Turing and Hopf Bifurcation for a Diffusive Nutrient–Microorganism System in Sediment

Abstract

The distribution of nutrients and microorganisms in sediments is not uniform, as it results from the complex interactions between chemical, biological, and physical transport processes. This study investigates, through a simplified model, the dynamics of a microbial population and its nutrients, accounting for the active metabolic state of the bacteria. Using the nutrient diffusion coefficient $d_2$ as a bifurcation parameter, we demonstrate that a Turing bifurcation, leading to spatial patterning, occurs at a critical value. Sufficient conditions for the stability of the resulting pattern are also established. Furthermore, by treating time delay as a bifurcation parameter, we prove the occurrence of Hopf bifurcation near the positive constant equilibrium solutions at a sequence of critical values, showing that time delay can induce spatially homogeneous and inhomogeneous periodic oscillatory patterns. Numerical simulations and solutions are provided to illustrate the theoretical findings.

1. Introduction

Biochemical processes in marine sediments have been a subject of sustained interest among scientists, ecologists, and mathematicians for several decades [1,2,3,4,5,6]. For instance, by analyzing extensive sediment data from the Santa Barbara Basin in California, Madani et al. [1] examined the coupled cycling of sulfur and iron, demonstrating the significant role of sulfur in organic matter degradation under anoxic conditions. Mudryk et al. [2] studied the abundance, distribution, and physiological activity of sulfate-reducing bacteria in the bottom sediments of the Gulf of Gdańsk. There exist many different microorganisms and chemical substances in the sediment. In [3], Baurmann and Feudel introduced a nutrient Cmicroorganism model in marine sediments, considering a single bacterial population (with density denoted as B (kgbiomass/m³)) that consumes one chemical species serving as a nutrient (with concentration denoted as N (mol/m³)). The system is governed by the following equations:

$$\left\{\begin{array}{cc}{\displaystyle \frac{dB}{d\widehat{t}}}=\alpha {\displaystyle \frac{B}{B+E}}BN-mB,\hfill & \\ {\displaystyle \frac{dN}{d\widehat{t}}}=\mathsf{\Theta }\left(\widehat{N}-N\right)-\beta {\displaystyle \frac{B}{B+E}}BN,\hfill \end{array}\right.$$

(1)

where $\alpha$ denotes the rate of conversion into biomass of bacteria, m the bacterial mortality rate, $\beta$ the nutrient uptake rate, and E the half saturation constant. The parameter $\widehat{N}$ denotes the concentration of N in the overlying seawater, and $\mathsf{\Theta }$ quantifies the diffusive exchange rate between the sediment and the seawater. All parameters are positive. The ecological background and derivation of system (1) are detailed in [3]. Introducing the following dimensionless variables and parameters

$$N={\displaystyle \frac{m}{\alpha }}v,\widehat{N}={\displaystyle \frac{m}{\alpha }}b,B={\displaystyle \frac{\mathsf{\Theta }}{\beta }}u,E={\displaystyle \frac{\mathsf{\Theta }}{\beta }}K,\widehat{t}={\displaystyle \frac{t}{\mathsf{\Theta }}},a={\displaystyle \frac{m}{\mathsf{\Theta }}}$$

and taking into account that the biomass of bacteria and nutrients move randomly—described as Brownian random motion—the authors extended system (1) into the following diffusive system:

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial u}{\partial t}}-d_1\Delta u=au\left({\displaystyle \frac{uv}{u+K}}-1\right),\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ {\displaystyle \frac{\partial v}{\partial t}}-d_2\Delta v=b-v-{\displaystyle \frac{u^{2}v}{u+K}},\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ {\partial }_{\eta }u={\partial }_{\eta }v=0,\hfill & x\in \partial \mathsf{\Omega },t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,v(x,0)=v_0\left(x\right)\ge 0,\hfill & x\in \mathsf{\Omega },\hfill \end{array}\right.$$

(2)

where $d_1$ and $d_2$ are the diffusion coefficients of u and v, respectively. The domain $\mathsf{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain with smooth boundary $\partial \mathsf{\Omega }$ and $\eta$ is the outward unit normal on $\partial \mathsf{\Omega }.$ The no-flux boundary condition indicates that the system is closely relative to the external environment. Apparently, $\left(0,b\right)$ is a nonnegative constant equilibrium solution of system (2). If $b>1+2\sqrt{K}$, then (2) admits two positive constant equilibrium solutions: ${\tilde{\mathbf{u}}}_1=({\tilde{u}}_1,{\tilde{v}}_1),$ ${\tilde{\mathbf{u}}}_2=({\tilde{u}}_2,{\tilde{v}}_2),$ with

$$\begin{array}{ccc}\hfill {\tilde{u}}_1& =& {\displaystyle \frac{1}{2}}\left(b-1-\sqrt{{(1-b)}^2-4K}\right),\hfill \\ \hfill {\tilde{u}}_2& =& {\displaystyle \frac{1}{2}}\left(b-1+\sqrt{{(1-b)}^2-4K}\right),\hfill \end{array}$$

where ${\tilde{v}}_1={\displaystyle \frac{{\tilde{u}}_1+K}{{\tilde{u}}_1}},$ ${\tilde{v}}_2={\displaystyle \frac{{\tilde{u}}_2+K}{{\tilde{u}}_2}}.$ If $b=1+2\sqrt{K}$, then (2) admits a unique positive constant equilibrium solution ${\tilde{\mathbf{u}}}_3=({\tilde{u}}_3,{\tilde{v}}_3)$ where ${\tilde{u}}_3=\sqrt{K},$ ${\tilde{v}}_3=1+\sqrt{K}.$ Baurmann and Feudel have conducted biological and numerical investigations of this system. More recently, Cao and Wu [5] examined the local stability of the nonnegative constant equilibrium solutions and analyzed the local and global bifurcation structures emanating from the two positive constant equilibrium solutions, denoted as ${\tilde{\mathbf{u}}}_1$ and ${\tilde{\mathbf{u}}}_2.$ In a subsequent study, Cao et al. [6] established steady state bifurcation with two dimensional kernel by employing space decomposition and the implicit function theorem. Furthermore, by treating a as a bifurcation parameter, they demonstrated the occurrence of Hopf bifurcations in the positive constant equilibrium solution.

Since the seminal work of [7], the formation of Turing patterns has been the subject of extensive research, as evidenced by studies such as [8,9,10,11,12,13,14,15,16,17]. Beyond their role in generating spatial structures within ecological systems, pattern solutions have also seen substantial theoretical progress regarding their stability and dispersion properties. In particular, Turing bifurcations have been widely examined in the context of reaction–diffusion models [18,19,20,21,22,23]. An interesting question is whether or not Turing bifurcation can be captured in system (2). Our first objective is to investigate the existence of Turing bifurcation.

Time delays are frequently incorporated into biological models to account for factors such as resource regeneration times, maturation periods, and reaction lags, as explored in numerous studies (see, e.g., [24,25,26]). The existence of time delay in the interaction process between bacteria and nutrients will affect the density of bacteria. To study the effects of time delays on model (2), we introduce a discrete delay into (2), which leads to the following delayed diffusive nutrient–microorganism model:

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial u}{\partial t}}-d_1\Delta u=au\left({\displaystyle \frac{uv}{u+K}}-1\right),\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ {\displaystyle \frac{\partial v}{\partial t}}-d_2\Delta v=b-v-{\displaystyle \frac{u^2(x,t-\tau )v}{u(x,t-\tau )+K}},\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ {\partial }_{\eta }u={\partial }_{\eta }v=0,\hfill & x\in \partial \mathsf{\Omega },t>0,\hfill \\ u(x,t)=u_0(x,t)\ge 0,v(x,t)=v_0(x,t)\ge 0,\hfill & x\in \mathsf{\Omega },t\in [-\tau ,0].\hfill \end{array}\right.$$

(3)

Here the time delay $\tau$ stands for the maturation period of bacteria, indicating that after entering the environment, bacteria require $\tau$ units of time to reach maturity and subsequently accomplish reproductive growth.

It is widely recognized that periodic solutions (both spatially homogeneous and inhomogeneous) emerging from Hopf bifurcations in delayed reaction–diffusion systems constitute an important mechanism for spatiotemporal pattern formation, a subject thoroughly explored in prior works (see, e.g., [27,28,29,30,31]). Accordingly, in studying system (3), our second objective is to establish the existence of both spatially homogeneous and spatially inhomogeneous periodic oscillations induced by Hopf bifurcation.

The key distinction and advancements of our work are as follows:

1.

Relation to Existing Studies: Prior research, including the works cited (e.g., [5,6]), has primarily focused on analyzing the stability, steady-state bifurcation, and Hopf bifurcations of the base model (2).

2.

Identification of Research Gap and Our Core Contributions: Our work addresses a distinct gap by investigating two critical aspects beyond the existing analysis: (1) Spatial Patterning via Turing Bifurcation: We establish the conditions for the existence of Turing bifurcation in model (2), leading to spatially inhomogeneous stationary patterns. Furthermore, we provide a stability analysis for the resulting patterns. (2) Delay-Induced Hopf bifurcation: We analyze the stability and Hopf bifurcations induced by the delay for model (2). The results reveal how time delays can destabilize positive constant equilibrium solutions and generate spatially homogeneous and inhomogeneous periodic patterns.

3.

Novelty and Incremental Advance: The key novelty of our work lies in clarifying how the interplay between time delay, introduced as a new bifurcation parameter, and diffusion governs stability and spatiotemporal pattern formation. We demonstrate that diffusion-driven Turing bifurcation occurs, accompanied by numerical simulations that confirm the emergence of spatially oscillatory patterns. Furthermore, time delay is shown to induce a Hopf bifurcation, giving rise to both homogeneous and inhomogeneous periodic oscillation dynamics in the model.

The rest of this paper is organized as follows. In Section 2, by taking diffusion coefficients $d_2$ as the bifurcation parameter in system (2), we show that at the critical value of the bifurcation parameter $d_2$, a Turing bifurcation occurs, i.e., a spatially inhomogeneous steady state solution; in other words, a pattern arises. In Section 3 a detailed stability analysis of the pattern is demonstrated. In Section 4, we analyze the occurrence of Hopf bifurcation in system (3) at the positive constant equilibrium solutions ${\tilde{\mathbf{u}}}_2$ by taking $\tau$ as a bifurcation parameter. In Section 5, we perform some numerical simulations to support our theoretical results. We end with a brief conclusion in Section 6.

