Spatial model for oncolytic virotherapy with lytic cycle delay

Abstract

We formulate a mathematical model of functional partial differential equations for oncolytic virotherapy which incorporates virus diffusivity, tumor cell diffusion, and the viral lytic cycle based on a basic oncolytic virus dynamics model. We conduct a detailed analysis for the dynamics of the model and carry out numerical simulations to demonstrate our analytic results. Particularly, we establish the positive invariant domain for the ω limit set of the system, and show that the model has three spatially homogenous equilibrium solutions. We prove that the spatially uniform virus-free steady state is globally asymptotically stable for any viral lytic period delay and diffusion coefficients of tumor cells and viruses when the viral burst size is smaller than a critical value. We obtain the conditions, for example, the ratio of virus diffusion coefficient to that of tumor cells is greater than a value and the viral lytic cycle is greater than a critical value, under which the spatially uniform positive steady state is locally asymptotically stable. We also obtain conditions under which the system undergos Hopf bifurcations, and stable periodic solutions occur. We point out medical implications of our results which are difficult to obtain from models without combining diffusive properties of viruses and tumor cells with viral lytic cycles.

1. Introduction

Oncolytic viruses are genetically altered viruses that can infect and multiply in cancer cells but leave healthy normal cells intact. Oncolytic viruses are of two types: oncolytic wild viruses that naturally occur preferentially in human cancer cells, and gene-modified viruses that are engineered to achieve selective oncolysis. Wild type viruses have shown a limited oncolytic efficacy in some preclinical trials, while gene-modified viruses seem to have a great potency of oncolysis [1–3]. Before 1990s, case studies and small trials of various viruses in cancer therapy were reported [4]. Since 1990s, genetic engineering began to be used for oncolytic viruses [5]. So far, many types of viruses have been modified for experiments [6], and some oncolytic viruses have been approved for human clinic trials [7]. However, the full potential of oncolytic viruses seem still not to have been reached [8]. One major challenge is how to make viruses spread thoroughly into solid tumors [9]. Understanding of the spreading dynamics of oncolytic viruses through tumors can help to overcome this difficulty, and possibly to develop strategies for clinical applications.

Mathematical modeling can allow us to explore the whole spectrum of possible outcomes, and provide rationales to optimize treatments. Several attempts have been made to understand and characterize viral dynamics by mathematical models, see for example [10, 11, 15, 17]. These mathematical models can be divided into two classes roughly. One class uses ordinary differential equations (ODEs) including delay differential equations (DDEs) [12, 13, 16, 18–28], and the other uses partial differential equations (PDEs) [11, 14, 17, 29–32]. For PDE models of oncolytic virotherapy, most of them use ideas from fluid dynamics to model solid tumor growth where tumor cells convect in the fluid velocity field within the tumor and viruses simply diffuse within the tumor. All these modeling studies have provided certain insights to viral therapy. One study by Jain and colleagues particularly emphasizes the importance of the diffusive property of viruses [9]. However, it is well known that the growth of solid tumors, particularly, brain tumor gliomas, also shows characters of cell diffusion [33]. It is, therefore, appropriate to combine diffusive tumor growth with virus diffusion into one mathematical model.

The viral lytic cycle is the time period of the intracellular viral life cycle starting from the time when a virus enters a cell to the time when a certain number (viral burst size) of newly replicated viruses come out upon the cell lysis. It is an important parameter of viral dynamics. The article [19] was the first to incorporate the viral lytic cycle as a delay parameter in mathematical model for oncolytic viral therapy.

In this study, building on a basic mathematical model for oncolytic viral dynamics [18, 19], we combine the viral lytic cycle, tumor cell diffusion, and virus diffusion into a system of functional partial differential equations as a spatial model for oncolytic viral therapy. Our spatial model considers a brain as a domain in ${ℝ}^3$ where a brain tumor grows and three populations, tumor cells, infected tumor cells, and intercellular free viruses, interact with each other. We conduct a detailed analysis and computational demonstrations.

Although our analysis focuses on long-term (asymptotical) dynamical behaviors, we also obtain short-term information about virus and cell distributions which may also give insights to experiments and treatments. Comparing with non-spatial and non-delay models, our model reveals more relevant details of viral therapy. For example, when the ratio of diffusion coefficients between viruses and tumor cells is greater than a particular value, the spatially homogeneous equilibrium solution is stable for any length of the viral lytic cycle. A possible medical implication is that if we can genetically make oncolytic viruses diffuse faster than tumor cell diffuse, the time period of the viral lytic cycle will not affect the stabilization of treatments and may improve over all efficacy of treatments. This mathematical result does not contradict the fact that there is a functional relation between the viral lytic cycle and burst size which reveals that at a critical lytic cycle time, Hopf bifurcations occur in the model system [18, 19]. In fact, for our spatial model, we also obtain a similar functional relation of bifurcation diagram for different range of parameter values. In addition, the diffusion coefficients of viruses and tumor cells with viral lytic cycle length have a combined effect on dramatic changes of the solutions, for example, the occurrence of Hopf bifurcations and occurrence of stable periodic solutions. This theoretical result may explain various outcomes of oncolytic viral therapies in the current experimental research. Therefore, combining diffusive property of viruses and tumor cells with viral lytic cycle into one mathematical model is necessary to further our understanding for oncolytic viral therapy.

The rest of the paper is organized as follows. In section 2, we review some basic mathematical models for oncolytic viral therapy, and introduce our model in terms of delay reaction-diffusion equations. In section 3, we mathematically analyze the dynamics of our functional PDE model for a solid tumor in ${ℝ}^3$. In section 4, we demonstrate some numerical simulations in a two dimensional spatial domain to support the theoretical results. In section 5, we highlight our results which cannot be obtained without virus and cell diffusion, explain some medical implications of our results, and discuss and compare several revelent results in the literature. In Appendix, we carry out analysis for properties of periodic solutions arising from Hopf bifurcations of functional partial differential equations which provides some guidance for numerical simulations, and also briefly describe numerical methods we use to solve our model.

2. Mathematical Model

In the paper [18], a basic mathematical model for oncolytic virotherapy was proposed as follows

$$\{\begin{array}{l}\frac{\text{d}u(t)}{\text{d}t}=\alpha u(t)(1-\frac{u(t)+w(t)}{K})-\beta u(t)v(t),\hfill \\ \frac{\text{d}w(t)}{\text{d}t}=\beta u(t)v(t)-\delta w(t),\hfill \\ \frac{\text{d}v(t)}{\text{d}t}=b\delta w(t)-\beta u(t)v(t)-\gamma v(t).\hfill \end{array}$$ (2.1)

Here, u(t) stands for the uninfected tumor cell population at time t, w(t) stands for the infected tumor cell population, and v(t) stands for the free virus population. Tumor growth is modeled by logistic growth, and K is the maximum tumor population. The coefficient β represents infectivity of the virus. The infected tumor cells die with a rate δw. The parameter b is the burst size of the virus, the parameter γv is the clearance rate of free virus particles by various mechanisms including non-specific binding and generation of the defective interfering particles.

Viruses need time to complete their life cycle within cells. Incorporating the viral lytic cycle within infected tumor cells, in Wang et al [19], the following delay differential equation model was proposed.

$$\{\begin{array}{l}\frac{\text{d}u(t)}{\text{d}t}=\alpha u(t)(1-\frac{u(t)+w(t)}{K})-\beta u(t)v(t),\hfill \\ \frac{\text{d}w(t)}{\text{d}t}=\beta u(t-\tau )v(t-\tau )-\delta w(t),\hfill \\ \frac{\text{d}v(t)}{\text{d}t}=b\delta w(t)-\beta u(t)v(t)-\gamma v(t),\hfill \end{array}$$ (2.2)

where τ stands for the mean time period of the viral lytic cycle.

Now we assume that tumor cells diffuse with diffusion coefficient D₁, and viruses diffuse with diffusion coefficient D₂. It is an assumption to consider cell movement to only be by diffusion [38, 39]. Based on the mathematical models above, we propose a spatial mathematical model for oncolytic viral therapy as follows:

$$\{\begin{array}{l}\frac{\partial u(x,t)}{\partial t}=D_1\mathrm{\Delta }u(x,t)+\alpha u(x,t)(1-\frac{u(x,t)+w(x,t)}{K})-\beta u(x,t)v(x,t),\hfill \\ \frac{\partial w(x,t)}{\partial t}=D_1\mathrm{\Delta }w(x,t)+\beta u(x,t-\tau )v(x,t-\tau )-\delta w(x,t),\hfill \\ \frac{\partial v(x,t)}{\partial t}=D_2\mathrm{\Delta }v(x,t)+b\delta w(x,t)-\beta u(x,t)v(x,t)-\gamma v(x,t),\hfill \end{array}$$ (2.3)

for which x ∈ Ω, t > 0, where Ω is the brain tissue which is a domain in ${ℝ}^3$. To complete the model, we give the no-flux boundary conditions and initial conditions as follows:

$$\{\begin{array}{l}\frac{\partial u(x,t)}{\partial \mathbf{n}}=\frac{\partial w(x,t)}{\partial \mathbf{n}}=\frac{\partial v(x,t)}{\partial \mathbf{n}}=0,x\in \partial \mathrm{\Omega },t\ge 0;\hfill \\ u(x,t)=u_0(x,t)\ge 0,w(x,t)=w_0(x,t)\ge 0,\text{and}\hfill \\ v(x,t)=v_0(x,t)\ge 0,x\in \overline{\mathrm{\Omega }},-\tau \le t\le 0.\hfill \end{array}$$ (2.4)

In the boundary and initial conditions, ∂Ω stands for the brain tissue boundary, n stands for its normal direction, $\overline{\mathrm{\Omega }}$ stands for the brain tissue with it boundary, and u₀(x, t), w₀(x, t), v₀(x, t) are initial densities. The zero-flux boundary condition means that there are no tumor cells or viruses crossing the brain tissue boundary. The zero-flux boundary condition for tumor cells has biological significance and has been applied in many diffusion type modeling of brain tumors [38, 39]. This is also biologically meaningful for viruses since oncolytic viruses do not infect normal cells by definition which are confined within the tumor [4].

