11}\\ v_{11}\\ \end{array} \right) W_{j}^{2}, \, \, \left( \begin{array}{l} U_{ij}\\ V_{ij}\\ \end{array} \right) = \left( \begin{array}{l} \bar{u}^{*}\\ \bar{v}^{*}\\ \end{array} \right) W_i\overline{W_j}, \\ \end{aligned} \end{eqnarray*}$

where

$\begin{eqnarray} \left(\begin{array}{c} \bar{u}_{00} \\ \bar{v}_{00} \end{array}\right) & = \frac{-2}{a_{11}a_{22} - a_{21} a_{12}} \left(\begin{array}{c} a_{22} \eta_1 - a_{12} \eta_2 \\ a_{11} \eta_2 - a_{21} \eta_1 \end{array}\right), \end{eqnarray}$

(3.19)

$\begin{eqnarray} \left(\begin{array}{c} \bar{u}_{11} \\ \bar{v}_{11} \end{array}\right) & = \left(\begin{array}{c} \frac{a_{12} \eta_2 - (a_{22} - 4 d_3 q_c^2) \eta_1}{(a_{11} - 4 d_1 q_c^2)(a_{22} - 4 d_3 q_c^2) - (a_{21} - 4 d ^{T}_2 q_c^2) a_{12}} \\ \frac{(a_{21} - 4 d^{T}_2 q_c^2) \eta_1 - (a_{11} - 4 d_1 q_c^2) \eta_2}{(a_{11} - 4 d_1 q_c^2)(a_{22} - 4 {d}_3 q_c^2) - (a_{21} - 4 d^{T}_2 q_c^2) a_{12}} \end{array}\right), \end{eqnarray}$

(3.20)

$\begin{eqnarray} \left(\begin{array}{c} \bar{u}_{*} \\ \bar{v}_{*} \end{array}\right) & = 2 \left(\begin{array}{c} \frac{a_{12} \eta_2 - (a_{22} - 3 d_3 q_c^2) \eta_1}{(a_{11} - 3 d_1 q_c^2)(a_{22} - 3 d_3 q_c^2) - (a_{21} - 3 d^{T}_2 q_c^2) a_{12}} \\ \frac{(a_{21} - 3 d^{T}_2 q_c^2) \eta_1 - (a_{11} - 3 d_1 q_c^2) \eta_2}{(a_{11} - 3 d_1 q_c^2)(a_{22} - 3 d_3 q_c^2) - (a_{21} - 3 d^{T}_2 q_c^2) a_{12}} \end{array}\right). \end{eqnarray}$

(3.21)

From the orthogonal condition, we obtain:

$\begin{eqnarray} \left\{ \begin{array}{l} \left( \varphi +\psi \right) \left( \frac{\partial Y_1}{\partial T_1}+\frac{\partial W_1}{\partial T_2} \right) = \eta_0\left( d^{(2)} _2W_1+d^{(1)} _2Y_1 \right) -\left[ \left( Q_1+\psi R_1 \right) \left| W_1 \right|^2+\left( Q_2+\psi R_2 \right) \left( \left| W_2 \right|^2+\left| W_3 \right|^2 \right) \right] W_1\\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, +2\left( \eta_1+\psi \eta_2 \right) \left( \overline{W_2}\overline{Y_3}+\overline{W_3}\overline{Y_2} \right) , \\ \left( \varphi +\psi \right) \left( \frac{\partial Y_2}{\partial T_1}+\frac{\partial W_2}{\partial T_2} \right) = \eta_0\left( d^{(2)} _2W_2+d^{(1)} _2Y_2 \right) -\left[ \left( Q_1+\psi R_1 \right) \left| W_2 \right|^2+\left( Q_2+\psi R_2 \right) \left( \left| W_1 \right|^2+\left| W_3 \right|^2 \right) \right] W_2\\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, +2\left( \eta_1+\psi \eta_2 \right) \left( \overline{W_1}\overline{Y_3}+\overline{W_3}\overline{Y_1} \right) , \\ \left( \varphi +\psi \right) \left( \frac{\partial Y_3}{\partial T_1}+\frac{\partial W_3}{\partial T_2} \right) = \eta_0\left( d^{(2)} _2W_3+d^{(1)} _2Y_3 \right) -\left[ \left( Q_1+\psi R_1 \right) \left| W_3 \right|^2+\left( Q_2+\psi R_2 \right) \left( \left| W_2 \right|^2+\left| W_1 \right|^2 \right) \right] W_3\\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, +2\left( \eta_1+\psi \eta_2 \right) \left( \overline{W_2}\overline{Y_1}+\overline{W_1}\overline{Y_2} \right) , \end{array} \right. \end{eqnarray}$

