Hopf Bifurcation Analysis of a Class of Saperda populnea Infectious Disease Model with Delay

Abstract

Under the background of double carbon, it is important to study forest pests and diseases to improve forest carbon sink. In this paper, we establish a delayed model associated with the larvae and adults of Saperda populnea, susceptible poplars, and infected poplars. First, we analyze the existence and stability of the equilibrium of the model. Second, we study the existence of Hopf bifurcation near the equilibrium and obtain the normal form of Hopf bifurcation by the multiple time scales method. Then, we analyze the direction and stability of Hopf bifurcating periodic solutions. Third, we analyze and conjecture some parameter values based on official data, and carry out numerical simulations to verify our results. Finally, we give some suggestions on the prevention and control of Saperda populnea.

1. Introduction

The international community generally believes that excessive carbon dioxide emissions are the main factor causing climate change. Greenhouse gases such as carbon dioxide emitted by human activities lead to global warming, exacerbating the instability of the climate system and increasing the probability of related extreme weather events [1]. At present, the world has formed a trend of low-carbon energy and industrial development, and countries have introduced carbon neutrality programs. China has achieved remarkable emission reduction results in recent years, and its carbon emission intensity in 2019 was 48.4% lower than that in 2005. China’s “double carbon” goal will open a historic turning point in carbon emission reduction, which is also an inevitable choice to promote the upgrading of China’s energy and related industries and achieve long-term healthy and sustainable development of the national economy [2]. To achieve the goal of “double carbon” is not to completely ban carbon dioxide emissions, but to promote carbon dioxide absorption while reducing carbon dioxide emissions, and offset emissions with absorption, so as to gradually transform the energy structure from high carbon to low carbon or even no carbon. In recent studies, scholars have used differential decomposition, average division index, index decomposition, mathematical modeling, and other methods to analyze the factors affecting carbon emissions [3,4,5,6].

As the largest carbon reservoir in terrestrial ecosystems, forests play a unique and important role in reducing the concentration of greenhouse gases in the atmosphere and slowing down global warming. Forest carbon sinks mean that forest plants absorb carbon dioxide from the atmosphere and fix it in vegetation or soil, thereby reducing the concentration of the gas in the atmosphere. Although forest areas account for only $1/3$ of the total land area, the carbon storage of forest vegetation areas accounts for almost half of the total land carbon storage. Forests are important in a two-carbon context because they are directly linked to climate change. Trees absorb large amounts of carbon dioxide from the atmosphere through photosynthesis, slowing the greenhouse effect. Expanding forest cover is an economically viable and cost-effective mitigation measure for the next 30 to 50 years. Hubau et al. [7] found that the carbon sink of aboveground biomass in intact African tropical forests remained stable in the three decades up to 2015, while the carbon sink of aboveground biomass in Amazon forests decreased for a long time. This difference is mainly due to the carbon loss caused by the death of trees. Obviously, the saturation and continuous decline of carbon sinks in tropical forests had an impact on the policy of stabilizing the earth’s climate. Therefore, many countries and international organizations are actively using forest carbon sinks to combat climate change [8].

It is clear that controlling forest pests and diseases plays a key role in increasing the carbon sink potential of forests [9]. In Ref. [10], Lierop et al. conducted a global analysis of forest areas affected by fires, major pest outbreaks, diseases, and severe weather reported by countries. The study showed that from 2003 to 2012, pests affected more than 380,000 hectares of forest, seriously harming forest resources. Canelles et al. [11] explored the mechanism of interactions between pest outbreaks and other forest disturbances, identified interactions that occur in current disturbance dynamics, and emphasized the role of simulation models in dynamically, mechanically, and spatially exploring these interactions. In Ref. [12], researchers proposed a multi-combination multivariate linear regression model and a multi-combination multivariate linear regression visual analysis method. Using this method, researchers could quickly evaluate the correlation between forest pest indicators and factors that may affect the occurrence of pests and diseases, so as to determine the influencing factors and find the relevant laws.

In 2022, China’s forest coverage reached 24.02%, and the forest area reached 231 million hectares. Because of China’s high forest coverage, the situation of forest pests and diseases is more serious than that of other countries. Therefore, controlling forest pests and diseases is the top priority of China’s current ecological development. Zhang et al. [13] dealt with the impulse delayed epidemic model with a stage structure and the general form of pest control strategy incidence. Using the discrete dynamic system determined by the stroboscopic diagram, they found that in addition to the release of infectious pests, the incidence, delay, and impulsive period had a greater impact on system dynamics. Arafa et al. [14] introduced a Filippov prey–predator (pest–natural enemy) model with time delay that could provide new insights into pest control models. Refs. [15,16,17] showed that the delay differential equation could better describe the development trend of forest pests and diseases, and provide a strategic basis for pest management. In Refs. [18,19,20,21,22], scholars analyzed the stability of delay differential equations. At the same time, Hopf bifurcation is also an important nonlinear dynamic phenomenon [23].

