Hyphanthia cunea (also called Fall webworm, H. cunea for short) is a kind of quarantine pest that poses a danger to forestry and agriculture. H. cunea originated in North America, and it was accidentally introduced to central Europe and eastern Asia in the early 1940s [1] . Now it has spread to more than 30 countries, including South Africa, Argentina, China, Belarus, etc., and has been classified as an invasive and quarantine insect by organizations such as APPPC, COSA VE and EAEU EPPO (from Global Data Base: https://gd.eppo.int/taxon/HYPHCU/distribution ). H. cunea has more than 600 recorded host plant species, which are widely distributed in several countries such as the United States, Japan, Korea, and China [2] . Due to its wide distribution across various host plants, H. cunea causes substantial economic losses globally. With the onset of global warming, H. cunea is increasingly spreading toward higher latitudes and elevations. Since it was first discovered in 1979 in Dandong City, Liaoning Province, China [3] , the pest has spread to the Yangtze River basin, involving more than 10 provinces (cities and autonomous regions).

Both larvae and adults of the H. cunea will gnaw on leaves and even trunks. The species is highly reproductive, spreads rapidly, and causes significant damage to forests [4] [5] . H. cunea can threaten the survival of local species, diminish biodiversity and endanger ecological security. Figure 1 shows the condition of trees infected by the H. cunea. The leaves of the trees become transparent after being eaten by H. cunea (as in Figure 1(a) and Figure 1(b) ), and the trunks of the trees turn yellow and wilt after being eaten (as in Figure 1(c) and Figure 1(d) , Figure 1(c) is a partially enlarged image of Figure 1(d) ) (The four images in Figure 1 are from: https://baike.baidu.com ).

The infestation of H. cunea mainly has the following control methods:

(1) Physical control method: This method primarily involves manual interventions such as cutting net screens, capturing adults, and digging pupae manually.

(2) Chemical pharmaceutical control method: This method mainly involves spraying susceptible trees with pesticides.

(3) Biological control method: This method involves releasing natural enemies of H. cunea, with the most effective and widely used being C. cunea.

The first two methods are effective, however, they either require a lot of manpower and material resources or can potentially damage the ecosystem. Biological control is the most environmentally friendly and economical method. In order to explore the biological control method of the H. cunea, Yang et al. [3] investigated the parasitic natural enemies of the H. cunea continuously from 1984 to 1986, and finally found a new natural enemy in the pupae of the H. cunea in Shanxi province. C. cunea is a dominant parasitic wasp with high parasitism and high fecundity. It is sensitive to lepidopteran pests such as H. cunea. C. cunea can insert its ovipositor into the pupae of insect pests, where it develops and grows, consuming all the nutrients and eventually causing the demise of H. cunea. H. cunea progresses through four life stages: egg, larva, pupa, and adult. The female wasps can lay eggs on the same day after eclosion under suitable temperature conditions. Female and male wasps mate within the host pupae and then emerge to seek a new host pupa for egg-laying. The release rate of C. cunea is determined by the density of H. cunea [6] . Figure 2 shows the reproductive mechanism of the coexistence of the H. cunea and the C. cunea in nature, with the red arrow indicating the artificial release of C. cunea. The left circle of Figure 2 shows the reproduction mechanism of the H. cunea, and the right circle shows the reproduction mechanism of the C. cunea. The intersection of the two circles is the H. cunea pupa.

In recent years, some scholars have used ecological niche models (ENMs) to combine H. cunea occurrence records with environmental data. By creating relevant models to predict their ecological demand and potential geographic distribution. These experiments show that incorporating global data can substantially increase the forecasting precision of ENMs. This will provide valuable insights for the control of wedge nematode disease in China [7] . Several studies have investigated how temperature and season affect the behaviour of H. cunea populations and have created population dynamics models for its different life stages [8] . In addition, single population models with time delay were created [2] . Therefore, it is valuable to describe the dynamics of H. cunea and C. cunea by applying mathematical models.

