Hopf Bifurcation in a Delayed Equation with Diffusion Driven by Carrying Capacity

Abstract

In this paper, a delayed reaction–diffusion equation with carrying capacity-driven diffusion is investigated. The stability of the positive equilibrium solutions and the existence of the Hopf bifurcation of the equation are considered by studying the principal eigenvalue of an associated elliptic operator. The properties of the bifurcating periodic solutions are also obtained by using the normal form theory and the center manifold reduction. Furthermore, some representative numerical simulations are provided to illustrate the main theoretical results.

1. Introduction

Reaction–diffusion equation models have frequently been used in population dynamics and biological problems. A widely used example is the classical logistic equation,

$$\left\{\begin{array}{cc}\frac{\partial u}{\partial t}=d\Delta u+m\left(x\right)u\left(1-\frac{u(t,x)}{K\left(x\right)}\right),\hfill & t>0,x\in \Omega ,\hfill \\ \frac{\partial u}{\partial \mathbf{n}}(t,x)=0,\hfill & t>0,x\in \partial \Omega ,\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in \Omega .\hfill \end{array}\right.$$

(1)

The spatial domain $\Omega$ is a region in ${\mathbb{R}}^N$ for $N\ge 1$, where $\mathbf{n}$ denotes the unit outward normal on the $\partial \Omega .$ The variables t and x represent time and location separately in the function $u(t,x)$ of (1). The function $m\left(x\right)$, under the assumption that $m\left(x\right)>0$ in $\Omega$, stands for the intrinsic growth rate, and the function $K\left(x\right)>0$, for all x in $\Omega$, stands for the carrying capacity.

Since the carrying capacity of the environment is spatially heterogeneous, i.e., K is not a constant function, system (1) may assume the migration of populations from resource-rich to resource-poor places. To improve on this aspect of diffusion, Korobenko and Braverman [1] considered the case where an ideal free distribution is a positive state that matches the local per capita growth rate. This led to the following initial boundary value problem:

$$\left\{\begin{array}{cc}\frac{\partial u}{\partial t}=d\Delta \frac{u}{K\left(x\right)}+m\left(x\right)u\left(1-\frac{u(t,x)}{K\left(x\right)}\right),\hfill & (t,x)\in (0,+\infty )\times \Omega ,\hfill \\ \frac{\partial (u/K(x\left)\right)}{\partial \mathbf{n}}=0,\hfill & t>0,x\in \partial \Omega ,\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in \Omega .\hfill \end{array}\right.$$

(2)

One can see that the function $K\left(x\right)$ is a positive equilibrium solution of system (2). Moreover, for any $u_0\left(x\right)\ge 0$, with $u_0\left(x\right)\not\equiv 0$, we have ${lim}_{t\to +\infty }u(t,x)=K\left(x\right)$ uniformly present in $x\in \overline{\Omega }$. For more details on this, we refer the reader to [2,3,4] and the references therein.

The effect of dispersal on species interaction is a very important topic in spatial ecology. In many cases, the ideal free distribution (IFD), introduced in [5], is evolutionary stable [6,7,8,9,10,11,12]. Since most species have a maturation time or a reproduction period, we include a time delay and consider the following model in this present work.

$$\left\{\begin{array}{cc}\frac{\partial u}{\partial t}=d\Delta \left(\frac{u}{K\left(x\right)}\right)+m\left(x\right)u\left(1-\frac{u(t-\tau ,x)}{K\left(x\right)}\right),\hfill & (t,x)\in (0,+\infty )\times \Omega ,\hfill \\ \frac{\partial (u/K(x\left)\right)}{\partial \mathbf{n}}=0,\hfill & t>0,x\in \partial \Omega ,\hfill \\ u(\theta ,x)=u_0(\theta ,x),\hfill & (\theta ,x)\in [-\tau ,0]\times \Omega ,\hfill \end{array}\right.$$

(3)

where $\tau >0$ represents the time delay, and $u_0\in C([-\tau ,0],\mathbb{Y})$ with $\mathbb{Y}=L^2(\Omega )$.

The model above involves heterogeneity in the environment conditions. Hence, the positive equilibrium is spatially non-homogeneous. Moreover, the characteristic equation is no longer an algebraic equation. This makes the study of dynamical behavior near the spatially non-homogeneous positive equilibrium difficult. It is worth noting that after the pioneering work of Busenberg and Huang [13], many authors investigated the Hopf bifurcation of models with heterogeneity in the environmental conditions [14,15,16,17,18,19,20,21,22,23,24,25,26].

We thus set

$$\lambda =\frac{1}{d},t=\tilde{t}/d,\tilde{\tau }=d.\tau \mathrm{and}\upsilon (t,x)=u/K\left(x\right)=\tilde{\upsilon }(\tilde{t},x),\frac{u_0(\theta ,x)}{K\left(x\right)}={\varphi }_0(\theta ,x).$$

Dropping the tilde sign, system (3) becomes

$$\left\{\begin{array}{cc}K\left(x\right)\frac{\partial \upsilon }{\partial t}=\Delta \upsilon +\lambda m\left(x\right)K\left(x\right)\upsilon (1-\upsilon (t-\tau ,x)),\hfill & (t,x)\in (0,+\infty )\times \Omega ,\hfill \\ \frac{\partial \upsilon }{\partial \mathbf{n}}=0,\hfill & t>0,x\in \partial \Omega ,\hfill \\ \upsilon (\theta ,x)={\varphi }_0(\theta ,x),\hfill & (\theta ,x)\in [-\tau ,0]\times \Omega .\hfill \end{array}\right.$$

(4)

The advantage of this rescale is that the unique positive equilibrium of (4) becomes equal to 1.

In this paper, we mainly study the influence of the delay $\tau$ on the stability of the positive equilibrium solution of model (4). Throughout the paper, we always assume that $m\left(x\right)$ and $K\left(x\right)$ satisfy the following hypothesis:

Hypothesis 1.

$m,K\in C^{1+\alpha }(\Omega )forsome\alpha \in (0,1),m\left(x\right)>0andk\left(x\right)>0forallx\in \Omega .$

2. Stability and Hopf Bifurcation

Before studying the stability of the delayed nonlinear system (4), we consider the following eigenvalue problem, which is critical for the determination of the properties of the solutions to the nonlinear Equation (4):

$$\left\{\begin{array}{cc}\frac{\Delta \varphi }{K\left(x\right)}+\lambda m\left(x\right)\varphi =0,\hfill & x\in \Omega ,\hfill \\ \frac{\partial \varphi }{\partial \mathbf{n}}=0,\hfill & x\in \partial \Omega .\hfill \end{array}\right.$$

(5)

Due to the Neumann boundary conditions, the principal eigenvalue of (5) is ${\lambda }_{*}=0$. The corresponding eigenfunction (the principal eigenfunction) is strictly positive and unique up to its multiplication by a positive constant. Thus, one may take the principal eigenfunction as $\varphi =1$, without loss of generality.

Throughout the paper, we will follow an approach that was recently used in [27] to study the Hopf bifurcation in a different reaction–advection–diffusion problem with time delay. We let $\mathbb{X}=H^2(\Omega )\cap H_0^1(\Omega )$, and $\mathbb{Y}=L^2(\Omega )$. Meanwhile, ${\mathbb{X}}_c:=\mathbb{X}\oplus i\mathbb{X}=\{a+ib|a,b\in \mathbb{X}\}$ denotes the complexification of a linear space $\mathbb{X}$, $\mathfrak{D}\left(L\right)$ denotes the domain of a linear operator L, $\mathfrak{N}\left(L\right)$ is the kernel of L, and $\mathfrak{R}\left(L\right)$ is the range of L. Furthermore, we define the inner product of the Hilbert space ${\mathbb{Y}}_c$ as $<u,v>={\int }_{\Omega }\overline{u}\left(x\right)v\left(x\right)dx$. $C=C\left(\right[-\tau ,0],\mathbb{Y})$ and $C^1=C^1([-\tau ,0],\mathbb{Y})$ denote the Banach space of continuous and differentiable mappings from $[-\tau ,0]$ into $\mathbb{Y}$, respectively.

For convenience of analysis, the spaces $\mathbb{X}$ and $\mathbb{Y}$ are decomposed as

$$\mathbb{X}=\mathcal{N}\oplus {\mathbb{X}}_1,\mathbb{Y}=\mathcal{N}\oplus {\mathbb{Y}}_1,$$

where

$$\mathcal{N}=\mathrm{span}\left\{1\right\},{\mathbb{X}}_1=\left\{y\in \mathbb{X}|{\int }_{\Omega }y\left(x\right)dx=0\right\},\mathrm{and}{\mathbb{Y}}_1=\left\{y\in \mathbb{Y}|{\int }_{\Omega }y\left(x\right)dx=0\right\}.$$

The linearization of (4) at $\upsilon =1$ is given by

$$\left\{\begin{array}{cc}K\left(x\right)\frac{\partial \omega }{\partial t}=\Delta \omega -\lambda m\left(x\right)K\left(x\right)\omega (t-\tau ),\hfill & (t,x)\in (0,+\infty )\times \Omega ,\hfill \\ \frac{\partial \omega }{\partial \mathbf{n}}=0,\hfill & t>0,x\in \partial \Omega .\hfill \end{array}\right.$$

(6)

We know, from [28], for example, that the solution semi-group of problem (6) has an infinitesimal generator $A_{\tau }$ that satisfies

$$A_{\tau }\left(\lambda \right)\phi =\dot{\phi }$$

on the domain

$$\mathfrak{D}\left(A_{\tau }\left(\lambda \right)\right)=\{\phi \in C_{\mathcal{C}}\cap C_{\mathcal{C}}^1,\mathrm{such}\mathrm{that}\dot{\phi }\left(0\right)=\frac{1}{K\left(x\right)}\Delta \phi -\lambda m\left(x\right)\phi (-\tau )\},$$

where $C_{\mathcal{C}}^1=C^1([-\tau ,0],{\mathbb{Y}}_{\mathcal{C}})$. From this formulation of (6), the spectrum of $A_{\tau }\left(\lambda \right)$ is

$$\sigma \left(A_{\tau }\left(\lambda \right)\right)=\left\{\mu \in \mathcal{C},\Lambda (\lambda ,\mu ,\tau )\phi =0,\phi \in {\mathbb{X}}_{\mathcal{C}}\backslash \left\{0\right\}\right\},$$

where

$$\Lambda (\lambda ,\mu ,\tau )\phi =\frac{1}{K\left(x\right)}\Delta \phi -\lambda m\left(x\right)\phi e^{-\mu \tau }-\mu \phi .$$

Lemma 1.