We non-dimensionalize the system (2.3)–(2.4) by setting

$$\overline{t}=\delta t,\overline{\tau }=\delta \tau ,u(x,t)=K\overline{u}(x,\overline{t}),w(x,t)=K\overline{w}(x,\overline{t}),v(x,t)=K\overline{v}(x,\overline{t}),$$

$$r=\frac{\alpha }{\delta },a=\frac{\beta K}{\delta },c=\frac{\gamma }{\delta },d_1=\frac{D_1}{\delta },d_2=\frac{D_2}{\delta },$$

$$u_0(x,t)=K{\overline{u}}_0(x,\overline{t}),w_0(x,t)=K{\overline{w}}_0(x,\overline{t}),v_0(x,t)=K{\overline{v}}_0(x,\overline{t}).$$

For convenience, dropping all bars over the variables, we have

$$\{\begin{array}{l}\frac{\partial u(x,t)}{\partial t}=d_1\mathrm{\Delta }u(x,t)+ru(x,t)(1-u(x,t)-w(x,t))-au(x,t)v(x,t),\hfill \\ \frac{\partial w(x,t)}{\partial t}=d_1\mathrm{\Delta }w(x,t)+au(x,t-\tau )v(x,t-\tau )-w(x,t),\hfill \\ \frac{\partial v(x,t)}{\partial t}=d_2\mathrm{\Delta }v(x,t)+bw(x,t)-au(x,t)v(x,t)-cv(x,t),x\in \mathrm{\Omega },t>0;\hfill \\ \frac{\partial u(x,t)}{\partial \mathbf{n}}=\frac{\partial w(x,t)}{\partial \mathbf{n}}=\frac{\partial v(x,t)}{\partial \mathbf{n}}=0,x\in \partial \mathrm{\Omega },t\ge 0;\hfill \\ u(x,t)=u_0(x,t)\ge 0,w(x,t)=w_0(x,t)\ge 0,\text{and}\hfill \\ v(x,t)=v_0(x,t)\ge 0,x\in \overline{\mathrm{\Omega }},-\tau \le t\le 0.\hfill \end{array}$$ (2.5)

3. Analysis of the model

In this section, we study the properties of spatial homogenous equilibrium solutions and periodic solutions arising from Hopf bifurcations. The system (2.5) always has two spatially uniform steady states E₀ = (0, 0, 0) and E₁ = (1, 0, 0). When $b>b_0=1+\frac{c}{a}$, the system (2.5) has three spatially uniform steady states E₀, E₁ and E∗ = (u∗, w∗, v∗), where

$$u*=\frac{c}{a(b-1)},w*=\frac{rc(ab-a-c)}{a(b-1)(rc+ab-a)},v*=\frac{r(ab-a-c)}{a(rc+ab-a)}.$$

In Subsection 3.1, we establish positive invariant region as a sub-functional space for the ω limit sets of our model, and prove that the spatially uniform steady state E₁ is globally asymptotically stable for any viral lytic period τ ≥ 0 when the viral burst size b < b₀. In Subsection 3.2, we obtain the conditions, for example, the ratio of diffusion coefficients $\frac{d_2}{d_1}$ is greater than some value and the viral lytic cycle is greater than a critical value, under which the spatially uniform positive steady state is locally asymptotically stable. We also obtain some conditions under which the system undergos Hopf bifurcations, and stable periodic solutions occur.

In what follows, we choose Ω be a open set in ${ℝ}^3$, and define the real-valued Sobolev space

$$X=\left\{{\left(u_1,u_2,u_3\right)}^{\text{T}}\in {\left(W^{2,2}(\mathrm{\Omega })\right)}^3:\frac{\partial u_1}{\partial \mathbf{n}}=\frac{\partial u_2}{\partial \mathbf{n}}=\frac{\partial u_3}{\partial \mathbf{n}}=0,\text{at}x\in \partial \mathrm{\Omega }\right\}.$$

3.1. Stability of the steady state of E₁

To simplify notations, we denote u₁(t) = u(x, t), u₂(t) = w(x, t), u₃(t) = v(x, t) and U(t) = (u₁(t), u₂(t), u₃(t))^T. Then, the system (2.5) can be rewritten as an abstract differential equation in the phase space $\mathcal{C}=C([-\tau ,0],X)$,

$$\dot{U}(t)=D\mathrm{\Delta }U(t)+G\left(U_t\right)$$ (3.1)

where U_t(·) = U(t + ·), D = diag(d₁, d₁, d₂) and $G:\mathcal{C}\to X$ are defined by

$$G\left(U_t\right)=\left(\begin{array}{c}r{u}_1(t)\left(1-u_1(t)-u_2(t)\right)-a{u}_1(t)u_3(t)\\ a{u}_1(t-\tau )u_3(t-\tau )-u_2(t)\\ b{u}_2(t)-a{u}_1(t)u_3(t)-c{u}_3(t)\end{array}\right).$$

Let N be a operator satisfies

$$N\left({\phi }_t\right)(x)=\frac{\partial \phi (x,t)}{\partial t}-D\mathrm{\Delta }\phi (x,t)-G\left({\phi }_t\right)(x),$$

where φ_t(θ)(x) = φ(x, t + θ) = (φ₁(x, t + θ), φ₂(x, t + θ), φ₃(x, t + θ)). Then N(0, 0, 0) = 0 ≤ 0 = N(u_t, w_t, v_t)(x), where u_t(θ)(x) = u(x, t + θ), w_t(θ)(x) = w(x, t + θ), v_t(θ)(x) = v(x, t + θ). Notices the homogeneous Neumann boundary and nonnegative initial value of system (2.5), we get (0, 0, 0) is a lower solution of system (2.5), and by the comparison principle we know that the solution (u, w, v) of system (2.5) satisfies u(x, t) ≥ 0, w(x, t) ≥ 0, v(x, t) ≥ 0 for t ≥ 0.

From the non-negativity of the solution, we have

$$\frac{\partial u}{\partial t}\le d_1\mathrm{\Delta }u+ru(1-u).$$

The standard comparison theorem for parabolic boundary-value problems implies that u(x, t) ≤ U₁(z, t) in $\overline{\mathrm{\Omega }}\times [0,\infty ]$, where U₁(x, t) is the positive solution of

$$\{\begin{array}{ll}\frac{\partial U_1}{\partial t}=d_1\mathrm{\Delta }{U}_1+r{U}_1\left(1-U_1\right),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ \frac{\partial U_1}{\partial \mathbf{n}}=0,\hfill & x\in \partial \mathrm{\Omega },t>0,\hfill \\ U_1(x,0)=u_0(x,0)>0,\hfill & x\in \overline{\mathrm{\Omega }}.\hfill \end{array}$$

It is well known that U₁(x, t) → 1 as t → ∞. So we have $\lim {\sup }_{t\to \infty }u(x,t)\le 1$. For any ε > 0, there exists a T₁ > 0 such that u(x, t) ≤ 1 + ε in $\overline{\mathrm{\Omega }}\times \left(T_1,\infty \right)$. When t > T₁, we have

$$\begin{gathered} \frac{\partial (u(x,t-\tau )+w(x,t))}{\partial t} \\ =d_1\mathrm{\Delta }(u(x,t-\tau )+w(x,t))+ru(x,t-\tau )(1-u(x,t-\tau )-w(x,t-\tau ))-w(x,t) \\ \le d_1\mathrm{\Delta }(u(x,t-\tau )+w(x,t))+r(1+\epsilon )(1-u(x,t-\tau ))-w(x,t) \\ \le d_1\mathrm{\Delta }(u(x,t-\tau )+w(x,t))+r(1+\epsilon )-r_1(u(x,t-\tau )+w(x,t)) \end{gathered}$$

where r₁ = min{1, r}. Similar to the previous discussion and noticed the sufficiently small properties of ε, we get $\lim {\sup }_{t\to \infty }(u(x,t-\tau )$,$w(x,t))\le \frac{r}{r_1}$. Since u(x, t − τ) ≥ 0, we have $\lim {\sup }_{t\to \infty }w(x,t)\le \frac{r}{r_1}$. There exists a T₂ > T₁ + τ such that $w(x,t)\le \frac{r}{r_1}+\epsilon$ in $\overline{\mathrm{\Omega }}\times \left(T_2,\infty \right)$. Then from the third equation of (2.5)

$$\frac{\partial v(x,t)}{\partial t}\le d_2\mathrm{\Delta }v+bw(x,t)-cv(x,t)\le d_2\mathrm{\Delta }v+b(\frac{r}{r_1}+\epsilon )-cv(x,t),t>T_2.$$

Similar to the previous discussion, we get $\lim {\sup }_{t\to \infty }\le \frac{br}{c{r}_1}$.

We define ${\mathcal{C}}^{+}=C\left(\overline{\mathrm{\Omega }}\times [-\tau ,0],{ℝ}^{+}\right)$ with norm $\Vert {\phi }_t\Vert ={\sup }_{(x,\theta )\in \overline{\mathrm{\Omega }}\times [-\tau ,0]}\phi (x,t+\theta )$. From the pervious discussion, we have the following result.

Lemma 3.1.

The solution of the system (2.5) are ultimately uniformly bounded in ${\mathcal{C}}^{+}\times {\mathcal{C}}^{+}\times {\mathcal{C}}^{+}$.

The ω limit set of the system (2.5) is contained in the bounded region:

$$R=\{\left(u_t,w_t,v_t\right)\in {\mathcal{C}}^{+}\times {\mathcal{C}}^{+}\times {\mathcal{C}}^{+}:\Vert u_t\Vert \le 1,\Vert w_t\Vert \le \frac{r}{r_1},\Vert v_t\Vert \le \frac{br}{c{r}_1}\}.$$

It is easy to verify that the region R is positively invariant for the system (2.5). In the rest of this paper, we will discuss the system (2.5) in region R.

From the above discussion, ($\overline{u}$,$\overline{w}$, $\overline{v}$) is a upper solution when the initial value is in the domain R, where $\overline{u}$,$\overline{w}$, $\overline{v}$ is the solution of the following system, respectively,

$$\{\begin{array}{l}\frac{\partial \overline{u}(x,t)}{\partial t}=d_1\mathrm{\Delta }\overline{u}(x,t)+r\overline{u}(x,t)(1-\overline{u}(x,t)),x\in \mathrm{\Omega },t>0;\hfill \\ \frac{\partial \overline{u}(x,t)}{\partial \nu }=0,x\in \partial \mathrm{\Omega },t\ge 0;\hfill \\ \overline{u}(x,t)=u_0(x,t)\ge 0,x\in \overline{\mathrm{\Omega }},-\tau \le t\le 0,\hfill \end{array}$$

$$\{\begin{array}{l}\frac{\partial \overline{w}(x,t)}{\partial t}=d_1\mathrm{\Delta }\overline{w}(x,t)+\frac{abr}{c{r}_1}-\overline{w}(x,t),x\in \mathrm{\Omega },t>0;\hfill \\ \frac{\partial \overline{w}(x,t)}{\partial \nu }=0,x\in \partial \mathrm{\Omega },t\ge 0;\hfill \\ \overline{w}(x,t)=w_0(x,t)\ge 0,x\in \overline{\mathrm{\Omega }},-\tau \le t\le 0,\hfill \end{array}$$

$$\{\begin{array}{l}\frac{\partial \overline{v}(x,t)}{\partial t}=d_2\mathrm{\Delta }\overline{v}(x,t)+\frac{br}{r_1}-c\overline{v}(x,t),x\in \mathrm{\Omega },t>0;\hfill \\ \frac{\partial \overline{v}(x,t)}{\partial \nu }=0,x\in \partial \mathrm{\Omega },t\ge 0;\hfill \\ \overline{v}(x,t)=v_0(x,t)\ge 0,x\in \overline{\mathrm{\Omega }},-\tau \le t\le 0.\hfill \end{array}$$

Furthermore, G(φ) satisfies a Lipschitz condition, therefore the system (2.5) has a unique classical solution when initial value is in the domain R.