Saperda populnea is one of the most important pests to poplars in north China. Saperda populnea generally reproduces one generation per year, but there are also individual regions of Saperda populnea that take 2 years to reproduce one generation. The damage of Saperda populnea to plants is the most serious in the larval stage. In one year, the larvae of Saperda populnea generally begin to move and pupate in April, and pupate adults appear in May, and then begin to emerge one after another. After a few days, a large number of eggs are produced. After the eggs hatch, new larvae will be produced and eat the phloem of the poplar. When they are slightly larger, they will eat around the branches, and then go deep into the xylem to form an expanded “insect gall”. After the fall, the larvae gradually stop infecting poplars and enter the overwintering state. Saperda populnea likes the sunlight very much, so it will deliberately select such branches when laying eggs. Therefore, the trunks and branches of poplars can easily become the source of Saperda populnea. The damage caused by Saperda populnea not only affects the growth of trees, but also directly endangers the growth of economic forests and the ecological benefits of trees to society.

At present, scholars at home and abroad study the biological characteristics, occurrence pattern, and control measures of Saperda populnea, without involving the different effects of Saperda populnea on poplars at different stages. Therefore, we simulate the damage of Saperda populnea larvae and adults on poplars by building a delay differential equation and conducting stability analysis.

The rest of this article is arranged as follows. In Section 2, according to the growth law of Saperda populnea, a delay differential equation model is established. In Section 3, we analyze the existence of equilibrium for the delay differential equation model and study the existence of Hopf bifurcation. In Section 4, we obtain the normal form of Hopf bifurcation of the above model by using the multiple time scales method, and analyze the stability of the bifurcating periodic solutions. In Section 5, we combine the official data to carry out numerical simulations to verify the correctness of our analysis, and give suggestions for the prevention and control of Saperda populnea. Finally, we conclude in Section 6.

2. Mathematical Modeling

Based on the above analysis, we build the following delay differential equation for the damage of Saperda populnea larvae and adults to poplars:

$$\left\{\begin{array}{c}\frac{dX}{dt}={\theta }_{1}Y(1-\frac{Y}{N})-\alpha X-{\theta }_{2}X(t-\tau ),\hfill \\ \frac{dY}{dt}={\theta }_{2}X(t-\tau )-\beta Y,\hfill \\ \frac{dW}{dt}=R-{\theta }_{3}XW-{\theta }_{4}YW+PZ,\hfill \\ \frac{dZ}{dt}={\theta }_{3}XW+{\theta }_{4}YW-PZ-dZ,\hfill \end{array}\right.$$

(1)

where ${\theta }_1,$${\theta }_2$, ${\theta }_3$, ${\theta }_4$, $\alpha$, $\beta$, P, N, R, and d are positive constants; X, Y, W, and Z are variables; and $\tau$ is the time delay from larvae to adults of Saperda populnea. The specific meanings of variables and parameters are shown in the following

Table 1. Description of variables and parameters in system (1).

Symbol	Description	Unit
X	Number of Saperda populnea larvae	${10}^9$ PCS
Y	Number of Saperda populnea adults	${10}^9$ PCS
W	Number of susceptible poplars	${10}^8$ trees
Z	Number of infected poplars	${10}^8$ trees
${\theta }_1$	Survival rate from egg to larva of Saperda populnea	-
${\theta }_2$	Survival rate from larva to adult of Saperda populnea	-
${\theta }_3$	Infection rate of Saperda populnea larvae to susceptible poplars	-
${\theta }_4$	Infection rate of Saperda populnea adults to susceptible poplars	-
R	Number of poplars in artificial afforestation	${10}^8$ trees
P	Recovery rate of infected poplars after human intervention	-
N	Environmental capacity of Saperda populnea adults	${10}^9$ PCS
$\alpha$	Natural mortality rate of Saperda populnea larvae	-
$\beta$	Natural mortality rate of Saperda populnea adults	-
d	Mortality rate of infected poplars	-
$\tau$	Time delay from larvae to adults of Saperda populnea	Week

.