(3.22)

where

$\begin{eqnarray*} Q_1 = -\left( 2\varphi k_{20}+k_{11} \right) \left( u_{00}+u_{11} \right) -\varphi k_{11}\left( v_{00}+v_{11} \right) -3k_{30}\varphi ^3-3k_{21}\varphi ^2, \\ Q_2 = -\left( 2\varphi k_{20}+k_{11} \right) \left( u_{00}+u_* \right) -\varphi k_{11}\left( v_{00}+v_* \right) -6k_{30}\varphi ^3-6k_{21}\varphi ^2, \\ R_1 = -\left( 2\varphi m_{20}+m_{11} \right) \left( u_{00}+u_{11} \right) -\varphi m_{11}\left( v_{00}+v_{11} \right) -3m_{30}\varphi ^3-3m_{21}\varphi ^2, \\ R_2 = -\left( 2\varphi m_{20}+m_{11} \right) \left( u_{00}+u_* \right) -\varphi m_{11}\left( v_{00}+v_* \right) -6m_{30}\varphi ^3-6m_{21}\varphi ^2. \end{eqnarray*}$

The amplitude $A_{j}$ is the coefficient of $e^{i\boldsymbol{q}_j\cdot r}$ at each level, so

$\begin{eqnarray} A_{j} = \varepsilon W_j+\varepsilon ^2Y_j+\mathcal{O}\left(\varepsilon^3 \right) . \end{eqnarray}$

(3.23)

Substituting Eqs (3.17) and (3.22) into Eq (3.23), we get the following amplitude equations:

$\begin{eqnarray} \left\{ \begin{array}{c} \tau _0\frac{\partial Z_1}{\partial t} = \eta Z_1+\chi \bar{Z}_2\bar{Z}_3-\left[ g_1|Z_1|^2+g_2\left( |Z_2|^2+|Z_3|^2 \right) \right] Z_1, \\ \tau _0\frac{\partial Z_2}{\partial t} = \eta Z_2+\chi \bar{Z}_1\bar{Z}_3-\left[ g_1|Z_2|^2+g_2\left( |Z_1|^2+|Z_3|^2 \right) \right] Z_2, \\ \tau _0\frac{\partial Z_3}{\partial t} = \eta Z_3+\chi \bar{Z}_1\bar{Z}_2-\left[ g_1|Z_3|^2+g_2\left( |Z_1|^2+|Z_2|^2 \right) \right] Z_3, \\ \end{array} \right. \end{eqnarray}$

(3.24)

where

$\begin{gather*} \eta = \frac{d _2 -d ^{T}_2}{d ^{T}_2}, \\ \tau _0 = \frac{\varphi +\psi}{d ^{T}_2\left[ \left( \varphi b_{11}+b_{12} \right) +\psi \left( \varphi b_{21}+b_{22} \right) \right]}, \\ \chi = \frac{k_{20}\varphi ^2+2k_{11}\varphi +m_{20}\varphi ^2\psi +2m_{11}\varphi \psi}{d ^{T}_2\left[ \left( \varphi b_{11}+b_{12} \right) +\psi \left( \varphi b_{21}+b_{22} \right) \right]}, \\ g_1 = \frac{Q_1+\psi R_1}{d ^{T}_2\left[ \left( \varphi b_{11}+b_{12} \right) +\psi \left( \varphi b_{21}+b_{22} \right) \right]}, \\ g_2 = \frac{Q_2+\psi R_2}{d ^{T}_2\left[ \left( \varphi b_{11}+b_{12} \right) +\psi \left( \varphi b_{21}+b_{22} \right) \right]}. \end{gather*}$

Since each amplitude $A_j = \omega _je^{i\theta _j}\left(j = 1, 2, 3 \right)$ in Eq (3.24) can be decomposed into mode $\omega _j = |A_j|$ and phase angle $\theta_j$ , substituting $A_{j}$ into Eq (3.24) to separate the real and imaginary parts yields the following equation:

$\begin{eqnarray} \left\{ \begin{array}{l} \tau _0\frac{\partial \theta}{\partial t} = -\chi \frac{\omega _{1}^{2}\omega _{2}^{2}+\omega _{1}^{2}\omega _{3}^{2}+\omega _{2}^{2}\omega _{3}^{2}}{\omega _1\omega _2\omega _3}\sin \theta, \\ \tau _0\frac{\partial \omega _1}{\partial t} = \eta \omega _1+\chi \omega _2\omega _3\cos \theta -\left[ g_1\omega _{1}^{3}+g_2\left( \omega _{2}^{2}+\omega _{3}^{2} \right) \omega _1 \right], \\ \tau _0\frac{\partial \omega _1}{\partial t} = \eta \omega _1+\chi \omega _2\omega _3\cos \theta -\left[ g_1\omega _{1}^{3}+g_2\left( \omega _{2}^{2}+\omega _{3}^{2} \right) \omega _1 \right] , \\ \tau _0\frac{\partial \omega _1}{\partial t} = \eta \omega _1+\chi \omega _2\omega _3\cos \theta -\left[ g_1\omega _{1}^{3}+g_2\left( \omega _{2}^{2}+\omega _{3}^{2} \right) \omega _1 \right], \\ \end{array} \right. \end{eqnarray}$

(3.25)

where, $\theta = \theta _1+\theta _2+\theta _3$ . From Eq (3.25), it can be deduced that the solution is stable under the conditions $\chi > 0, \psi = 0$ and $\chi < 0, \psi = \pi$ . The solutions of Eq (3.25) are shown in Table 4.

Table 4. The relationship between pattern shape and steady-state solutions.

Conditions	Solution	Pattern shape
—	$\omega_1=\omega_{2}=\omega_3=0$	Stationary state
—	$\omega_1=\sqrt{\dfrac{\eta}{g_1}}, \omega_2=\omega_3=0$	Strip pattern
$\eta > \eta_1=\dfrac{-\chi^2}{4(g_1+2g_2)}$	$\omega_1=\omega_{2}=\omega_3=\dfrac{\|\chi\|\pm\sqrt{\chi^2+4(g_1+2g_2)\eta}}{2(g_1+2g_2)}$	Hexagon pattern
$g_2 > g_1, \eta > g_1\omega^2_1$	$\omega_1=\dfrac{\|\chi\|}{g_2-g_1}, \omega_2=\omega_3=\sqrt{\dfrac{\eta-g_1\omega^2_1}{g_1+g_2}}$	Mixed state

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4. Numerical simulation

In this section, we conduct numerical simulations to verify the theoretical analysis and observe its pattern dynamic behavior. In order to observe the dynamic behavior, we introduce the numerical method of the PZEM model (2.1). If $0 < \alpha_i < 1 (i = 1, 2)$ , the time derivative is expressed by formula (2.11), and the space derivative is expressed by the following formula (4.2). If $\alpha_i = 1(i = 1, 2)$ , we employ the Euler discretization approach for conducting numerical simulations in a two-dimensional domain $\varOmega = \left[0, L_x \right] \times \left[0, L_y \right]$ . We select $L_x = 250, L_y = 250$ , a time increment $\varDelta t = 0.2$ , and a spatial increment $\varDelta h = 0.69$ . We denote $u_{pq}^{n} = u\left(x_p, y_q, n\varDelta t \right)$ and $v_{pq}^{n} = v\left(x_p, y_q, n\varDelta t \right)$ . The model (2.1) is discretized using the Euler method, as outlined below:

$\begin{eqnarray} \left\{ \begin{array}{l} \frac{\bar{u}_{pq}^{n+1} - \bar{u}_{pq}^{n}}{\Delta t} = \bar{u}_{pq}^{n} \left(1 - \frac{\bar{u}_{pq}^{n}}{\kappa}\right) - \frac{\bar{u}_{pq}^{n} \bar{v}_{pq}^{n}}{1 + \alpha \zeta + \bar{u}_{pq}^{n}} + d_{1} \nabla^{2} \bar{u}_{pq}^{n}, \\ \frac{\bar{v}_{pq}^{n+1} - \bar{v}_{pq}^{n}}{\Delta t} = \frac{\tau \left(\bar{u}_{pq}^{n} + \zeta\right) \bar{v}_{pq}^{n}}{1 + \alpha \zeta + \bar{u}_{pq}^{n}} - m \bar{v}_{pq}^{n} - \mu \frac{\bar{u}_{pq}^{n} \bar{v}_{pq}^{n}}{G + \bar{u}_{pq}^{n}} + d_{2} \nabla^{2} \bar{u}_{pq}^{n} + d_{3} \nabla^{2} \bar{v}_{pq}^{n}, \end{array} \right. \end{eqnarray}$