If $\tau =0$, then the steady state $\upsilon =1$ in (4) is locally asymptotically stable for any $\lambda >0$. Moreover, $0\notin A_{\tau }\left(\lambda \right)$ for any $\tau >0.$

Proof. 

When $\tau =0$, computing the spectrum of $A_{\tau }\left(\lambda \right)$ leads us to the equation

$$\Lambda (\lambda ,\mu ,0)\phi =\frac{1}{K\left(x\right)}\Delta \phi -\lambda m\left(x\right)\phi -\mu \phi ,$$

which in turn leads to the investigation of the following linear eigenvalue problem:

$$\left\{\begin{array}{cc}\Delta \phi -\lambda m\left(x\right)K\left(x\right)\phi =\mu K\left(x\right)\phi ,\hfill & x\in \Omega ,\hfill \\ \frac{\partial \phi }{\partial \mathbf{n}}=0,\hfill & x\in \partial \Omega .\hfill \end{array}\right.$$

(7)

Then, ${\mu }_1$ has the following variational characterization:

$${\mu }_1=\underset{\phi \in {\mathbb{X}}_{\mathcal{C}},\phi \ne 0}{max}\left[\frac{-{\int }_{\Omega }[{(\nabla \phi )}^2+\lambda m\left(x\right)K\left(x\right){\phi }^2]dx}{{\int }_{\Omega }K\left(x\right){\phi }^2}\right].$$

(8)

This yields ${\mu }_1\le -\lambda {min}_{x\in \Omega }m\left(x\right)<0$. Therefore, the steady state $K\left(x\right)$ of (3) is locally asymptotically stable when the delay $\tau =0$.

We can similarly show $0\notin \sigma \left(A_{\tau }\left(\lambda \right)\right)$ for any $\tau >0$. □

Next, we prove that the eigenvalues of $A_{\tau }\left(\lambda \right)$ will pass through the imaginary axis for some $\tau >0$, which is a necessary condition for the occurrence of the Hopf bifurcation. Indeed, $A_{\tau }\left(\lambda \right)$ has a purely imaginary eigenvalue $\mu =i\nu (\nu >0)$, for some $\tau >0$, if and only if the equation

$$\Delta \phi -\lambda m\left(x\right)K\left(x\right)\phi e^{-i\theta }-i\nu K\left(x\right)\phi =0$$

(9)

is solvable for some values such that $\nu >0$, $\theta \in [0,2\pi )$, $\nu \tau =\theta$, $\phi \in {\mathbb{X}}_{\mathcal{C}},\phi \ne 0$. We are now in place to state the following lemmas.

Lemma 2.

If there exist $({\nu }_{\lambda },{\theta }_{\lambda },{\phi }_{\lambda })\in {\mathbb{R}}^{+}\times \mathbb{R}\times {\mathbb{X}}_{\mathcal{C}}\backslash \left\{0\right\}$ that solve system (9), then $\frac{{\nu }_{\lambda }}{\lambda }$ is uniformly bounded.

Proof. 

After substituting $({\nu }_{\lambda },{\theta }_{\lambda },{\phi }_{\lambda })$ into system (9) and multiplying ${\overline{\phi }}_{\lambda }$, an integration of the result over $\Omega$ leads to

$$\begin{array}{c}\hfill \langle {\phi }_{\lambda },\Delta {\phi }_{\lambda }\rangle -\lambda {\int }_{\Omega }m\left(x\right)K\left(x\right)|{\phi }_{\lambda }{|}^2{e}^{-i\theta }-i{\nu }_{\lambda }{\int }_{\Omega }K\left(x\right){\left|{\phi }_{\lambda }\right|}^2=0.\end{array}$$

(10)

After separating the real and imaginary parts of (10), we obtain

$$\begin{array}{c}\hfill \lambda sin{\theta }_{\lambda }{\int }_{\Omega }m\left(x\right)K\left(x\right)|{\phi }_{\lambda }{|}^2={\nu }_{\lambda }{\int }_{\Omega }K\left(x\right){|{\phi }_{\lambda }|}^2.\end{array}$$

(11)

Hence,

$$\begin{array}{c}\hfill \frac{{\nu }_{\lambda }}{\lambda }=\frac{sin{\theta }_{\lambda }{\int }_{\Omega }m\left(x\right)K\left(x\right){\left|{\phi }_{\lambda }\right|}^2}{{\int }_{\Omega }K\left(x\right){\left|{\phi }_{\lambda }\right|}^2}\le \underset{\Omega }{max}m\left(x\right).\end{array}$$

(12)

□

Lemma 3.

Let $\mathbf{L}:=\Delta$. If $\eta \in {\mathbb{X}}_c$ and $\langle \eta ,1\rangle =0$, then

$$\left|\langle \mathbf{L}\eta ,\eta \rangle \right|\ge {\gamma }_2{\parallel \eta \parallel }_{{\mathbb{Y}}_c}^2,$$

where ${\gamma }_2$ is the second eigenvalue of operator $-\mathbf{L}$.

Proof. 

Under zero Neumann boundary conditions, it is well-known that the operator $-\mathbf{L}$ defined on $\Omega$ has a sequence of eigenvalues ${\left\{{\gamma }_i\right\}}_{i=1}^{\infty }$, such that

$$0={\gamma }_1<{\gamma }_2\le {\gamma }_3\le \cdots \mathrm{and}\underset{n\to \infty }{lim}{\gamma }_n=\infty .$$

Moreover, the corresponding eigenfunctions ${\left\{{\varphi }_i\right\}}_{i=1}^{\infty }$ form an orthogonal basis for ${\mathbb{Y}}_{\mathcal{C}}$. The principal eigenfunction is ${\varphi }_1=1.$ In particular, for each $\eta \in {\mathbb{X}}_{\mathcal{C}}$ that satisfies $\langle \eta ,1\rangle =0$, there is a sequence of real numbers ${\left\{c_n\right\}}_{n=2}^{\infty }$, such that $\eta ={\sum }_{n=2}^{\infty }{c}_n{\varphi }_n$. Therefore,

$${\displaystyle \mathbf{L}\eta =\sum _{n=2}^{\infty }{c}_n\mathbf{L}{\varphi }_n=-\sum _{n=2}^{\infty }{c}_n{\gamma }_n{\varphi }_n.}$$

(13)

It thus follows from the above equality that

$$\begin{array}{c}\hfill \left|\langle \mathbf{L}\eta ,\eta \rangle \right|={\gamma }_n\sum _{n=2}^{\infty }{c}_n^2{∥{\varphi }_n∥}_{L^2}^2\ge {\gamma }_2\sum _{n=2}^{\infty }{c}_n^2{∥{\varphi }_n∥}_{Y_{\mathbb{C}}}^2={\gamma }_2{∥\eta ∥}_{Y_{\mathbb{C}}}^2.\end{array}$$

(14)

□

Observe that $\mathbb{X}=\mathcal{N}\oplus {\mathbb{X}}_1.$ Hence, if $(\nu ,\theta ,\phi )$ satisfies system (9), then we ignore a scalar factor and have $\phi =\beta +\lambda \eta$, where $\beta >0$, $\eta \in {\mathbb{X}}_{1\mathcal{C}}$ and ${∥\phi ∥}_{{\mathbb{Y}}_{\mathcal{C}}}^2=\left|\Omega \right|$. By letting $\nu =\lambda h$ and substituting these into (9), we obtain

$$\left\{\begin{array}{c}{f}_1(\eta ,\beta ,\theta ,h,\lambda ):=\Delta \eta -m\left(x\right)K\left(x\right)(\beta +\lambda \eta )e^{-i\theta }-ihK\left(x\right)(\beta +\lambda \eta )=0,\hfill \\ f_2(\eta ,\beta ,\theta ,h,\lambda ):=({\beta }^2-1)|\Omega |+{\lambda }^2{∥\eta ∥}_{{\mathbb{Y}}_{\mathcal{C}}}^2=0.\hfill \end{array}\right.$$

(15)

Define $F:=(f_1,f_2)$ from ${\mathbb{X}}_{1\mathcal{C}}\times {\mathbb{R}}^4\to {\mathbb{Y}}_{\mathcal{C}}\times \mathbb{R}$. We confirm in the following lemma that $F(\eta ,\beta ,\theta ,h,\lambda )=0$ is uniquely solvable when $\lambda \to 0$.