For system (1), we have the following explanation:

(i)

X and Y denote the number of larvae and adults of Saperda populnea, respectively. We believe that it takes a certain time for the larvae to grow into adults. Therefore, there will be a time delay $\tau$ to indicate the time required for the larvae to grow into adults. Moreover, because the eggs of Saperda populnea are related to the adults, we consider the effect of Saperda populnea adult environmental capacity on Saperda populnea larvae. When the number of adults is Y and the environmental capacity is N, the relationship between the number of larvae X and the number of adults Y can be obtained by logistic equation. Therefore, we will obtain the relationship between adults and larvae.

(ii)

Suppose that susceptible poplars become infected poplars after Saperda populnea infection, and infected poplars become susceptible poplars after artificial treatment. ${\theta }_3$ and ${\theta }_4$ are the probability of Saperda populnea larvae and adults infecting poplars, respectively. ${\theta }_{3}XW$ and ${\theta }_{4}XW$ represent the number of susceptible poplars that become infected poplars after being infected by larvae and adults, respectively. P is the recovery rate of infected poplars after artificial treatment. In addition, we also consider the mortality of poplars after infection.

3. Stability Analysis of Equilibrium and Existence of Hopf Bifurcation

In this section, considering system (1), we show the following assumption:

$${\displaystyle \mathbf{(}}\mathbf{A}\mathbf{1} {\displaystyle \mathbf{)}}\frac{{\theta }_1{\theta }_2}{\beta }-\alpha -{\theta }_2>0.$$

Under the assumption (A1), system (1) has a unique positive equilibrium:

$$\begin{array}{cc}\hfill & E=(X^{*},Y^{*},W^{*},Z^{*}),\hfill \end{array}$$

where $X^{*}=\frac{{\beta }^{2}N}{{\theta }_1{\theta }_2^2}{\gamma }_1$, $Y^{*}=\frac{\beta N}{{\theta }_1{\theta }_2}{\gamma }_1$, $W^{*}=\frac{RP+Rd}{d{\gamma }_1{\gamma }_2}$, $Z^{*}=\frac{R}{d}$, with ${\gamma }_1=\frac{{\theta }_1{\theta }_2}{\beta }-\alpha -{\theta }_2$, ${\gamma }_2=\frac{{\theta }_3{\beta }^{2}N+{\theta }_2{\theta }_4\beta N}{{\theta }_1{\theta }_2^2}$.

The characteristic equation of system (1) at equilibrium E is as follows:

$$\begin{array}{c}\hfill [{\lambda }^2+(P+d+{\theta }_4{Y}^{*}+{\theta }_3{X}^{*})\lambda +({\theta }_4{Y}^{*}+{\theta }_3{X}^{*})d][{\lambda }^2+(\alpha +{\theta }_2{\mathtt{e}}^{-\lambda \tau }+\beta )\lambda +\\ \hfill ({\theta }_2\beta -{\theta }_1{\theta }_2+\frac{2{\theta }_1{Y}^{*}}{N}{\theta }_2){\mathtt{e}}^{-\lambda \tau }+\alpha \beta ]=0.\end{array}$$

(2)

When $\tau =0$, the characteristic Equation (2) becomes

$$\begin{array}{c}\hfill [{\lambda }^2+(P+d+{\theta }_4{Y}^{*}+{\theta }_3{X}^{*})\lambda +({\theta }_4{Y}^{*}+{\theta }_3{X}^{*})d][{\lambda }^2+\\ \hfill (\alpha +{\theta }_2+\beta )\lambda +({\theta }_2\beta -{\theta }_1{\theta }_2+\frac{2{\theta }_1{Y}^{*}}{N}{\theta }_2)+\alpha \beta ]=0.\end{array}$$

(3)

Under the assumption (A1), $X^{*}>0$, $Y^{*}>0$, and ${\theta }_3$, ${\theta }_4$ are positive constants, so ${\theta }_3{X}^{*}+{\theta }_4{Y}^{*}>0$ holds. According to the Vieta theorem, under the assumption (A1), all the roots of Equation (3) have negative real parts, and the equilibrium $E=(X^{*},Y^{*},W^{*},Z^{*})$ is locally asymptotically stable when $\tau =0$.