Linearizing system (3.1) at E₁, we have

$$\dot{U}(t)=D\mathrm{\Delta }U(t)+L_{E_1}\left(U_t\right),$$ (3.2)

where

$$L_{E_1}\left(U_t\right)=\left(\begin{array}{ccc}-r& -r& -a\\ 0& -1& 0\\ 0& b& -a-c\end{array}\right)\left(\begin{array}{l}{u}_1(t)\hfill \\ u_2(t)\hfill \\ u_3(t)\hfill \end{array}\right)+\left(\begin{array}{lll}0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & a\hfill \\ 0\hfill & 0\hfill & 0\hfill \end{array}\right)\left(\begin{array}{l}{u}_1(t-\tau )\hfill \\ u_2(t-\tau )\hfill \\ u_3(t-\tau )\hfill \end{array}\right).$$

From Wu [37], the corresponding integral equation of (3.1) is

$$U(t)=T(t)U(0)+{\int }_0^{t}T(t-s)L_{E_1}\left(U_s\right)\text{d}s,$$ (3.3)

where T(t) is a C₀ semigroup generated by DΔ. And its characteristic equation is given by

$$\lambda y-D\mathrm{\Delta }y-L_{E_1}\left(e^{\lambda \cdot }y\right)=0,$$ (3.4)

where y ∈ dom(Δ) \ {0}, dom(Δ) ⊂ X and e^λ·(θ)y = e^λθy, for θ ∈ [−τ, 0]. It is known that the eigenvalue problem

$$\{\begin{array}{ll}-\mathrm{\Delta }\varphi =\mu \varphi ,\hfill & x\in \mathrm{\Omega },\hfill \\ \frac{\partial \varphi }{\partial \mathbf{n}}=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$$

has eigenvalues μ_n, $n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$, with corresponding eigenfunctions φ_n(x), and μ_n satisfy 0 = μ₀ < μ₁ ≤ μ₂ ≤ ··· ≤ μ_n ≤ ···. By using the Fourier expansion in (3.4),

$$y=\sum _{n=0}^{\infty }\left(\begin{array}{l}{a}_n\hfill \\ b_n\hfill \\ c_n\hfill \end{array}\right){\varphi }_n(x),$$

where a_n, b_n, $c_n\in ℂ$, we know that the characteristic equation (3.4) is equivalent to

$$\det \left(\begin{array}{ccc}\lambda +d_1{\mu }_n+r& r& a\\ 0& \lambda +d_1{\mu }_n+1& -a{\text{e}}^{-\lambda \tau }\\ 0& -b& \lambda +d_2{\mu }_n+a+c\end{array}\right)=0,n\in {\mathbb{N}}_0.$$

That is, each eigenvalue λ is a root of the equation

$$\left(\lambda +d_1{\mu }_n+r\right)\left({\lambda }^2+T_n\lambda +D_n-ab{\text{e}}^{-\lambda \tau }\right)=0,n\in {\mathbb{N}}_0,$$ (3.5)

where T_n = (d₁ + d₂)μ_n + a + c + 1,$D_n=d_1{d}_2{\mu }_n^2+\left[d_1(a+c)+d_2\right]{\mu }_n+a+c$. By discussing the distribution of the roots of (3.5), we have the following Lemma.

Lemma 3.2.

E₁ is locally asymptotically stable when $b<1+\frac{c}{a}$. E₁ is unstable when $b>1+\frac{c}{a}$.

Proof.

Obviously, λ_1,n = −d₁μ_n − r < 0, $n\in {\mathbb{N}}_0$, are roots of (3.5). We only need to discuss the roots of

$${\lambda }^2+T_n\lambda +D_n-ab{\text{e}}^{-\lambda \tau }=0,n\in {\mathbb{N}}_0.$$ (3.6)

When τ = 0, noticed that $T_n^2-4\left(D_n-ab\right)={\left(d_1-d_2\right)}^2{\mu }_n^2+2(a+c+1)\left(d_1+d_2\right){\mu }_n+{(a+c-1)}^2+4ab>0$, we get equation (3.6) has two series real roots

$${\lambda }_{2,n}=-\frac{1}{2}{T}_n-\frac{1}{2}\sqrt{T_n^2-4\left(D_n-ab\right)},n\in {\mathbb{N}}_0,$$

$${\lambda }_{3,n}=-\frac{1}{2}{T}_n+\frac{1}{2}\sqrt{T_n^2-4\left(D_n-ab\right)},n\in {\mathbb{N}}_0.$$

Clearly, λ_2,n < 0 for $n\in {\mathbb{N}}_0$. And λ_3,n < 0 for all $n\in {\mathbb{N}}_0$ if and only if $b<1+\frac{c}{a}$. We claimed that all the roots have negative real parts when $b<1+\frac{c}{a}$ and τ > 0. Otherwise, we assume (3.6) have a pure imaginary root iν_n for some τ > 0, then we have

$$-{\nu }_n^2+\text{i}{T}_n{\nu }_n+D_n-ab\left(\cos {\nu }_n\tau -\text{i}\sin {\nu }_n\tau \right)=0.$$

Separating the real and imaginary parts gives

$$\{\begin{array}{l}-{\nu }_2+D_n=ab\cos {\nu }_n\tau ,\hfill \\ -T_n{\nu }_n=ab\sin {\nu }_n\tau .\hfill \end{array}$$

Furthermore, we need

$${\nu }_n^4+\left(T_n^2-2D_n\right){\nu }_n^2+D_n^2-a^2{b}^2=0.$$ (3.7)

Denote $Q_n(z)=z^2+\left(T_n^2-2D_n\right)z+D_n^2-a^2{b}^2$. Obviously,

$$T_n^2-2D_n=\left(d_1^2+d_2^2\right){\mu }_n^2+\left[2d_1+2(a+c)d_2\right]{\mu }_n+{(a+c)}^2+1>0.$$

If $b<1+\frac{c}{a}$, then

$$\begin{gathered} D_n^2-a^2{b}^2={\left\{d_1{d}_2{\mu }_n^2+\left[d_1(a+c)+d_2\right]{\mu }_n\right\}}^2 \\ +2(a+c)\left\{d_1{d}_2{\mu }_n^2+\left[d_1(a+c)+d_2\right]{\mu }_n\right\}+{\left(a+c\right)}^2-a^2{b}^2>0. \end{gathered}$$

So Q_n(z) has no positive roots when $b<1+\frac{c}{a}$. We know that (3.7) is always not valid when $b<1+\frac{c}{a}$. This is a contradiction. □

Remark 3.3.

The spatially uniform steady state E₀ of the system (2.5) is always unstable because there has an eigenvalue r > 0.

Theorem 3.4.

If $b<b_0=1+\frac{c}{a}$, then the spatially uniform steady state E₁ of the system (2.5) is globally asymptotically stable for τ ≥ 0.

Proof.

Define a Lyapunov functional on the region R by

$$V\left({\varphi }_t\right)(x)={\int }_{\mathrm{\Omega }}\left(b{\phi }_t^2(x,0)+{\phi }_t^3(x,0)+ba{\int }_{-\tau }^0{\phi }_t^1(x,\theta ){\phi }_t^3(x,\theta )\text{d}\theta \right)\text{d}x$$

where ${\varphi }_t(\theta )=\left({\phi }_t^1(\theta ),{\phi }_t^2(\theta ),{\phi }_t^3(\theta )\right)$, ${\phi }_t^i(\theta )(x)={\phi }^i(x,t+\theta )$, i = 1, 2, 3. When b < b₀, by virtue of the Neumann boundary condition, we obtain that

$$\begin{gathered} \dot{V}\left({\varphi }_t\right)={\int }_{\mathrm{\Omega }}\left[a(b-1){\phi }_t^1(x,0)-c\right]{\phi }_t^3(x,0)\text{d}x \\ \le {\int }_{\mathrm{\Omega }}\left[a(b-1)-c\right]{\phi }_t^3(x,0)\text{d}x\le 0. \end{gathered}$$

And $\dot{V}\left({\varphi }_t\right)=0$ if and only if ${\phi }_t^3(x,0)=0$. Furthermore, we can get the maximum invariant set in {φ ∈ R : $\dot{V}\left({\varphi }_t\right)=0$} is the single point set {E₁}. The Lasalles invariance principle implies that E₁ is globally attractive. By Lemma 3.2, we get E₁ is global asymptotically stable for τ ≥ 0. □

3.2. Stability of the positive steady state E^∗ and Hopf bifurcations

If $b>b_0=1+\frac{c}{a}$, the system (2.5) has a spatially uniform positive steady state E∗. In this subsection, we always assume that $b>b_0=1+\frac{c}{a}$ holds.

Linearizing the system (2.5) at E∗, we get

$$\dot{U}(t)=D\mathrm{\Delta }U(t)+L_{E^{*}}\left(U_t\right),$$

where

$$L_{E^{*}}=L_1\varphi (0)+L_2\varphi (-\tau )$$

with

$$L_1=\left(\begin{array}{ccc}-ru*& -ru*& -au*\\ 0& -1& 0\\ -av*& b& -au*-c\end{array}\right)\text{and}{L}_2=\left(\begin{array}{ccc}0& 0& 0\\ av*& 0& au*\\ 0& 0& 0\end{array}\right),$$

for φ(θ) = (φ₁(θ), φ₂(θ), φ₃(θ))^T. Similar to the previous subsection, we find that the characteristic equation is equivalent to

$${\lambda }^3+a_{2,n}{\lambda }^2+a_{1,n}\lambda +a_{0,n}+\left(b_{1,n}\lambda +b_{0,n}\right){\text{e}}^{-\lambda \tau }=0,n\in {\mathbb{N}}_0,$$ (3.8)

where

$$a_{2,n}=\left(2d_1+d_2\right){\mu }_n+\frac{rc+abc+ab-a}{a(b-1)},$$

$$a_{1,n}=\left(d_1^2+2d_1{d}_2\right){\mu }_n^2+\left[\frac{rc+2abc+a^{2}b-a^2}{a(b-1)}{d}_1+\frac{rc+ab-a}{a(b-1)d_2}\right]{\mu }_n+\frac{rc(bc+b-1)}{a{(b-1)}^2}+\frac{bc}{b-1}-\frac{rc(ab-a-c)}{(b-1)(rc+ab-a)},$$

$$\begin{gathered} a_{0,n}=d_1^2{d}_2{\mu }_n^3+\left[\frac{bc}{b-1}{d}_1^2+\frac{rc+ab-a}{a(b-1)}{d}_1{d}_2\right]{\mu }_n^2 \\ +\left[\left(\frac{bc(rc+ab-a)}{a{(b-1)}^2}-\frac{rc(ab-a-c)}{a(b-1)(rc+ab-a)}\right)d_1+\frac{rc}{a(b-1)}{d}_2\right]{\mu }_n+\frac{rb{c}^2}{a{(b-1)}^2}-\frac{rc(ab-a-c)}{(b-1)(rc+ab-a)}, \end{gathered}$$

$$b_{1,n}=\frac{r^{2}c(ab-a-c)}{(b-1)(rc+ab-a)}-\frac{bc}{b-1},$$

$$b_{0,n}=\left[\frac{r^{2}c(ab-a-c)}{a(b-1)(rc+ab-a)}{d}_2-\frac{bc}{b-1}{d}_1\right]{\mu }_n-\frac{rb{c}^2}{a{(b-1)}^2}+\frac{rc(ab-a-c)(ab+rc)}{a(b-1)(rc+ab-a)}.$$

When τ = 0, the characteristic equation (3.8) becomes the following sequence of equations

$${\lambda }^3+a_{2,n}{\lambda }^2+\left(a_{1,n}+b_{1,n}\right)\lambda +a_{0,n}+b_{0,n}=0,n\in {\mathbb{N}}_0.$$ (3.9)

By the Routh-Hurwitz Criterion, all roots of (3.9) have negative real parts if and only if

$$H_{1,n}=a_{2,n}>0,H_{2,n}=a_{2,n}\left(a_{1,n}+b_{1,n}\right)-a_{0,n}-b_{0,n}>0,H_{3,n}=\left(a_{0,n}+b_{0,n}\right)H_{2,n}>0$$

for $n\in {\mathbb{N}}_0$. When b > b₀, we can see that a_2,n > 0 and a_0,n + b_0,n > 0 for $n\in {\mathbb{N}}_0$. So when

$$(\mathcal{H})b>b_0,a_{2,n}\left(a_{1,n}+b_{1,n}\right)-a_{0,n}-b_{0,n}>0,$$

all roots of (3.9) have negative real parts. We get the following conclusion.