Lemma 4.

The equation

$$\begin{array}{c}F(\eta ,\beta ,\theta ,h,0)=0,\mathrm{for}\eta \in {\mathbb{X}}_{\mathcal{C}},\beta \ge 0,\theta \in [0,\frac{\pi }{2}]\mathrm{and}h\ge 0\hfill \end{array}$$

(16)

has a unique solution $({\eta }_0,{\beta }_0,{\theta }_0,h_0)$ as $\lambda \to 0$. Here, ${\beta }_0=1$, ${\theta }_0=\frac{\pi }{2}$, $h_0=\frac{{\int }_{\Omega }m\left(x\right)K\left(x\right)}{{\int }_{\Omega }K\left(x\right)}$ and ${\eta }_0$ satisfies the following equation

$$\Delta {\eta }_0+im\left(x\right)K\left(x\right)-i{h}_{0}K\left(x\right)=0.$$

(17)

Proof. 

We see from the second equation of (15) that $f_2(\eta ,\beta ,\theta ,h,0)=0$ (with $\beta \ge 0$) if and only if ${\beta }_0=1$. Substituting ${\beta }_0=1$ into the first equation of (15), we see that ${\eta }_0$ satisfies

$$\begin{array}{c}\hfill \Delta {\eta }_0-m\left(x\right)K\left(x\right)e^{-i{\theta }_0}-i{h}_{0}K\left(x\right)=0.\end{array}$$

(18)

Integrating (18) over $\Omega$ and then separating the imaginary and real parts, we have

$$\left\{\begin{array}{c}cos{\theta }_0{\int }_{\Omega }m\left(x\right)K\left(x\right)=0,\hfill \\ sin{\theta }_0{\int }_{\Omega }m\left(x\right)K\left(x\right)=h_0{\int }_{\Omega }K\left(x\right).\hfill \end{array}\right.$$

(19)

Setting ${\theta }_0=\frac{\pi }{2}$, we then obtain

$$h_0=\frac{{\displaystyle {\int }_{\Omega }m\left(x\right)K\left(x\right)}}{{\displaystyle {\int }_{\Omega }K\left(x\right)}}.$$

□

Next, we prove that there exists ${\lambda }_{*}>0$, such that $F(\eta ,\beta ,\theta ,h,\lambda )=0$ has solutions for all $\lambda \in (0,{\lambda }_{*})$.

Lemma 5.

There exists ${\lambda }_{*}>0$ and a unique continuously differentiable mapping $\lambda \mapsto ({\eta }_{\lambda },{\beta }_{\lambda },{\theta }_{\lambda },h_{\lambda })$ from $(0,{\lambda }_{*})$ to ${\mathbb{X}}_{1\mathcal{C}}\times {\mathbb{R}}^3$, such that $F({\eta }_{\lambda },{\beta }_{\lambda },{\theta }_{\lambda },h_{\lambda },\lambda )=0$.

Proof. 

We define $T=(T_1,T_2):$ ${\mathbb{X}}_{1\mathcal{C}}\times {\mathbb{R}}^3\to {\mathbb{Y}}_{\mathcal{C}}\times \mathbb{R}$ to be the Fréchet derivative of F with respect to $(\eta ,\beta ,\theta ,h)$ at the point $({\eta }_0,{\beta }_0,{\theta }_0,h_0)$, then

$$\left\{\begin{array}{c}{T}_1({\eta }_{ϵ},{\beta }_{ϵ},{\theta }_{ϵ},h_{ϵ})=\Delta {\eta }_{ϵ}+i(m\left(x\right)-h_0)K\left(x\right){\beta }_{ϵ}+m\left(x\right)K\left(x\right){\theta }_{ϵ}-iK\left(x\right)h_{ϵ}\hfill \\ T_2({\eta }_{ϵ},{\beta }_{ϵ},{\theta }_{ϵ},h_{ϵ})=2{\beta }_{ϵ}.\hfill \end{array}\right.$$

(20)

We can easily check that T is bijective from ${\mathbb{X}}_{1\mathcal{C}}\times {\mathbb{R}}^3$ to ${\mathbb{Y}}_{\mathcal{C}}\times \mathbb{R}$. Thus, by the implicit function theorem, there exists ${\lambda }_{*}>0$ and a continuously differentiable mapping $\lambda \mapsto ({\eta }_{\lambda },{\beta }_{\lambda },{\theta }_{\lambda },h_{\lambda })$ from $(0,{\lambda }_{*})$ to ${\mathbb{X}}_{1\mathcal{C}}\times {\mathbb{R}}^3$, such that $F({\eta }_{\lambda },{\beta }_{\lambda },{\theta }_{\lambda },h_{\lambda },\lambda )=0$.

To prove uniqueness, we appeal to the implicit function theorem, which states that it is sufficient to show that if

$${\eta }^{\lambda }\in {\mathbb{X}}_{1\mathcal{C}}, {\beta }^{\lambda },h^{\lambda }>0, {\theta }^{\lambda }\in [0,2\pi ), \mathrm{and}F({\eta }^{\lambda },{\beta }^{\lambda },{\theta }^{\lambda },h^{\lambda },\lambda )=0,$$

then $({\eta }^{\lambda },{\beta }^{\lambda },{\theta }^{\lambda },h^{\lambda })\to ({\eta }_0,{\beta }_0,{\theta }_0,h_0)$ as $\lambda \to 0$ in the norm of ${\mathbb{X}}_{1\mathcal{C}}\times {\mathbb{R}}^3$.

We can easily obtain the boundedness of the sequences $\left\{{\beta }^{\lambda }\right\}$, $\left\{{\theta }^{\lambda }\right\}$, and $\left\{h^{\lambda }\right\}$ from the definition, hypothesis, and Lemma 2, respectively. Based on the first equation of (15) and Lemma 3, we have

$$\begin{array}{c}{∥{\eta }^{\lambda }∥}_{{\mathbb{Y}}_c}^2\le \frac{1}{{\gamma }_2}\left|\langle \mathbf{L}{\eta }^{\lambda },{\eta }^{\lambda }\rangle \right|\mathrm{and}\\ \frac{1}{{\gamma }_2}\left|\langle m\left(x\right)K\left(x\right)({\beta }^{\lambda }+\lambda {\eta }^{\lambda })e^{-i{\theta }^{\lambda }}+i{h}^{\lambda }K\left(x\right)({\beta }^{\lambda }+\lambda {\eta }^{\lambda }),{\eta }^{\lambda }\rangle \right|.\end{array}$$

(21)

The boundedness of $m\left(x\right)$, $K\left(x\right)$, and $\left\{h^{\lambda }\right\}$ implies that there exists a constant $M_1$, such that

$$\frac{1}{{\gamma }_2}{∥m\left(x\right)K\left(x\right)e^{-i{\theta }^{\lambda }}+i{h}^{\lambda }K\left(x\right)∥}_{\infty }\le M_1.$$

This allows us to get

$${∥{\eta }^{\lambda }∥}_{{\mathbb{Y}}_{\mathcal{C}}}^2\le M_1|{\beta }^{\lambda }|{∥{\eta }^{\lambda }∥}_{{\mathbb{Y}}_{\mathcal{C}}}+\lambda M_1{∥{\eta }^{\lambda }∥}_{{\mathbb{Y}}_{\mathcal{C}}}^2.$$

Therefore, if ${\lambda }_{*}$ is sufficiently small, we have $\lambda M_1\le \frac{1}{2}$. Hence,

$${∥{\eta }^{\lambda }∥}_{{\mathbb{Y}}_{\mathcal{C}}}\le 2M_1\left|{\beta }^{\lambda }\right|{∥{\eta }^{\lambda }∥}_{{\mathbb{Y}}_{\mathcal{C}}}.$$

As a result, $\left\{{\eta }^{\lambda }\right\}$ is bounded in ${\mathbb{Y}}_{\mathcal{C}}$ for $\lambda \in (0,{\lambda }_{*})$. In addition, since the operator ${\mathbf{L}}^{-1}$ is bounded, we see that $\left\{{\eta }^{\lambda }\right\}$ is also bounded in ${\mathbb{X}}_{\mathcal{C}}$, which implies that $({\eta }^{\lambda },{\beta }^{\lambda },{\theta }^{\lambda },h^{\lambda })$ is pre-compact in ${\mathbb{Y}}_{\mathcal{C}}\times {\mathbb{R}}^3$ for $\lambda \in (0,{\lambda }_{*})$. Therefore, there exists a subsequence $({\eta }^{{\lambda }_i},{\beta }^{{\lambda }_i},{\theta }^{{\lambda }_i},h^{{\lambda }_i})$, such that

$$({\eta }^{{\lambda }_i},{\beta }^{{\lambda }_i},{\theta }^{{\lambda }_i},h^{{\lambda }_i})\to ({\eta }^0,{\alpha }^0,{\theta }^0,h^0)\mathrm{in}{\mathbb{Y}}_{\mathcal{C}}\times {\mathbb{R}}^3\mathrm{and}{\lambda }_i\to 0,\mathrm{as}i\to \infty .$$

Taking the limit of the equation ${\mathbf{L}}^{-1}{f}_1({\eta }^{{\lambda }_i},{\beta }^{{\lambda }_i},{\theta }^{{\lambda }_i},h^{{\lambda }_i},{\lambda }_i)=0$ as $i\to \infty$, we see that

$$({\eta }^{{\lambda }_i},{\beta }^{{\lambda }_i},{\theta }^{{\lambda }_i},h^{{\lambda }_i})\to ({\eta }^0,{\alpha }^0,{\theta }^0,h^0)\mathrm{in}{\mathbb{X}}_{\mathcal{C}}\times {\mathbb{R}}^3\mathrm{and}{\lambda }_i\to 0,\mathrm{as}i\to \infty .$$

Moreover, since $F(\eta ,\beta ,\theta ,h,0)=0$ has a unique solution, we obtain

$$({\eta }^0,{\beta }^0,{\theta }^0,h^0)=({\eta }_0,{\beta }_0,{\theta }_0,h_0).$$

The proof is completed. □

Remark 1.