When $\tau >0$, we analyze the existence of Hopf bifurcation near the equilibrium E. We take $\lambda =\mathtt{i}\omega$ $(\omega >0)$ into Equation (2) and separate the real part and the imaginary part. Then, we can obtain the following equation:

$$\left\{\begin{array}{c}{\omega }^2-\alpha \beta =({\theta }_2\beta -{\theta }_1{\theta }_2+\frac{2{\theta }_1{\theta }_2{Y}^{*}}{N})cos\left(\omega \tau \right)+{\theta }_2\omega sin\left(\omega \tau \right),\hfill \\ -\omega \alpha -\omega \beta =(-{\theta }_2\beta +{\theta }_1{\theta }_2-\frac{2{\theta }_1{\theta }_2{Y}^{*}}{N})sin\left(\omega \tau \right)+{\theta }_2\omega cos\left(\omega \tau \right),\hfill \end{array}\right.$$

That is,

$$\left\{\begin{array}{c}sin\left(\omega \tau \right)=\frac{{\theta }_2\omega ({\omega }^2-\alpha \beta )+\omega (\alpha +\beta )({\theta }_1{\theta }_2-{\theta }_2\beta -2\alpha \beta )}{{\left({\theta }_2\omega \right)}^2+{({\theta }_1{\theta }_2-{\theta }_2\beta -2\alpha \beta )}^2},\hfill \\ cos\left(\omega \tau \right)=\frac{-{\theta }_2{\omega }^2(\alpha +\beta )-({\omega }^2-\alpha \beta )(-{\theta }_1{\theta }_2+{\theta }_2\beta +2\alpha \beta )}{{\left({\theta }_2\omega \right)}^2+{({\theta }_1{\theta }_2-{\theta }_2\beta -2\alpha \beta )}^2}.\hfill \end{array}\right.$$

(4)

Let $z={\omega }^2$; we obtain

$$h\left(z\right)=z^2+({\alpha }^2+{\beta }^2-{\theta }_2^2)z-M=0,$$

(5)

where $M={\theta }_1{\theta }_2({\theta }_1{\theta }_2-2{\theta }_2\beta -4\alpha \beta )+{\beta }^2({\theta }_2^2+3{\alpha }^2+4{\theta }_2\alpha )$.

For the case of the root of Equation (5), we make the following analysis:

(1)

If $M>0$, then Equation (5) has a positive root and a negative root.

(2)

If ${\alpha }^2+{\beta }^2-{\theta }_2^2<0$, $-\frac{{({\alpha }^2+{\beta }^2-{\theta }_2^2)}^2}{4}<M<0$, then Equation (5) has two positive roots.

(3)

If $M=-\frac{{({\alpha }^2+{\beta }^2-{\theta }_2^2)}^2}{4}$, then Equation (5) has two equal real roots $z_1=z_2=-\frac{{\alpha }^2+{\beta }^2-{\theta }_2^2}{2}$.

We hypothesize that Equation (5) has k ($k=1,2$) positive roots and denote them as $z_1$ and $z_2$. Substituting ${\omega }_k=\sqrt{z_k}$ ($k=1,2$) into Equation (4), we obtain the expression of $\tau$:

$${\tau }_k^{\left(j\right)}=\left\{\begin{array}{c}\frac{1}{{\omega }_k}[arccos\left(P_k\right)+2j\pi ],Q_k\ge 0,\hfill \\ \frac{1}{{\omega }_k}[2\pi -arccos\left(P_k\right)+2j\pi ],Q_k<0,\hfill \end{array}\right.k=1,2,j=0,1,2,\cdots ,$$

(6)

where

$$\begin{array}{cc}\hfill & Q_k=sin\left({\omega }_k{\tau }_k^{\left(j\right)}\right)=\frac{{\theta }_2{\omega }_k({\omega }_k^2-\alpha \beta )+{\omega }_k(\alpha +\beta )({\theta }_1{\theta }_2-{\theta }_2\beta -2\alpha \beta )}{{\left({\theta }_2{\omega }_k\right)}^2+{({\theta }_1{\theta }_2-{\theta }_2\beta -2\alpha \beta )}^2},\hfill \\ \hfill & P_k=cos\left({\omega }_k{\tau }_k^{\left(j\right)}\right)=\frac{-{\theta }_2{\omega }_k^2(\alpha +\beta )-({\omega }_k^2-\alpha \beta )(-{\theta }_1{\theta }_2+{\theta }_2\beta +2\alpha \beta )}{{\left({\theta }_2{\omega }_k\right)}^2+{({\theta }_1{\theta }_2-{\theta }_2\beta -2\alpha \beta )}^2}.\hfill \end{array}$$

If (A1) holds, and $\sqrt{z_k}={\omega }_k,h^{\prime }\left(z_k\right)\ne 0$. We get the following transversality condition:

$$\begin{array}{c}\mathtt{Re}{\left(\frac{d\lambda }{d\tau }\right)}^{-1}{{|}_{{\tau }_k^{\left(\mathtt{j}\right)}}=\mathtt{Re}\left(\frac{d\tau }{d\lambda }\right)|}_{{\tau }_k^{\left(\mathtt{j}\right)}}=\frac{{\omega }^2{h}^{\prime }\left(z\right)}{{\left({\omega }^2{\theta }_2\right)}^2+{({\theta }_1{\theta }_2-2\alpha \beta -\beta {\theta }_2)}^2}\ne 0,k=1,2,j=0,1,2,\cdots .\hfill \end{array}$$

Theorem 1.