2. Turing Bifurcation Analysis of System (2)

In the remainder of this paper, without loss of generality, we choose the domain $\mathsf{\Omega }=(0,l\pi )$ $(l\in {\mathbb{R}}^{+})$, since the eigenvalue and characteristic function of the Laplace operator are clear. We denote $U={\left(u,v\right)}^T,$ $D=diag(d_1,d_2)$ and the vector

$$F\left(U\right)=\left(\begin{array}{c}f(u,v)\\ g(u,v)\end{array}\right)=\left(\begin{array}{c}au\left({\displaystyle \frac{uv}{u+K}}-1\right)\hfill \\ b-v-{\displaystyle \frac{u^{2}v}{u+K}}\hfill \end{array}\right).$$

Then system (2) becomes

$$\left\{\begin{array}{cc}{U}_t-D{U}_{xx}=F\left(U\right),\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ U_x\left(0,t\right)=U_x\left(l\pi ,t\right)=0,\hfill & t>0,\hfill \\ U(x,0)=U_0\left(x\right),\hfill & x\in \mathsf{\Omega }.\hfill \end{array}\right.$$

(4)

Let $V=\left(V_1,V_2\right)=\left(u-{\tilde{u}}_2,v-{\tilde{v}}_2\right),$ then the linearized system of (4) at the positive constant equilibrium solution ${\tilde{\mathbf{u}}}_2$ assumes the form:

$$\left\{\begin{array}{cc}{V}_t-D{V}_{xx}=AV,\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ V_x\left(0,t\right)=V_x\left(l\pi ,t\right)=0,\hfill & t>0,\hfill \\ V(x,0)=V_0\left(x\right),\hfill & x\in \mathsf{\Omega },\hfill \end{array}\right.$$

(5)

where

$$A=F_U\left({\tilde{\mathbf{u}}}_2\right)=\left(\begin{array}{cc}{\displaystyle \frac{aK}{{\tilde{u}}_2+K}}\hfill & {\displaystyle \frac{a{\tilde{u}}_2^2}{{\tilde{u}}_2+K}}\hfill \\ -2+{\displaystyle \frac{{\tilde{u}}_2}{{\tilde{u}}_2+K}}\hfill & -1-{\displaystyle \frac{{\tilde{u}}_2^2}{{\tilde{u}}_2+K}}\hfill \end{array}\right)=\left(\begin{array}{cc}a(1-{\displaystyle \frac{1}{{\tilde{v}}_2}})\hfill & a{\displaystyle \frac{{\tilde{u}}_2}{{\tilde{v}}_2}}\hfill \\ -(2-{\displaystyle \frac{1}{{\tilde{v}}_2}})\hfill & -(1+{\displaystyle \frac{{\tilde{u}}_2}{{\tilde{v}}_2}})\hfill \end{array}\right).$$

(6)

Suppose $V(x,t)=y\left(t\right)\phi \left(x\right)$ is the solution of (5). $y:[0,\infty )\to {\mathbb{R}}^2,$ $\phi :\left[0,l\pi \right]\to \mathbb{R}$ are to satisfy

$${\displaystyle \frac{dy}{dt}}=\left(A-\mu D\right)y$$

(7)

and

$$-{\phi }^{″}=\mu \phi ,{\phi }^{\prime }\left(0\right)={\phi }^{\prime }\left(l\pi \right)=0.$$

(8)

The eigenvalues of (8) are

$${\mu }_j={\left({\displaystyle \frac{j}{l}}\right)}^2,j\in {\mathbb{N}}_0=\{0,1,\dots \}$$

and the associated eigenfunctions

$${\phi }_j\left(x\right)=cos{\displaystyle \frac{jx}{l}}.$$

We substitute these eigenvalues into (7). Let $y_{1j},y_{2j}$ be the two independent solutions of (7) for $\mu ={\mu }_j$, The solution to system (5) admits a representation of the form

$$V\left(x,t\right)={\sum }_{j=0}^{\infty }\left[a_{1j}{y}_{1j}\left(t\right)+a_{2j}{y}_{2j}\left(t\right)\right]cos{\displaystyle \frac{jx}{l}},$$

and here $a_{ij}$ $\left(i=1,2;j=0,1,2,\dots \right)$ are to be determined according to the initial condition $V\left(x,0\right).$ For example, if $y_{1j}\left(0\right)=(1,0),$ $y_{2j}\left(0\right)=(0,1)$ for $j\in {\mathbb{N}}_0,$ then

$$\left(\begin{array}{c}{a}_{10}\\ a_{20}\end{array}\right)={\displaystyle \frac{1}{l\pi }}{\int }_0^{l\pi }V\left(x,0\right)dx$$

$$\left(\begin{array}{c}{a}_{1k}\\ a_{2k}\end{array}\right)={\displaystyle \frac{2}{l\pi }}{\int }_0^{l\pi }V\left(x,0\right)cos{\displaystyle \frac{kx}{l}}dx(k=1,2,\dots ).$$

We will now derive the conditions for a Turing bifurcation in the linear system (5) of (2), treating the nutrient diffusion coefficient $d_2$ as the bifurcation parameter.

Theorem 1.

Assume that $a<{\displaystyle \frac{{\tilde{u}}_2^2+{\tilde{u}}_2+K}{K}}$ holds.

(i) If

$$d_1\ge {\displaystyle \frac{aK}{\left({\tilde{u}}_2+K\right){\mu }_1}}$$

(9)

then the zero solution of system (5) is asymptotically stable for all $d_2>0$.

(ii) If

$${\displaystyle \frac{aK}{\left({\tilde{u}}_2+K\right){\mu }_2}}\le d_1<{\displaystyle \frac{aK}{\left({\tilde{u}}_2+K\right){\mu }_1}}$$

(10)

then at

$$d_2=d_2^{*}={\displaystyle \frac{detA+\frac{\left({\tilde{u}}_2^2+{\tilde{u}}_2+K\right)d_1{\mu }_1}{{\tilde{u}}_2+K}}{\frac{aK{\mu }_1}{{\tilde{u}}_2+K}-d_1{\mu }_1^2}}$$

(11)

the zero solution of system (5) undergoes a Turing bifurcation.

Proof. 

(i) Denote

$$\mathcal{B}\left(\mu \right)=A-\mu D,{\mathcal{B}}_j=\mathcal{B}\left({\mu }_j\right)=A-{\mu }_{j}D.$$

(12)

It is easy to obtain that

$$TrA={\displaystyle \frac{aK-\left({\tilde{u}}_2^2+{\tilde{u}}_2+K\right)}{{\tilde{u}}_2+K}}$$

(13)

$$detA={\displaystyle \frac{a\left({\tilde{u}}_2^2-K\right)}{{\tilde{u}}_2+K}}>0$$

(14)

$$Tr{\mathcal{B}}_j=TrA-\left(d_1+d_2\right){\mu }_j$$

(15)

$$\begin{array}{ccc}\hfill det{\mathcal{B}}_j& =& det\left(\begin{array}{cc}{\displaystyle \frac{aK}{{\tilde{u}}_2+K}}-d_1{\mu }_j\hfill & {\displaystyle \frac{a{\tilde{u}}_2^2}{{\tilde{u}}_2+K}}\hfill \\ -2+{\displaystyle \frac{{\tilde{u}}_2}{{\tilde{u}}_2+K}}\hfill & -1-{\displaystyle \frac{{\tilde{u}}_2^2}{{\tilde{u}}_2+K}}-d_2{\mu }_j\hfill \end{array}\right)\hfill \\ & =& detA+d_1{d}_2{\mu }_j^2+{\displaystyle \frac{d_1\left({\tilde{u}}_2^2+{\tilde{u}}_2+K\right)-d_{2}aK}{{\tilde{u}}_2+K}}{\mu }_j\hfill \\ & =& \left({\mu }_j{d}_1-{\displaystyle \frac{aK}{{\tilde{u}}_2+K}}\right)\left({\mu }_j{d}_2+{\displaystyle \frac{{\tilde{u}}_2^2+{\tilde{u}}_2+K}{{\tilde{u}}_2+K}}\right)+{\displaystyle \frac{a{\tilde{u}}_2^2\left({\tilde{u}}_2+K\right)}{{\left({\tilde{u}}_2+K\right)}^2}}.\hfill \end{array}$$

(16)

Since $a<{\displaystyle \frac{{\tilde{u}}_2^2+{\tilde{u}}_2+K}{K}}$, we have $TrA<0$. Consequently, the zero solution of the corresponding ODE system is asymptotically stable. Equation (9) implies $det{\mathcal{B}}_j>0$ for all $j\in {\mathbb{N}}_0$. From (15) we see that $Tr{\mathcal{B}}_j<0$ for all $j\in {\mathbb{N}}_0,$ and hence the zero solution of system (5) is asymptotically stable.

(ii) If $d_1$ satisfies (10) and $d_2$ is chosen according to (11) then $det{\mathcal{B}}_1=0$. Apparently, $det{\mathcal{B}}_1>0$ for $0<d_2<d_2^{*}$ and $det{\mathcal{B}}_1<0$ for $d_2>d_2^{*}$. In all these cases $det{\mathcal{B}}_j>0$ for $j\ne 1$. Thus, the zero solution is asymptotically stable for $0<d_2<d_2^{*}$ and unstable for $d_2>d_2^{*}.$ If $d_2=d_2^{*}$, one eigenvalue of ${\mathcal{B}}_1$ is zero and the other is negative. Denote the eigenvector corresponding to the zero eigenvalue by $y_{11}={\left({\xi }_1,{\eta }_1\right)}^T,$ i.e.,

$${\mathcal{B}}_1{y}_{11}=\left(A-{\mu }_{1}D\right)y_{11}=0,y_{11}\ne 0.$$

Then the function

$$v_1\left(x\right)=\left(\begin{array}{c}u\left(x\right)\\ v\left(x\right)\end{array}\right)=y_{11}{\phi }_1\left(x\right)=\left(\begin{array}{c}{\xi }_1\\ {\eta }_1\end{array}\right)cos{\displaystyle \frac{x}{l}}$$

(17)

is a nonconstant steady state solution of the linearized system (5). □

Remark 1.

If

$${\displaystyle \frac{aK}{\left({\tilde{u}}_2+K\right){\mu }_{j+1}}}\le d_1<{\displaystyle \frac{aK}{\left({\tilde{u}}_2+K\right){\mu }_j}}$$

for $j=2,3,\dots ,$ and the critical value (18) of $d_2$ is to be replaced by

$$d_2=d_2^{*}={\displaystyle \frac{detA+\frac{\left({\tilde{u}}_2^2+{\tilde{u}}_2+K\right)d_1{\mu }_j}{{\tilde{u}}_2+K}}{\frac{aK{\mu }_j}{{\tilde{u}}_2+K}-d_1{\mu }_j^2}}$$

(18)

then a similar result holds.

Next, we apply the Crandall–Rabinowitz bifurcation theorem from a simple eigenvalue [32] to extend the Turing bifurcation result obtained for the linearized system (5) to the original nonlinear system (4) (i.e., system (2)). In what follows, the function space $\mathbf{X}$ is taken as

$$\mathbf{X}=\{V\in C^2([0,l\pi ],{\mathbb{R}}^2):V_x\left(0\right)=V_x\left(l\pi \right)=0\}$$

equipped with the usual supremum norm that includes the first and second derivatives. The space $\mathbf{Y}=C^0([0,l\pi ],{\mathbb{R}}^2)$ is endowed with the standard supremum norm. We are now in a position to prove the following result.

Theorem 2.

Suppose that $a<{\displaystyle \frac{{\tilde{u}}_2^2+{\tilde{u}}_2+K}{K}}$ holds.

(i) If (9) holds then $({\tilde{u}}_2,{\tilde{v}}_2)$ of (4) is asymptotically stable.

(ii) Under the conditions that ${\left(0,{\eta }_1\right)}^T$ is not aligned with the second eigenvector $y_{21}$ of ${\mathcal{B}}_1$ and $d_1$ meets (10), the positive constant equilibrium solution $({\tilde{u}}_2,{\tilde{v}}_2)$ experiences a Turing bifurcation at the critical value $d_2=d_2^{*}$.

Proof. 

(i) This can be deduced immediately from the asymptotic stability of the zero solution of (5).