From Lemma 5, we conclude that for each $\lambda \in (0,{\lambda }_{*})$, the eigenvalue problem $\Lambda (\lambda ,i\nu ,\tau )\phi =0$ (where $\nu >0$, $\tau \ge 0$ and $\phi (\ne 0)\in {\mathbb{X}}_{\mathcal{C}}$) has a solution $(\nu ,\tau ,\phi )$ if and only if

$$\nu ={\nu }_{\lambda }=\lambda h_{\lambda },\phi =c{\phi }_{\lambda }\mathrm{and}\tau ={\tau }_n=\frac{{\theta }_{\lambda }+2n\pi }{{\nu }_{\lambda }},\mathrm{for}n=0,1,2,\dots .$$

Here, ${\phi }_{\lambda }={\beta }_{\lambda }+\lambda {\eta }_{\lambda }$.

For convenience, we will always assume $\lambda \in (0,{\lambda }_{*})$ in what follows. In fact, considering that further perturbation arguments are used, the interval of $\lambda$ might be smaller.

Lemma 6.

Assume that $0<\lambda <{\lambda }_{*}$. Then,

$$S_n\left(\lambda \right)={\int }_{\Omega }K\left(x\right){\phi }_{\lambda }^2\left[(1-{\tau }_n\lambda m\left(x\right)e^{-i{\theta }_{\lambda }})\right]\ne 0.$$

Proof. 

Since ${lim}_{\lambda \to 0}{\theta }_{\lambda }=\frac{\pi }{2}$, we have ${lim}_{\lambda \to 0}{\phi }_{\lambda }=1$, ${lim}_{\lambda \to 0}\lambda {\tau }_n=\frac{(\frac{\pi }{2}+2n\pi ){\int }_{\Omega }K\left(x\right)}{{\int }_{\Omega }m\left(x\right)K\left(x\right)}$.

Therefore, ${lim}_{\lambda \to 0}{S}_n\left(\lambda \right)={\int }_{\Omega }K\left(x\right)(1+i(\frac{\pi }{2}+2n\pi ))\ne 0$. □

Theorem 1.

For $\lambda \in (0,{\lambda }_{*})$, there exists a neighborhood of $({\tau }_n,i{\nu }_{\lambda },{\phi }_{\lambda })$ such that $A_{{\tau }_n}\left(\lambda \right)$ has a simple eigenvalue $\mu \left({\tau }_n\right)=a\left({\tau }_n\right)+ib\left({\tau }_n\right)$. Moreover, $a\left({\tau }_n\right)=0$, $b\left({\tau }_n\right)={\nu }_{\lambda }$.

Proof. 

We know that $\mathcal{N}[A_{{\tau }_n}\left(\lambda \right)-i{\nu }_{\lambda }]=\mathrm{span}\left\{e^{i{\nu }_{\lambda }\theta }{\phi }_{\lambda }\right\}$, where $\theta \in [-{\tau }_n,0]$. If ${\varphi }_1\in \mathcal{N}{[A_{{\tau }_n}\left(\lambda \right)-i{\nu }_{\lambda }]}^2$, then $\mathcal{N}[A_{{\tau }_n}\left(\lambda \right)-i{\nu }_{\lambda }]{\varphi }_1\in \mathcal{N}[A_{{\tau }_n}\left(\lambda \right)-i{\nu }_{\lambda }]=\mathrm{span}\left\{e^{i{\nu }_{\lambda }\theta }{\phi }_{\lambda }\right\}.$ Thus, there is a constant a such that

$$\mathcal{N}[A_{{\tau }_n}\left(\lambda \right)-i{\nu }_{\lambda }]{\varphi }_1=a{e}^{i{\nu }_{\lambda }\theta }{\phi }_{\lambda }.$$

Hence,

$$\left\{\begin{array}{c}{\dot{\varphi }}_1\left(\theta \right)=i{\nu }_{\lambda }{\varphi }_1\left(\theta \right)+a{e}^{i{\nu }_{\lambda }\theta }{\phi }_{\lambda },\theta \in [-{\tau }_n,0],\hfill \\ {\dot{\varphi }}_1\left(0\right)=\frac{1}{K\left(x\right)}\Delta {\varphi }_1\left(0\right)-\lambda m\left(x\right){\varphi }_1(-{\tau }_n).\hfill \end{array}\right.$$

(22)

The first equation of (22) yields

$$\left\{\begin{array}{c}{\varphi }_1\left(\theta \right)={\varphi }_1\left(0\right)e^{i{\nu }_{\lambda }\theta }+a\theta e^{i{\nu }_{\lambda }\theta }{\phi }_{\lambda },\hfill \\ {\dot{\varphi }}_1\left(0\right)=i{\nu }_{\lambda }{\varphi }_1\left(0\right)+a{\phi }_{\lambda }.\hfill \end{array}\right.$$

(23)

From (22) and (23), we have

$$\begin{array}{c}\hfill K\left(x\right)\Lambda (\lambda ,i{\nu }_{\lambda },{\tau }_n){\varphi }_1\left(0\right)=\Delta {\varphi }_1\left(0\right)-\lambda m\left(x\right)K\left(x\right){\varphi }_1\left(0\right)e^{-i{\theta }_{\lambda }}-i{\nu }_{\lambda }K\left(x\right){\varphi }_1\left(0\right)\\ \hfill =aK\left(x\right)({\phi }_{\lambda }-{\tau }_n\lambda m\left(x\right)e^{-i{\theta }_{\lambda }}{\phi }_{\lambda }).\end{array}$$

(24)

Hence,

$$\begin{array}{c}\hfill 0={\int }_{\Omega }{\varphi }_1\left(0\right)[K\left(x\right)\Lambda (\lambda ,i{\nu }_{\lambda },{\tau }_n){\phi }_{\lambda }]={\int }_{\Omega }{\phi }_{\lambda }[K\left(x\right)\Lambda (\lambda ,i{\nu }_{\lambda },{\tau }_n){\varphi }_1\left(0\right)]=\\ \hfill a{\int }_{\Omega }K\left(x\right){\phi }_{\lambda }^2(1-{\tau }_n\lambda m\left(x\right)e^{-i{\theta }_{\lambda }}).\end{array}$$

(25)

By Lemma 6, we obtain that $a=0$. This implies that $\mu =i{\nu }_{\lambda }$ is a simple eigenvalue of $A_{{\tau }_n}\left(\lambda \right)$. By the implicit function theorem, there is a neighborhood of $({\tau }_n,i{\nu }_{\lambda },{\phi }_{\lambda })$, such that $A_{\tau }\left(\lambda \right)$ has an eigenvalue $\mu \left(\tau \right)=a\left(\tau \right)+ib\left(\tau \right)$ for $\lambda \in (0,{\lambda }_{*})$. Moreover, $a\left({\tau }_n\right)=0$ and $b\left({\tau }_n\right)={\nu }_{\lambda }$. □

Now, we study the following transversality condition.

Theorem 2.

Assume that $\lambda \in (0,{\lambda }_{*})$. Then, we obtain

$$\mathcal{R}e\frac{d\mu \left({\tau }_n\right)}{d\tau }>0.$$

Proof. 