In some sense, the richer dynamic properties of the model may be induced by time delay. Martin and Ruan [9] [10] proposed three types of time delay, these terms manifest within the prey-specific growth period, the predator response component of the predator equation, and the interaction component of the predator equation. Jiang et al. [11] [12] developed a predator-prey model incorporating time delay and explored a variety of dynamic properties associated with it. Time delay is crucial for the emergence of oscillatory and periodic behaviours. Considering that C. cunea can inhibit the growth rate of H. cunea, but this inhibition is not instantaneous. Therefore, the inclusion of time delay produces richer dynamic properties. The research on current time delay models has been a hot topic in mathematics, physics, biology and other fields. Furthermore, the introduction of nonlocal competition and its impact on model stability has been a subject of intense discussion in the literature over the past few decades. The works of Britton et al. [13] [14] have made significant contributions to the advancement of nonlocal competition. In their respective papers, they considered nonlocal competition and incorporated spatial convolution integrals into single species models. Britton et al. suggested that resource utilization in a given area is influenced by two factors: the density of the local population and the weighted average populations in surrounding areas. Song et al. [15] - [18] incorporated nonlocal competition into a predator-prey model with time delay and discovered that the interplay between nonlocal competition and time delay causes instability in the equilibrium.

The structure of the paper is as follows: Section 2 describes the process of establishing the mathematical model. Section 3 investigates the existence and stability of the positive equilibria and the existence of Hopf and Turing bifurcations near the equilibria. In Section 4, we utilize the multiple time scales method to derive the normal form of Hopf bifurcation for the model incorporating nonlocal competition. Section 5 presents numerical simulations that demonstrate our main results. Section 6 provides the conclusion.

In this section, we show the mathematical modeling process and explain it. Firstly, as noted in Ref. [19] , the Lotka-Volterra and Nicholson-Bailey models were two primary frameworks traditionally employed to study predator-prey and parasitic-host interrelations. The Lotka-Volterra model results in neutrally stable cycles, while the Nicholson-Bailey system leads to diverging cycles and eventual extinction. Specifically, the Lotka-Volterra model can be expressed as follows:

$\{\begin{array}{l}\frac{\text{d}u}{\text{d}t}=ru\left(1-\frac{u}{K}\right)-{\alpha }^{\prime }uv\mathrm{,}\\ \frac{\text{d}v}{\text{d}t}={\alpha }^{\prime }{\beta }^{\prime }uv-m^{\prime }v\mathrm{,}\end{array}$ (1)

here, $u$ and $v$ represent the prey and predator populations, respectively. The parameters $r$ and $K$ denote the highest per capita growth rate and environmental capacity of prey, respectively. ${\alpha }^{\prime }$ represents the capture rate of prey. ${\beta }^{\prime }$ represents the efficiency of converting ingested prey into new predators. $m^{\prime }$ represents the per capita mortality rate or the per capita nutritional needs for sustaining and replacing predators. Taking into account both biological and physical control, we establish the H. cunea-C. cunea model as follows:

$\{\begin{array}{l}\frac{\text{d}u}{\text{d}t}=r\left(1-\frac{u}{K}\right)u-\alpha \beta uv-hu\mathrm{,}\\ \frac{\text{d}v}{\text{d}t}=\delta \alpha \beta uv-mv+cu\mathrm{,}\end{array}$ (2)

where $r$ and $K$ have the same meanings as in model (1). $u$ and $v$ represent the population density of H. cunea and C. cunea, respectively. $\alpha$ represents the probability of the H. cunea pupa being parasitized by the C. cunea. $\beta$ represents the proportionality coefficient. $h$ represents the mortality rate of H. cunea caused by physical control methods. $m$ represents the natural mortality rate of C. cunea. $\delta$ represents the probability of successful eclosion of the parasitized pupae. $c$ represents the release rate of C. cunea. This paper intends to study the method of reducing the density of H. cunea in more environmentally friendly ways. Therefore, the model (2) does not include the death rate of H. cunea due to pesticides. Reducing the population of H. cunea can be achieved by regularly releasing a quantified amount of C. cunea.

We also take into account the time delay, as the eclosion of H. cunea pupae is not instantaneous, regardless of parasitization. Since the consumption of environmental resources is affected by the flow of H. cunea, thus, the competition is not local. A plethora of researchers incorporated nonlocal competition by substituting the original $u/K$ with $\hat{u}/K$, where $\hat{u}={\displaystyle {\int }_{\Omega }G\left(x\mathrm{,}y\right)u\left(y\mathrm{,}t\right)\text{d}y}$, which reflects intraspecific competition within prey populations. Note that $\Omega =\left(\mathrm{0,}l\pi \right)$ with $l$ is a positive parameter that measures the length of the habitat, and $\Omega$ represents the habitat volume of H. cunea. We assume that $G\left(x\mathrm{,}y\right)=1\backslash l\pi$ as $\left|\Omega \right|=l\pi$, and we study the model (3) with Neumann boundary condition to facilitate the study of nonlocal competition. To sum up, we develop a H. cunea-C. cunea model with both nonlocal competition and time delay as follows:

$\{\begin{array}{l}\frac{\partial u\left(x,t\right)}{\partial t}=d_1\Delta u\left(x,t\right)+r\left(1-1/K{\displaystyle {\int }_{\Omega }G}\left(x,y\right)u\left(y,t\right)\text{d}y\right)u\left(x,t\right)\\ -\alpha \beta u\left(x,t-\tau \right)v\left(x,t-\tau \right)-hu\left(x,t\right),\mathrm{<mi>\; </mi>}x\in \Omega ,t>0,\\ \frac{\partial v\left(x,t\right)}{\partial t}=d_2\Delta v\left(x,t\right)+\delta \alpha \beta u\left(x,t-\tau \right)v\left(x,t-\tau \right)-mv\left(x,t\right)\\ +cu\left(x,t\right),\mathrm{<mi>\; </mi>}x\in \Omega ,t>0,\\ \frac{\partial u\left(x,t\right)}{\partial \stackrel{\to }{n}}=\frac{\partial v\left(x,t\right)}{\partial \stackrel{\to }{n}}=0,\mathrm{<mi>\; </mi>}x\in \partial \Omega ,t>0,\\ u\left(x,t\right)=u_0\left(x,t\right)\ge 0,\mathrm{<mi>\; </mi>}v\left(x,t\right)=v_0\left(x,t\right)\ge 0,\mathrm{<mi>\; </mi>}x\in \bar{\Omega },\mathrm{<mi>\; </mi>}t\in \left[-\tau ,0\right],\end{array}$ (3)

where $d_1>0$, $d_2>0$ represent the diffusion coefficients of H. cunea and C. cunea, respectively. The other parameters $r$, $K$, $\alpha$, $\beta$, $h$, $m$, $\delta$ and $c$ represent the same meanings as the parameters in model (2), and all of them are non-negative.

Next, we will analyze the dynamic properties of model (3).

In this section, we will consider the existence and stability of the equilibria for model (3), as well as the existence of Hopf and Turing bifurcations near the equilibria.

We know that model (3) has a trivial equilibrium $E_0=\left(\mathrm{0,0}\right)$, and we make the following hypothesis:

(H₁) $r>h,$

if (H₁) holds, then the model (3) has a positive equilibrium $E_{\mathrm{<mo>\; *\; </mo>}}=\left(u^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{v}^{\mathrm{<mo>\; *\; </mo>}}\right)$, where

$\begin{array}{l}{u}^{\mathrm{<mo>\; *\; </mo>}}=\frac{m{v}^{\mathrm{<mo>\; *\; </mo>}}}{\delta \alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}+c}\mathrm{,}\\ v^{\mathrm{<mo>\; *\; </mo>}}=\frac{I+\sqrt{I^2+4k^2\delta {\alpha }^2{\beta }^{2}c\left(r-h\right)}}{2k\delta {\alpha }^2{\beta }^2}\mathrm{,}\end{array}$

with $I=kr\delta \alpha \beta -kh\delta \alpha \beta +kc\alpha \beta +rm$, it is obvious that $u^{*}>0$, and $v^{\mathrm{<mo>\; *\; </mo>}}>0$ when $r>h$.

Remark 1. The death rates from physical insect control methods are generally low, such as traps, barriers, electric shocks, etc., which are often non-lethal. The objective of physical control typically aims to diminish insect populations rather than inducing immediate widespread death.

In this section, we derive the normal form of the Hopf bifurcation for the model (3) by using the multiple time scales method as referenced in ref. [20] - [22] . When $\tau ={\tau }_c$, the characteristic equation Eq. (5) has a pair of pure imaginary roots $\lambda =\pm \text{i}\omega$, at which model (3) undergoes $n_l$, $n_s$ or $n_T$-mode Hopf bifurcation at equilibrium $E_{\mathrm{<mo>\; *\; </mo>}}=\left(u^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{v}^{\mathrm{<mo>\; *\; </mo>}}\right)$. In the subsequent derivation, we simplify notation by using $n$ instead of $n_l$, $n_s$ or $n_T$. We consider the time delay $\tau$ serves as the bifurcation parameter in this paper, where $\tau ={\tau }_c+\epsilon \mu$, with $\tau ={\tau }_c$ being the critical value for Hopf bifurcation. The parameter $\epsilon$ is a dimensionless scaling factor, and $\mu$ is a perturbation parameter. We also introduce the transformations: $t\mapsto t/\tau$, $u\mapsto u-u^{\mathrm{<mo>\; *\; </mo>}}$, $v\mapsto v-v^{\mathrm{<mo>\; *\; </mo>}}$, then model (3) is transformed into