(ii) We aim to prove the existence of a nonconstant stationary solution within a certain neighborhood of $d_2^{*}$. Since $V=U-{\tilde{\mathbf{u}}}_2,$ the stationary system of (4) can be written as

$$\left\{\begin{array}{cc}D{V}_{xx}+AV+H\left(V\right)=0,\hfill & x\in \mathsf{\Omega },\hfill \\ V_x\left(0\right)=V_x\left(l\pi \right)=0,\hfill \end{array}\right.$$

(19)

and here

$$H\left(V\right)=F(V+{\tilde{\mathbf{u}}}_2)-AV,H\left(0\right)=0,{\left.{\displaystyle \frac{\partial H\left(V\right)}{\partial V}}\right|}_{V=\mathbf{0}}=0.$$

(20)

Denote the left-hand side of equation (19) by $T(d_2,V)$. This defines a single-parameter family of operators acting on the space $\mathbf{X}=\{V\in C^2([0,l\pi ],{\mathbb{R}}^2):V_x\left(0\right)=V_x\left(l\pi \right)=0\}$ and mapping into $\mathbf{Y}=C^0([0,l\pi ],{\mathbb{R}}^2).$ The spectrum of the linearized operator ${\mathcal{L}}^0=T_V(d_2^{*},\mathbf{0})={\displaystyle \frac{\partial T(d_2^{*},\mathbf{0})}{\partial V}}$ consists of the eigenvalues ${\rho }_{i,j}$ $\left(i=1,2,j\in {\mathbb{N}}_0\right)$ of the operator ${\mathcal{B}}_j$. The associated eigenfunctions take the form $y_{ij}{\phi }_j\left(x\right)$ $\left(i=1,2,j\in {\mathbb{N}}_0\right)$, where $y_{ij}$ denotes the eigenvector of the matrix ${\mathcal{B}}_j$ corresponding to ${\rho }_{i,j}$. Here all matrices ${\mathcal{B}}_j=A-{\mu }_{j}D$ are evaluated at $d_2=d_2^{*}$. According to the proof of Theorem 1 and relations (15) and (16), for $i=1,2,j=0,2,3,\dots ,$ all eigenvalues ${\rho }_{i,j}$ possess negative real parts. For $j=1$, one eigenvalue is zero (${\rho }_{1,1}=0$) while the other remains negative. The eigenfunction corresponding to ${\rho }_{1,1}=0$ is $v_1\left(x\right)=y_{11}cos{\displaystyle \frac{x}{l}}$ (see (17)). It follows that the operator ${\mathcal{L}}^0=T_V(d_2^{*},\mathbf{0})$ has a one-dimensional kernel, which is generated by $v_1\left(x\right).$ The range of this operator is described by

$$\begin{array}{ccc}\hfill \mathcal{R}\left({\mathcal{L}}^0\right)& =& \left\{\mathcal{W}\in \mathbf{Y}:\mathcal{W}’\mathrm{s}\mathrm{Fourier}\mathrm{expansion}\mathrm{does}\mathrm{not}\right.\hfill \\ & & \left.\mathrm{contain}cos{\displaystyle \frac{x}{l}}\mathrm{term}\right\}\cup \left\{y_{21}cos{\displaystyle \frac{x}{l}}\right\},\hfill \end{array}$$

which follows from the orthogonality and completeness of the functions ${\phi }_j\left(x\right)$ together with the linear independence of $y_{11}$ and $y_{21}$. Hence $\mathcal{R}\left({\mathcal{L}}^0\right)$ is of codimension one. Set

$${\mathcal{L}}^1={\displaystyle \frac{\partial T_V(d_2^{*},\mathbf{0})}{\partial d_2}},$$

then

$${\mathcal{L}}^1=\tilde{D}{\displaystyle \frac{{\partial }^2}{\partial x^2}},$$

where

$$\tilde{D}={\displaystyle \frac{\partial D}{\partial d_2}}=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right).$$

Clearly

$${\mathcal{L}}^1{v}_1\left(x\right)=-\tilde{D}{\displaystyle \frac{1}{l^2}}{y}_{11}cos{\displaystyle \frac{x}{l}}=-{\displaystyle \frac{1}{l^2}}\left(\begin{array}{c}0\\ {\eta }_1\end{array}\right)cos{\displaystyle \frac{x}{l}}.$$

Thus, ${\mathcal{L}}^1{v}_1\left(x\right)$ and $y_{21}cos{\displaystyle \frac{x}{l}}$ are not parallel, and

$$⟨v_1\left(x\right),{\mathcal{L}}^1{v}_1\left(x\right)⟩=-{\displaystyle \frac{1}{l^2}}{\int }_0^{l\pi }{\eta }_1^2{cos}^2{\displaystyle \frac{x}{l}}dx\ne 0$$

since ${\eta }_1\ne 0$. Consequently ${\mathcal{L}}^1{v}_1\left(x\right)\notin \mathcal{R}\left({\mathcal{L}}^0\right)$.

According to the bifurcation theorem in [32], the point $\left(d_2^{*},\mathbf{0}\right)$ is a bifurcation point. Hence, there exists $\delta >0$, a function $d_2:\left(-\delta ,\delta \right)\to \mathbb{R}$ and for each $s\in \left(-\delta ,\delta \right)$ a solution of (19) with $d_2=d_2\left(s\right)$ $\left(\left|s\right|<\delta \right)$ of the form

$$V\left(s,x\right)=s{y}_{11}cos{\displaystyle \frac{x}{l}}+s\varphi \left(s,x\right)$$

(21)

such that $d_2\left(0\right)=d_2^{*},$ $\varphi \left(0,x\right)=0,$ $d_2\in C^1,$ $\varphi \left(·,x\right)\in C^1$ and $\varphi \left(s,·\right)\in \mathcal{R}\left({\mathcal{L}}^0\right).$ The associated nonconstant stationary solution of system (2) is then given by

$$U\left(s,x\right)={\tilde{\mathbf{u}}}_2+s{y}_{11}cos{\displaystyle \frac{x}{l}}+\mathcal{O}\left(s^2\right).$$

(22)

□

3. Stability of the Nonconstant Stationary Solution

In this section, we utilize the findings from [33] to derive conditions for the asymptotic stability of a nonconstant stationary solution, i.e., the pattern. According to the reference,

$$\underset{s\to 0}{lim}{\displaystyle \frac{s{d}_2^{\prime }\left(s\right){\rho }_0^{\prime }\left(d_2^{*}\right)}{\rho \left(s\right)}}=-1,$$

(23)

where ${\rho }_0\left(d_2\right)$ denotes the principal eigenvalue of operator $T_V(d_2,0)={\displaystyle \frac{\partial T(d_2,0)}{\partial V}},$ and $\rho \left(s\right)$ is the principal eigenvalue associated with the linearization of the solution $\left(d_2\left(s\right),U(s,x)\right)$.

Our goal is to determine the signs of ${\rho }_0^{\prime }\left(d_2^{*}\right)$ and $d_2^{\prime }\left(s\right).$ One can readily demonstrate that ${\rho }_0\left(d_2\right)$ satisfies

$${\rho }_0^2\left(d_2\right)-Tr{\mathcal{B}}_1\left(d_2\right){\rho }_0\left(d_2\right)+det{\mathcal{B}}_1\left(d_2\right)=0.$$

Applying implicit differentiation gives

$$2{\rho }_0\left(d_2\right){\rho }_0^{\prime }\left(d_2\right)+{\mu }_1{\rho }_0\left(d_2\right)-Tr{\mathcal{B}}_1\left(d_2\right){\rho }_0^{\prime }\left(d_2\right)+{\mu }_1^2{d}_1-{\mu }_1{\displaystyle \frac{aK}{{\tilde{u}}_2+K}}=0.$$

We obtain

$${\rho }_0^{\prime }\left(d_2^{*}\right)={\displaystyle \frac{{\mu }_1\left({\mu }_1{d}_1-\frac{aK}{{\tilde{u}}_2+K}\right)}{TrA-\left(d_1+d_2^{*}\right){\mu }_1}}>0,$$

(24)

where (15) and (16) were used and (10) was taken into account.

Next, we examine the sign of $d_2^{\prime }\left(s\right)$. We substitute $d_2\left(s\right)$ and the solution

$$V\left(s,x\right)=s{y}_{11}cos{\displaystyle \frac{x}{l}}+s\varphi \left(s,x\right)$$

into (19). Note that $v_1\left(x\right)=y_{11}cos{\displaystyle \frac{x}{l}}$; we have

$$D\left(s\right)\left[-s{\displaystyle \frac{1}{l^2}}{v}_1\left(x\right)+s{\varphi }_{xx}(s,x)\right]+sA{v}_1\left(x\right)+sA\varphi (s,x)+H\left(s{v}_1\left(x\right)+s\varphi (s,x)\right)=0,$$

Dividing this identity by s, differentiating in s and then setting $s=0$ yields

$$D\left(0\right){\varphi }_{xx}^{\prime }(0,x)-D^{\prime }\left(0\right){\displaystyle \frac{1}{l^2}}{v}_1\left(x\right)+D^{\prime }\left(0\right){\varphi }_{xx}(0,x)+A{\varphi }^{\prime }(0,x)+H\left(v_1\left(x\right)\right)=0,$$

and

$$D^{\prime }\left(0\right)=\left(\begin{array}{cc}0& 0\\ 0& d_2^{\prime }\left(0\right)\end{array}\right).$$

(25)

Since $\varphi (0,x)=0$, this simplifies to

$$D\left(0\right){\varphi }_{xx}^{\prime }(0,x)-{\displaystyle \frac{1}{l^2}}{D}^{\prime }\left(0\right)v_1\left(x\right)+A{\varphi }^{\prime }(0,x)+H\left(v_1\left(x\right)\right)=0.$$

(26)

Then

$$\begin{array}{cc}\hfill & ⟨v_1\left(x\right),D\left(0\right){\varphi }_{xx}^{\prime }(0,x)⟩-{\displaystyle \frac{1}{l^2}}⟨v_1\left(x\right),D^{\prime }\left(0\right)v_1\left(x\right)⟩\hfill \\ \hfill & +⟨v_1\left(x\right),A{\varphi }^{\prime }(0,x)⟩+⟨v_1\left(x\right),H\left(v_1\left(x\right)\right)⟩=0.\hfill \end{array}$$

(27)

Performing two integrations by parts on the first term in (27) leads to

$$\begin{array}{ccc}\hfill ⟨v_1\left(x\right),D\left(0\right){\varphi }_{xx}^{\prime }(0,x)⟩& =& {\int }_0^{l\pi }{v}_1\left(x\right)D\left(0\right){\varphi }_{xx}^{\prime }(0,x)dx\hfill \\ & =& {\int }_0^{l\pi }{v}_1\left(x\right)D\left(0\right)d{\varphi }_x^{\prime }(0,x)\hfill \\ & =& {\displaystyle \frac{1}{l}}{\int }_0^{l\pi }D\left(0\right)y_{11}sin{\displaystyle \frac{x}{l}}d{\varphi }^{\prime }(0,x)\hfill \\ & =& -{\displaystyle \frac{1}{l^2}}{\int }_0^{l\pi }{v}_1\left(x\right)D\left(0\right){\varphi }^{\prime }(0,x)dx\hfill \\ & =& -{\mu }_1⟨v_1\left(x\right),D\left(0\right){\varphi }^{\prime }(0,x)⟩.\hfill \end{array}$$

We begin by observing that the sum of the first and third terms in (27) can be expressed as

$$⟨v_1\left(x\right),D\left(0\right){\varphi }_{xx}^{\prime }(0,x)⟩+⟨v_1\left(x\right),A{\varphi }^{\prime }(0,x)⟩=⟨v_1\left(x\right),{\mathcal{B}}_1{\varphi }^{\prime }(0,x)⟩,$$

(28)

where, from (12), ${\mathcal{B}}_1=A-{\mu }_{1}D\left(0\right)$. We expand ${\varphi }^{\prime }(0,x)\in \mathbf{X}$ and $H\left(v_1\left(x\right)\right)\in \mathbf{X}$ according to the decomposition $\mathbf{X}=span{v}_1\left(x\right)\oplus \mathcal{R}\left({\mathcal{L}}^0\right)$:

$${\varphi }^{\prime }(0,x)={\sigma }_1{v}_1\left(x\right)+r_1,H\left(v_1\left(x\right)\right)=\sigma v_1\left(x\right)+r$$

(29)

with ${\sigma }_1,$ $\sigma \in \mathbb{R},$ $r_1,$ $r\in \mathcal{R}\left({\mathcal{L}}^0\right)$. Substituting the expansion of ${\varphi }^{\prime }(0,x)$ from (29) into the right-hand side of (28) yields

$$⟨v_1\left(x\right),{\mathcal{B}}_1\left({\sigma }_1{v}_1\left(x\right)+r_1\right)⟩={\sigma }_1{\rho }_{1,1}⟨v_1\left(x\right),v_1\left(x\right)⟩+⟨v_1\left(x\right),{\mathcal{B}}_1{r}_1⟩=⟨v_1\left(x\right),{\mathcal{B}}_1{r}_1⟩$$

since ${\rho }_{1,1}=0.$ Substituting $H\left(v_1\left(x\right)\right)$ of (29) into the final term of (27) with its expansion from (29) gives

$$⟨v_1\left(x\right),\sigma v_1\left(x\right)+r⟩=\sigma ⟨v_1\left(x\right),v_1\left(x\right)⟩.$$