Lemma 3.5.

If $(\mathcal{H})$ holds, the spatially uniform positive steady state E∗ of the system (2.5) is locally asymptotically stable when τ = 0.

Now we look for sufficient conditions to ensure H_2,n > 0. By calculation, we have

$$H_{2,n}=2d_1{\left(d_1+d_2\right)}^2{\mu }_n^3+A_3{\mu }_n^2+\left(A_1{d}_1+A_2{d}_2\right){\mu }_n+H_{2,0}$$

where

$$A_1=\frac{(rc+abc+ab-a)\left(a^{2}b-a^2+2abc+rc\right)}{a^2{(b-1)}^2}+\frac{rc\left[2c(rc+ab)+(b-1)\left(3a+2ra-a^2\right)+c\right]}{a(b-1)(rc+ab-a)}-\frac{rc(b-2c)}{a{(b-1)}^2},$$

$$A_2=\frac{(rc+abc+ab-a)(rc+ab-a)}{a^2{(b-1)}^2}+\frac{rb{c}^2}{a{(b-1)}^2}-\frac{rc(ab-a-c)}{a(b-1)(rc+ab-a)},$$

$$A_3=\left[\frac{4abc+3rc}{a(b-1)}+2a+1\right]d_1^2+\frac{rc+ab-a}{a(b-1)}{d}_2^2+\left[\frac{4rc+2ac+2abc}{a(b-1)}+2c+a+3\right]d_1{d}_2,$$

$$H_{2,0}=\frac{rc(rc+abc+ab-a)}{a^2{(b-1)}^2}\left[\frac{rc(bc+b-1)}{(b-1)}+\frac{(ab-a-c)(r-a)}{rc+ab-a}\right]-\frac{rc(ab-a-c)}{a(b-1)}.$$

If b > b₀, then A₃ > 0. We define a function ψ(x) by

$$\begin{gathered} \psi (x)=-a^3{x}^4+\left[c^3+3c+r+1-(c-1)a^3\right]x^3 \\ +\left[\left(c^2+3c+r+1\right)rca-3(2c+r+1)a^2-(3-2c)a^3\right]x^2 \\ +\left[r^2{c}^3-(3c+2r+2)rca-\left(c^2-3c-3r-3\right)a^2-(c-1)a^3\right]x \\ +rc(r+1)a-(r+1)a^2. \end{gathered}$$

It is easy to see that

$$H_{2,0}=\frac{rc\psi (b)}{a{(b-1)}^3(rc+ab-a)}.$$

When b > b₀, sign{H_2,0} = sign{ψ(b)}. Since

$$\psi \left(b_0\right)=c^3(r+1)\left[r+a+2c+2+\frac{(c+1)(r+c+1)}{a}\right]>0$$

and ψ(x) → −∞ as x → ±∞, so ψ(x) has at least one root and at most three root greater than b₀. According to the distribution of the roots of ψ(x), we have the following two cases:

Case I: ψ(b) has only root b₁ > b₀ or has three roots b₁, b₂, b₃ which are greater than b₀. When it has three roots which greater than b₀, the three roots are satisfied b₁ < b₂ = b₃ or b₁ = b₂ = b₃.

Case II: ψ(b) has three roots b₁, b₂, b₃ that are greater than b₀, and the three roots are satisfied b₁ ≤ b₂ < b₃.

When Case I occurs, if b ∈ (b₀, b₁), then H_2,0 > 0. When Case II occurs, if $b\in \left(b_0,b_1\right)\cup \left(b_2,b_3\right)$, then H_2,0 > 0.

When b > b₀, we can see that A₃ > 0. So if b > b₀, H_2,0 > 0 and A₁d₁ +A₂d₂ ≥ 0, then H_2,n > 0 for $n\in {\mathbb{N}}_0$.

When A₂ > 0, if the diffusion coefficient d₂ is sufficiently large relative to the diffusion coefficient d₁, then A₁d₁ + A₂d₂ ≥ 0 can be established.

It is easy to see that if $\frac{rb{c}^2}{a{(b-1)}^2}-\frac{rc(ab-a-c)}{a(b-1)(rc+ab-a)}>0$, then A₂ > 0.$\frac{rb{c}^2}{a{(b-1)}^2}-\frac{rc(ab-a-c)}{a(b-1)(rc+ab-a)}>0$, is equivalent to

$$a(c-a)b^2+\left(r{c}^2+2a^2\right)b-a(c+a)>0.$$ (3.10)

When c = a, the (3.10) always holds since b > b₀. When c ≠ a, the equation a(c − a)b² + (rc² + 2a²)b − a(c + a) = 0 has two real roots

$$b_{4,5}=\frac{-\left(rc+2a^2\right)\pm \sqrt{r^2{c}^2+4r{a}^2+4a^2}}{2a(c-a)}.$$

If c > a, b₅ < 0 < b₄ and the (3.10) holds when b > b₄ or b < b₅. If c < a, 0 < b₄ < b₅ and the (3.10) holds when b₄ < b < b₅.

Through calculation, we get $A_2=\frac{a_3{y}^3+a_2{y}^2+a_{1}y+a_0}{a^2{y}^2(ay+rc)}$, where

$$a_0=(r+2a)c^3{r}^2>0,$$

$$a_1=(3r+2cr+4a)a{c}^{2}r>0,$$

$$a_2=[(1-r)a+3(c+1)r]a^{2}c,$$

$$a_3=(c+1)a^3>0,$$

$$y=b-1.$$

If r ≤ 1 or a ≤ 3(c + 1), then a₂ > 0. When r > 1 and a > 3(c + 1), a₂ > 0 if and only if $r<\frac{a}{a-3(c+1)}$. Obviously, if a₂ > 0, then A₂ > 0 for b > b₀.

From the discussion above and Lemma 3.5, we reach the following conclusions.

Corollary 3.6.

In Case I, if one of the following conditions holds

1. c > a, max{b₀, b₄} < b < b₁,

2. c < a, max{b₀, b₄} < b < min{b₁, b₅},

3. c = a, b₀ < b < b₁,

and $\frac{d_2}{d_1}>-\frac{A_1}{A_2}$, then the spatially uniform positive steady state E∗ of the system (2.5) is locally asymptotically stable when τ = 0.

Corollary 3.7.

In CaseII, if one of the following conditions holds

1. c > a, max{b₀, b₄} < b < b₁ or max{b₂, b₄} < b < b₃,

2. c < a, max{b₀, b₄} < b < min{b₁, b₅} or max{b₂, b₄} < b < min{b₃, b₅},

3. c = a, b₀ < b < b₁ or b₂ < b < b₃,

and $\frac{d_2}{d_1}>-\frac{A_1}{A_2}$, then the spatially uniform positive steady state E∗ of system (2.5) is locally asymptotically stable when τ = 0.

Corollary 3.8.

Suppose b > b ₀ . If one of the following conditions holds

1. r ≤ 1,

2. a ≤ 3(c + 1),

3. a > 3(c + 1) and $1<r<\frac{a}{a-3(c+1)}$

and $\frac{d_2}{d_1}>-\frac{A_1}{A_2}$, then the spatially uniform positive steady state E∗ of system (2.5) is locally asymptotically stable when τ = 0.

Now, we assume the condition $(\mathcal{H})$ holds. we would like to seek critical value τ such that there exists a pair of purely imaginary eigenvalue. Let iω(ω > 0) be the solution of the (n + 1)th equation of (3.8), we have

$$-\text{i}{\omega }^3-a_{2,n}{\omega }^2+\text{i}{a}_{1,n}\omega +a_{0,n}+\left(\text{i}{b}_{1,n}\omega +b_{0,n}\right)(\cos \omega \tau -\text{i}\sin \omega \tau )=0.$$

Separating the real and imaginary parts gives

$$\{\begin{array}{l}-{\omega }^3+a_{1,n}\omega =b_{0,n}\sin \omega \tau -b_{1,n}\omega \cos \omega \tau ,\hfill \\ a_{2,n}{\omega }^2-a_{0,n}=b_{0,n}\cos \omega \tau +b_{1,n}\omega \sin \omega \tau .\hfill \end{array}$$ (3.11)

Square each sides of (3.11) and plus, we have

$${\omega }^6+f_n{\omega }^4+g_n{\omega }^2+h_n=0$$ (3.12)

where $f_n=a_{2,n}^2-2a_{1,n}$, $g_n=a_{1,n}^2-2a_{2,n}{a}_{0,n}-b_{1,n}^2$,$h_n=a_{0,n}^2-b_{0,n}^2$.

Let z = ω², then (3.12) becomes

$$P_n(z)=z^3+f_n{z}^2+g_{n}z+h_n=0.$$ (3.13)

Lemma 3.9.

For the equation (3.13), we have the following conclusions.

1. If h_n < 0, the equation (3.13) has at least one positive roots.

2. If h_n ≥ 0 and $f_n^2-3g_n\le 0$, the equation (3.13) has no positive root.

3. If h_n ≥ 0 and $f_n^2-3g_n>0$, the equation (3.13) has positive roots if and only if $z_{1,n}^{*}>0$ and $P\left(z_{1,n}^{*}\right)\le 0$, where $z_{1,n}^{*}=\frac{-f_n+\sqrt{f_n^2-3g_n}}{3}$.

Proof.

Since P_n(z) → +∞ as z → +∞, we conclude that P_n(z) has at least one positive root if P_n(0) = h_n < 0. Differentiating P_n(z) with respect to z, we have $P_n^{\prime }(z)=3z^2+2f_{n}z+g_n$. If $f_n^2-3g_n\le 0$, then $P_n^{\prime }(z)\ge 0$ and P_n(z) is nondecreasing when z ≥ 0. Thus for h_n ≥ 0 and $f_n^2-3g_n\le 0$, the equation (3.13) has no positive root. When h_n ≥ 0 and $f_n^2-3g_n>0$, $P_n^{\prime }(z)$ has two roots $z_{1,n}^{*}=\frac{-f_n+\sqrt{f_n^2-3g_n}}{3}$ and $z_{2,n}^{*}=\frac{-f_n-\sqrt{f_n^2-3g_n}}{3}$. Clearly,

$$p_n^{\prime \prime }\left(z_{1,n}^{*}\right)=2\sqrt{f_n^2-3g_n}>0,p_n^{\prime \prime }\left(z_{2,n}^{*}\right)=-2\sqrt{f_n^2-3g_n}<0.$$

So $z_{1,n}^{*}$ is the locally minimum point of P_n(z) and $z_{2,n}^{*}$ is the locally maximum point of P_n(z). Noticing that h_n ≥ 0, $z_{1,n}^{*}$ is the local minimum of P_n(z) and P_n(z) → +∞ as z → +∞, we know that the sufficiency of (iii) is true. In what follows, we need to prove the necessity of (iii). Suppose that either $z_{1,n}^{*}\le 0$ or $z_{1,n}^{*}>0$ and $P_n\left(z_{1,n}^{*}\right)>0$. Since P_n(z) is increasing for z ≥ 0 and P_n(0) = h_n ≥ 0, we know that P_n(z) has no positive real roots for $z_{1,n}^{*}\le 0$. If $z_{1,n}^{*}>0$ and $P_n\left(z_{1,n}^{*}\right)>0$, since $h\left(x_1^{*}\right)$ is the local minimum value, it follows that P_n(0) = h_n > 0 that when $z_{1,n}^{*}$ and $P_n\left(z_{1,n}^{*}\right)>0$, P_n(z) has no positive real roots, too. This proves the lemma. □

Corollary 3.10.