$\{\begin{array}{l}\frac{\partial u}{\partial t}=\tau [d_1\Delta u+r\left(1-\frac{\hat{u}+u^{\mathrm{<mo>\; *\; </mo>}}}{K}\right)u-\alpha \beta u\left(t-1\right)v\left(t-1\right)\\ \begin{array}{c}\\ \end{array}-\alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}u\left(t-1\right)-\alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}v\left(t-1\right)-hu]\mathrm{,}\\ \frac{\partial v}{\partial t}=\tau [d_2\Delta v-mv+\delta \alpha \beta u\left(t-1\right)v\left(t-1\right)+\delta \alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}u\left(t-1\right)\\ +\delta \alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}v\left(t-1\right)+cu]\mathrm{.}\end{array}$ (12)

Let $g$ be the eigenvector of the linear operator corresponding to the eigenvalue $\lambda =\text{i}\omega \tau$ of the linearized system of Eq. (12) for equilibrium $E^{\mathrm{<mo>\; *\; </mo>}}$, and let $g^{\mathrm{<mo>\; *\; </mo>}}$ be the normalized eigenvector of the adjoint operator of the linear operator corresponding to the eigenvalue $\lambda =-\text{i}\omega \tau$, satisfying

$\langle g^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}g\rangle ={\overline{g^{\mathrm{<mo>\; *\; </mo>}}}}^{\text{T}}g=1$. By calculation, it can be found

$g={\left(g_{11}\mathrm{,}{g}_{12}\right)}^{\text{T}}={\left(\mathrm{1,}\frac{\text{i}\omega +n^2{d}_1-a_1-b_1{\text{e}}^{-\text{i}\omega \tau }-c_1+\frac{r{u}^{\mathrm{<mo>\; *\; </mo>}}}{K}{x}_n}{b_2{\text{e}}^{-\text{i}\omega \tau }}\right)}^{\text{T}}\mathrm{,}$

$g^{\mathrm{<mo>\; *\; </mo>}}=d^{\mathrm{<mo>\; *\; </mo>}}{\left(g_{21}\mathrm{,}{g}_{22}\right)}^{\text{T}}=d^{\mathrm{<mo>\; *\; </mo>}}{\left(\frac{b_2{\text{e}}^{-\text{i}\omega \tau }}{-\text{i}\omega +n^2{d}_2-a_1-b_1{\text{e}}^{\text{i}\omega \tau }-c_1}\mathrm{,1}\right)}^{\text{T}}\mathrm{,}$ (13)

where $d^{\mathrm{<mo>\; *\; </mo>}}={\left(g_{21}\overline{g_{11}}+\overline{g_{22}}{g}_{12}\right)}^{-1}$, and $x_n=\{\begin{array}{ll}\mathrm{1,}\hfill & \text{if}n=0\hfill \\ \mathrm{0,}\hfill & \text{if}n\in {\mathbb{N}}_{+}\hfill \end{array}$.

According to the multiple scales method, we can assume the solution of Eq. (12) as follows:

$u\left(x,t\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\epsilon }^k{u}_k\left(x,t_0,t_1,t_2,\cdots \right),$

$v\left(x,t\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\epsilon }^k{v}_k\left(x,t_0,t_1,t_2,\cdots \right),$ (14)

where $t_i={\epsilon }^{i}t\mathrm{,}i=\mathrm{1,2,3,}\cdots$, specially,

$u_{j,{\tau }_c}\left(x,t-1\right)=u_j\left(x,t_0-1,t_1,t_2,\cdots \right),$

$v_{j,{\tau }_c}\left(x,t-1\right)=v_j\left(x,t_0-1,t_1,t_2,\cdots \right),j=1,2,3,\cdots .$ (15)

By substituting (14) and (15) into (12), we compare the coefficients of $\epsilon \mathrm{,}{\epsilon }^2$ and ${\epsilon }^3$ on either side of the equation, resulting in the following expressions:

$\{\begin{array}{l}{D}_0{u}_1-{\tau }_c{d}_1\Delta u_1-{\tau }_{c}r{u}_1+\frac{r{\tau }_c{u}^{*}{u}_1}{K}+{\tau }_{c}h{u}_1+{\tau }_c\alpha \beta v^{*}{u}_{1{\tau }_c}+{\tau }_c\alpha \beta u^{*}{v}_{1{\tau }_c}=0,\\ D_0{v}_1-{\tau }_c{d}_2\Delta v_1+{\tau }_{c}m{v}_1-{\tau }_{c}c{u}_1-{\tau }_c\delta \alpha \beta v^{*}{u}_{1{\tau }_c}-{\tau }_c\delta \alpha \beta u^{*}{v}_{1{\tau }_c}=0.\end{array}$ (16)

$\{\begin{array}{l}{D}_0{u}_2-{\tau }_c{d}_1\Delta u_2-{\tau }_{c}r{u}_2+\frac{r{\tau }_c{u}^{*}{u}_2}{K}+{\tau }_{c}h{u}_2+{\tau }_c\alpha \beta v^{*}{u}_{2{\tau }_c}+{\tau }_c\alpha \beta u^{*}{v}_{2{\tau }_c}\\ =\mu d_1\Delta u_1+\mu r{u}_1-\frac{\mu r{u}^{*}{u}_1}{K}-\mu h{u}_1-\mu \alpha \beta v^{*}{u}_{1{\tau }_c}-\mu \alpha \beta u_{*}{v}_{1{\tau }_c}-\frac{{\tau }_{c}r{u}_1^2}{K}\\ -{\tau }_c\alpha \beta u_{1{\tau }_c}{v}_{1{\tau }_c}+{\tau }_c\alpha \beta v^{*}{D}_1{u}_{1{\tau }_c}+{\tau }_c\alpha \beta u^{*}{D}_1{v}_{1{\tau }_c}-D_1{u}_1,\\ D_0{v}_2-{\tau }_c{d}_2\Delta v_2+{\tau }_{c}m{v}_2-{\tau }_{c}c{u}_2-{\tau }_c\delta \alpha \beta v^{*}{u}_{2{\tau }_c}-{\tau }_c\delta \alpha \beta u^{*}{v}_{2{\tau }_c}\\ =\mu d_2\Delta v_1-\mu m{v}_1+\mu \delta \alpha \beta v^{*}{u}_{1{\tau }_c}+\mu \delta \alpha \beta v^{*}{v}_{1{\tau }_c}+{\tau }_c\delta \alpha \beta u_{1{\tau }_c}{v}_{1{\tau }_c}+\mu c{u}_1\\ -{\tau }_c\delta \alpha \beta v^{*}D-1u_{1{\tau }_c}-{\tau }_c\delta \alpha \beta u^{*}{D}_1{v}_{1{\tau }_c}-D_1{v}_1.\end{array}$ (17)

$\{\begin{array}{l}{D}_0{u}_3-{\tau }_c{d}_1\Delta u_3-{\tau }_{c}r{u}_3+\frac{r{\tau }_c{u}^{*}{u}_3}{K}+{\tau }_{c}h{u}_3+{\tau }_c\alpha \beta v^{*}{u}_{3{\tau }_c}+{\tau }_c\alpha \beta u^{*}{v}_{3{\tau }_c}\\ =-\mu \alpha \beta v^{*}{u}_{2{\tau }_c}+\mu \alpha \beta v^{*}{D}_1{u}_{1{\tau }_c}+\mu \alpha \beta u^{*}{D}_1{v}_{1{\tau }_c}-\frac{2{\tau }_{c}r{u}_1{u}_2}{K}\\ \mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}-{\tau }_c\alpha \beta u_{1{\tau }_c}{v}_{2{\tau }_c}-{\tau }_c\alpha \beta u_{2{\tau }_c}{v}_{1{\tau }_c}+{\tau }_c\alpha \beta D_2{u}_{1{\tau }_c}+{\tau }_c\alpha \beta v^{*}{D}_1{u}_{1{\tau }_c}\\ \mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}+{\tau }_c\alpha \beta u^{*}{D}_2{v}_{1{\tau }_c}+{\tau }_c\alpha \beta D_1{v}_{1{\tau }_c}-D_2{u}_1-D_1{u}_2+o\left(\mu \right),\\ D_0{v}_3-{\tau }_c{d}_2\Delta v_3+{\tau }_{c}m{v}_3-{\tau }_{c}c{u}_3-{\tau }_c\delta \alpha \beta v^{*}{u}_{3{\tau }_c}-{\tau }_c\delta \alpha \beta u^{*}{v}_{3{\tau }_c}\\ =\mu \delta \alpha \beta u_{1{\tau }_c}{v}_{1{\tau }_c}+\mu c{u}_2-\mu \delta \alpha \beta v^{*}{D}_1{u}_{1{\tau }_c}-\mu \delta \alpha \beta u^{*}{D}_1{v}_{1{\tau }_c}\\ \mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}+{\tau }_c\delta \alpha \beta u_{1{\tau }_c}{v}_{2{\tau }_c}+{\tau }_c\delta \alpha \beta u_{2{\tau }_c}{v}_{1{\tau }_c}-{\tau }_c\delta \alpha \beta v^{*}{D}_2{u}_{1{\tau }_c}-{\tau }_c\delta \alpha \beta v^{*}{D}_1{u}_{1{\tau }_c}\\ \mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}-{\tau }_c\delta \alpha \beta u^{*}{D}_2{v}_{1{\tau }_c}-{\tau }_c\delta \alpha \beta u^{*}{D}_1{v}_{1{\tau }_c}-D_2{v}_1-D_1{v}_2+o\left(\mu \right).\end{array}$ (18)