Since

$$\begin{array}{c}\hfill \Lambda (\lambda ,\mu \left(\tau \right),\tau )\phi \left(\tau \right)=\frac{1}{K\left(x\right)}\Delta \phi \left(\tau \right)-\lambda m\left(x\right)\phi \left(\tau \right)e^{-\mu \left(\tau \right)\tau }\\ \hfill -\mu \left(\tau \right)\phi \left(\tau \right)=0.\end{array}$$

(26)

Differentiating Equation (26) with respect to $\tau$ at $\tau ={\tau }_n$ yields

$$\begin{array}{c}\hfill \Lambda (\lambda ,i{\nu }_{\lambda },{\tau }_n)\frac{d\phi \left({\tau }_n\right)}{d\tau }+\frac{d\mu \left({\tau }_n\right)}{d\tau }(\lambda {\tau }_{n}m\left(x\right){\phi }_{\lambda }{e}^{-i{\theta }_{\lambda }}-{\phi }_{\lambda })\\ \hfill +i{\nu }_{\lambda }\lambda m\left(x\right){\phi }_{\lambda }{e}^{-i{\theta }_{\lambda }}=0.\end{array}$$

(27)

Then, multiplying Equation (27) by $K\left(x\right){\phi }_{\lambda }$ and integrating it over $\Omega$, we obtain

$$\begin{array}{c}\frac{d\mu \left({\tau }_n\right)}{d\tau }=\frac{{\int }_{\Omega }i{\nu }_{\lambda }\lambda m\left(x\right)K\left(x\right){\phi }_{\lambda }^2{e}^{-i{\theta }_{\lambda }}}{{\int }_{\Omega }K\left(x\right){\phi }_{\lambda }^2\left[(1-{\tau }_n\lambda m\left(x\right)e^{-i{\theta }_{\lambda }})\right]}\hfill \\ =\frac{1}{|S_n{\left(\lambda \right)|}^2}\left({\int }_{\Omega }i{\nu }_{\lambda }\lambda m\left(x\right)K\left(x\right){\phi }_{\lambda }^2{e}^{-i{\theta }_{\lambda }}\overline{{\int }_{\Omega }K\left(x\right){\phi }_{\lambda }^2}-i{\nu }_{\lambda }{\lambda }^2{\tau }_n{\left|{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }^2\right|}^2\right).\hfill \end{array}$$

(28)

Thus,

$$\begin{array}{cc}\underset{\lambda \to 0}{lim}\frac{1}{{\lambda }^2}\mathcal{R}e\frac{d\mu \left({\tau }_n\right)}{d\tau }\hfill & =\underset{\lambda \to 0}{lim}\frac{1}{{\lambda }^2}\mathcal{R}e\frac{1}{|S_n{\left(\lambda \right)|}^2}\left({\int }_{\Omega }i{\nu }_{\lambda }\lambda m\left(x\right)K\left(x\right){\phi }_{\lambda }^2{e}^{-i{\theta }_{\lambda }}\overline{{\int }_{\Omega }K\left(x\right){\phi }_{\lambda }^2}\right)\hfill \\ & =\frac{h_0{\int }_{\Omega }m\left(x\right)K\left(x\right){\int }_{\Omega }K\left(x\right)}{{lim}_{\lambda \to 0}{\left|S_n\left(\lambda \right)\right|}^2}=\frac{{\left({\int }_{\Omega }m\left(x\right)K\left(x\right)\right)}^2}{{lim}_{\lambda \to 0}{\left|S_n\left(\lambda \right)\right|}^2}>0.\hfill \end{array}$$

This completes the proof. □

On the basis of the lemmas and theorems above, we obtain the following results immediately.

Theorem 3.

The infinitesimal generator $A_{{\tau }_n}\left(\lambda \right)$, $\lambda \in (0,{\lambda }_{*})$, has exactly $2(n+1)$ eigenvalues with positive real parts when $\tau \in ({\tau }_n,{\tau }_{n+1}]$, where $n=0,1,2,\cdots$.

The above results and the theory on partial functional differential equations, explained in [28], lead us to the following local Hopf bifurcation theorem.

Theorem 4.

For fixed $\lambda \in (0,{\lambda }_{*})$, the positive steady state $\upsilon =1$ of (4) is locally asymptotically stable when $\tau \in [0,{\tau }_0)$ and is unstable when $\tau \in [{\tau }_0,\infty )$. Furthermore, system (4) exhibits a Hopf bifurcation at the positive steady state $\upsilon =1$ when $\tau ={\tau }_n$, $(n=0,1,2,\cdots )$.

3. The Direction of the Hopf Bifurcation

The results of Section 2 demonstrate that periodic solutions bifurcate from the positive steady-state solution $\upsilon =1$ as the delay $\tau$ passes through the critical value ${\tau }_n$, $n=0,1,2,\cdots$. In this section, with $\tau$ considered as a bifurcation parameter, we apply the normal form theory and center manifold reduction to analyze the direction of the Hopf bifurcation that occurs around the positive steady state.

We first transform the steady state $\upsilon =1$ of system (4) and the critical value ${\tau }_n$ into the origin via the translations $V(t,x)=\upsilon (t,x)-1$, $t=\tau \widehat{t}$ and $\tau ={\tau }_n+\varrho$. Then, dropping the tilde sign, system (4) becomes

$$\frac{dV\left(t\right)}{dt}=\frac{{\tau }_n}{K\left(x\right)}\Delta V\left(t\right)-{\tau }_n\lambda m\left(x\right)V(t-1)+F(V_t,\varrho ),$$

(29)

where

$$F(V_t,\varrho )=\varrho \frac{{\tau }_n}{K\left(x\right)}\Delta V\left(t\right)-\varrho \lambda m\left(x\right)V(t-1)-({\tau }_n+\varrho )\lambda m\left(x\right)V\left(t\right)V(t-1).$$

Similar to Section 2, we define that ${\mathbb{A}}_{{\tau }_n}\left(\lambda \right)$ is the infinitesimal generator of the linearized Equation (29). Thus, we have

$${\mathbb{A}}_{{\tau }_n}\left(\lambda \right)\phi =\dot{\phi },$$

with

$$\mathfrak{D}\left({\mathbb{A}}_{{\tau }_n}\left(\lambda \right)\right)=\{\phi \in C_{\mathcal{C}}\cap C_{\mathcal{C}}^1,\mathrm{such}\mathrm{that}\dot{\phi }\left(0\right)=\frac{{\tau }_n}{K\left(x\right)}\Delta \phi \left(0\right)-\lambda {\tau }_{n}m\left(x\right)\phi (-1)\},$$

where $C_{\mathcal{C}}^1=C^1([-1,0],{\mathbb{Y}}_{\mathcal{C}})$. Let us also define

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

Then, (29) can be rewritten as

$$\frac{d{V}_t}{dt}={\mathbb{A}}_{{\tau }_n}\left(\lambda \right)V_t+\mathbb{F}(V_t,\varrho )\left(\theta \right).$$

(30)

It follows from the previous section that ${\mathbb{A}}_{{\tau }_n}$ has only one pair of simple purely imaginary eigenvalues $\pm i{\nu }_{\lambda }{\tau }_n$. The eigenfunction associated with $i{\nu }_{\lambda }{\tau }_n$ (respectively, $-i{\nu }_{\lambda }{\tau }_n$) is $\gamma \left(\theta \right)={\phi }_{\lambda }{e}^{i{\nu }_{\lambda }{\tau }_n\theta }$ (respectively, $\overline{\gamma }\left(\theta \right)={\overline{\phi }}_{\lambda }{e}^{-i{\nu }_{\lambda }{\tau }_n\theta }$) for $\theta \in [-1,0]$, where ${\phi }_{\lambda }$ is defined as in Remark 1.

Based on [27], we introduce the following bilinear form: for $\Phi \in C_{\mathcal{C}}$ and $\tilde{\Phi }\in C_{\mathcal{C}}^{*}$,

$$\langle \langle \Phi ,\tilde{\Phi }\rangle \rangle ={\langle \Phi \left(0\right),\tilde{\Phi }\left(0\right)\rangle }_1-\lambda {\tau }_n{\int }_{-1}^0{\langle \Phi (s+1),m\left(x\right)\tilde{\Phi }\left(s\right)\rangle }_{1}ds,$$

where ${\langle u,v\rangle }_1={\int }_{\Omega }K\left(x\right)\overline{u}\left(x\right)v\left(x\right)dx$.

The following lemma is concerned with the formal adjoint operator of ${\mathbb{A}}_{{\tau }_n}$ under the bilinear product above.

Lemma 7.

Let

$${\mathbb{A}}_{{\tau }_n}^{*}\widehat{\Phi }\left(s\right)=\left\{\begin{array}{cc}\dot{\widehat{\Phi }}\left(s\right),& s\in (0,1],\hfill \\ \frac{{\tau }_n}{K\left(x\right)}\Delta \widehat{\Phi }\left(0\right)-\lambda {\tau }_{n}m\left(x\right)\widehat{\Phi }\left(1\right),& s=0,\hfill \end{array}\right.$$

for $\widehat{\Phi }\in C_{\mathcal{C}}^{*}\cap C_{\mathcal{C}}^{1*}$, where $C_{\mathcal{C}}^{*}=C([0,1],{\mathbb{Y}}_{\mathcal{C}})$. Then, ${\mathbb{A}}_{{\tau }_n}$ and ${\mathbb{A}}_{{\tau }_n}^{*}$ satisfy

$$\langle \langle \widehat{\Phi },{\mathbb{A}}_{{\tau }_n}\Phi \rangle \rangle =\langle \langle {\mathbb{A}}_{{\tau }_n}^{*}\widehat{\Phi },\Phi \rangle \rangle$$

Proof. 