When $\left(\mathbf{A}\mathbf{1}\right)$ holds, considering system (1), we draw the following conclusions.

(1) 

If $h\left(z\right)$ has no positive roots, when $\tau \ge 0$, the equilibrium E is locally asymptotically stable.

(2) 

If $h\left(z\right)$ has one positive root $z_1$, when $\tau \in [0,{\tau }_1^{\left(0\right)})$, the equilibrium E is locally asymptotically stable, and unstable when $\tau >{\tau }_1^{\left(0\right)}$, where ${\tau }_1^{\left(0\right)}$ is given in Equation (6).

(3) 

If $h\left(z\right)$ has two positive roots $z_1$ and $z_2$, we suppose $z_1<z_2$, then we have $h^{\prime }\left(z_1\right)<0,h^{\prime }\left(z_2\right)>0$. Then, ∃ m $\in N$ makes $0<{\tau }_2^{\left(0\right)}<{\tau }_1^{\left(0\right)}<{\tau }_2^{\left(1\right)}<{\tau }_1^{\left(1\right)}<\cdots <{\tau }_1^{(m-1)}<{\tau }_2^{\left(m\right)}<{\tau }_2^{(m+1)}<\cdots$. When $\tau \in [0,{\tau }_2^{\left(0\right)})\cup {\displaystyle \bigcup _{l=1}^m}({\tau }_1^{(l-1)},{\tau }_2^{\left(l\right)})$, the equilibrium E of the system (2.1) is locally asymptotically stable, and when $\tau \in {\displaystyle \bigcup _{l=0}^{m-1}}({\tau }_2^{\left(l\right)},{\tau }_1^{\left(l\right)})\cup ({\tau }_2^{\left(m\right)},+\infty )$, the equilibrium E is unstable, where ${\tau }_k^{\left(j\right)}$ ($k=1,2,j=0,1,2,\cdots$) is given in Equation (6).

4. Normal Form of Hopf Bifurcation

In this section, we use the multiple time scales method to obtain the normal form of Hopf bifurcation of system (1). In order to reflect the reality of Saperda populnea damage to poplars, we analyze the delay required for Saperda populnea larvae to grow to adults and the effect of this delay on poplar damage. When $\tau ={\tau }_k^{\left(j\right)}$, ${\tau }_k^{\left(j\right)}$ is given in Equation (6), the characteristic Equation (2) has the eigenvalue $\lambda =\mathtt{i}\omega$, and the system (1) undergoes Hopf bifurcation at the equilibrium E. Next, we translate the equilibrium to the origin and make a transformation on the time scale, that is, let $t\mapsto t/\tau$. Then the original system (1) becomes

$$\left\{\begin{array}{c}\frac{dX}{dt}=\tau [{\theta }_{1}Y(1-\frac{Y^{*}}{N})-\frac{{\theta }_{1}Y}{N}(Y+Y^{*})-\alpha X-{\theta }_{2}X(t-1)],\hfill \\ \frac{dY}{dt}=\tau [{\theta }_{2}X(t-1)-\beta Y],\hfill \\ \frac{dW}{dt}=\tau [-{\theta }_3(XW+X{W}^{*}+X^{*}W)-{\theta }_4(YW+Y{W}^{*}+Y^{*}W)+PZ],\hfill \\ \frac{dZ}{dt}=\tau [{\theta }_3(XW+X{W}^{*}+X^{*}W)+{\theta }_4(YW+Y{W}^{*}+Y^{*}W)-(P+d)Z].\hfill \end{array}\right.$$

(7)

Then, Equation (7) also can be written as

$$\dot{S\left(t\right)}=\tau N_{1}S\left(t\right)+\tau N_{2}S(t-1)+\tau F(S\left(t\right),S(t-1)),$$