Referring to the notations in Refs. [20] [21] , then Eq. (16) has the solution with following form:

$\left(\begin{array}{c}{u}_1\\ v_1\end{array}\right)=Gg{\text{e}}^{\text{i}\omega {\tau }_c{T}_0}+\bar{G}\bar{g}{\text{e}}^{-\text{i}\omega {\tau }_c{T}_0}\mathrm{.}$ (19)

Therefore, by substituting Eq. (19) into the right part of Eq. (17), then we define the coefficient vector of ${\text{e}}^{\text{i}\omega {\tau }_c{T}_0}$ as $m_1$. Therefore, according to the solvability condition, we have $\langle g^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}\left(m_1\mathrm{,}\cos nx\right)\rangle =0$. Then we can solve $\frac{\partial G}{\partial T_1}$, namely,

$\frac{\partial G}{\partial T_1}=M\mu G\mathrm{,}$ (20)

where

$M=\frac{\overline{g_{21}}{\varrho }_1+\overline{g_{22}}{\varrho }_2}{\overline{g_{21}}{\varrho }_3+\overline{g_{22}}{\varrho }_4}\mathrm{,}$

with

$\{\begin{array}{l}{\varrho }_1=\left(-d_1\frac{n^2}{l^2}+r-\frac{r{u}^{\mathrm{<mo>\; *\; </mo>}}}{K}-h-\alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-\text{i}\omega {\tau }_c}\right)g_{11}-\alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-\text{i}\omega {\tau }_c}{g}_{12}\mathrm{,}\\ {\varrho }_2=\left(-d_2\frac{n^2}{l^2}-m+\delta \alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-\text{i}\omega {\tau }_c}\right)g_{12}+\left(\delta \alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-\text{i}\omega {\tau }_c}+c\right)g_{11}\mathrm{,}\\ {\varrho }_3={\tau }_c\alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-\text{i}\omega {\tau }_c}{g}_{11}-{\tau }_c\alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-\text{i}\omega {\tau }_c}{g}_{12}-g_{11}\mathrm{,}\\ {\varrho }_4={\tau }_c\delta \alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-\text{i}\omega {\tau }_c}{g}_{12}+{\tau }_c\delta \alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-\text{i}\omega {\tau }_c}{g}_{12}+g_{12}\mathrm{.}\end{array}$

Suppose that the solution of (17) satisfies following form:

$\begin{array}{l}{u}_2={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}\left({\eta }_{0k}G\bar{G}+{\eta }_{1k}{\text{e}}^{2i\omega {\tau }_c{T}_0}{G}^2+\overline{{\eta }_{1k}}{\text{e}}^{-2\text{i}\omega {\tau }_c{T}_0}\overline{G^2}\right)\cos \frac{kx}{l}\mathrm{,}\\ v_2={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}\left({\xi }_{0k}G\bar{G}+{\xi }_{1k}{\text{e}}^{2\text{i}\omega {\tau }_c{T}_0}{G}^2+\overline{{\xi }_{1k}}{\text{e}}^{-2\text{i}\omega {\tau }_c{T}_0}\overline{G^2}\right)\cos \frac{kx}{l}\mathrm{,}\end{array}$ (21)

we substitute (21) into the left end of (17), it can be obtained by the method of undetermined coefficients, where