For $\Phi \in \mathfrak{D}\left({\mathbb{A}}_{{\tau }_n}\right)$ and $\widehat{\Phi }\in \mathfrak{D}\left({\mathbb{A}}_{{\tau }_n}^{*}\right)$, we have

$$\begin{array}{c}\langle \langle \widehat{\Phi },{\mathbb{A}}_{{\tau }_n}\Phi \rangle \rangle ={\langle \widehat{\Phi }\left(0\right),{\mathbb{A}}_{{\tau }_n}\Phi \left(0\right)\rangle }_1-\lambda {\tau }_n{\int }_{-1}^0{\langle \widehat{\Phi }(s+1),m\left(x\right){\mathbb{A}}_{{\tau }_n}\Phi \left(s\right)\rangle }_{1}ds\\ ={\langle \widehat{\Phi }\left(0\right),\frac{{\tau }_n}{K\left(x\right)}\Delta \Phi \left(0\right)-\lambda {\tau }_{n}m\left(x\right)\Phi (-1)\rangle }_1-\lambda {\tau }_n{\int }_{-1}^0{\langle \widehat{\Phi }(s+1),m\left(x\right)\dot{\Phi }\left(s\right)\rangle }_{1}ds\\ ={\langle \widehat{\Phi }\left(0\right),\frac{{\tau }_n}{K\left(x\right)}\Delta \Phi \left(0\right)\rangle }_1-\lambda {\tau }_n{\left[{\langle \widehat{\Phi }(s+1),m\left(x\right)\Phi \left(s\right)\rangle }_1\right]}_{-1}^0\\ +{\langle \widehat{\Phi }\left(0\right),-\lambda {\tau }_{n}m\left(x\right)\Phi (-1)\rangle }_1+\lambda {\tau }_n{\int }_{-1}^0{\langle \dot{\widehat{\Phi }}(s+1),m\left(x\right)\Phi \left(s\right)\rangle }_{1}ds\\ ={\langle ({\mathbb{A}}_{{\tau }_n}^{*}\widehat{\Phi })\left(0\right),\Phi \left(0\right)\rangle }_1-\lambda {\tau }_n{\int }_{-1}^0{\langle -\dot{\widehat{\Phi }}(s+1),m\left(x\right)\Phi \left(s\right)\rangle }_{1}ds\\ =\langle \langle {\mathbb{A}}_{{\tau }_n}^{*}\widehat{\Phi },\Phi \rangle \rangle .\end{array}$$

□

Similarly, we see that ${\mathbb{A}}_{{\tau }_n}^{*}$ has only one pair of simple purely imaginary eigenvalues $\pm i{\nu }_{\lambda }{\tau }_n$. The eigenfunction associated with $-i{\nu }_{\lambda }{\tau }_n$ (respectively, $i{\nu }_{\lambda }{\tau }_n$) is $\widehat{\gamma }\left(s\right)={\overline{\phi }}_{\lambda }{e}^{i{\nu }_{\lambda }{\tau }_{n}s}$ (respectively, $\overline{\widehat{\gamma }}\left(s\right)={\phi }_{\lambda }{e}^{-i{\nu }_{\lambda }{\tau }_{n}s}$) for $s\in [0,1],$ where ${\phi }_{\lambda }$ is defined as in Remark 1. The center subspace of () is $P=\mathrm{span}\{\gamma \left(\theta \right),\overline{\gamma }\left(\theta \right)\}$. Furthermore, the basis of the eigenfunction space of the adjoint operator ${\mathbb{A}}_{{\tau }_n}^{*}$ associated with the eigenvalues $\pm i{\nu }_{\lambda }{\tau }_n$ is $P^{*}=\mathrm{span}\{\widehat{\gamma }\left(s\right),\overline{\widehat{\gamma }}\left(s\right)\}$. Meanwhile, $P^{*}$ is the formal adjoint subspace of P. As usual, $C_{\mathcal{C}}$ can be decomposed as follows:

$$C_{\mathcal{C}}=P⨁Q,$$

where

$$Q=\{\psi \in C_{\mathcal{C}}|\langle \langle \widehat{\psi },\psi \rangle \rangle =0,\mathrm{for}\mathrm{all}\widehat{\psi }\in P^{*}\}.$$

Set

$${\Phi }_{\gamma }=(\gamma \left(\theta \right),\overline{\gamma }\left(\theta \right))\mathrm{for}\theta \in [-1,0],\mathrm{and}{\Psi }_{\widehat{\gamma }}=(\frac{\widehat{\gamma }\left(s\right)}{S_n\left(\lambda \right)},\frac{\overline{\widehat{\gamma }}\left(s\right)}{{\overline{S}}_n\left(\lambda \right)})\mathrm{for}s\in [0,1].$$

One can check that $\langle \langle {\Phi }_{\gamma },{\Psi }_{\widehat{\gamma }}\rangle \rangle =I$, where I is the identity matrix in ${\mathbb{R}}^{2\times 2}$. For the bifurcation direction and stability, since the formulas to be built are all only relative to $\varrho =0$, we let $\varrho =0$ in Equation (30) and define

$$z\left(t\right)=\frac{1}{S_n\left(\lambda \right)}\langle \langle \widehat{\gamma },V_t\rangle \rangle .$$

Let

$$W(z,\overline{z})=W_{20}\left(\theta \right)\frac{z^2}{2}+W_{11}\left(\theta \right)z\overline{z}+W_{02}\left(\theta \right)\frac{{\overline{z}}^2}{2}+{\mathbf{O}\left(\right|z|}^3)$$

be the center manifold with a range Q. Then, the flow of (30) on the center manifold can be written as follows:

$$V_t={\Phi }_{\gamma }·{(z\left(t\right),\overline{z}\left(t\right))}^T+W(z\left(t\right),\overline{z}\left(t\right)).$$

Since $\varrho =0$, we have

$$\dot{z}\left(t\right)=\frac{1}{S_n\left(\lambda \right)}\frac{d\langle \langle \widehat{\gamma },V_t\rangle \rangle }{dt}=i{\nu }_{\lambda }{\tau }_{n}z\left(t\right)+g(z\left(t\right),\overline{z}\left(t\right)),$$

where

$$\begin{array}{cc}g(z\left(t\right),\overline{z}\left(t\right))\hfill & =\frac{1}{S_n\left(\lambda \right)}{\langle \widehat{\gamma }\left(0\right),F(V_t,0)\rangle }_1\hfill \\ & =\frac{1}{S_n\left(\lambda \right)}{\langle \widehat{\gamma }\left(0\right),F({\Phi }_{\gamma }·{(z\left(t\right),\overline{z}\left(t\right))}^T+W(z\left(t\right),\overline{z}\left(t\right)),0)\rangle }_1\hfill \\ & :=g_{20}\frac{z^2}{2}+g_{11}z\overline{z}+g_{02}\frac{{\overline{z}}^2}{2}+g_{21}\frac{z^2\overline{z}}{2}+\cdots .\hfill \end{array}$$

A straight forward calculation leads to

$$\begin{array}{cc}{g}_{20}=\hfill & -\frac{2\lambda {\tau }_n}{S_n\left(\lambda \right)}{e}^{-i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }^{3}dx,\hfill \\ g_{11}=\hfill & -\left[\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}(e^{i{\nu }_{\lambda }{\tau }_n}+e^{-i{\nu }_{\lambda }{\tau }_n})\right]{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }\mid {\phi }_{\lambda }{\mid }^{2}dx,\hfill \\ g_{02}=\hfill & -\frac{2\lambda {\tau }_n}{S_n\left(\lambda \right)}{e}^{i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }{\overline{\phi }}_{\lambda }^{2}dx,\hfill \\ \mathrm{and}\hfill \\ g_{21}=\hfill & -\frac{2\lambda {\tau }_n}{S_n\left(\lambda \right)}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }^2{W}_{11}(-1)dx\hfill \\ \hfill & -\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}{\int }_{\Omega }m\left(x\right)K\left(x\right)\mid {\phi }_{\lambda }{\mid }^2{W}_{20}(-1)dx\hfill \\ \hfill & -\frac{2\lambda {\tau }_n}{S_n\left(\lambda \right)}{e}^{-i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }^2{W}_{11}\left(0\right)dx\hfill \\ \hfill & -\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}{e}^{i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right)\mid {\phi }_{\lambda }{\mid }^2{W}_{20}\left(0\right)dx.\hfill \end{array}$$

(31)

In order to obtain the stability of bifurcating periodic orbits and the bifurcation directions, we compute the quantities as follows:

$${\mathcal{C}}_1\left(0\right)=\frac{i}{2{\omega }_{\lambda }{\tau }_n}\left(g_{11}{g}_{20}-2{\left|g_{11}\right|}^2-\frac{|g_{02}{|}^2}{3}\right)+\frac{g_{21}}{2},$$

$${\mu }_2=-\frac{\mathcal{R}e\left\{{\mathcal{C}}_1\left(0\right)\right\}}{\mathcal{R}e\left\{{\mu }^{\prime }\left({\tau }_n\right)\right\}},{\beta }_2=2\mathcal{R}e\left\{{\mathcal{C}}_1\left(0\right)\right\},$$

and

$$T_2=-\frac{\mathcal{I}m\left\{{\mathcal{C}}_1\left(0\right)\right\}+{\mu }_2\mathcal{I}m\left\{{\mu }^{\prime }\left({\tau }_n\right)\right\}}{{\tau }_n}.$$

Motivated by [28,29], we have the following results.

Lemma 8.

For $\lambda \in (0,{\lambda }_{*})$, Equation (4) creates the Hopf bifurcation at the positive steady state $\upsilon =1$; moreover,

(i) 

${\mu }_2$ determines the direction of the Hopf bifurcation: If ${\mu }_2>0$ (${\mu }_2<0$), then the bifurcating periodic solutions exist for $\tau >{\tau }_n$ (resp. $\tau <{\tau }_n$), and the bifurcation is called forward (resp. backward).

(ii) 

${\beta }_2$ determines the stability of bifurcating periodic solutions: The bifurcating periodic solutions are orbitally asymptotically stable (resp. unstable) if ${\beta }_2<0$ (resp. ${\beta }_2>0$).

(iii) 

$T_2$ determines the period of the bifurcating periodic solutions: The period increases (resp. decreases) if $T_2>0$ (resp. $T_2<0$).