If $b_0<b<b_6=1+\frac{3c}{2a}+\frac{\sqrt{9c^2+8ac}}{2a}$, then h₀ < 0.

Proof.

When b > b₀, we have a_0,0 + b_0,0 > 0. So $h_0=a_{0,0}^2-b_{0,0}^2=\left(a_{0,0}+b_{0,0}\right)(a_{0,0}-b_{0,0})<0$ if a_0,0 − b_0,0 < 0 and b > b₀.

$$\begin{gathered} a_{0,0}-b_{0,0}=\frac{2rb{c}^2}{a{(b-1)}^2}-\frac{rc(ab-a-c)(ab+rc+a)}{(b-1)(rc+ab-a)} \\ =\frac{rc}{a(b-1)}\left(\frac{2bc}{b-1}-(ab-a-c)-\frac{2a(ab-a-c)}{rc+ab-a}\right) \end{gathered}$$

If b > b₀ and $\frac{2bc}{b-1}-(ab-a-c)<0$, then a_0,0 − b_0,0 < 0. When b > b₀,

$$\frac{2bc}{b-1}-(ab-a-c)<0$$

$$\iff a{b}^2-(2a+3c)b+a+c>0$$

$$\iff 1+\frac{3c}{2a}-\frac{\sqrt{9c^2+8ac}}{2a}<b<1+\frac{3c}{2a}+\frac{\sqrt{9c^2+8ac}}{2a}.$$

Because of $1+\frac{3c}{2a}-\frac{\sqrt{9c^2+8ac}}{2a}<b_0<1+\frac{3c}{2a}+\frac{\sqrt{9c^2+8ac}}{2a}$, we get h < 0 if b₀ < b < b₆. □

Denote

$$\mathcal{D}=\left\{n\in {\mathbb{N}}_0:h_n<0;\text{or}{h}_n\ge 0,f_n^2-3g_n>0,z_{1,n}^{*}0,\text{and}{P}_n\left(z_{1,n}^{*}\right)\le 0\right\}.$$

From Lemma 3.9, we know that the equation (3.13) has positive roots if and only if $n\in \mathcal{D}$. Without loss of generality, we assume the equation (3.13) has three positive roots z_1,n, z_1,n and z_3,n when $n\in \mathcal{D}$. We denote ${\omega }_{k,n}=\sqrt{z_{k,n}}$ and

$${\tau }_{k,n}^{(j)}=\{\begin{array}{l}\frac{1}{{\omega }_{k,n}}\left[\arccos \frac{b_{0,n}\left(a_{2,n}{\omega }_{k,n}^2-a_{0,n}\right)+b_{1,n}{\omega }_{k,n}\left({\omega }_{k,n}^2-a_{1,n}{\omega }_{k,n}\right)}{b_{1,n}^2\omega +k,n^2+b_{0,n}^2}+2j\pi \right],\sin {\omega }_{k,n}{\tau }_{k,n}^{(0)}\ge 0,\hfill \\ \frac{1}{{\omega }_{k,n}}\left[-\arccos \frac{b_{0,n}\left(a_{2,n}{\omega }_{k,n}^2-a_{0,n}\right)+b_{1,n}{\omega }_{k,n}\left({\omega }_{k,n}^2-a_{1,n}{\omega }_{k,n}\right)}{b_{1,n}^2\omega +k,n^2+b_{0,n}^2}+2(j+1)\pi \right],\sin {\omega }_{k,n}{\tau }_{k,n}^{(0)}<0,\hfill \end{array}$$

for k = 1, 2, 3, $j\in {\mathbb{N}}_0$ and $n\in \mathcal{D}$. When $\tau ={\tau }_{k,n}^{(j)}$, the equation (3.8) has a pair of purely imaginary roots ±iω_k,n.

For some $n\in \mathcal{D}$, let λ(τ) = α_n(τ) + iω be a root of the (n + 1)th equation of (3.8) which satisfies ${\omega }_n\left({\tau }_{k,n}^{(j)}\right)={\omega }_{k,n}$ and ${\alpha }_n\left({\tau }_{k,n}^{(j)}\right)=0$ for k = 1, 2, 3 and $j\in {\mathbb{N}}_0$.

Lemma 3.11.

$$sign\left\{{\alpha }_n^{\prime }\left({\tau }_{k,n}^{(j)}\right)\right\}=sign\left\{P_n^{\prime }\left(z_{k,n}\right)\right\}$$

Proof.

Differentiating the both side of the equation (3.8) with respect to τ, it follows that

$${\left(\frac{\text{d}\lambda }{\text{d}\tau }\right)}^{-1}=\frac{\left(3{\lambda }^2+2a_{2,n}\lambda +a_{1,n}\right){\text{e}}^{\lambda \tau }}{\lambda \left(b_{1,n}\lambda +b_{0,n}\right)}+\frac{b_{1,n}}{\lambda \left(b_{1,0}\lambda +b_{0,n}\right)}-\frac{\tau }{\lambda }.$$

From (3.11), we have

$${\alpha }_n^{\prime }{\left({\tau }_{k,n}^{(j)}\right)}^{-1}={\mathrm{Re}{\left(\frac{\text{d}\lambda }{\text{d}\tau }\right)}^{-1}|}_{\tau ={\tau }_{k,n}^{(j)}}=\frac{P_n^{\prime }\left(z_{k,n}\right)}{b_{1,n}^2{\omega }_{k,n}^2+b_{0,n}^2}.$$

This proves the lemma. □

Let

$${\tau }^{*}={\tau }_{k_0,n_0}^{(0)}=\underset{k=1,2,3,n\in \mathcal{D}}{\min }\left\{{\tau }_{k,n}^{(0)}\right\}.$$

From Lemma 3.5, 3.9 and 3.11, we have the following conclusion.

Theorem 3.12.

Suppose the condition $(\mathcal{H})$ holds.

1. If h ≥ 0, $f_n^2-3g_n\le 0$, or $f_n^2-3g_n>0$, $z_{1,n}^{*}<0$, or $f_n^2-3g_n>0$, $z_{1,n}^{*}>0$, $P_n\left(z_{1,n}^{*}\right)>0$ for all $n\in {\mathbb{N}}_0$, the spatially uniform steady state E∗ of the system (2.5) is locally asymptotically stable for all τ > 0.

2. If h_n < 0, or h_n ≥ 0, $f_n^2-3g_n>0$, $z_{1,n}^{*}>0$ and $P_n\left(z_{1,n}^{*}\right)\le 0$ for some $n\in {\mathbb{N}}_0$, the spatially uniform steady state E∗ of the system (2.5) is locally asymptotically stable for τ ∈ [0, τ∗).

3. If all conditions as stated in (ii) and $P_n^{\prime }\left(z_{k,n}\right)\ne 0$ hold, the system (2.5) undergoes a Hopf bifurcation at E∗ when $\tau ={\tau }_{k,n}^{(j)}$, k = 1, 2, 3,$j\in {\mathbb{N}}_0$.

4. Numerical simulations

For the convenience of presentation, our numerical simulations are performed for the tumor site as a domain in ${ℝ}^2$. We describe briefly the numerical method we use to solve the system (2.5) at the end of the appendix. In the following, we demonstrate several typical cases which correspond to our analysis.

Let Ω = (0, π) × (0, π) (i.e. p = q = 1), and choose

$$r=0.1,a=0.01,c=0.08,d_1=0.02,d_2=\mathrm{0.2.}$$

For this set of parameter values we can get $b_0=1+\frac{c}{a}=9$, b₁ ≈ 135.5540. And when 9 < b < 135.5540, (H) holds. So when τ = 0, the positive steady state E∗ is stable if 9 < b < 135.5540. From our calculation, we find only for b ≥ 24.4105 and n = 0, (3.12) has a positive root, and we get a bifurcation curve about τ∗ and b (see Figure 1).

Figure 1:

A bifurcation curve τ∗(b).

If we choose

$$r=0.1,a=0.01,b=4,c=0.08,d_1=0.02,d_2=0.2,\tau =2.$$

For this set of parameter values we observe that $b<1+\frac{c}{a}$. So from Theorem 3.4, the steady stete E₁ = (1, 0, 0) is globally asymptotically stable (see Figure 2). The initial conditions of Figure 3 are u(x, t) = 0.5000 + 0.3000 cos 2x cos y, w(x, t) = 0.0500 + 0.0300 cos 2x cos y, v(x, t) = 8.0000 + 2.0000 cos 2x cos y, (x, y, t) ∈ [0, π] × [0, π] × [−τ, 0]. In Figure 2, we take x = π/3 and y = π/3 respectively.

Figure 2:

The steady stete E₁ = (1, 0, 0) is globally asymptotically stable.

Figure 3:

The positive equilibrium is asymptotically stable when τ ∈ [0, τ∗), where τ = 0.9 < τ∗ ≈ 1.8518.

If we chose b = 60, then by calculation, we have

$${\omega }_{1,0}=\approx 0.0743,{\tau }_{1,0}^{(j)}+\approx 1.8518+84.5804j,\text{for}j\in {\mathbb{N}}_0.$$

So $\tau *={\tau }_{1,0}^{(0)}\approx 1.8518$. Furthermore, from Appendix A, we have c₁(0) ≈ ≈ −0.5369 − 2.2209i,$P^{\prime }\left({\omega }_{1,0}^2\right)=0.0118>0$. From Theorem 3.12 and Appendix A, we know that positive equilibrium E∗ = (0.1356, 0.0116, 8.5284) is locally asymptotically stable when τ ∈ [0, τ∗) (see Figure 3), and the direction of the Hopf bifurcation is supercritical when τ = τ∗, that is the bifurcating periodic solutions exist for τ > τ∗, and they are orbitally stable(see Figure 4). The initial conditions of Figure 3 are u(x, t) = 0.1356 + 0.1000 cos 2x cos y, w(x, t) = 0.0116 + 0.0100 cos 2x cos y, v(x, t) = 8.5284 + 3.0000 cos 2x cos y, (x, y, t) ∈ [0, π]×[0, π]×[−τ, 0] and the initial conditions of Figure 4 are u(x, t) = 0.2000 + 0.1000 cos 2x cos y, w(x, t) = 0.0200 + 0.0100 cos 2x cos y, v(x, t) = 8.5000 + 3.0000 cos 2x cos y, (x, y, t) ∈ [0, π] × [0, π] × [−τ, 0]. In Figure 3 and Figure 3, we take x = π/3 and y = π/3 respectively.

Figure 4:

The bifurcating periodic solution is stable, where τ = 2 > τ∗ ≈ 1.8518.