(4.1)

where

$\begin{eqnarray} \left\{ \begin{array}{l} \nabla^{2} \bar{u}_{pq} = \frac{\bar{u}_{p+1, q+1} + \bar{u}_{p-1, q-1} + \bar{u}_{p+1, q-1} + \bar{u}_{p-1, q+1} + 4 \left( \bar{u}_{p+1, q} + \bar{u}_{p-1, q} + \bar{u}_{p, q+1} + \bar{u}_{p, q-1} \right) - 20 \bar{u}_{pq}}{6h^{2}}, \\ \nabla^{2} \bar{v}_{pq} = \frac{\bar{v}_{p+1, q+1} + \bar{v}_{p-1, q-1} + \bar{v}_{p+1, q-1} + \bar{v}_{p-1, q+1} + 4 \left( \bar{v}_{p+1, q} + \bar{v}_{p-1, q} + \bar{v}_{p, q+1} + \bar{v}_{p, q-1} \right) - 20 \bar{v}_{pq}}{6h^{2}}. \end{array} \right. \end{eqnarray}$

(4.2)

Next, we select the parameters shown in Table 5 and use Eq (4.1) to conduct numerical simulations.

Table 5. Parameter values.

$\alpha$	$\tau$	$\zeta$	$G$	$n$	$\kappa$	$m$	$\kappa_{c}$	$d_{1}$	$d_{2}$	$d_{3}$
$0.55$	$4.1$	$0.72$	$8.43$	$0.79$	$2.38$	$2.45$	$1.91$	$0.0036$	$0.00158$	$0.0141$

| Show Table

DownLoad: CSV

The parameter values are shown in Table 5, and the calculation results are as follows:

$\begin{gather*} E^{*} = \left( 0.3131, 1.4843 \right) , \chi = -0.4312, \eta = 0.2461, \tau_{0} = 0.3270, g_1 = 1.2869, g_2 = 1.7464. \end{gather*}$

The initial conditions are specified as follows:

$\begin{eqnarray*} u\left( x, y, 0 \right) = u_*\left( 1+0.1\left( rand-0.5 \right) \right) , v\left( x, y, 0 \right) = v_*\left( 1+0.1\left( rand-0.5 \right) \right). \end{eqnarray*}$

The outcomes of the numerical simulation indicate that speckle and mixed-structure patterns emerge in the graphical representation with this particular set of parameters, as illustrated in Figure 5.

Figure 5. Density distribution pattern of system (2.1) at

$\alpha = 0.55, \tau = 4.1, \zeta = 0.72, G = 8.43, n = 0.79, \kappa = 2.38, m = 2.45, d_{1} = 0.0036, d_{2} = 0.00158, d_{3} = 0.0141$ , and

$\alpha_1 = \alpha_2 = 1$ .

DownLoad: Full-Size Img PowerPoint

Now, we use the parameters in and symmetric initial conditions to perform numerical simulations by changing parameters $d_{1}$ , $d_{2}$ , and $d_{3}$ . Numerical simulation was carried out with other parameters remaining unchanged, and the results showed that there is a symmetric mixed structure solution. The subsequent figure illustrates the intricate pattern evolution of the symmetric mixed pattern across parameters. We select the following symmetric initial conditions:

$\begin{eqnarray*} u\left( x, y, 0 \right) = \left\{ \begin{array}{l} u^*, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, x, y\in \left( 80, 120 \right) , \\ u^*-0.001, \, \, other, \\ \end{array} \right. \\ v\left( x, y, 0 \right) = \left\{ \begin{array}{l} v^*, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, x, y\in \left( 80, 120 \right) , \\ v^*-0.001, \, \, other.\\ \end{array} \right. \end{eqnarray*}$

shows the spatial distribution patterns of phytoplankton and zooplankton at different diffusion coefficients, $d_{2}$ . The diffusion coefficient $d$ reflects the influence of phytoplankton on the distribution of zooplankton. When $d_{2} = 0.003$ , phytoplankton and zooplankton form highly clustered patterns, with zooplankton closely following the prey. When $d_{2} = 0.005$ , phytoplankton and zooplankton are more dispersed, and the zooplankton pattern is more complex. This indicates that the diffusion ability of phytoplankton affects the hunting efficiency of zooplankton, and zooplankton tend to migrate to areas with a high density of phytoplankton. When $d_{2} = 0.004$ , the distribution densities of phytoplankton and zooplankton are between the distribution densities of phytoplankton and zooplankton at $d_{2} = 0.003$ and $d_{2} = 0.005$ .

Figure 6. Population density distribution pattern of phytoplankton and zootoplankton with with different parameters

${d}_{1} = 0.002$ and

${d}_{3} = 0.018$ .