To compute $g_{21}$, we need to figure out $W_{11}\left(\theta \right)$ and $W_{20}\left(\theta \right)$. Note that $W(z,\overline{z})$ satisfies

$$\dot{W}={\mathbb{A}}_{{\tau }_n}W+H_{20}\frac{z^2}{2}+H_{11}z\overline{z}+H_{02}\frac{{\overline{z}}^2}{2}+\cdots .$$

(32)

We see that $W_{11}\left(\theta \right)$, $W_{20}\left(\theta \right)$, $H_{11}\left(\theta \right)$, and $H_{20}\left(\theta \right)$ satisfy

$$\left\{\begin{array}{c}-{\mathbb{A}}_{{\tau }_n}{W}_{11}\left(\theta \right)=H_{11}\left(\theta \right),\hfill \\ (2i{\nu }_{\lambda }{\tau }_n-{\mathbb{A}}_{{\tau }_n})W_{20}\left(\theta \right)=H_{20}\left(\theta \right),\hfill \\ H_{11}\left(\theta \right)=-(g_{11}\gamma \left(\theta \right)+{\overline{g}}_{11}\overline{\gamma }\left(\theta \right)),\hfill & \theta \in [-1,0),\hfill \\ H_{20}\left(\theta \right)=-(g_{20}\gamma \left(\theta \right)+{\overline{g}}_{02}\overline{\gamma }\left(\theta \right)),\hfill & \theta \in [-1,0),\hfill \\ H_{11}\left(0\right)=-(g_{11}\gamma \left(0\right)+{\overline{g}}_{11}\overline{\gamma }\left(0\right))-\lambda {\tau }_{n}m\left(x\right)(e^{i{\nu }_{\lambda }{\tau }_n}+e^{-i{\nu }_{\lambda }{\tau }_n}){\left|{\phi }_{\lambda }\right|}^2,\hfill \\ H_{20}\left(0\right)=-(g_{20}\gamma \left(0\right)+{\overline{g}}_{02}\overline{\gamma }\left(0\right))-2\lambda {\tau }_{n}m\left(x\right)e^{-i{\nu }_{\lambda }{\tau }_n}{\phi }_{\lambda }^2.\hfill \end{array}\right.$$

(33)

From (33), we have

$$\left\{\begin{array}{c}{W}_{20}\left(\theta \right)=\frac{i{g}_{20}}{{\nu }_{\lambda }{\tau }_n}\gamma \left(\theta \right)+\frac{i{\overline{g}}_{02}}{3{\nu }_{\lambda }{\tau }_n}\overline{\gamma }\left(\theta \right)+E{e}^{2i{\nu }_{\lambda }{\tau }_n\theta },\hfill \\ W_{11}\left(\theta \right)=\frac{i{g}_{11}}{{\nu }_{\lambda }{\tau }_n}\gamma \left(\theta \right)+\frac{i{\overline{g}}_{11}}{{\nu }_{\lambda }{\tau }_n}\overline{\gamma }\left(\theta \right)+F.\hfill \end{array}\right.$$

(34)

In what follows, we solve for E and F. From (33) and (34), together with the definition of ${\mathbb{A}}_{{\tau }_n}$, we see that E and F satisfy

$$\left\{\begin{array}{c}\Lambda (\lambda ,2i{\nu }_{\lambda },{\tau }_n)E=2\lambda m\left(x\right)e^{-i{\nu }_{\lambda }{\tau }_n}{\phi }_{\lambda }^2,\hfill \\ \Lambda (\lambda ,0,{\tau }_n)F=\lambda m\left(x\right)(e^{i{\nu }_{\lambda }{\tau }_n}+e^{-i{\nu }_{\lambda }{\tau }_n}){\left|{\phi }_{\lambda }\right|}^2.\hfill \end{array}\right.$$

(35)

Lemma 9.

Assume that E and F satisfy (35). Then, $E=c_{\lambda }+{\vartheta }_{\lambda }$, $F={\tilde{\vartheta }}_{\lambda }$, where ${\vartheta }_{\lambda }$ and ${\tilde{\vartheta }}_{\lambda }$ satisfy $\langle 1,{\vartheta }_{\lambda }\rangle =0$, ${lim}_{\lambda \to 0}{∥{\vartheta }_{\lambda }∥}_{{\mathbb{Y}}_c}=0$, ${lim}_{\lambda \to 0}{∥{\tilde{\vartheta }}_{\lambda }∥}_{{\mathbb{Y}}_c}=0$. Moreover, the constant $c_{\lambda }$ satisfies ${lim}_{\lambda \to 0}{c}_{\lambda }=\frac{2i}{(2i-1)}$.

Proof. 

We only give the estimate for E since the estimate for F can be obtained similarly. Substituting $E=c_{\lambda }+{\vartheta }_{\lambda }$ into the first equation of system (35), we get

$$\frac{\Delta (c_{\lambda }+{\vartheta }_{\lambda })}{K\left(x\right)}-\lambda m\left(x\right)(c_{\lambda }+{\vartheta }_{\lambda })e^{-2i{\nu }_{\lambda }\tau }-2i{\nu }_{\lambda }(c_{\lambda }+{\vartheta }_{\lambda })=2\lambda m\left(x\right)e^{-i{\nu }_{\lambda }{\tau }_n}{\phi }_{\lambda }^2.$$

Hence,

$$\begin{array}{c}\hfill \frac{\Delta {\vartheta }_{\lambda }}{K\left(x\right)}-\lambda m\left(x\right)(c_{\lambda }+{\vartheta }_{\lambda })e^{-2i{\nu }_{\lambda }{\tau }_n}-2i{\nu }_{\lambda }(c_{\lambda }+{\vartheta }_{\lambda })=2\lambda m\left(x\right)e^{-i{\nu }_{\lambda }{\tau }_n}{\phi }^2.\end{array}$$

(36)

Multiplying both sides of the Equation (36) by $K\left(x\right)$ and then integrating the result over $\Omega$,

$$\begin{array}{c}\lambda c_{\lambda }{e}^{-2i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right)dx+2i{\nu }_{\lambda }{c}_{\lambda }{\int }_{\Omega }K\left(x\right)dx\hfill \\ =-\lambda e^{-2i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\vartheta }_{\lambda }dx-2i{\nu }_{\lambda }{\int }_{\Omega }K\left(x\right){\vartheta }_{\lambda }dx\hfill \\ -2\lambda e^{-i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }^{2}dx.\hfill \end{array}$$

(37)

Multiplying this by $K\left(x\right){\overline{\vartheta }}_{\lambda }$ and integrating the result over $\Omega$, we have

$$\begin{array}{c}\langle {\vartheta }_{\lambda },\Delta {\vartheta }_{\lambda }\rangle -\lambda c_{\lambda }{e}^{-2i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\overline{\vartheta }}_{\lambda }dx-2i{\nu }_{\lambda }{c}_{\lambda }{\int }_{\Omega }K\left(x\right){\overline{\vartheta }}_{\lambda }dx\hfill \\ =\lambda e^{-2i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right)|{\vartheta }_{\lambda }{|}^{2}dx+2i{\nu }_{\lambda }{\int }_{\Omega }K\left(x\right){\left|{\vartheta }_{\lambda }\right|}^{2}dx\hfill \\ +2\lambda e^{-i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }^2{\overline{\vartheta }}_{\lambda }dx.\hfill \end{array}$$

(38)

From the lemma above, we then get

$${\phi }_{\lambda }\to 1,\frac{{\nu }_{\lambda }}{\lambda }\to h_0\mathrm{and}{\nu }_{\lambda }{\tau }_n\to \frac{\pi }{2}+2n\pi ,\mathrm{as}\lambda \to 0.$$

Hence, it follows from Equation (37) that there exist constants ${\lambda }_{*}>0,$$M_0$, and $M_1>0$, such that for any $\lambda \in (0,{\lambda }_{*})$,

$$\begin{array}{c}\hfill |c_{\lambda }|\le M_0{∥{\vartheta }_{\lambda }∥}_{{\mathbb{Y}}_c}+M_1.\end{array}$$

(39)

Furthermore, from Equations (38) and (39), combined with Lemma 3, we obtain that there exist constants $M_2,M_3>0$, such that for any $\lambda \in (0,{\lambda }_{*})$,

$$\begin{array}{c}\hfill {\gamma }_2·{∥{\vartheta }_{\lambda }∥}_{{\mathbb{Y}}_c}^2\le \lambda M_2{∥{\vartheta }_{\lambda }∥}_{{\mathbb{Y}}_c}^2+\lambda M_3{∥{\vartheta }_{\lambda }∥}_{{\mathbb{Y}}_c}.\end{array}$$

(40)

Therefore, we have ${lim}_{\lambda \to 0}{∥{\vartheta }_{\lambda }∥}_{{\mathbb{Y}}_c}=0$. This, together with Equation (37), implies that

$$\underset{\lambda \to 0}{lim}{c}_{\lambda }=\frac{2i}{(2i-1)}.$$

□

With arguments similar to those in [14,27], one can obtain the following result.

Theorem 5.

For $\lambda \in (0,{\lambda }_{*})$, Equation (4) creates the Hopf bifurcation at the positive steady state $\upsilon =1$, near ${\tau }_n$ ($n\in \mathbb{N}\cup \left\{0\right\}$). Moreover, the direction of the Hopf bifurcation at $\tau ={\tau }_n$ is forward, and the bifurcating periodic solution from $\tau ={\tau }_0$ is orbitally asymptotically stable.