${\eta }_{0k}=\frac{{\zeta }_1{\zeta }_2-{\zeta }_3{\zeta }_4}{{\zeta }_6{\zeta }_2}\mathrm{,\; <mi>\; </mi>}{\eta }_{1k}=\frac{Q_1{Q}_2-Q_3{Q}_4}{Q_5{Q}_2-Q_3{Q}_4}\mathrm{,}$

${\xi }_{0k}=\frac{{\zeta }_5{\zeta }_1+{\zeta }_3{\zeta }_6}{{\zeta }_2{\zeta }_6+{\zeta }_4{\zeta }_5}\mathrm{,\; <mi>\; </mi>}{\xi }_{1k}=\frac{Q_1{Q}_6-Q_5{Q}_3}{Q_5{Q}_2+Q_3{Q}_4{Q}_6}\mathrm{,}$ with

$\{\begin{array}{l}{Q}_1=-\frac{{\tau }_{c}r}{K}{g}_{11}^2-{\tau }_c\alpha \beta {\text{e}}^{-2\text{i}\omega {\tau }_c}{g}_{11}{g}_{12}\mathrm{,}\\ Q_2=2\text{i}\omega {\tau }_c+{\tau }_c{d}_2\frac{k^2}{l^2}+{\tau }_{c}m-{\tau }_c\delta \alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-2\text{i}\omega {\tau }_c}\mathrm{,}\\ Q_3={\tau }_c\delta \alpha \beta g_{11}{g}_{12}{\text{e}}^{-2\text{i}\omega {\tau }_c}\mathrm{,}\\ Q_4={\tau }_c\alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-2\text{i}\omega {\tau }_c}\mathrm{,}\\ Q_5=2\text{i}\omega {\tau }_c+{\tau }_c{d}_1\frac{k^2}{l^2}+{\tau }_c\alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-2\text{i}\omega {\tau }_c}-{\tau }_{c}r+\frac{{\tau }_{c}r{u}^{\mathrm{<mo>\; *\; </mo>}}}{K}+{\tau }_{c}h\mathrm{,}\\ Q_6=-{\tau }_c\delta \alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}{\text{e}}^{-2\text{i}\omega {\tau }_c}-c{\tau }_c\mathrm{,}\end{array}$ and

$\{\begin{array}{l}{\zeta }_1=-\frac{2r{\tau }_c}{K}{g}_{11}\overline{g_{11}}-{\tau }_c\alpha \beta g_{11}\overline{g_{12}}-{\tau }_c\alpha \beta g_{12}\overline{g_{11}},\\ {\zeta }_2={\tau }_c{d}_2\frac{k^2}{l^2}+{\tau }_{c}m-{\tau }_c\delta \alpha \beta u^{*},\\ {\zeta }_3={\tau }_c\delta \alpha \beta g_{11}\overline{g_{12}},\\ {\zeta }_4={\tau }_c\alpha \beta u^{*},\\ {\zeta }_5=-{\tau }_c\delta \alpha \beta v^{*}-c{\tau }_c,\\ {\zeta }_6={\tau }_c{d}_1\frac{k^2}{l^2}+{\tau }_c\alpha \beta v^{*}-{\tau }_{c}r+\frac{{\tau }_{c}r{u}^{*}}{K}+{\tau }_{c}h.\end{array}$

Next, substitute (19) and (21) into (18), then compare the coefficients, and denoting the coefficient vector of ${\text{e}}^{\text{i}\omega {\tau }_c{T}_0}$ as $m_2$. Therefore, according to the solvability condition, we have $\langle g^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}\left(m_2\mathrm{,}\cos nx\right)\rangle =0$. Specially, $\mu$ is a perturbation parameter, and ${\mu }^2$ has minimal influence of small unfolding parameters. Thus, we have

$\frac{\partial G}{\partial T_2}=X{G}^2\bar{G}\mathrm{,}$ (22)

where

$X=\frac{\left[\overline{g_{21}}{\iota }_1-\overline{g_{22}}{\iota }_2\right]}{\left[\overline{g_{21}}{\iota }_3+\overline{g_{22}}{\iota }_4\right]D_k}\mathrm{,}$