5. Conclusion and discussion

Incorporating virus diffusivity, tumor cell diffusion, and the viral lytic cycle into a basic oncolytic virus dynamics model, we formulate a mathematical model in terms of delay partial differential equations for oncolytic virotherapy. We conduct a detailed study of the dynamics of our model. We find the positive invariant domain for the omega limit set and three spatially uniform equilibrium solutions of our model. Comparing with the basic models without diffusions of virus and tumor cells and the delay of the viral lytic cycle, we find a first critical value of the viral burst size, and when the viral burst size is smaller than this first critical value, the spatially uniform virus-free steady state is globally asymptotically stable for any viral lytic period delay and any diffusion coefficients of tumor cells and viruses. When the viral burst size is greater than the first critical value, we characterize the stability of the spatially uniform positive steady state by three major parameters: the ratio of virus diffusion coefficient to that of tumor cells, the length of the viral lytic cycle, and the viral burst size with six different values that are greater than the first critical value. These results are summarized in Lemma 3.5 with Corollaries 3.6, 3.7, and 3.8; lemma 3.9 with Corollary 3.10, Lemma 3.11; and Theorem 3.12. These detailed information cannot be obtained from the models without diffusions of virus and tumor cells and delays of viral lytic cycle. For instant, if we don’t consider the viral lytic cycle meaning the viral lytic cycle length is zero, in order to have a locally asymptotical stability of the spatially uniform positive steady state, the ratio between diffusion coefficients must be greater than a particular value and the viral burst size must be greater than the first critical value but smaller than other values depending on the tumor growth rate and virus clearance rate. If we consider the viral lytic cycle as a delay parameter, we find a critical value for the length of the viral lytic cycle. Combining conditions for the ratio between diffusion coefficients and viral burst size, the spatially uniform positive steady state is locally asymptotical stable when the viral lytic cycle length is smaller than the critical value while the system undergoes Hopf bifurcation at the critical value of the viral lytic cycle length and has stable periodic solutions when the viral lytic cycle length is greater than the critical value. For these results, we give some biological implications as follows.

From our analysis, we know that the parameters including the viral burst size, the viral lytic cycle, the diffusion coefficients of viruses and cells play important roles in the dynamics of oncolytic viral therapy. Our model has three spatially homogeneous equilibrium solutions, E₀, E₁, and E∗ which are related to long-term behaviors of general solutions. They may have some medical implications. It is reasonable that the zero solution E₀ is always unstable for any parameter values since the tumor always grows in its beginning. Theorem 3.4 gives a critical value for the viral burst size, and below that value, the viral therapy fails no matter what values of any other parameters are chosen. This confirms that the potency of oncolytic viruses is important, and the viral burst size greater than that critical value is necessary condition for the treatment to work. It is hoped oncolytic viruses will eradicate the tumor completely. However, this is not always the case. When viruses cannot completely eradicate the tumor, we may hope the virus treatment will be stabilized at a level where the tumor cells, infected tumor cells, and viruses are constant. We find some conditions under which the viral therapy can be stabilized. Under the condition $(\mathcal{H})$ which includes the viral burst size is greater than the critical value and the ratio of virus diffusion coefficient to cell diffusion coefficient is greater than certain values in Corollary 3.6–3.8, and the condition states in Theorem 3.12, the spatially uniform equilibrium solution E∗ which is constant solution is stable. We also obtain a critical value of the viral lytic cycle below that value E∗ is stable while at that value point, the system undergoes a Hopf bifurcation. Spatially homogeneous periodic solutions arise from Hopf bifurcations. We numerically show these periodic solutions are stable. We may consider the stable spatially homogeneous periodic solution as one type of treatment stabilization. In addition, we find a functional relationship between the viral burst size and lytic cycle shown in Figure 1 which may implicate that the viral lytic cycle length will change if the viral burst size changes. Since we can genetically change viral genome for different burst sizes, we accordingly change their lytic cycles. As long as the condition in Theorem 3.12 is satisfied, the stabilization of treatments can be achieved.

As mentioned in Introduction, there have been many studies on oncolytic virotherapy experimentally and theoretically. Recently, Wodarz et al [29] conducted an in vitro experiment with recombinant adenovirus type-5 in a two-dimensional setting and used an agent-based stochastic computational model to simulate experimental observations. They observed three spatial patterns, “hollow ring structure”, “filled ring structure”, and “disperse pattern”. They found that both the filled ring structure and disperse pattern of initial expansion were indicative of treatment failure. The hollow ring structure was associated with either target cell extinction or low-level persistence, both of which can be viewed as treatment success. It seems that the filled ring structure and disperse pattern correspond to the spatial positive steady state in our model, and the hollow ring structure corresponds the spatial virus-free steady state in our model. However, our study gives more detailed information about the spatial positive equilibriums and possibilities of stable periodic behaviors of the spatial patterns. They also found that equilibrium properties of ordinary differential equations describing the dynamics in local neighborhoods in the agent-based model can predict the outcome of the spatial virus-cell dynamics. This finding agrees with our results since our model is based on local interactions. A more recent study by Wang et al [30] incorporated nonlocal reaction terms in their delay reaction-diffusion system for oncolytic virotherapy. They gave different treatment strategies according to different gene mutations, which has theoretical interests. Friedman and Lai [31] presented a computational study about combination therapy for cancer with oncolytic virus and checkpoint inhibitor. This study did not focus on spatial patterns of oncolytic virus spreading within tumors. Timalsina et al [32] proposed a detailed reaction-diffusion model for virotherapy and conducted a computational study. This model focused on the different functions of innate and adaptive immune systems in oncolytic viral therapies.

It is clear that our model incorporates basic components of oncolytic viral therapy although it reveals important features of oncolytic viral therapy. There are several components which are not included, for example, immune cells. The immune system has two different functions in oncolytic viral therapies. The immune cells can be incorporated to our model. However, it is beyond the scope of our current work. This may be considered in our future study.

A. Stability and direction of the Hopf bifurcations

Theory of functional reaction-diffusion systems says that a family of spatially homogeneous or inhomogeneous periodic solutions may bifurcate from the positive homogeneous equilibrium state E∗ of the system (2.5) when τ crosses through the critical value τ∗. In Appendix, we investigate the stability and direction of Hopf bifurcations by using the center manifold theorem and the normal formal theory of partial functional differential equation [34, 37]. Basically, the system (2.5) firstly is represented as an abstract ODE system. Secondly, at the center manifold of the ODE system corresponding to E∗, the normal form or Taylor expansion of the ODE system is computed. Then, the coefficients of the first 4 terms of the normal form will reveal all the properties of the periodical solutions [35]. At the end, we briefly describe the numerical method we use to solve our system.

Let u₁(·, t) = u(·, τt) − u∗, u₂(·, t) = w(·, τt) − w∗, u₃(·, t) = v(·, τt) − v∗ and U(t) = (u₁(·, t), u₂(·, t), u₃(·, t))^T. Then the system (2.5) can be written as an equation in the function space $\mathcal{C}=C([-1,0],X)$ :

$$\frac{\text{d}U(t)}{\text{d}t}=\tau D\mathrm{\Delta }U(t)+L(\tau )\left(U_t\right)+f\left(U_t,\tau \right),$$ (A.1)

where D = diag(d₁, d₁, d₂), $L(\tau )(\cdot ):\mathcal{C}\to X$ and $f:\mathcal{C}\times ℝ\to X$ are given, respectively, by

$$L(\tau )(\phi )=\tau L_1\phi (0)+\tau L_2\phi (-1),$$

$$f(\phi ,\tau )=\tau {\left(f_1(\phi ,\tau ),f_2(\phi ,\tau ),f_3(\phi ,\tau )\right)}^T,$$

with

$$f_1(\phi ,\tau )=-r{\phi }_1^2(0)-r{\phi }_1(0){\phi }_2(0)-a{\phi }_1(0){\phi }_3(0),$$

$$f_2(\phi ,\tau )=a{\phi }_1(-\tau ){\phi }_3(-\tau ),$$

$$f_3(\phi ,\tau )=-a{\phi }_1(0){\phi }_3(0),$$

for $\phi ={\left({\phi }_1,{\phi }_2,{\phi }_3\right)}^{\text{T}}\in \mathcal{C}$. Let τ = τ∗ + σ, then (A.1) can be rewritten as

$$\frac{\text{d}U(t)}{\text{d}t}=\tau *D\mathrm{\Delta }U(t)+L\left(\tau *\right)\left(U_t\right)+F\left(U_t,\sigma \right),$$ (A.2)

where

$$F(\phi ,\sigma )=\sigma D\mathrm{\Delta }\phi (0)+L(\sigma )(\phi )+f\left(\phi ,\tau *+\sigma \right),$$

for $\phi \in \mathcal{C}$.

From the previous subsection, when σ = 0 (i.e. τ = τ∗) the system (A.2) undergoes Hopf bifurcation at the equilibrium (0, 0, 0). It is also clear that ±iω∗τ∗ are simply purely imaginary eigenvalues of the linearized system of (A.2) at the origin

$$\frac{\text{d}U(t)}{\text{d}t}=\left(\tau *+\sigma \right)D\mathrm{\Delta }U(t)+L\left(\tau *+\sigma \right)\left(U_t\right),$$ (A.3)

with σ = 0 and all other eigenvalues of (A.3) at σ = 0 have negative real parts.

The eigenvalues of τDΔ on X are −τd₁μ_n (the number of multiples is two) and −τd₂μ_n, $n\in {\mathbb{N}}_0$, with corresponding eigenfunctions ${\beta }_n^1(x)={\left({\gamma }_n(x),0,0\right)}^T$, ${\beta }_n^2(x)={\left(0,{\gamma }_n(x),0\right)}^T$ and ${\beta }_n^3(x)={\left(0,0,{\gamma }_n(x)\right)}^T$, where ${\gamma }_n(x)=\frac{{\varphi }_n(x)}{{\left({\int }_{\omega }{\varphi }_n(x)\text{d}x\right)}^{\frac{1}{2}}}$.