Proof. 

Since ${lim}_{\lambda \to 0}\lambda {\tau }_n=\frac{(\frac{\pi }{2}+2n\pi ){\int }_{\Omega }K\left(x\right)}{{\int }_{\Omega }m\left(x\right)K\left(x\right)}$, ${lim}_{\lambda \to 0}{S}_n\left(\lambda \right)={\int }_{\Omega }K\left(x\right)(1+i(\frac{\pi }{2}+2n\pi ))\ne 0$. Then,

$$\begin{array}{c}\hfill \underset{\lambda \to 0}{lim}{g}_{20}=-\frac{2\lambda {\tau }_n}{S_n\left(\lambda \right)}{e}^{-i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }^{3}dx\\ \hfill =\frac{i(\pi +4n\pi )}{1+i(\frac{\pi }{2}+2n\pi )},\end{array}$$

(41)

$$\begin{array}{c}\hfill \underset{\lambda \to 0}{lim}{g}_{11}=-\left[\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}(e^{i{\nu }_{\lambda }{\tau }_n}+e^{-i{\nu }_{\lambda }{\tau }_n})\right]{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }\mid {\phi }_{\lambda }{\mid }^{2}dx=0,\end{array}$$

(42)

$$\begin{array}{c}\hfill \underset{\lambda \to 0}{lim}{g}_{02}=-\frac{2\lambda {\tau }_n}{S_n\left(\lambda \right)}{e}^{i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }{\overline{\phi }}_{\lambda }^{2}dx=-\frac{i(\pi +4n\pi )}{1+i(\frac{\pi }{2}+2n\pi )}.\end{array}$$

(43)

Former conclusions give

$$\begin{array}{c}\hfill W_{20}\left(\theta \right)=\frac{i{g}_{20}}{{\nu }_{\lambda }{\tau }_n}\gamma \left(\theta \right)+\frac{i{\overline{g}}_{02}}{3{\nu }_{\lambda }{\tau }_n}\overline{\gamma }\left(\theta \right)+E{e}^{2i{\nu }_{\lambda }{\tau }_n\theta },\\ \hfill W_{11}\left(\theta \right)=\frac{i{g}_{11}}{{\nu }_{\lambda }{\tau }_n}\gamma \left(\theta \right)+\frac{i{\overline{g}}_{11}}{{\nu }_{\lambda }{\tau }_n}\overline{\gamma }\left(\theta \right)+F\end{array}$$

(44)

and $\gamma \left(\theta \right)={\phi }_{\lambda }{e}^{i{\nu }_{\lambda }{\tau }_n\theta }$, $\overline{\gamma }\left(\theta \right)={\overline{\phi }}_{\lambda }{e}^{-i{\nu }_{\lambda }{\tau }_n\theta }.$ Hence,

$$\begin{array}{c}\hfill W_{20}(-1)=\frac{i{g}_{20}}{{\nu }_{\lambda }{\tau }_n}{\phi }_{\lambda }{e}^{-i{\nu }_{\lambda }{\tau }_n}+\frac{i{\overline{g}}_{02}}{3{\nu }_{\lambda }{\tau }_n}{\overline{\phi }}_{\lambda }{e}^{i{\nu }_{\lambda }{\tau }_n}+(c_{\lambda }+{\vartheta }_{\lambda })e^{-2i{\nu }_{\lambda }{\tau }_n},\\ \hfill e^{i{\nu }_{\lambda }{\tau }_n}{W}_{20}\left(0\right)=\frac{i{g}_{20}}{{\nu }_{\lambda }{\tau }_n}{\phi }_{\lambda }{e}^{i{\nu }_{\lambda }{\tau }_n}+\frac{i{\overline{g}}_{02}}{3{\nu }_{\lambda }{\tau }_n}{\overline{\phi }}_{\lambda }{e}^{i{\nu }_{\lambda }{\tau }_n}+(c_{\lambda }+{\vartheta }_{\lambda })e^{i{\nu }_{\lambda }{\tau }_n},\end{array}$$

(45)

$$\mathrm{where}\left\{\begin{array}{c}\underset{\lambda \to 0}{lim}\frac{i{g}_{20}}{{\nu }_{\lambda }{\tau }_n}{\phi }_{\lambda }{e}^{-i{\nu }_{\lambda }{\tau }_n}=-\underset{\lambda \to 0}{lim}\frac{i{g}_{20}}{{\nu }_{\lambda }{\tau }_n}{\phi }_{\lambda }{e}^{i{\nu }_{\lambda }{\tau }_n}=\frac{(\pi +4n\pi )+i2}{1+\frac{\pi }{2}+2n\pi },\hfill \\ \underset{\lambda \to 0}{lim}\frac{i{\overline{g}}_{02}}{3{\nu }_{\lambda }{\tau }_n}\overline{\phi }{e}^{i{\nu }_{\lambda }{\tau }_n}=\frac{(\pi +4n\pi )-i2}{3(1+\frac{\pi }{2}+2n\pi )},\hfill \\ \underset{\lambda \to 0}{lim}(c_{\lambda }+{\vartheta }_{\lambda })e^{-2i{\nu }_{\lambda }{\tau }_n}=\frac{-4+2i}{5},\hfill \\ \underset{\lambda \to 0}{lim}(c_{\lambda }+{\vartheta }_{\lambda })e^{i{\nu }_{\lambda }{\tau }_n}=\frac{i4+2}{5}.\hfill \end{array}\right.$$

(46)

Then,

$$\begin{array}{c}\underset{\lambda \to 0}{lim}{g}_{21}=-\frac{2\lambda {\tau }_n}{S_n\left(\lambda \right)}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }^2{W}_{11}(-1)dx-\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}{\int }_{\Omega }m\left(x\right)K\left(x\right)\mid {\phi }_{\lambda }{\mid }^2{W}_{20}(-1)dx\hfill \\ -\frac{2\lambda {\tau }_n}{S_n\left(\lambda \right)}{e}^{-i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right){\phi }_{\lambda }^2{W}_{11}\left(0\right)dx-\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}{e}^{i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right)\mid {\phi }_{\lambda }{\mid }^2{W}_{20}\left(0\right)dx\hfill \\ =\underset{\lambda \to 0}{lim}\left(-\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}{\int }_{\Omega }m\left(x\right)K\left(x\right)\mid {\phi }_{\lambda }{\mid }^2{W}_{20}(-1)dx\right)\hfill \\ -\underset{\lambda \to 0}{lim}\left(\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}{e}^{i{\nu }_{\lambda }{\tau }_n}{\int }_{\Omega }m\left(x\right)K\left(x\right)\mid {\phi }_{\lambda }{\mid }^2{W}_{20}\left(0\right)dx\right)\hfill \\ =\underset{\lambda \to 0}{lim}\left[-\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}{\int }_{\Omega }m\left(x\right)K\left(x\right)\left(\frac{i2{\overline{g}}_{02}}{3{\nu }_{\lambda }{\tau }_n}\overline{\phi }{e}^{i{\nu }_{\lambda }{\tau }_n}+(c_{\lambda }+{\vartheta }_{\lambda })e^{-2i{\nu }_{\lambda }{\tau }_n}+(c_{\lambda }+{\vartheta }_{\lambda })e^{i{\nu }_{\lambda }{\tau }_n}\right)\right]\hfill \\ =\left[-\frac{\lambda {\tau }_n}{S_n\left(\lambda \right)}{\int }_{\Omega }m\left(x\right)K\left(x\right)\left(\frac{(2\pi +8n\pi )-i4}{3(1+\frac{\pi }{2}+2n\pi )}+\frac{-4+2i}{5}+\frac{i4+2}{5}\right)\right]\hfill \\ =-\frac{\frac{\pi }{2}+2n\pi }{1+i(\frac{\pi }{2}+2n\pi )}\left(\frac{(2\pi +8n\pi )-i4}{3(1+\frac{\pi }{2}+2n\pi )}+\frac{i6-2}{5}\right).\hfill \end{array}$$

We can verify that ${lim}_{\lambda \to 0}\mathcal{R}e\left\{g_{21}\right\}<0$. This in turn implies that ${lim}_{\lambda \to 0}\mathcal{R}e\left\{{\mathcal{C}}_1\left(0\right)\right\}<0$. Using Lemma 8, the proof of this theorem is complete. □

4. The Numerical Simulations and Conclusions

In this paper, we studied a delayed reaction–diffusion equation that has a carrying capacity-driven diffusion. The local asymptotic stability of the positive equilibrium solution $\upsilon =\frac{u}{K\left(x\right)}=1$ was considered by studying the associated eigenvalue problem. We determined a threshold value ${\tau }_0$ for the time delay $\tau$. It was demonstrated that the positive equilibrium solution $\upsilon =1$ is asymptotically stable when $\tau <{\tau }_0$, and unstable when $\tau >{\tau }_0$. In addition, we also showed that the system can undergo the Hopf bifurcation when $\tau$ passes through a sequence of critical values ${\tau }_n$.

In this section, some numerical simulations for model (4) are carried out to validate our theoretical results. It follows from Figure 1 and Figure 2 that the unique positive steady state $\upsilon =1$ is stable when $\tau \in [0,{\tau }_0)$, unstable when $\tau \in [{\tau }_0,+\infty )$. From model (4), a Hopf bifurcation occurs at $\upsilon =1$ when $\tau ={\tau }_0$ (See Figure 3).