(8)

where

$$\begin{array}{cc}\hfill & S\left(t\right)={(X\left(t\right),Y\left(t\right),W\left(t\right),Z\left(t\right))}^T,S(t-1)={(X(t-1),Y(t-1),W(t-1),Z(t-1))}^T,\hfill \\ \hfill & F(S\left(t\right),S(t-1))={(\frac{-{\theta }_{1}Y{\left(t\right)}^2}{N},0,-{\theta }_{3}X\left(t\right)W\left(t\right)-{\theta }_{4}Y\left(t\right)W\left(t\right),{\theta }_{3}X\left(t\right)W\left(t\right)+{\theta }_{4}Y\left(t\right)W\left(t\right))}^T,\hfill \end{array}$$

and

$$N_1=\left(\begin{array}{cccc}-\alpha & {\theta }_1-\frac{{\theta }_1{Y}^{*}}{N}& 0& 0\\ 0& -\beta & 0& 0\\ -{\theta }_3{W}^{*}& -{\theta }_4{W}^{*}& -{\theta }_3{X}^{*}-{\theta }_4{Y}^{*}& P\\ {\theta }_3{W}^{*}& {\theta }_4{W}^{*}& {\theta }_3{X}^{*}+{\theta }_4{Y}^{*}& -P-d\end{array}\right),$$

$$N_2=\left(\begin{array}{cccc}-{\theta }_2& 0& 0& 0\\ {\theta }_2& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$

The eigenvector corresponding to the eigenvalue $\lambda =\mathtt{i}\omega \tau$ of the linear system of Equation (8) is h, and the eigenvector corresponding to the eigenvalue $\lambda =-\mathtt{i}\omega \tau$ of the adjoint matrix of the linear system of Equation (8) is $h^{*}$, and satisfies $<h^{*},h>={\overline{h^{*}}}^{T}h=1$.

Through calculation, we obtain

$$\begin{array}{cc}\hfill & h={(h_1,h_2,h_3,h_4)}^T\hfill \\ \hfill & ={(\frac{\mathtt{i}\omega +\beta }{{\theta }_2{\mathtt{e}}^{-\mathtt{i}\omega \tau }},1,\frac{p_1-{\theta }_4{W}^{*}}{\mathtt{i}\omega +{\theta }_3{X}^{*}+{\theta }_4{Y}^{*}+p_2},\frac{-\mathtt{i}\omega (p_1-{\theta }_4{W}^{*})}{(\mathtt{i}\omega +d)(\mathtt{i}\omega +{\theta }_3{X}^{*}+{\theta }_4{Y}^{*}+p_2\mathtt{i}\omega +d)})}^T,\hfill \\ \hfill & h^{*}=f{(h_1^{*},h_2^{*},h_3^{*},h_4^{*})}^T={(1,\frac{p_3}{-\mathtt{i}\omega -\beta },0,0)}^T,\hfill \end{array}$$

(9)

where $f=\frac{{\theta }_2[cos\left(\omega \tau \right)(\omega \mathtt{i}-\beta )-sin\left(\omega \tau \right)(\omega +\beta \mathtt{i})]}{{\omega }^2+2\beta \omega \mathtt{i}-{\beta }^2+[cos\left(\omega \tau \right)+sin\left(\omega \tau \right)\mathtt{i}]({\theta }_1{\theta }_2-2\alpha \beta -2\beta {\theta }_2)}$, $p_1=\frac{-{\theta }_3{W}^{*}\mathtt{i}\omega +\beta }{{\theta }_2{\mathtt{e}}^{-\mathtt{i}\omega \tau }}$, $p_2=\frac{P\mathtt{i}\omega }{\mathtt{i}\omega +d}$, $p_3={\theta }_1-\frac{2\alpha \beta }{{\theta }_2}-2\beta .$

We take $\tau$ as a bifurcation parameter. Let $\tau ={\tau }_c+\epsilon \mu$, where ${\tau }_c={\tau }_k^{\left(j\right)}$$(j=0,1,2,\cdots )$ is obtained by Equation (6). $\mu$ is the perturbation parameter, and $\epsilon$ is the dimensionless scaling parameter. Therefore, we can assume that the solution for Equation (8) is as follows:

$$S\left(t\right)=S(T_0,T_1,T_2,...)=\sum _{k=1}^{+\infty }{\epsilon }^k{S}_k(T_0,T_1,T_2,\cdots ),$$

(10)

where

$$\begin{array}{cc}\hfill & S(T_0,T_1,T_2,\cdots )=\hfill \\ \hfill & {(X(T_0,T_1,T_2,\cdots ),Y(T_0,T_1,T_2,\cdots ),W(T_0,T_1,T_2,\cdots ),Z(T_0,T_1,T_2,\cdots ))}^T,\hfill \\ \hfill & S_k={(X_k(T_0,T_1,T_2,\cdots ),Y_k(T_0,T_1,T_2,\cdots ),W_k(T_0,T_1,T_2,\cdots ),Z_k(T_0,T_1,T_2,\cdots ))}^T.\hfill \end{array}$$

We transform the derivative with respect to t:

$$\begin{array}{cc}\hfill & \frac{d}{dt}=\frac{\partial }{\partial T_0}+\epsilon \frac{\partial }{\partial T_1}+{\epsilon }^2\frac{\partial }{\partial T_2}+\cdots =D_0+\epsilon D_1+{\epsilon }^2{D}_2+\cdots ,\hfill \end{array}$$

where $D_i=\frac{\partial }{\partial T_i}$, $i=0,1,2,\cdots$.