For the second term in (27), using (17) and (25) we obtain

$$-{\displaystyle \frac{1}{l^2}}⟨v_1\left(x\right),D^{\prime }\left(0\right)v_1\left(x\right)⟩=-{\mu }_1{\eta }_1^2{d}_2^{\prime }\left(0\right){\int }_0^{l\pi }{cos}^2{\displaystyle \frac{x}{l}}dx={\displaystyle \frac{-{\mu }_1{\eta }_1^2{d}_2^{\prime }\left(0\right)l\pi }{2}}.$$

Summing up these results, (27) becomes

$${\displaystyle \frac{-{\mu }_1{\eta }_1^2{d}_2^{\prime }\left(0\right)l\pi }{2}}+\sigma ⟨v_1\left(x\right),v_1\left(x\right)⟩+⟨v_1\left(x\right),{\mathcal{B}}_1{r}_1⟩=0.$$

Therefore

$$sgn{d}_2^{\prime }\left(0\right)=sgn\left\{\sigma ⟨v_1\left(x\right),v_1\left(x\right)⟩+⟨v_1\left(x\right),{\mathcal{B}}_1{r}_1⟩\right\}.$$

(30)

The coefficient $\sigma$ can be written explicitly in terms of known quantities. We choose the coordinates ${\xi }_1$ and ${\eta }_1$ of $y_{11}$ according to (17). Because ${\mathcal{B}}_1{y}_{11}=0$ and $det{\mathcal{B}}_1=0,$ it follows that

$$\left[{\displaystyle \frac{aK}{{\tilde{u}}_2+K}}-{\displaystyle \frac{1}{l^2}}{d}_1\right]{\xi }_1-{\displaystyle \frac{a{\tilde{u}}_2^2}{{\tilde{u}}_2+K}}{\eta }_1=0,$$

so that, for example, we may choose

$${\xi }_1=-{\displaystyle \frac{a{\tilde{u}}_2^2}{{\tilde{u}}_2+K}},{\eta }_1={\displaystyle \frac{aK}{{\tilde{u}}_2+K}}-{\displaystyle \frac{1}{l^2}}{d}_1.$$

(31)

Then from the definition (20) of $H\left(V\right)$, we have

$$H\left(v_1\left(x\right)\right)=F\left({\tilde{u}}_2+{\xi }_{1}cos{\displaystyle \frac{x}{l}},{\tilde{v}}_2+{\eta }_{1}cos{\displaystyle \frac{x}{l}}\right)-A\left(\begin{array}{c}{\xi }_1\\ {\eta }_1\end{array}\right)cos{\displaystyle \frac{x}{l}}.$$

If we let $H_1\left(x\right)$ and $H_2\left(x\right)$ be the first and second coordinates (respectively) of the last expression for $H\left(v_1\left(x\right)\right)$, then by (29) we have

$$\left(\begin{array}{c}{H}_1\left(x\right)\\ H_2\left(x\right)\end{array}\right)=\sigma \left(\begin{array}{c}{\xi }_1\\ {\eta }_1\end{array}\right)cos{\displaystyle \frac{x}{l}}+\dots ,$$

Forming the inner product of both sides with $v_1\left(x\right)$ gives

$$\sigma ={\displaystyle \frac{2}{l\pi \left({\xi }_1^2+{\eta }_1^2\right)}}{\int }_0^{l\pi }\left({\xi }_1{H}_1\left(x\right)+{\eta }_1{H}_2\left(x\right)\right)cos{\displaystyle \frac{x}{l}}dx.$$

Theorem 3.

Suppose that the conditions of Theorem 2 (ii) hold, and that the eigenvalue $\rho \left(s\right)$ associated with the nonconstant stationary solution bifurcating from the critical eigenvalue ${\rho }_{1,1}=\rho \left(0\right)=0$ is nonzero for small $\left|s\right|\ne 0.$ If expression (30) is positive (respectively, negative), then the nonconstant stationary solution of (2) is asymptotically stable (respectively, unstable) for $s>0$, and unstable (respectively, asymptotically stable) for $s<0$.

Proof. 

If (30) is positive, continuity implies $d_2^{\prime }\left(s\right)>0$ for sufficiently small $\left|s\right|$. Consequently, ${\rho }_0^{\prime }\left(d_2^{*}\right)>0.$ Hence $\rho \left(s\right)<0$ when $s>0,$ and $\rho \left(s\right)>0$ when $s<0.$ For small $\left|s\right|$, the remaining eigenvalues remain close to ${\rho }_{i,j}$ $\left(i=1,2,j=0,2,3,\dots \right)$ and to ${\rho }_{2,1}$, all lying in the left half-plane. Therefore, the bifurcating solution is asymptotically stable for $s>0$ and unstable for $s<0$. The case where (30) is negative follows analogously. □

Remark 2.

It is noteworthy that under homogeneous Neumann boundary conditions, the Turing bifurcation is expected to be a pitchfork bifurcation when $d_2^{\prime }\left(0\right)=0$. In such a case, the stability of nonconstant steady-state solutions remains the same independently of the sign of s, as shown, for instance, in [12,34].

Corollary 1.

Under the conditions specified in Theorem 3, provided that expression (30) does not vanish, one can choose a sufficiently small $\delta >0$ such that for every $d_2\in \left(d_2^{*}-\delta ,d_2^{*}+\delta \right)$ system (2) admits a nonconstant stationary solution. This solution is asymptotically stable whenever $d_2>d_2^{*}$, and becomes unstable when $d_2<d_2^{*}$.

Proof. 

If (30) is positive, then $d_2^{\prime }\left(s\right)>0$, implying $d_2\left(s\right)>d_2^{*}$ for $s>0$ and $d_2\left(s\right)<d_2^{*}$ for $s<0$. Conversely, if (30) is negative, then $d_2^{\prime }\left(s\right)<0$, so $d_2\left(s\right)>d_2^{*}$ for $s<0$ and $d_2\left(s\right)<d_2^{*}$ for $s>0$. □

Remark 3.

The above result shows that in system (2), when the nutrient diffusion rate $d_2$ exceeds the critical value $d_2^{*}$ and the positive constant equilibrium solution $({\tilde{u}}_2,{\tilde{v}}_2)$ loses stability, a stable small-amplitude pattern emerges.

4. Hopf Bifurcation Analysis of System (3)

In this section we consider Hopf bifurcation at ${\tilde{\mathbf{u}}}_2=({\tilde{u}}_2,{\tilde{v}}_2)$ of (3) by regarding $\tau$ as a bifurcation parameter. In addition, we investigate the direction of spatially homogeneous Hopf bifurcation and the stability of the resulting periodic solutions by using the normal from the theory and center manifold reduction for PFDEs developed by Wu [35].

4.1. Existence of Hopf Bifurcations

For notational convenience, we denote $\left(u^{*},v^{*}\right)=({\tilde{u}}_2,{\tilde{v}}_2)$; let $\widehat{u}=u-u^{*},$ $\widehat{v}=v-v^{*}$ and drop the hats for the sake of simplicity. System (3) can be rewritten as

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial u}{\partial t}}-d_1{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=a(u+u^{*})\left({\displaystyle \frac{(u+u^{*})(v+v^{*})}{u+u^{*}+K}}-1\right),\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ {\displaystyle \frac{\partial v}{\partial t}}-d_2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}=b-(v+v^{*})-{\displaystyle \frac{{(u_{\tau }+u^{*})}^2(v+v^{*})}{u_{\tau }+u^{*}+K}},\hfill & x\in \mathsf{\Omega },t>0,\hfill \\ {\displaystyle \frac{\partial u(x,t)}{\partial x}}={\displaystyle \frac{\partial v(x,t)}{\partial x}}=0,\hfill & x\in \partial \mathsf{\Omega },t>0,\hfill \\ u(x,t)=u_0(x,t)\ge 0,v(x,t)=v_0(x,t)\ge 0,\hfill & x\in \mathsf{\Omega },t\in [-\tau ,0].\hfill \end{array}\right.$$

(32)

where $\mathsf{\Omega }=(0,l\pi )$ $(l\in {\mathbb{R}}^{+})$, $u=u\left(x,t\right),$ $u_{\tau }=u\left(x,t-\tau \right),$ $v=v\left(x,t\right).$

Since system (32) is already partial functional differential equations, we define the functional space $\mathcal{X}$ by

$$\mathcal{X}=\left\{u\in W^{2,2}(0,l\pi ):u_x\left(0\right)=u_x\left(l\pi \right)=0\right\}.$$

Now let $U\left(t\right)={(u_1\left(t\right),u_2\left(t\right))}^T={(u_1(t,·),u_2(t,·))}^T$. Then (32) can be rewritten as an abstract functional differential equation in the phase space $\mathcal{C}=C\left([-\tau ,0],\mathcal{X}\right)$ of the form

$${\displaystyle \frac{dU\left(t\right)}{dt}}=D\Delta U\left(t\right)+L\left(U_t\right)+G\left(U_t\right),$$

(33)

and $L:\mathcal{C}\to \mathcal{X}$, $G:\mathcal{C}\to \mathcal{X}$ are given, respectively, by

$$L\left(\phi \right)=\left(\begin{array}{c}a\left(1-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1\left(0\right)+{\displaystyle \frac{a{u}^{*}}{v^{*}}}{\phi }_2\left(0\right)\\ -\left(2-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1(-\tau )-\left(1+{\displaystyle \frac{u^{*}}{v^{*}}}\right){\phi }_2\left(0\right)\end{array}\right)$$

and

$$G\left(\phi \right)=\left(\begin{array}{c}-a{u}^{*}-a\left(2-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1\left(0\right)-{\displaystyle \frac{a{u}^{*}}{v^{*}}}{\phi }_2\left(0\right)+{\displaystyle \frac{a{({\phi }_1\left(0\right)+u^{*})}^2({\phi }_2\left(0\right)+v^{*})}{{\phi }_1\left(0\right)+u^{*}{v}^{*}}}\\ b-v^{*}+{\displaystyle \frac{u^{*}}{v^{*}}}{\phi }_2\left(0\right)+\left(2-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1(-\tau )-{\displaystyle \frac{{({\phi }_1(-\tau )+u^{*})}^2({\phi }_2\left(0\right)+v^{*})}{{\phi }_1(-\tau )+u^{*}{v}^{*}}}\end{array}\right),$$

for $\phi ={({\phi }_1,{\phi }_2)}^T\in \mathcal{C}$.

By direct calculation, one finds that the characteristic equation obtained from linearizing (33) is equivalent to a sequence of transcendental equations.

$${\lambda }^2+A_k\lambda +B{e}^{-\lambda \tau }+C_k=0,k\in {\mathbb{N}}_0,$$

(34)

where

$$\begin{array}{ccc}\hfill A_k& =& \left(d_1+d_2\right){\left({\displaystyle \frac{k}{l}}\right)}^2+1+{\displaystyle \frac{u^{*}}{v^{*}}}-a\left(1-{\displaystyle \frac{1}{v^{*}}}\right),\hfill \\ \hfill B& =& {\displaystyle \frac{a{u}^{*}}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right),\hfill \\ \hfill C_k& =& \left[d_1{\left({\displaystyle \frac{k}{l}}\right)}^2-a\left(1-{\displaystyle \frac{1}{v^{*}}}\right)\right]\left[d_2{\left({\displaystyle \frac{k}{l}}\right)}^2+1+{\displaystyle \frac{u^{*}}{v^{*}}}\right].\hfill \end{array}$$

Let $\lambda =\pm i\omega$ $\left(\omega >0\right)$ be a pair of purely imaginary root of (34) for $k\in {\mathbb{N}}_0.$ It follows that

$$-{\omega }^2+A_k\omega i+B(cos\omega \tau -isin\omega \tau )+C_k=0,$$

which implies that

$$\left\{\begin{array}{c}{\omega }^2-C_k=Bcos\omega \tau ,\hfill \\ A_k\omega =Bsin\omega \tau .\hfill \end{array}\right.$$

(35)

From (35), we have

$${\omega }^4+(A_k^2-2C_k){\omega }^2+C_k^2-B^2=0,$$

(36)

where

$$\begin{array}{ccc}\hfill A_k^2-2C_k& =& {\left[\left(d_1+d_2\right){\left({\displaystyle \frac{k}{l}}\right)}^2+1+{\displaystyle \frac{u^{*}}{v^{*}}}-a\left(1-{\displaystyle \frac{1}{v^{*}}}\right)\right]}^2\hfill \\ & & +2\left[a\left(1-{\displaystyle \frac{1}{v^{*}}}\right)-d_1{\left({\displaystyle \frac{k}{l}}\right)}^2\right]\left[d_2{\left({\displaystyle \frac{k}{l}}\right)}^2+1+{\displaystyle \frac{u^{*}}{v^{*}}}\right],\hfill \end{array}$$

(37)

$$C_k^2-B^2={\left[d_1{\left({\displaystyle \frac{k}{l}}\right)}^2-a\left(1-{\displaystyle \frac{1}{v^{*}}}\right)\right]}^2{\left[d_2{\left({\displaystyle \frac{k}{l}}\right)}^2+1+{\displaystyle \frac{u^{*}}{v^{*}}}\right]}^2-{\left[{\displaystyle \frac{a{u}^{*}}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)\right]}^2.$$

(38)

Assume that $C_0^2-B^2<0$ holds. Since ${\displaystyle \underset{k\to \infty }{lim}}\left(C_k^2-B^2\right)=+\infty ,$ then there exists a minimal $K_0\ge 0$ such that the Equation (36) has no positive root for $k>K_0$ and the Equation (36) has one positive root at most for $0\le k\le K_0$. In addition, we suppose $d_2\ge d_1$ to ensure that $A_k^2-2C_k>0$ is always satisfied.