with

$\{\begin{array}{l}{\iota }_1=\frac{2r{\tau }_c}{K}{g}_{11}{\eta }_{00}{C}_0+\frac{2r{\tau }_c}{K}\overline{g_{11}}{\eta }_{10}{C}_0+{\tau }_c\alpha \beta g_{12}\left({\eta }_{00}{C}_0+{\eta }_{02n}{C}_{2n}\right)\\ +{\tau }_c\alpha \beta \overline{g_{12}}\left({\eta }_{10}{C}_0+{\eta }_{12n}{C}_{2n}\right)+{\tau }_c\alpha \beta g_{11}{\text{e}}^{-\text{i}\omega {\tau }_c}\left({\xi }_{00}{C}_0+{\xi }_{02n}{C}_{2n}\right)\\ +{\tau }_c\alpha \beta \overline{g_{11}}{\text{e}}^{-\text{i}\omega {\tau }_c}\left({\xi }_{10}{C}_0+{\xi }_{12n}{C}_{2n}\right)\mathrm{,}\\ {\iota }_2={\tau }_c\delta \alpha \beta g_{11}{\text{e}}^{-\text{i}\omega {\tau }_c}\left({\xi }_{00}{C}_0+{\xi }_{02n}{C}_{2n}\right)+{\tau }_c\delta \alpha \beta \overline{g_{11}}{\text{e}}^{-\text{i}\omega {\tau }_c}\left({\xi }_{10}{C}_0+{\xi }_{12n}{C}_{2n}\right)\\ +{\tau }_c\delta \alpha \beta g_{12}{\text{e}}^{-\text{i}\omega {\tau }_c}\left({\eta }_{00}{C}_0+{\eta }_{02n}{C}_{2n}\right)+{\tau }_c\delta \alpha \beta \overline{g_{12}}{\text{e}}^{-\text{i}\omega {\tau }_c}\left({\eta }_{10}{C}_0+{\eta }_{12n}{C}_{2n}\right)\mathrm{,}\\ {\iota }_3={\tau }_c\alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}{g}_{11}{\text{e}}^{-\text{i}\omega {\tau }_c}+{\tau }_c\alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}{g}_{12}{\text{e}}^{-\text{i}\omega {\tau }_c}-g_{11}\mathrm{,}\\ {\iota }_4=-{\tau }_c\delta \alpha \beta v^{\mathrm{<mo>\; *\; </mo>}}{g}_{11}{\text{e}}^{-\text{i}\omega {\tau }_c}-{\tau }_c\delta \alpha \beta u^{\mathrm{<mo>\; *\; </mo>}}{g}_{12}{\text{e}}^{-\text{i}\omega {\tau }_c}-g_{12}\mathrm{,}\end{array}$

and

$C_k={\displaystyle {\int }_0^{l\pi }{\cos }^2\frac{nx}{l}\text{d}x\cos \frac{kx}{l}\text{d}x}=\{\begin{array}{l}\frac{l\pi }{2}\mathrm{,}k=\mathrm{0,}\\ \frac{l\pi }{4}\mathrm{,\; <mi>\; </mi>}k=2n\ne \mathrm{0,}\\ \mathrm{0,}k\ne 2n\ne 0.\end{array}$

$D_k={\displaystyle {\int }_0^{l\pi }\cos \frac{nx}{l}\cos \frac{nx}{l}\text{d}x}=\{\begin{array}{l}l\pi \mathrm{,}k=\mathrm{0,}\\ \frac{l\pi }{2}\mathrm{,}k\ne 0.\end{array}$

We drive the normal form of Hopf bifurcation for model (3) reduce on the center manifold as follows:

$\stackrel{\dot{}}{G}=M\mu G+X{G}^2\bar{G}\mathrm{,}$ (23)

where $M$ and $X$ are defined by (20) and (22), respectively. Substituting $G=r{\text{e}}^{\text{i}\theta }$ into the Eq. (23), and we derive the amplitude equation on the center manifold as follows:

$\{\begin{array}{l}\stackrel{\dot{}}{r}=\text{Re}\left(M\right)\mu r+\text{Re}\left(X\right)r^3\mathrm{,}\\ \stackrel{\dot{}}{\theta }=\text{Im}\left(M\right)\mu +\text{Im}\left(X\right)r^2\mathrm{,}\end{array}$

then we state the following theorem.

Theorem 4. If the condition $\frac{\text{Re}\left(M\right)\mu }{\text{Re}\left(X\right)}<0$ holds, the model (3) generates periodic solutions near the equilibrium $E_{\mathrm{<mo>\; *\; </mo>}}$. If $\text{Re}\left(M\right)\mu <0$ $\left(\text{Re}\left(M\right)\mu >0\right)$, the bifurcating periodic solutions are unstable (stable). Additionally, the direction of bifurcating periodic solution is forward (backward) if $\mu >0$ ( $\mu <0$).

In this section, we first analyze the parameter ranges, then simulations are performed numerically.