Then, we obtain

$$\dot{S\left(t\right)}=\epsilon D_0{S}_1+{\epsilon }^2{D}_1{S}_1+{\epsilon }^3{D}_2{S}_1+{\epsilon }^2{D}_0{S}_2+{\epsilon }^3{D}_1{S}_2+{\epsilon }^3{D}_0{S}_3+\cdots .$$

(11)

We expand $X(T_0-1,\epsilon (T_0-1),{\epsilon }^2(T_0-1),\cdots )$ at $X(T_0-1,T_1,T_2,\cdots )$ by the Taylor expansion, and we can obtain

$$X(t-1)=\epsilon X_{1,{\tau }_c}+{\epsilon }^2{X}_{2,{\tau }_c}+{\epsilon }^3{X}_{3,{\tau }_c}-{\epsilon }^2{D}_1{X}_{1,{\tau }_c}-{\epsilon }^3{D}_2{X}_{1,{\tau }_c}-{\epsilon }^3{D}_1{X}_{2,{\tau }_c}+\cdots ,$$

(12)

where $X_{j,{\tau }_c}=X_j(T_0-1,T_1,T_2,...)$, $j=1,2,\cdots$.

We put Formulas (10)–(12) into Equation (8), and then the coefficients before $\epsilon$, ${\epsilon }^2$, and ${\epsilon }^3$ on both sides of the equation are equal, and the following expression is obtained:

$$\left\{\begin{array}{c}{D}_0{X}_1-{\tau }_c[{\theta }_1(1-\frac{2Y^{*}}{N})Y_1-\alpha X_1-{\theta }_2{X}_{1,{\tau }_c}]=0,\hfill \\ D_0{Y}_1-{\tau }_c({\theta }_2{X}_{1,{\tau }_c}-\beta Y_1)=0,\hfill \\ D_0{W}_1-{\tau }_c[-{\theta }_3(W^{*}{X}_1+X^{*}{W}_1)-{\theta }_4(W^{*}{Y}_1+Y^{*}{W}_1)+P{Z}_1=0],\hfill \\ D_0{Z}_1-{\tau }_c[{\theta }_3(W^{*}{X}_1+X^{*}{W}_1)+{\theta }_4(W^{*}{Y}_1+Y^{*}{W}_1)-(P+d)Z_1]=0.\hfill \end{array}\right.$$

(13)

$$\left\{\begin{array}{c}{D}_0{X}_2-{\tau }_c[{\theta }_1(1-\frac{2Y^{*}}{N})Y_2-\alpha X_2-{\theta }_2{X}_{2,{\tau }_c}]\hfill \\ =-D_1{X}_1+{\tau }_c(-\frac{{\theta }_1}{N}{Y}_1^2+{\theta }_2{D}_1{X}_{1,{\tau }_c})+\mu [{\theta }_1(1-\frac{2Y^{*}}{N})Y_1-\alpha X_1-{\theta }_2{X}_{1,{\tau }_c}],\hfill \\ D_0{Y}_2-{\tau }_c({\theta }_2{X}_{2,{\tau }_c}-\beta Y_2)=-D_1{Y}_1+\mu ({\theta }_2{X}_{1,{\tau }_c}-\beta Y_1)-{\tau }_{c)}{\theta }_2{D}_1{X}_{1,{\tau }_c},\hfill \\ D_0{W}_2-{\tau }_c[-{\theta }_3(W^{*}{X}_2+X^{*}{W}_2)-{\theta }_4(W^{*}{Y}_2+Y^{*}{W}_2+P{Z}_2)]\hfill \\ =-D_1{W}_1+{\tau }_c(-{\theta }_3{X}_1{W}_1-{\theta }_4{Y}_1{W}_1)+\mu [-{\theta }_3(W^{*}{X}_1+X^{*}{W}_1)\hfill \\ - {\theta }_4(W^{*}{Y}_1+Y^{*}{W}_1)+P{Z}_1],\hfill \\ D_0{Z}_2-{\tau }_c[-{\theta }_3(W^{*}{X}_2+X^{*}{W}_2)+{\theta }_4(W^{*}{Y}_2+Y^{*}{W}_2-(P+d)Z_2)]\hfill \\ =-D_1{Z}_1+{\tau }_c({\theta }_3{X}_1{W}_1+{\theta }_4{Y}_1{W}_1)+\mu [{\theta }_3(W^{*}{X}_1+X^{*}{W}_1)\hfill \\ + {\theta }_4(W^{*}{Y}_1+Y^{*}{W}_1)-(P+d)Z_1].\hfill \end{array}\right.$$