For $0\le k\le K_0$, Equation (36) has a positive root ${\omega }_k$ satisfying

$${\omega }_k^2={\displaystyle \frac{-\left(A_k^2-2C_k\right)+\sqrt{{\left(A_k^2-2C_k\right)}^2-4\left(C_k^2-B^2\right)}}{2}}.$$

(39)

Therefore, (36) admits a pair of imaginary roots $\pm i{\omega }_k$ when

$$\tau ={\tau }_k^j={\tau }_0^j+{\displaystyle \frac{2j\pi }{{\omega }_k}},j=0,1,2,\dots ,$$

(40)

where ${\tau }_0^j$ satisfies

$${\tau }_0^j=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\omega }_k}}arccos{\displaystyle \frac{{\omega }_k^2-C_k}{B}},\hfill & A_k\ge 0,\hfill \\ {\displaystyle \frac{1}{{\omega }_k}}\left(2\pi -arccos{\displaystyle \frac{{\omega }_k^2-C_k}{B}}\right),\hfill & A_k<0.\hfill \end{array}\right.$$

(41)

Let ${\lambda }_k\left(\tau \right)={\alpha }_k\left(\tau \right)+i{\omega }_k\left(\tau \right)$ be the root of (36) satisfying ${\alpha }_k\left({\tau }_k^j\right)=0$ and ${\omega }_k\left({\tau }_k^j\right)={\omega }_k$ when $\tau$ is close to ${\tau }_k^j.$ Consequently, we arrive at the following transversality condition.

Lemma 1.

The transversality condition holds, that is

$${\left.{\displaystyle \frac{d{\alpha }_k}{d\tau }}\right|}_{\tau ={\tau }_k^j}>00\le k\le K_0,j\in {\mathbb{N}}_0.$$

Proof. 

Differentiating both sides of Equation (34) with respect to $\tau$ yields

$${\displaystyle \frac{d{\lambda }_k}{d\tau }}={\displaystyle \frac{B{\lambda }_k{e}^{-{\lambda }_k\tau }}{2{\lambda }_k-B\tau e^{-{\lambda }_k\tau }+A_k}}.$$

(42)

Substituting $\tau ={\tau }_k^j$ into (42) yields

$${\left.{\displaystyle \frac{d{\lambda }_k}{d\tau }}\right|}_{\tau ={\tau }_k^j}={\displaystyle \frac{Bi{\omega }_k{e}^{-i{\omega }_k{\tau }_k^j}}{2i{\omega }_k-B{\tau }_k^j{e}^{-i{\omega }_k{\tau }_k^j}+A_k}}.$$

(43)

Since

$${\left.{\displaystyle \frac{d{\alpha }_k}{d\tau }}\right|}_{\tau ={\tau }_k^j}=\mathrm{Re}\left({\left.{\displaystyle \frac{d{\lambda }_k}{d\tau }}\right|}_{\tau ={\tau }_k^j}\right)$$

and

$$\mathrm{Re}\left({\left.{\displaystyle \frac{d{\lambda }_k}{d\tau }}\right|}_{\tau ={\tau }_k^j}\right)\ne 0\iff \mathrm{Re}{\left({\left.{\displaystyle \frac{d{\lambda }_k}{d\tau }}\right|}_{\tau ={\tau }_k^j}\right)}^{-1}\ne 0,$$

then from (43) we obtain

$$\mathrm{Re}{\left({\left.{\displaystyle \frac{d{\lambda }_k}{d\tau }}\right|}_{\tau ={\tau }_k^j}\right)}^{-1}={\displaystyle \frac{2\left({\omega }_k^2-C_k\right)+A_k^2}{B^2}}={\displaystyle \frac{\sqrt{{\left(A_k^2-2C_k\right)}^2-4\left(C_k^2-B^2\right)}}{B^2}}>0.$$

Hence the proof is completed. □

It is easy to see from (40) that ${\tau }_k^{j+1}>{\tau }_k^j$, and the following proposition shows that ${\tau }_{k+1}^j>{\tau }_k^j$ is established under some conditions.

Lemma 2.

Let ${\tau }_k^j$ be defined as in (40) and (41). If $d_2\ge d_1$, $b>1+2\sqrt{K},$ $0<a<{\displaystyle \frac{u^{*2}+u^{*}+K}{K}}$, then there exists a $\widehat{K}>0$ such that ${\tau }_{k+1}^j>{\tau }_k^j$ for $\widehat{K}\le k\le K_0,$ $j\in {\mathbb{N}}_0.$

Proof. 

From (39) we have

$$\begin{array}{ccc}\hfill {\omega }_k^2& =& {\displaystyle \frac{-\left(A_k^2-2C_k\right)+\sqrt{{\left(A_k^2-2C_k\right)}^2-4\left(C_k^2-B^2\right)}}{2}}\hfill \\ & =& {\displaystyle \frac{2}{\sqrt{\frac{{\left(A_k^2-2C_k\right)}^2}{{\left(B^2-C_k^2\right)}^2}+\frac{4}{B^2-C_k^2}}+\frac{A_k^2-2C_k}{B^2-C_k^2}}}.\hfill \end{array}$$

Since when $d_2\ge d_1,$ $A_k^2-2C_k$ is strictly increasing in k and there exists an integer $\widehat{K}>0$ such that $B^2-C_k^2$ is strictly decreasing in k for $\widehat{K}\le k\le K_0,$ then we obtain ${\omega }_{k+1}^2>{\omega }_k^2.$ Since $0<a<{\displaystyle \frac{u^{*2}+u^{*}+K}{K}}$, that is $a\left(1-{\displaystyle \frac{1}{v^{*}}}\right)<1+{\displaystyle \frac{u^{*}}{v^{*}}},$ then $A_k>0,$ and consequently from (41) we have ${\tau }_k^0={\displaystyle \frac{1}{{\omega }_k}}arccos{\displaystyle \frac{{\omega }_k^2-C_k}{B}},$ and hence we can obtain ${\tau }_{k+1}^0>{\tau }_k^0,$ $\widehat{K}\le k\le K_0.$ Since ${\omega }_{k+1}>{\omega }_k$, then from (40) we obtain that ${\tau }_{k+1}^j>{\tau }_k^j$, $j\in {\mathbb{N}}_0,$ $\widehat{K}\le k\le K_0.$ Thus, the proof is completed. □

According to Lemma 2, the smallest bifurcation value in the sequence {${\tau }_k^j$} is ${\tau }_0^0$. We have the following conclusion.

Theorem 4.

Assume that $d_2\ge d_1$, $b>1+2\sqrt{K},$ $0<a<{\displaystyle \frac{u^{*2}+u^{*}+K}{K}}$, then the following statements hold.

(i) If $\tau \in [0,{\tau }_0^0),$ then the positive constant equilibrium solution $\left(u^{*},v^{*}\right)$ is locally asymptotically stable.

(ii) If $\tau \in ({\tau }_0^0,\infty ),$ then the positive constant equilibrium solution $\left(u^{*},v^{*}\right)$ is unstable.

(iii) System (3) undergoes a homogeneous Hopf bifurcation near the steady state $\left(u^{*},v^{*}\right)$ at $\tau ={\tau }_0^j$ for $j\in {\mathbb{N}}_0$. Moreover, it undergoes inhomogeneous Hopf bifurcations near $\left(u^{*},v^{*}\right)$ at $\tau ={\tau }_k^j$ for $0<k\le K_0,$ $j\in {\mathbb{N}}_0$.

4.2. Properties of Hopf Bifurcations

In this subsection, we investigate the direction of spatially homogeneous Hopf bifurcation and the stability of the resulting periodic solutions when $\tau ={\tau }_0^0:={\tau }_0$ (${\tau }_0^j,$ $j\in \mathbb{N}$ can be treated in a similar way). Equation (34) possesses a simple pair of purely imaginary roots $\pm i{\omega }_0$ when $\tau ={\tau }_0$. Set $\mu =\tau -{\tau }_0,$ normalizing the delay $\tau$ by the time-scaling $t\to t/\tau$. Then (33) can be rewritten in the fixed phase space $\mathcal{C}=C\left([-1,0],\mathcal{X}\right)$ as

$${\displaystyle \frac{d}{dt}}U\left(t\right)={\tau }_{0}D\Delta U\left(t\right)+{\tau }_0{L}_0\left(U_t\right)+G(U_t,\mu ),$$

(44)

where

$$L_0\left(\phi \right)=\left(\begin{array}{c}a\left(1-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1\left(0\right)+{\displaystyle \frac{a{u}^{*}}{v^{*}}}{\phi }_2\left(0\right)\\ -\left(2-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1(-1)-\left(1+{\displaystyle \frac{u^{*}}{v^{*}}}\right){\phi }_2\left(0\right)\end{array}\right),$$

$$G(\phi ,\mu )=\mu D\Delta \phi \left(0\right)+\mu L_0\left(\phi \right)+({\tau }_0+\mu )F_0\left(\phi \right),$$

with

$$F_0\left(\phi \right)=\left(\begin{array}{c}-a{u}^{*}-a\left(2-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1\left(0\right)-{\displaystyle \frac{a{u}^{*}}{v^{*}}}{\phi }_2\left(0\right)+{\displaystyle \frac{a{({\phi }_1\left(0\right)+u^{*})}^2({\phi }_2\left(0\right)+v^{*})}{{\phi }_1\left(0\right)+u^{*}{v}^{*}}}\\ b-v^{*}+\left(2-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1(-1)+{\displaystyle \frac{u^{*}}{v^{*}}}{\phi }_2\left(0\right)-{\displaystyle \frac{{({\phi }_1(-1)+u^{*})}^2({\phi }_2\left(0\right)+v^{*})}{{\phi }_1(-1)+u^{*}{v}^{*}}}\end{array}\right).$$

(45)

The linear system of (44) is

$${\displaystyle \frac{d}{dt}}U\left(t\right)={\tau }_{0}D\Delta U\left(t\right)+{\tau }_0{L}_0\left(U_t\right).$$

(46)

The Riesz representation theorem guarantees the existence of a $2\times 2$ matrix function $\eta (\theta ,{\tau }_0)$ with elements of bounded variation, defined on the interval $[-1,0],$ such that

$$({\tau }_0+\mu )L_0\left(\phi \right)={\int }_{-1}^{0}d\left[\eta (\theta ,\mu )\right]\varphi \left(\theta \right)\mathrm{for}\varphi \in C([-1,0],{\mathbb{R}}^2).$$