We have that the solution operator of (A.3) is a C₀ semigroup, and the infinitesimal generator A_σ is given by

$$A_{\sigma }\varphi =\{\begin{array}{ll}\dot{\varphi }(\theta ),\hfill & \theta \in [-r,0),\hfill \\ D\mathrm{\Delta }\varphi (0)+L\left(\tau *+\sigma \right)(\varphi ),\hfill & \theta =0.\hfill \end{array}$$ (A.4)

and the domain dom(A_σ) of A_σ is

$$\mathrm{dom}\left(A_{\sigma }\right):=\left\{\varphi \in \mathcal{C}:\dot{\varphi }\in \mathcal{C},\varphi (0)\in \mathrm{dom}(\mathrm{\Delta }),\dot{\varphi }(0)=D\mathrm{\Delta }\varphi (0)+L\left(\tau *+\sigma \right)(\varphi )\right\},$$

Hence, equation (A.2) can be rewritten as the abstract ODE in $\mathcal{C}$:

$${\dot{U}}_t=A_{\sigma }{U}_t+R\left(\sigma ,U_t\right),$$ (A.5)

where

$$R\left(\sigma ,U_t\right)(\theta )=\{\begin{array}{ll}0,\hfill & \theta \in [-1,0),\hfill \\ F\left(\sigma ,U_t\right),\hfill & \theta =0.\hfill \end{array}$$

We denote

$${\beta }_n=\left\{{(1,0,0)}^{\text{T}}{\gamma }_n,{(0,1,0)}^{\text{T}}{\gamma }_n,{(0,0,1)}^{\text{T}}{\gamma }_n\right\}.$$

For $\varphi ={\left({\varphi }^{(1)},{\varphi }^{(2)}\right)}^T\in \mathcal{C}$, denote

$${\varphi }_n=\langle \varphi ,{\beta }_n\rangle ={\left(\langle {\varphi }^{(1)},{\gamma }_n\rangle ,\langle {\varphi }^{(2)},{\gamma }_n\rangle ,\langle {\varphi }^{(3)},{\gamma }_n\rangle \right)}^{\text{T}}.$$

Define A_σ,n as

$$A_{\sigma ,n}\left({\varphi }_n(\theta ){\gamma }_n\right)=\{\begin{array}{ll}{\dot{\varphi }}_n(\theta ){\gamma }_n,\hfill & \theta \in [-1,0),\hfill \\ {\int }_{-r}^0\text{d}{\eta }_n(\sigma ,\theta ){\varphi }_n(\theta ){\gamma }_n,\hfill & \theta =0.\hfill \end{array}$$ (A.6)

Furthermore, we have

$$L_{\sigma ,n}\left({\varphi }_n\right)=\left(\tau *+\sigma \right)L_1{\varphi }_n(0)+\left(\tau *+\sigma \right)L_2{\varphi }_n(-1),$$

and

$$-{\mu }_{n}D{\varphi }_n(0)+L_{\sigma ,n}\left({\varphi }_n\right)={\int }_{-1}^0\text{d}{\eta }_n(\sigma ,\theta ){\varphi }_n(\theta ),$$

where

$${\eta }_n(\sigma ,\theta )=\{\begin{array}{ll}-\left(\tau *+\sigma \right)L_2,\hfill & \theta =-1,\hfill \\ 0,\hfill & \theta \in (-1,0),\hfill \\ \left(\tau *+\sigma \right)\left(L_1-{\mu }_{n}D\right),\hfill & \theta =0.\hfill \end{array}$$

Define $\mathcal{C}*=C([0,1];X)$ and a bilinear form (·,·) on $\mathcal{C}*\times \mathcal{C}$

$$(\psi ,\varphi )=\sum _{k,j=0}^{\infty }{\left({\psi }_k,{\varphi }_j\right)}_c{\int }_{\mathrm{\Omega }}{\gamma }_k{\gamma }_j\text{d}x,$$

where (·, ·)_c is the bilinear form defined on C∗ × C

$${\left({\psi }_n,{\varphi }_n\right)}_c={\overline{\psi }}_n^T(0){\varphi }_n(0)-{\int }_{-1}^0{\int }_{\xi =0}^{\theta }{\overline{\psi }}_n^T(\xi -\theta )\text{d}{\eta }_n(0,\theta ){\varphi }_n(\xi )\text{d}\xi ,$$

and

$$\psi =\sum _{n=0}^{\infty }{\psi }_n{\gamma }_n\in \mathcal{C}*,\varphi =\sum _{n=0}^{\infty }{\varphi }_n{\gamma }_n\in \mathcal{C},$$

with

$${\varphi }_n\in C=C\left([-1,0],{ℝ}^2\right),{\psi }_n\in C*=\left([0,1],{ℝ}^2\right).$$

Notice that

$${\int }_{\mathrm{\Omega }}{\gamma }_k{\gamma }_j\text{d}x=0\text{for}k\ne j,$$

we have

$$(\psi ,\varphi )=\sum _{n=0}^{\infty }{\left({\psi }_n,{\varphi }_n\right)}_c,$$

We define the adjoint operator A∗ of A₀

$$A^{*}\psi (s)=\{\begin{array}{ll}-\dot{\psi }(s),\hfill & s\in (0,1],\hfill \\ {\sum }_{n=0}^{\infty }{\int }_{-1}^0\text{d}{\eta }_n^T(0,t){\psi }_n(-t){\gamma }_n,\hfill & s=0.\hfill \end{array}$$

Let

$$q(\theta ){\gamma }_{n_0}=q(0)e^{i{\omega }^{*}{\tau }^{*}\theta }{\gamma }_{n_0},q^{*}(s){\gamma }_{n_0}=q^{*}(0)e^{i{\omega }^{*}{\tau }^{*}s}{\gamma }_{n_0}$$

be the eigenfunctions of A₀, and A∗ corresponds to iω∗τ∗ and −iω∗τ∗, respectively. By direct calculations, we chose

$$q(0)={\left(1,C_1,C2\right)}^{\text{T}},q*(0)=M{\left(1,C_3,C_4\right)}^{\text{T}}$$

where

$$C_1=\frac{a^{2}u*v*-\left(\text{i}\omega *+d_1{\mu }_{n_0}+ru*\right)\left(\text{i}\omega *+d_2{\mu }_{n_0}+au*+c\right)}{abu*+ru*\left(\text{i}\omega *+d_2{\mu }_{n_0}+au*+c\right)},$$

$$C_2=-\frac{\left(\text{i}\omega *+d_1{\mu }_{n_0}+ru*\right)b+rau*v*}{abu*+ru*\left(\text{i}\omega *+d_2{\mu }_{n_0}+au*+c\right)},$$

$$C_3=\frac{\left(-\text{i}\omega *+d_1{\mu }_{n_0}+ru*\right)\left(-\text{i}\omega *+d_2{\mu }_{n_0}+au*+c\right)-a^{2}u*v*}{av*{\text{e}}^{-\text{i}{\omega }^{*}{\tau }^{*}}\left(-\text{i}\omega *+d_2{\mu }_{n_0}+c\right)},$$

$$C_4=\frac{\left(-\text{i}\omega *+d_1{\mu }_{n_0}+ru*\right)u*-au*v*}{v*\left(-\text{i}\omega *+d_2{\mu }_{n_0}+c\right)}$$

and

$$\overline{M}=\frac{1}{1+C_1\overline{C_3}+C_2\overline{C_4}+\tau *a\overline{C_3}\left(v*+u*C_2\right){\text{e}}^{-\text{i}\omega {\tau }^{*}}}.$$

Obviously, (q∗, q)_c = 1. Then we decompose $\mathcal{C}$ by

$$\Lambda =\left\{\pm i\omega *\tau *\right\},$$

$\mathcal{C}=P\oplus Q$, where

$$P=\left\{zq{\gamma }_{n_0}+\overline{zq}{\gamma }_{n_0}\mid z\in ℂ\right\},$$

$$Q=\left\{\varphi \in \mathcal{C}\mid \left(q*{\gamma }_{n_0},\varphi \right)=0\text{and}\left(\overline{q}*{\gamma }_{n_0},\varphi \right)=0\right\}.$$

Thus, system (A.5) could be rewritten as

$$U_t=z(t)q(\cdot ){\gamma }_{n_0}+\overline{z}(t)\overline{q}(\cdot ){\gamma }_{n_0}+W(t,\cdot ),$$

where

$$W(t,\cdot )\in Q.$$

As in [35], we have

$$z(t)=\left(q*{\gamma }_{n_0},U_t\right),W(t,\theta )=U_t(\theta )-2\mathrm{Re}\left\{z(t)q(\theta ){\gamma }_{n_0}\right\}.$$ (A.7)

Then it follows that

$$\dot{z}(t)=i{\omega }_{0}z(t)+{\overline{q}}^{*\text{T}}(0)\langle F\left(0,U_t\right),{\beta }_{n_0}\rangle ,$$ (A.8)

where

$$\langle F,{\beta }_n\rangle :={\left(\langle F_1,{\gamma }_n\rangle ,\langle F_2,{\gamma }_n\rangle ,\langle F_3,{\gamma }_n\rangle \right)}^{\text{T}},$$

With the center manifold theorem [36], there exists a center manifold ${\mathcal{C}}_0$ and on ${\mathcal{C}}_0$ we have

$$W(t,\theta )=W(z(t),\overline{z}(t),\theta )=W_{20}(\theta )\frac{z^2}{2}+W_{11}(\theta )z\overline{z}+W_{02}(\theta )\frac{{\overline{z}}^2}{2}+\cdots ,$$ (A.9)

where z and $\overline{z}$ are local coordinates for center manifold ${\mathcal{C}}_0$ in the direction of $q{\gamma }_{n_0}$ and $\overline{q}{\gamma }_{n_0}$, respectively. For solution $U_t\in {\mathcal{C}}_0$, we denote

$${F\left(0,U_t\right)|}_{{\mathcal{C}}_0}=\tilde{F}(0,z,\overline{z}),$$

and

$$\tilde{F}(0,z,\overline{z})={\tilde{F}}_{zz}^{\prime \prime }\frac{z^2}{2}+{\tilde{F}}_{z\overline{z}}^{\prime \prime }z\overline{z}+{\tilde{F}}_{\overline{z}\overline{z}}^{\prime \prime }\frac{{\overline{z}}^2}{2}+{\tilde{F}}_{zz\overline{z}}^{\prime \prime }\frac{z^2\overline{z}}{2}+\cdots .$$

For convenience, we rewrite (A.8) as

$$\dot{z}(t)=i{\omega }_{0}z(t)+g(z,\overline{z}),$$

and denote

$$g(z,\overline{z})=g_{20}\frac{z^2}{2}+g_{11}z\overline{z}+g_{02}\frac{{\overline{z}}^2}{2}+g_{21}\frac{z^2\overline{z}}{2}+\cdots .$$

From direct calculation, we get

$$g_{20}=2\overline{M}\tau *{\int }_{\mathrm{\Omega }}{\gamma }_{n_0}^3\text{d}x\left(-r-r{C}_1-a{C}_2+a{C}_2{\text{e}}^{-2\text{i}{\omega }^{*}{\tau }^{*}}\overline{C_3}-a{C}_2\overline{C_4}\right),$$

$$g_{11}=\overline{M}\tau *{\int }_{\mathrm{\Omega }}{\gamma }_{n_0}^3\text{d}x\left[-2r-r\left(C_1+\overline{C_1}\right)-a\left(C_2+\overline{C_2}\right)+a\left(C_2+\overline{C_2}\right)\overline{C_3}-a\left(C_2+\overline{C_2}\right)\overline{C_4}\right],$$

$$g_{02}=2\overline{M}\tau *{\int }_{\mathrm{\Omega }}{\gamma }_{n_0}^3\text{d}x\left(-r-r{\overline{C}}_1-a{\overline{C}}_2+a{\overline{C}}_2{\text{e}}^{2\text{i}{\omega }^{*}{\tau }^{*}}\overline{C_3}-a{\overline{C}}_2\overline{C_4}\right)$$

and

$$\begin{gathered} g_{21}=\overline{M}\tau *{\int }_{\mathrm{\Omega }}\{[-r\left(2w_{20}^{(1)}(0)+4w_{11}^{(1)}(0)\right)-r\left(w_{20}^{(2)}(0)+2w_{11}^{(2)}(0)+{\overline{C}}_1{w}_{20}^{(1)}(0)+2C_1{w}_{11}^{(1)}(0)\right) \\ -a\left(w_{20}^{(3)}(0)+2w_{11}^{(3)}(0)+{\overline{C}}_2{w}_{20}^{(1)}(0)+2C_2{w}_{11}^{(1)}(0)\right)]{\gamma }_{n_0}^2 \\ +a\left(w_{20}^{(3)}(-1){\text{e}}^{\text{i}{\omega }^{*}{\tau }^{*}}+2w_{11}^{(3)}(-1){\text{e}}^{-\text{i}{\omega }^{*}{\tau }^{*}}+{\overline{C}}_2{w}_{20}^{(1)}(-1){\text{e}}^{\text{i}{\omega }^{*}{\tau }^{*}}+2C_2{w}_{11}^{(1)}(-1){\text{e}}^{-\text{i}{\omega }^{*}{\tau }^{*}}\right){\gamma }_{n_0}^2\overline{C_3} \\ -a\left(w_{20}^{(3)}(0)+2w_{11}^{(3)}(0)+{\overline{C}}_2{w}_{20}^{(1)}(0)+2C_2{w}_{11}^{(1)}(0)\right){\gamma }_{n_0}^2\overline{C_4}\}\text{d}x. \end{gathered}$$