DownLoad: Full-Size Img PowerPoint

Let $d_{1} = 0.002, d_{2} = 0.004$ , and $d_{3} = 0.015$ through the observation of the distribution pattern of plankton population density in . At $t = 6000$ , the system presents an incomplete pattern structure. Phytoplankton begins to form an irregular pattern distribution, while zooplankton shows an aggregation pattern following predation. At $t = 8000$ , the pattern structure gradually forms a stable state. At $t = 10000$ , the system reaches a stable state and forms a predator aggregation area that is misaligned with the plant pattern. We find that as time goes by, $t$ , the distribution pattern of plankton population density becomes more complex. This indicates that the system presents a complex competitive process over time. This reflects the process by which the ecosystem reaches a stable state through dynamic adjustment.

Figure 7. The distribution patterns of phytoplankton and zooplankton population densities at different time periods.

DownLoad: Full-Size Img PowerPoint

Let parameter $d_{1}=0.002, d_{2}=0.004$ , and $d_{3}=0.017$ , the patterns of phytoplankton and zooplankton at different times are shown in and . For diffusion coefficients at $d_{1}=0.002, d_{2}=0.006$ , and $d_{3}=0.017$ , the patterns of phytoplankton and zooplankton at different times are shown in Figure 8 $b_{1}-b_{4}$ and Figure 8 $d_{1}-d_{4}$ . In Figure 8, the density of phytoplankton gradually increases over time. It can be seen from the pattern that the difference between the high-density area and the low-density area significantly increases. The internal structure of the pattern becomes more complex, showing more details and layers. The distribution pattern of zooplankton shows a significantly misaligned predation aggregation area with that of plants, and the density of predation hotspots continues to increase over time. It can be seen from the pattern that the spatial distribution of phytoplankton and zooplankton shows complex dynamic behaviors.

Figure 8. Population density distribution patterns of phytoplankton and zootoplankton with parameters

${d}_{1} = 0.002$ and

${d}_{3} = 0.017$ .

DownLoad: Full-Size Img PowerPoint

Through the analysis of the patterns of phytoplankton and zooplankton, we see that, as time goes by, the patterns of zooplankton evolve from simple patterns to complex ones, demonstrating the complexity of this ecosystem. The diffusion coefficient is highly sensitive to the generation of patterns. A smaller diffusion coefficient leads to more regular and symmetrical patterns, while a larger diffusion coefficient results in more complex and irregular patterns, significantly affecting the distribution pattern of plankton. The initial conditions and diffusion coefficient significantly impact the long-term ecological behavior of the model. Moreover, initial conditions affect the formation of the pattern, while the parameters affect the distribution structure of the ecosystem.

5. Conclusions

We combine theoretical analysis with numerical simulation to deeply explore the dynamic behavioral characteristics of the fractional-order PZEM. In theoretical analysis, we conduct stability, Turing instability, Hopf bifurcation, and nonlinear analysis for the PZEM. In numerical methods, we develop a high-precision numerical method for fractional-order PZEM without diffusion terms and verify the superiority of this method through comparison with other methods. Moreover, an effective discretization method is established for the model with diffusion terms. The numerical simulation results not only verify the correctness of the theoretical analysis, but also demonstrate complex dynamic behaviors at different diffusion coefficients. Moreover, the system presents significantly different spatial pattern evolution characteristics. The results reveal that the diffusion coefficient is crucial to the generation of patterns. A bigger diffusion coefficient will lead to more complex and irregular pattern structures, which directly affects the spatial distribution patterns of plankton populations. Furthermore, the initial conditions and the diffusion coefficient have a decisive influence on the long-term ecological behavior of the model: The initial conditions affect the formation process of the pattern, while the diffusion coefficient affects the dynamic evolution of the pattern structure. In future work, we will explore the response mechanism of the system at the coupling effect of multiple factors, as well as the effect of dynamic behavior for fractional-order.

Author contributions

Conceptualization, Methodology, Software, Data, Formal analysis and Funding acquisition, Writing-original draft and writing-review and editing: Shuai Zhang, Hao Lu Zhang, Yu Lan Wang and Zhi Yuan Li. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This paper is supported by Doctoral research start-up Fund of Inner Mongolia University of Technology (DC2300001252) and the Natural Science Foundation of Inner Mongolia (2024LHMS06025).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The authors thank Alexander Monroe for critical reading of the manuscript. This work was supported by the Deutsche Forschungsgemeinschaft grant TA320-7-1 and the BMBF grant 031B0358A.

Conflict of interest

The authors declare no conflict of interest.

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