We then define

$$\eta (\theta ,\mu )=({\tau }_0+\mu )\left(\begin{array}{cc}a\left(1-{\displaystyle \frac{1}{v^{*}}}\right)& {\displaystyle \frac{a{u}^{*}}{v^{*}}}\\ 0& -\left(1+{\displaystyle \frac{u^{*}}{v^{*}}}\right)\end{array}\right)\delta \left(\theta \right)-({\tau }_0+\mu )\left(\begin{array}{cc}0& 0\\ -\left(2-{\displaystyle \frac{1}{v^{*}}}\right)& 0\end{array}\right)\delta (\theta +1).$$

For $\delta \left(\theta \right):[-1,0]\to (\mathcal{X},\mathcal{X}),$ we have

$$\delta \left(\theta \right)=\left\{\begin{array}{cc}0,\hfill & \theta \in [-1,0),\hfill \\ 1,\hfill & \theta =0.\hfill \end{array}\right.$$

Define $A\left(0\right)$ and $A^{*}$ as

$$A\left(0\right)\left(\varphi \left(\theta \right)\right)=\left\{\begin{array}{cc}{\displaystyle \frac{d\varphi \left(\theta \right)}{d\theta }},\hfill & \theta \in [-1,0),\hfill \\ {\int }_{-1}^{0}d\left[\eta (\theta ,0)\right]\varphi \left(\theta \right),\hfill & \theta =0,\hfill \end{array}\right.$$

$$A^{*}\left(\psi \left(s\right)\right)=\left\{\begin{array}{cc}-{\displaystyle \frac{d\psi \left(s\right)}{ds}},\hfill & s\in (0,1],\hfill \\ {\int }_{-1}^{0}d\left[\eta (\theta ,0)\right]\psi \left(-s\right),\hfill & s=0,\hfill \end{array}\right.$$

(47)

where $\varphi \in C([-1,0],{\mathbb{R}}^2),$ $\psi \in C([0,1],{\mathbb{R}}^2).$ For $\mathbf{u}=\left(u_1,u_2\right)\in \mathcal{X},$ $\mathbf{v}=\left(v_1,v_2\right)\in \mathcal{X}$, we defined

$$⟨\mathbf{u},\mathbf{v}⟩={\displaystyle \frac{1}{l\pi }}{\int }_0^{l\pi }{u}_1{\overline{v}}_{1}dx+{\displaystyle \frac{1}{l\pi }}{\int }_0^{l\pi }{u}_2{\overline{v}}_{2}dx$$

Then $A^{*}$ is the formal adjoint of $A\left(0\right)$ under the bilinear pairing

$${⟨\psi ,\varphi ⟩}_0=\overline{\psi }\left(0\right)\varphi \left(0\right)-{\int }_{-1}^0{\int }_0^{\theta }\overline{\psi }(\xi -\theta )d\left[\eta (\theta ,0)\right]\varphi \left(\xi \right)d\xi .$$

Note that $\pm i{\omega }_0{\tau }_0$ are eigenvalues of both $A\left(0\right)$ and $A^{*}.$ Let $q\left(\theta \right)$ be an eigenvector of A corresponding to $i\omega {\tau }_0$, and let $q^{*}\left(s\right)$ be an eigenvector of $A^{*}$ corresponding to $-i{\omega }_0{\tau }_0.$ From the definitions of $A\left(0\right)$ and $A^{*},$ the eigenvectors take the form

$$\begin{array}{cc}q\left(\theta \right)={(q_1,q_2)}^T{e}^{i{\omega }_0{\tau }_0\theta },\hfill & \theta \in [-1,0],\hfill \\ q^{*}\left(s\right)={\displaystyle \frac{1}{{\overline{D}}^{*}}}(q_1^{*},q_2^{*})e^{-i{\omega }_0{\tau }_{0}s},\hfill & s\in [0,1].\hfill \end{array}$$

Using (47), we obtain

$$\begin{array}{ccc}\hfill (q_1,q_2)& =& \left(1,{\displaystyle \frac{a(1-v^{*})+i{\omega }_0{v}^{*}}{a{u}^{*}}}\right),\hfill \\ \hfill (q_1^{*},q_2^{*})& =& \left(1,{\displaystyle \frac{a(v^{*}-1)+i{\omega }_0{v}^{*}}{2v^{*}-1}}\right).\hfill \end{array}$$

Since $q\left(\theta \right)$ and $q^{*}\left(s\right)$ satisfy ${⟨q^{*}\left(s\right),q\left(\theta \right)⟩}_0=1$ and ${⟨q^{*}\left(s\right),\overline{q}\left(\theta \right)⟩}_0=0,$ we have

$$D^{*}=1-{\displaystyle \frac{{[a(1-v^{*})+i{\omega }_0{v}^{*}]}^2}{a{u}^{*}(2v^{*}-1)}}-\left[a\left(1-{\displaystyle \frac{1}{v^{*}}}\right)-i{\omega }_0\right]{\tau }_0{e}^{-i{\omega }_0{\tau }_0}.$$

Let $Y=\{(zq\left(\theta \right)+\overline{z}\overline{q}\left(\theta \right)):z\in \mathcal{C}\}$ be the center subspace of (46). Setting $\mu =0$ in system (44), a center manifold can be expressed as

$$W\left(z,\overline{z}\right)=W_{20}\left(\theta \right){\displaystyle \frac{z^2}{2}}+W_{11}\left(\theta \right)z\overline{z}+W_{02}\left(\theta \right){\displaystyle \frac{{\overline{z}}^2}{2}}+\dots .$$

(48)

Assume that $\mathsf{\Phi }=(q\left(\theta \right),\overline{q}\left(\theta \right)),$ $\mathsf{\Psi }={(q^{*}\left(s\right),{\overline{q}}^{*}\left(s\right))}^T.$ The flow on the center manifold in system (44) can be written as

$$\begin{array}{ccc}\hfill U_t& =& \mathsf{\Phi }{(z\left(t\right),\overline{z}\left(t\right))}^T+W(z\left(t\right),\overline{z}\left(t\right))\hfill \\ & =& zq\left(\theta \right)+\overline{z}\overline{q}\left(\theta \right)+W_{20}\left(\theta \right){\displaystyle \frac{z^2}{2}}+W_{11}\left(\theta \right)z\overline{z}+W_{02}\left(\theta \right){\displaystyle \frac{{\overline{z}}^2}{2}}+\dots .\hfill \end{array}$$

According to the inner product formula, it can be obtained that $⟨\varphi ,f_0⟩={\left(⟨\varphi ,f_0^1⟩,⟨\varphi ,f_0^2⟩\right)}^T,$ where $\varphi \in C\left(\right[-1,0],\mathcal{X})$ $f_0={\left(f_0^1,f_0^2\right)}^T$, $f_0^1={(1,0)}^T,$ $f_0^1={(0,1)}^T.$ Letting $\mu =0,$ we have

$${\displaystyle \frac{\partial z}{\partial t}}=i{\omega }_0{\tau }_{0}z\left(t\right)+{\overline{q}}^{*}\left(0\right)⟨G(U_t,0),f_0⟩=i{\omega }_0{\tau }_{0}z\left(t\right)+g(z,\overline{z}),$$

where

$$g(z,\overline{z})={\overline{q}}^{*}\left(0\right)⟨G(U_t,0),f_0⟩=g_{20}{\displaystyle \frac{z^2}{2}}+g_{11}z\overline{z}+g_{02}{\displaystyle \frac{{\overline{z}}^2}{2}}+g_{21}{\displaystyle \frac{z^2\overline{z}}{2}}+\dots .$$

(49)

From the expression for $G(\phi ,\mu )$, we know that $G(\phi ,0)={\tau }_0{F}_0\left(\phi \right)={\tau }_0{(G_1,G_2)}^T,$ where

$$\begin{array}{ccc}\hfill G_1& =& {\displaystyle \frac{a}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{\phi }_1^2\left(0\right)+{\displaystyle \frac{a}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1\left(0\right){\phi }_2\left(0\right)-{\displaystyle \frac{a}{{\left(u^{*}\right)}^2{v}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{\phi }_1^3\left(0\right)\hfill \\ & & +{\displaystyle \frac{a}{u^{*}{v}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{\phi }_1^2\left(0\right){\phi }_2\left(0\right)+O\left({∥(u,v)∥}^4\right),\hfill \end{array}$$

(50)

$$\begin{array}{ccc}\hfill G_2& =& -{\displaystyle \frac{1}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{\phi }_1^2(-1)-{\displaystyle \frac{1}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right){\phi }_1(-1){\phi }_2\left(0\right)+{\displaystyle \frac{1}{{\left(u^{*}\right)}^2{v}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{\phi }_1^3(-1)\hfill \\ & & -{\displaystyle \frac{1}{u^{*}{v}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{\phi }_1^2(-1){\phi }_2\left(0\right)+O\left({∥(u,v)∥}^4\right).\hfill \end{array}$$

(51)

Combining (49)–(51), we obtain the following equations

$$\begin{array}{ccc}\hfill g_{20}& =& {\displaystyle \frac{2{\overline{q}}_1^{*}{\tau }_0}{D^{*}}}\left[{\displaystyle \frac{a}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1^2+{\displaystyle \frac{a}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)q_1{q}_2\right]\hfill \\ & & +{\displaystyle \frac{2{\overline{q}}_2^{*}{\tau }_0}{D^{*}}}\left[-{\displaystyle \frac{1}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1^2{e}^{-2i{\omega }_0{\tau }_0}-{\displaystyle \frac{1}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)q_1{q}_2{e}^{-i{\omega }_0{\tau }_0}\right],\hfill \\ \hfill g_{02}& =& {\overline{g}}_{20},\hfill \\ \hfill g_{11}& =& {\displaystyle \frac{{\overline{q}}_1^{*}{\tau }_0}{D^{*}}}\left[{\displaystyle \frac{2a}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1{\overline{q}}_1+{\displaystyle \frac{a}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)(q_1{\overline{q}}_2+{\overline{q}}_1{q}_2)\right]\hfill \\ & & -{\displaystyle \frac{{\overline{q}}_2^{*}{\tau }_0}{D^{*}}}\left[{\displaystyle \frac{2}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1{\overline{q}}_1+{\displaystyle \frac{1}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)(q_1{\overline{q}}_2{e}^{-i{\omega }_0{\tau }_0}+{\overline{q}}_1{q}_2{e}^{i{\omega }_0{\tau }_0})\right],\hfill \end{array}$$

$$\begin{array}{c}{g}_{21}={\displaystyle \frac{2a{\overline{q}}_1^{*}{\tau }_0}{D^{*}{u}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2(2W_{11}^1\left(0\right)q_1+W_{20}^1\left(0\right){\overline{q}}_1)\hfill \\ +{\displaystyle \frac{a{\overline{q}}_1^{*}{\tau }_0}{D^{*}{v}^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)(2W_{11}^2\left(0\right)q_1+W_{20}^2\left(0\right){\overline{q}}_1+W_{20}^1\left(0\right){\overline{q}}_2+2W_{11}^1\left(0\right)q_2)\hfill \\ -{\displaystyle \frac{6a{\overline{q}}_1^{*}{\tau }_0}{D^{*}{\left(u^{*}\right)}^2{v}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1^2{\overline{q}}_1+{\displaystyle \frac{2a{\overline{q}}_1^{*}{\tau }_0}{D^{*}{u}^{*}{v}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2(q_1^2{\overline{q}}_2+2q_1{q}_2{\overline{q}}_1)\hfill \\ -{\displaystyle \frac{2{\overline{q}}_2^{*}{\tau }_0}{D^{*}{u}^{*}{v}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2(q_1^2{\overline{q}}_2{e}^{-2i{\omega }_0{\tau }_0}+2q_1{q}_2{\overline{q}}_1)\hfill \\ -{\displaystyle \frac{{\overline{q}}_2^{*}{\tau }_0}{D^{*}{v}^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)(2W_{11}^1(-1)q_2+W_{20}^1(-1){\overline{q}}_2+W_{20}^2\left(0\right){\overline{q}}_1{e}^{i{\omega }_0{\tau }_0}+2W_{11}^2\left(0\right)q_1{e}^{-i{\omega }_0{\tau }_0})\hfill \\ +{\displaystyle \frac{6{\overline{q}}_2^{*}{\tau }_0}{D^{*}{\left(u^{*}\right)}^2{v}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1^2{\overline{q}}_1{e}^{-i{\omega }_0{\tau }_0}-{\displaystyle \frac{2{\overline{q}}_2^{*}{\tau }_0}{D^{*}{u}^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2(2W_{11}^1(-1)q_1{e}^{-i{\omega }_0{\tau }_0}+W_{20}^1(-1){\overline{q}}_1{e}^{i{\omega }_0{\tau }_0}).\hfill \end{array}$$