Here w₁₁ and w₂₀ are need to be computed. From (A.7), we have

$$\begin{gathered} \dot{W}={\dot{U}}_t-\dot{z}q{\gamma }_{n_0}-\dot{\overline{z}}\overline{q}{\gamma }_{n_0} \\ =\{\begin{array}{ll}AW-2\mathrm{Re}\{g(z,\overline{z})q(\theta )\}{\gamma }_{n_0},\hfill & \theta \in [-r,0),\hfill \\ AW-2\mathrm{Re}\{g(z,\overline{z})q(\theta )\}{\gamma }_{n_0}+\tilde{F},\hfill & \theta =0,\hfill \end{array} \\ \doteq AW+H(z,\overline{z},\theta ), \end{gathered}$$ (A.10)

where

$$H(z,\overline{z},\theta )=H_{20}(\theta )\frac{z^2}{2}+H_{11}(\theta )z\overline{z}+H_{02}(\theta )\frac{{\overline{z}}^2}{2}+\cdots .$$

Obviously,

$$\begin{gathered} H_{20}(\theta )=\{\begin{array}{ll}-g_{20}q(\theta ){\gamma }_{n_0}-{\overline{g}}_{02}\overline{q}(\theta ){\gamma }_{n_0},\hfill & \theta \in [-r,0),\hfill \\ -g_{20}q(0){\gamma }_{n_0}-{\overline{g}}_{02}\overline{q}(0){\gamma }_{n_0}+{\tilde{F}}_{zz}^{\prime \prime },\hfill & \theta =0,\hfill \end{array} \\ {H}_{11}(\theta )=\{\begin{array}{ll}-g_{11}q(\theta ){\gamma }_{n_0}-{\overline{g}}_{11}\overline{q}(\theta ){\gamma }_{n_0},\hfill & \theta \in [-r,0),\hfill \\ -g_{11}q(0){\gamma }_{n_0}-{\overline{g}}_{11}\overline{q}(0){\gamma }_{n_0}+{\tilde{F}}_{z\overline{z}}^{\prime \prime },\hfill & \theta =0,\hfill \end{array} \\ ···. \end{gathered}$$

Comparing the coefficients of (A.10) with the derived function of (A.9), we obtain

$$\left(A_0-2\text{i}{\omega }_{0}I\right)W_{20}(\theta )=-H_{20}(\theta ),A_0{W}_{11}(\theta )=-H_{11}(\theta ),\cdots .$$ (A.11)

From (A.4) and (A.11), for θ ∈ [−1, 0), we have

$$\begin{gathered} W_{20}(\theta )=-\frac{g_{20}}{\text{i}\omega *\tau *}q(\theta ){\gamma }_{n_0}-\frac{{\overline{g}}_{02}}{3\text{i}\omega *\tau *}\overline{q}(\theta ){\gamma }_{n_0}+E_1{\text{e}}^{2\text{i}{\omega }^{*}{\tau }^{*}\theta }, \\ {W}_{11}(\theta )=\frac{g_{11}}{\text{i}\omega *\tau *}q(0){\text{e}}^{\text{i}{\omega }^{*}{\tau }^{*}\theta }{\gamma }_{n_0}-\frac{{\overline{g}}_{11}}{\text{i}\omega *\tau *}\overline{q}(0){\text{e}}^{-\text{i}{\omega }^{*}{\tau }^{*}\theta }{\gamma }_{n_0}+E_2, \end{gathered}$$ (A.12)

where E₁ and E₂ are both 3-dimensional vectors in X, and can be determined by setting θ = 0 in H. In fact, set θ = 0 and by (A.11) and (A.12), we obtain

$${\left(A_0-2\text{i}\omega *\tau *I\right)E_1{\text{e}}^{2\text{i}{\omega }^{*}{\tau }^{*}\theta }|}_{\theta =0}=-{\tilde{F}}_{zz}^{\prime \prime },{A_0{E}_2|}_{\theta =0}=-{\tilde{F}}_{z\overline{z}}^{\prime \prime }.$$ (A.13)

The terms ${\tilde{F}}_{zz}^{\prime \prime }$ and ${\tilde{F}}_{z\overline{z}}^{\prime \prime }$ are elements in the space $\mathcal{C}$, and

$${\tilde{F}}_{zz}^{\prime \prime }=\sum _{n=1}^{\infty }\langle {\tilde{F}}_{zz}^{\prime \prime },{\beta }_n\rangle {\gamma }_n,{\tilde{F}}_{z\overline{z}}^{\prime \prime }=\sum _{n=1}^{\infty }\langle {\tilde{F}}_{z\overline{z}}^{\prime \prime },{\beta }_n\rangle {\gamma }_n.$$

Denote

$$E_1=\sum _{n=0}^{\infty }{E}_1^n{\gamma }_n,E_2=\sum _{n=0}^{\infty }{E}_2^n{\gamma }_n,$$

then from (A.13) we have

$$\begin{gathered} {\left(A_0-2\text{i}{\omega }^{*}{\tau }^{*}I\right)E_1^n{\gamma }_n{\text{e}}^{2\text{i}{\omega }^{*}{\tau }^{*}\theta }|}_{\theta =0}=-\langle {\tilde{F}}_{zz}^{\prime \prime },{\beta }_n\rangle {\gamma }_n, \\ {A_0{E}_2^n{\gamma }_n|}_{\theta =0}=-\langle {\tilde{F}}_{z\overline{z}}^{\prime \prime },{\beta }_n\rangle {\gamma }_n, \\ n=0,1,\cdots . \end{gathered}$$

Thus, $E_1^n$ and $E_2^n$ could be calculated by

$$\begin{gathered} E_1^n={\left(2\text{i}\omega *\tau *I-{\int }_{-1}^0{\text{e}}^{2\text{i}{\omega }^{*}{\tau }^{*}\theta }\text{d}{\eta }_n(0,\theta )\right)}^{-1}\langle {\tilde{F}}_{zz}^{\prime \prime },{\beta }_n\rangle , \\ {E}_2^n=-{\left({\int }_{-1}^0\text{d}{\eta }_n(0,\theta )\right)}^{-1}\langle {\tilde{F}}_{z\overline{z}}^{\prime \prime },{\beta }_n\rangle , \\ n=0,1,\cdots , \end{gathered}$$

where

$${\tilde{F}}_{20}=2\tau *\left(\begin{array}{c}-r-r{C}_1-a{C}_2\\ a{C}_2{\text{e}}^{-2\text{i}{\omega }^{*}{\tau }^{*}}\\ -a{C}_2\end{array}\right),$$

$${\tilde{F}}_{11}=\tau *\left(\begin{array}{c}-2r-r\left(C_1+{\overline{C}}_1\right)-a\left(C_2+{\overline{C}}_2\right)\\ a\left(C_2+{\overline{C}}_2\right)\\ -a\left(C_2+{\overline{C}}_2\right)\end{array}\right).$$

Hence, g₂₁ could be represented explicitly.

Denote

$$\begin{gathered} c_1(0)=\frac{i}{2\omega *\tau *}\left(g_{20}{g}_{11}-2{\left|g_{11}\right|}^2-\frac{1}{3}{\left|g_{02}\right|}^2\right)+\frac{1}{2}{g}_{21}, \\ \mu =-\frac{\mathrm{Re}\left(c_1(0)\right)}{\tau *\mathrm{Re}\left({\lambda }^{\prime }\left(\tau *\right)\right)},{\beta }_2=2\mathrm{Re}\left(c_1(0)\right), \\ {T}_2=-\frac{1}{\omega *\tau *}(\mathrm{Im}\left(c_1(0)\right)+{\mu }_2\left(\omega *+\tau *\mathrm{Im}\left({\lambda }^{\prime }\left(\tau *\right)\right)\right). \end{gathered}$$ (A.14)

Then by the general Hopf bifurcation theory (see [35]), we know that μ determines the directions of the Hopf bifurcation: if μ > 0(< 0), then the direction of the Hopf bifurcation is forward (backward), that is the bifurcating periodic solutions exist when a > 0(< 0); β₂ determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are orbitally stable(unstable) if β₂ < 0(> 0); and T₂ determines the period of the bifurcation periodic solutions: the period increases(decreases) if T₂ > 0(< 0).

We now describe briefly the numerical method we use to solve the system (2.5) as follows. For Ω = (0, pπ) × (0, qπ), let x_i = ih_x, i = 1, 2, ···, l, $h_x=\frac{p\pi }{l}$, y_j = jh_y, j = 1, 2, ···, m, $h_y=\frac{q\pi }{m}$, t_k = kh_t, $h_t=\frac{\tau }{N}$(N is a positive integer), and u(i, j, k) = u(x_i, y_j, t_k), v(i, j, k) = v(x_i, y_j, t_k), w(i, j, k) = w(x_i, y_j, t_k). We replace $\frac{\partial u\left(x_i,y_j,t_k\right)}{\partial t}$ with a first-order difference $\frac{u(i,j,k+1)-u(i,j,k)}{h_t}$, and replace $\frac{{\partial }^{2}u\left(x_i,y_j,t_k\right)}{\partial x^2}+\frac{{\partial }^{2}u\left(x_i,y_j,t_k\right)}{\partial y^2}$ with a second-order difference $\frac{u(i+1,j,k)-2u(i,j,k)+u(i-1,j,k)}{h_x}+\frac{u(i,j+1,k)-2u(i,j,k)+u(i,j-1,k)}{h_y}$. Similarly, we take differences of the partial derivative of v and w. Then, we get a system of difference equations:

$$\{\begin{array}{l}lu(i,j,k+1)=u(i,j,k)+h_t{d}_1\left(\frac{u(i+1,j,k)-2u(i,j,k)+u(i-1,j,k)}{h_x}+\frac{u(i,j+1,k)-2u(i,j,k)+u(i,j-1,k)}{h_y}\right)\\ +h_t(ru(i,j,k)(1-u(i,j,k)-w(i,j,k)-au(i,j,k)v(i,j,k),\\ w(i,j,k+1)=w(i,j,k)+h_t{d}_1\left(\frac{w(i+1,j,k)-2w(i,j,k)+w(i-1,j,k)}{h_x}+\frac{w(i,j+1,k)-2w(i,j,k)+w(i,j-1,k)}{h_y}\right)\\ +h_t(au(i,j,k-N)v(i,j,k-N)-w(i,j,k),\\ v(i,j,k+1)=v(i,j,k)+h_t{d}_2\left(\frac{v(i+1,j,k)-2v(i,j,k)+v(i-1,j,k)}{h_x}+\frac{v(i,j+1,k)-2v(i,j,k)+v(i,j-1,k)}{h_y}\right)\\ +h_t(bw(i,j,k)-au(i,j,k)v(i,j,k-cv(i,j,k).\end{array}$$

We implement this method in Matlab, and obtain numerical solutions of the system (2.5).