Since $W_{20}\left(\theta \right)$ and $W_{11}\left(\theta \right)$ are included in $g_{21}$, we still need to compute them. By Wu [35], $W(z,\overline{z})$ satisfies

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial W}{\partial t}}& =& A\left(0\right)W+X_{0}G(U_t,0)-\mathsf{\Phi }{⟨\mathsf{\Psi },⟨X_{0}G(U_t,0),f_0⟩⟩}_0·f_0\hfill \\ & =& A\left(0\right)W+H_{20}{\displaystyle \frac{z^2}{2}}+H_{11}z\overline{z}+H_{02}{\displaystyle \frac{{\overline{z}}^2}{2}}+\dots ,\hfill \end{array}$$

(52)

where

$$X_0\left(\theta \right)=\left\{\begin{array}{c}0,\theta \in [-1,0),\hfill \\ I,\theta =0.\hfill \end{array}\right.$$

Through the chain rule, we can obtain

$${\displaystyle \frac{\partial W}{\partial t}}={\displaystyle \frac{\partial W(z,\overline{z})}{\partial z}}{\displaystyle \frac{\partial z}{\partial t}}+{\displaystyle \frac{\partial W(z,\overline{z})}{\partial \overline{z}}}{\displaystyle \frac{\partial \overline{z}}{\partial t}},$$

then

$$H_{20}=[2i{\omega }_0{\tau }_0-A\left(0\right)]W_{20},H_{11}=-A\left(0\right)W_{11},H_{02}=[-2i{\omega }_0{\tau }_0-A\left(0\right)]W_{02}.$$

(53)

For $\theta \in [-1,0),$ from (52) we have

$$-\mathsf{\Phi }{⟨\mathsf{\Psi },⟨X_{0}G(U_t,0),f_0⟩⟩}_0·f_0=H_{20}{\displaystyle \frac{z^2}{2}}+H_{11}z\overline{z}+H_{02}{\displaystyle \frac{{\overline{z}}^2}{2}}+\dots .$$

Thus

$$\left\{\begin{array}{c}{H}_{20}\left(\theta \right)=-g_{20}q\left(\theta \right)-{\overline{g}}_{02}\overline{q}\left(\theta \right),\\ H_{11}\left(\theta \right)=-g_{11}q\left(\theta \right)-{\overline{g}}_{11}\overline{q}\left(\theta \right).\end{array}\right.$$

(54)

According to (53) and (54), we have

$$\left\{\begin{array}{cc}{\displaystyle \frac{d{W}_{20}\left(\theta \right)}{d\theta }}=2i{\omega }_0{\tau }_0{W}_{20}\left(\theta \right)+g_{20}q\left(\theta \right)+{\overline{g}}_{02}\overline{q}\left(\theta \right),\hfill & \\ {\displaystyle \frac{d{W}_{11}\left(\theta \right)}{d\theta }}=g_{11}q\left(\theta \right)+{\overline{g}}_{11}\overline{q}\left(\theta \right).\hfill \end{array}\right.$$

(55)

The solution to Equation (55) is

$$\begin{array}{ccc}\hfill W_{20}\left(\theta \right)& =& {\displaystyle \frac{i{g}_{20}q\left(\theta \right)}{{\omega }_0{\tau }_0}}+{\displaystyle \frac{i{\overline{g}}_{02}\overline{q}\left(\theta \right)}{3{\omega }_0{\tau }_0}}+E_1{e}^{2i{\omega }_0{\tau }_0\theta },\hfill \\ \hfill W_{11}\left(\theta \right)& =& -{\displaystyle \frac{i{g}_{11}q\left(\theta \right)}{{\omega }_0{\tau }_0}}+{\displaystyle \frac{i{\overline{g}}_{11}\overline{q}\left(\theta \right)}{{\omega }_0{\tau }_0}}+E_2.\hfill \end{array}$$

Hence for $\theta =0,$ (53) yields

$$\begin{array}{ccc}\hfill H_{20}\left(0\right)& =& -g_{20}q\left(0\right)-{\overline{g}}_{02}\overline{q}\left(0\right)+{\tau }_0{E}_1^{\left(2\right)},\hfill \\ \hfill H_{11}\left(0\right)& =& -g_{11}q\left(0\right)-{\overline{g}}_{11}\overline{q}\left(0\right)+{\tau }_0{E}_2^{\left(2\right)}.\hfill \end{array}$$

Then

$$E_1=E_1^{\left(1\right)}·E_1^{\left(2\right)},E_2=E_2^{\left(1\right)}·E_2^{\left(2\right)},$$

where

$$\begin{array}{ccc}\hfill E_1^{\left(1\right)}& =& {\left(\begin{array}{cc}2i{\omega }_0-a\left(1-{\displaystyle \frac{1}{v^{*}}}\right)& -{\displaystyle \frac{a{u}^{*}}{v^{*}}}\\ \left(2-{\displaystyle \frac{1}{v^{*}}}\right)e^{-2i{\omega }_0{\tau }_0}& 2i{\omega }_0+\left(1+{\displaystyle \frac{u^{*}}{v^{*}}}\right)\end{array}\right)}^{-1},\hfill \\ \hfill E_1^{\left(2\right)}& =& \left(\begin{array}{c}{\displaystyle \frac{a}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1^2+{\displaystyle \frac{a}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)q_1{q}_2\\ -{\displaystyle \frac{1}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1^2{e}^{-2i{\omega }_0{\tau }_0}-{\displaystyle \frac{1}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)q_1{q}_2{e}^{-i{\omega }_0{\tau }_0}\end{array}\right),\hfill \\ \hfill E_2^{\left(1\right)}& =& {\left(\begin{array}{cc}-a\left(1-{\displaystyle \frac{1}{v^{*}}}\right)& -{\displaystyle \frac{a{u}^{*}}{v^{*}}}\\ 2-{\displaystyle \frac{1}{v^{*}}}& 1+{\displaystyle \frac{u^{*}}{v^{*}}}\end{array}\right)}^{-1},\hfill \\ \hfill E_2^{\left(2\right)}& =& \left(\begin{array}{c}{\displaystyle \frac{2a}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1{\overline{q}}_1+{\displaystyle \frac{a}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)(q_1{\overline{q}}_2+{\overline{q}}_1{q}_2)\\ -{\displaystyle \frac{2}{u^{*}}}{\left(1-{\displaystyle \frac{1}{v^{*}}}\right)}^2{q}_1{\overline{q}}_1-{\displaystyle \frac{1}{v^{*}}}\left(2-{\displaystyle \frac{1}{v^{*}}}\right)(q_1{\overline{q}}_2{e}^{-i{\omega }_0{\tau }_0}+{\overline{q}}_1{q}_2{e}^{i{\omega }_0{\tau }_0})\end{array}\right).\hfill \end{array}$$

So far, $g_{21}$ can be expressed in terms of the parameters of system (3). Consequently, the properties of the bifurcating periodic solutions at the critical value ${\tau }_0$ are determined by the sign of ${\mu }_2,$ ${\sigma }_2,$ $T_2$ where

$$\left\{\begin{array}{c}{c}_1\left({\tau }_0\right)={\displaystyle \frac{i}{2{\omega }_0{\tau }_0}}\left(g_{11}{g}_{20}-2{\left|g_{11}\right|}^2-{\displaystyle \frac{{\left|g_{02}\right|}^2}{3}}\right)+{\displaystyle \frac{g_{21}}{2}},\hfill \\ {\mu }_2=-{\displaystyle \frac{\mathrm{Re}\left(c_1\left({\tau }_0\right)\right)}{\mathrm{Re}\left({\lambda }^{\prime }\left({\tau }_0\right)\right)}},\hfill \\ {\sigma }_2=2\mathrm{Re}\left(c_1\left({\tau }^{*}\right)\right),\hfill \\ T_2=-{\displaystyle \frac{\mathrm{Im}\left(c_1\left({\tau }_0\right)\right)+{\mu }_2\mathrm{Im}\left({\lambda }^{\prime }\left({\tau }_0\right)\right)}{{\omega }_0{\tau }_0}},\hfill \end{array}\right.$$

Following the results in [35], the properties of the Hopf bifurcation are characterized by ${\mu }_2$: if ${\mu }_2>0$ (respectively ${\mu }_2<0$), the bifurcation is supercritical (respectively subcritical) and the periodic solutions exist for $\tau >{\tau }_0$ (respectively $\tau <{\tau }_0$). The stability of the bifurcating periodic solutions on the center manifold is determined by ${\sigma }_2$: they are stable if ${\sigma }_2<0$ and unstable if ${\sigma }_2>0$. Moreover, the period increases if $T_2>0$ and decreases if $T_2<0$.

5. Numerical Simulations

In this section we present some numerical simulations and computations which complement the theoretical results in the previous sections.

5.1. Numerical Simulations of Turing Bifurcation

Example 1.

Let $a=5,b=3.5,K=1$ in system (2) and fix the habitat length as $l={\displaystyle \frac{10}{\pi }}$. Then the interval given in (10) becomes $0.006<d_1<0.01689$. We choose $d_1=0.06$ within this range. Then the critical value of diffusion coefficient $d_2$ at which the Turing bifurcation takes place is $d_2^{*}\approx 0.6018$ by (18). As predicted by Theorem 2, system (2) exhibits pattern formation (see Figure 1).

Figure 1. Numerical simulations of system (2) with the parameters specified in Example 1. System (2) generates a spatial oscillation pattern.

Remark 4.

Regarding the specific pattern shown in Figure 1, the spatial oscillation pattern can be interpreted as representing heterogeneous distributions of microbial biomass and nutrients within the sediment. The diffusion coefficients ($d_1=0.06$ for bacteria, $d_2\approx 0.6018$ for nutrient) satisfy the Turing condition ($d_2>d_1$), implying that the nutrient diffuses much faster than the bacteria. This is ecologically reasonable: dissolved chemical species (v) can diffuse through pore water, while bacteria (u) are often attached to sediment particles or move very slowly, leading to much lower effective dispersal.

5.2. Numerical Simulations of Homogeneous Hopf Bifurcation

In this subsection, we give some numerical simulations and computations of homogeneous Hopf bifurcation.

Example 2.

Consider system (3) with $\mathsf{\Omega }=(0,2\pi )$ and parameter values $(a,b,K,d_1,d_2)=(1,3,0.3,0.1,5).$ The steady state is computed as $\left(u^{*},v^{*}\right)=({\tilde{u}}_2,{\tilde{v}}_2)=\left(1.8367,1.1633\right)$. Equation (36) gives only one ${\omega }_0\approx 0.6615$. By using (41) we have ${\tau }_0^j\approx 1.6787+{\displaystyle \frac{2j\pi }{{\omega }_0}}$. According to Theorem 4, the equilibrium $\left(1.8367,1.1633\right)$ is asymptotically stable for $0<\tau <{\tau }_0^0\approx 1.6787$ and becomes unstable when $\tau >{\tau }_0^0\approx 1.6787.$ As τ increases through ${\tau }_0^0\approx 1.6787$, a spatially homogeneous periodic solution bifurcates from the equilibrium, as illustrated in Figure 2 and Figure 3.

Figure 2. Numerical simulations of system (3) with the parameters given in Example 2, $\tau =1.5$ and initial values $u(x,t)=1.5\sin x$, $v(x,t)=\sin x$, $(x,t)\in [0,2\pi ]\times [-1.5,0]$. The positive constant equilibrium solution $\left(1.8367,1.1633\right)$ is asymptotically stable.

Figure 3. Numerical simulations of system (3) with the parameters given in Example 2, $\tau =1.7$ and initial values $u(x,t)=1.5\sin x$, $v(x,t)=\sin x$, $(x,t)\in [0,2\pi ]\times [-1.7,0]$. The point $\left(1.8367,1.1633\right)$ is unstable and system (3) exhibits periodic solutions.

Further numerical simulations are carried out with mesh sizes $\Delta x={\displaystyle \frac{\pi }{100}}$ in space and $\Delta t=0.05$ in time. Let $U_{\mathbf{j}}^{\mathbf{n}}\approx u(x_{\mathbf{j}},t_{\mathbf{n}})$ and $V_{\mathbf{j}}^{\mathbf{n}}\approx v(x_{\mathbf{j}},t_{\mathbf{n}}),$ where $x_{\mathbf{j}}=\mathbf{j}\Delta x,$ $t_{\mathbf{n}}=\mathbf{n}\Delta t,$ $\mathbf{j}=0,1,\dots ,200,$ $\mathbf{n}=1,2,3,\dots .$ For $\tau =1.5$, we focus on the spatial point $x_{100}=100\Delta x=\pi .$ Applying an implicit difference scheme, the numerical results at $x_{100}=\pi$ and numerical solutions $U_{100}^{\mathbf{n}}$ and $V_{100}^{\mathbf{n}}$ when $\mathbf{n}=100\mathbf{k}$ $(\mathbf{k}=1,2,\dots ,30)$ are given in Figure 4 and Table 1 and Table 2, respectively. These confirm the asymptotic stability of $\left(1.8367,1.1633\right)$ located in $x_{100}=\pi$. When $\tau =1.7>{\tau }_0\approx 1.6787$, simulations at $x_{100}=\pi$ are presented in Figure 5, and the corresponding numerical solutions $U_{100}^{\mathbf{n}}$ and $V_{100}^{\mathbf{n}}$ for $\mathbf{n}=10\mathbf{k}$ $(\mathbf{k}=259,260,\dots ,300)$ appear in Table 3 and Table 4. The results indicate the emergence of a periodic solution near $\left(1.8367,1.1633\right)$ at $x_{100}=\pi$.

Figure 4. Numerical simulations of system (3) with the parameters given in Example 2 at mesh point $x_{100}=\pi$ plotted against $t_{\mathbf{n}}=\mathbf{n}\Delta t$ with $\tau =1.5$.

Table 1. Numerical solution $U_{100}^{\mathbf{n}}$ when $\mathbf{n}=100\mathbf{k}$ $(\mathbf{k}=1,2,\dots ,30)$.

solution	100	$2\times 100$	$3\times 100$	$4\times 100$	$5\times 100$	$6\times 100$
$U_{100}^{\mathbf{n}}$	1.6001	1.8770	1.8277	1.8299	1.8462	1.8274
solution	$7\times 100$	$8\times 100$	$9\times 100$	$10\times 100$	$11\times 100$	$12\times 100$
$U_{100}^{\mathbf{n}}$	1.8440	1.8314	1.8404	1.8343	1.8383	1.8357
solution	$13\times 100$	$14\times 100$	$15\times 100$	$16\times 100$	$17\times 100$	$18\times 100$
$U_{100}^{\mathbf{n}}$	1.8373	1.8364	1.8369	1.8366	1.8368	1.8367
solution	$19\times 100$	$20\times 100$	$21\times 100$	$22\times 100$	$23\times 100$	$24\times 100$
$U_{100}^{\mathbf{n}}$	1.8367	1.8367	1.8367	1.8367	1.8367	1.8367
solution	$25\times 100$	$26\times 100$	$27\times 100$	$28\times 100$	$29\times 100$	$30\times 100$
$U_{100}^{\mathbf{n}}$	1.8367	1.8367	1.8367	1.8367	1.8367	1.8367

Table 2. Numerical solution $V_{100}^{\mathbf{n}}$ when $\mathbf{n}=100\mathbf{k}$ $(\mathbf{k}=1,2,\dots ,30)$.

solution	100	$2\times 100$	$3\times 100$	$4\times 100$	$5\times 100$	$6\times 100$
$U_{100}^{\mathbf{n}}$	1.4157	1.1407	1.1827	1.1459	1.1759	1.1544
solution	$7\times 100$	$8\times 100$	$9\times 100$	$10\times 100$	$11\times 100$	$12\times 100$
$U_{100}^{\mathbf{n}}$	1.1692	1.1595	1.1657	1.1618	1.1642	1.1628
solution	$13\times 100$	$14\times 100$	$15\times 100$	$16\times 100$	$17\times 100$	$18\times 100$
$U_{100}^{\mathbf{n}}$	1.1636	1.1631	1.1634	1.1633	1.1633	1.1633
solution	$19\times 100$	$20\times 100$	$21\times 100$	$22\times 100$	$23\times 100$	$24\times 100$
$U_{100}^{\mathbf{n}}$	1.1633	1.1633	1.1633	1.1633	1.1633	1.1633
solution	$25\times 100$	$26\times 100$	$27\times 100$	$28\times 100$	$29\times 100$	$30\times 100$
$U_{100}^{\mathbf{n}}$	1.1633	1.1633	1.1633	1.1633	1.1633	1.1633

Figure 5. Numerical simulations of system (3) with the parameters given in Example 2 at mesh point $x_{100}=\pi$ plotted against $t_{\mathbf{n}}=\mathbf{n}\Delta t$ with $\tau =1.7$.

Table 3. Numerical solution $U_{100}^{\mathbf{n}}$ when $\mathbf{n}=10\mathbf{k}$ $(\mathbf{k}=259,260,\dots ,300)$.

solution	2590	2600	2610	2620	2630	2640	2650
$U_{100}^{\mathbf{n}}$	3.4270	2.4710	1.5479	0.9404	0.6039	0.4410	0.4130
solution	2660	2670	2680	2690	2700	2710	2720
$U_{100}^{\mathbf{n}}$	0.6703	1.7639	3.2513	3.8012	3.7064	2.9904	1.9760
solution	2730	2740	2750	2760	2770	2780	2790
$U_{100}^{\mathbf{n}}$	1.2059	0.7454	0.5059	0.4079	0.4808	1.0745	2.5841
solution	2800	2810	2820	2830	2840	2850	2860
$U_{100}^{\mathbf{n}}$	3.6334	3.8217	3.4301	2.4754	1.5512	0.9423	0.6049
solution	2870	2880	2890	2900	2910	2920	2930
$U_{100}^{\mathbf{n}}$	0.4415	0.4127	0.6680	1.7572	3.2469	3.8005	3.7080
solution	2940	2950	2960	2970	2980	2990	3000
$U_{100}^{\mathbf{n}}$	2.9946	1.9799	1.2084	0.7468	0.5065	0.4081	0.4798

Table 4. Numerical solution $V_{100}^{\mathbf{n}}$ when $\mathbf{n}=10\mathbf{k}$ $(\mathbf{k}=259,260,\dots ,300)$.

solution	2590	2600	2610	2620	2630	2640	2650
$U_{100}^{\mathbf{n}}$	2.6165	1.9771	1.2655	0.8298	0.6225	0.5368	0.5090
solution	2660	2670	2680	2690	2700	2710	2720
$U_{100}^{\mathbf{n}}$	0.5246	0.6341	1.0385	1.9449	2.6405	2.3588	1.5942
solution	2730	2740	2750	2760	2770	2780	2790
$U_{100}^{\mathbf{n}}$	1.0134	0.7053	0.5698	0.5177	0.5103	0.5598	0.7779
solution	2800	2810	2820	2830	2840	2850	2860
$U_{100}^{\mathbf{n}}$	1.4414	2.3932	2.6178	1.9805	1.2680	0.8311	0.6230
solution	2870	2880	2890	2900	2910	2920	2930
$U_{100}^{\mathbf{n}}$	0.5370	0.5090	0.5244	0.6332	1.0357	1.9406	2.6396
solution	2940	2950	2960	2970	2980	2990	3000
$U_{100}^{\mathbf{n}}$	2.6317	1.5973	1.1053	0.7061	0.5702	0.5178	0.5103

6. Conclusions

In this paper, we have investigated a minimal sediment model that describes the interaction between a bacterial population and its nutrients. In this model we take into account that bacteria can live in an active state, in which they reproduce, consuming nutrients, as well as in a passive state of dormancy. Our analysis focused on the pattern formation and stability of patterns for system (2) and Hopf bifurcation induced by diffusion and time delay. The key findings are summarized as follows:

1.

Diffusion-driven pattern (Turing bifurcation): The condition $a<{\displaystyle \frac{{\tilde{u}}_2^2+{\tilde{u}}_2+K}{K}}$ in Theorem 1 is fundamental for the existence of stable, positive constant equilibrium solutions $({\tilde{u}}_2,{\tilde{v}}_2)$ before diffusion is considered. Ecologically speaking, parameter $a=m/\mathsf{\Theta }$ represents the ratio of the bacterial mortality rate (m) to the water–sediment exchange rate ($\mathsf{\Theta }$). It can be interpreted as a dimensionless net loss rate for the bacteria. The term $({\tilde{u}}_2^2+{\tilde{u}}_2+K)/K$ is a threshold. The condition $a<{\displaystyle \frac{{\tilde{u}}_2^2+{\tilde{u}}_2+K}{K}}$ essentially states that the bacterial net loss rate (a) must not be too high relative to a threshold by the equilibrium bacterial density (${\tilde{u}}_2$) and half-saturation (K). In such conditions, if the biomass of bacteria diffusion rate $d_1$ is relatively high compared, e.g., to the square of the length of the spatial domain (condition (9)) then $({\tilde{u}}_2,{\tilde{v}}_2)$ is locally asymptotically stable. However, when $d_1$ is lower (condition (10)), then one may increase the nutrient diffusion rate $d_2$ to a value (18) higher than the biomass of bacteria diffusion rate at which $({\tilde{u}}_2,{\tilde{v}}_2)$ loses its stability. To be sure, $({\tilde{u}}_2,{\tilde{v}}_2)$ without diffusion stays stable, i.e., a so-called diffusional instability occurs quite naturally, contrary to the general belief that diffusion usually stabilizes a system. At the critical value (18) of the bifurcation parameter (the nutrient rate) a local pattern (a nonconstant steady state solution) emerges. A first order approximation of this pattern (22) is explicitly given. A quantity (30) is derived, and this condition predicts the stability of the formed spatial patterns. A positive $d_2^{\prime }\left(0\right)$ indicates that as we move into the Turing parameter region (increasing nutrient diffusivity $d_2$ beyond the critical one $d_2^{*}$), the pattern is stable. This means the microbial community will maintain this spatial structure over time. Conversely, a negative value suggests the pattern still exists but becomes unstable.

2.

Time-delay-induced oscillations (Hopf bifurcation): By taking time delay as the bifurcation parameter, we demonstrate the occurrence of a Hopf bifurcation at the positive constant equilibrium solution $({\tilde{u}}_2,{\tilde{v}}_2)$. The results show that $({\tilde{u}}_2,{\tilde{v}}_2)$ is asymptotically stable when discrete delay is less than a certain critical value and unstable when the discrete delay is greater than this critical value, and the model undergoes a Hopf bifurcation at $({\tilde{u}}_2,{\tilde{v}}_2)$ when the discrete delay crosses through the above critical value.

3.

Future Work: On the one hand, a particularly promising and theoretically rich direction is the systematic study of Turing–Hopf bifurcations induced by the joint variation of diffusion ($d_2$) and time delay ($\tau$). In the parameter plane $(d_2,\tau )$, the critical curve for Turing instability may intersect the Hopf bifurcation curve, giving rise to a codimension-2 bifurcation point. Near such a point, the system can exhibit a wealth of complex dynamical behaviors. On the other hand, in the study of delayed reaction–diffusion models, common types of delays include discrete delay, distributed delay, and nonlocal delay. This paper focuses on discrete delay, while distributed delay and nonlocal delay will be considered in subsequent research.