(14)

$$\left\{\begin{array}{c}{D}_0{X}_3-{\tau }_c[{\theta }_1(1-\frac{2Y^{*}}{N})Y_3-\alpha X_3+{\theta }_2{X}_{3,{\tau }_c}]\hfill \\ =-D_2{X}_1-D_1{X}_2+{\tau }_c[-\frac{2{\theta }_1}{N}{Y}_1{Y}_2+{\theta }_2(D_2{X}_{1,{\tau }_c}+D_1{X}_{2,{\tau }_c})]\hfill \\ + \mu [{\theta }_1(1-\frac{2Y^{*}}{N}{Y}_2-\frac{{\theta }_1}{N})Y_1^2-\alpha X_2+{\theta }_2(-X_{2,{\tau }_c}+D_1{X}_{1,{\tau }_c})],\hfill \\ D_0{Y}_3-{\tau }_c({\theta }_2{X}_{3,{\tau }_c}-\beta Y_3)\hfill \\ =-D_2{Y}_1-D_1{Y}_2-{\tau }_c\left[{\theta }_2(D_2{X}_{1,{\tau }_c}+D_1{X}_{2,{\tau }_c})\right]+\mu [{\theta }_2(X_{2,{\tau }_c}-D_1{X}_{1,{\tau }_c})-\beta Y_2],\hfill \\ D_0{W}_3-{\tau }_c[-{\theta }_3(W^{*}{X}_3+X^{*}{W}_3)-{\theta }_4(W^{*}{Y}_3+Y^{*}{W}_3)+P{Z}_3]\hfill \\ =-D_2{Z}_1-D_1{Z}_2+{\tau }_c[-{\theta }_3(X_1{W}_2+X_2{W}_1)-{\theta }_4(Y_1{W}_2+Y_2{W}_1)]\hfill \\ + \mu [-{\theta }_3(X_1{W}_1+W^{*}{X}_2+X^{*}{W}_2)-{\theta }_4(Y_1{W}_1+W^{*}{Y}_2+Y^{*}{W}_2)+P{Z}_2],\hfill \\ D_0{Z}_3-{\tau }_c[{\theta }_3(W^{*}{X}_3+X^{*}{W}_3)+{\theta }_4(W^{*}{Y}_3+Y^{*}{W}_3)-(P+d)Z_3]\hfill \\ =-D_2{Z}_1-D_1{Z}_2+{\tau }_c[{\theta }_3(X_1{W}_2+X_2{W}_1)+{\theta }_4(Y_1{W}_2+Y_2{W}_1)]\hfill \\ + \mu [{\theta }_3(X_1{W}_1+W^{*}{X}_2+X^{*}{W}_2)+{\theta }_4(Y_1{W}_1+W^{*}{Y}_2+Y^{*}{W}_2)-(P+d)Z_2].\hfill \end{array}\right.$$

(15)

Equation (13) has the following form of solution:

$$S_1=Gh{\mathtt{e}}^{\mathtt{i}\omega {\tau }_c{T}_0}+\overline{G}\overline{h}{\mathtt{e}}^{-\mathtt{i}\omega {\tau }_c{T}_0},$$

(16)

where h is given by (9). Thus, substituting solution (16) into the right part of Equation (14), and noting the coefficient vector of ${\mathtt{e}}^{\mathtt{i}\omega {\tau }_c{T}_0}$ as $m_1$, by $<h^{*},m_1>=0$, we can solve $\frac{\partial G}{\partial T_1}$, namely,

$$\begin{array}{c}\hfill \frac{\partial G}{\partial T_1}=K\mu G,\end{array}$$

(17)

where $K=\frac{-k_1-a{k}_2}{k_3+h_4^{*}{k}_4}$, with $h_4^{*}$, $h_1$, $p_3$ given in Equation (9), and

$$\begin{array}{cc}\hfill & k_1={\theta }_1(1-\frac{2Y^{*}}{N})-\alpha h_1-{\theta }_2{h}_1{\mathtt{e}}^{-\mathtt{i}\omega {\tau }_c},k_2={\theta }_2{h}_1{\mathtt{e}}^{-\mathtt{i}\omega {\tau }_c}-\beta ,\hfill \\ \hfill & k_3=-h_1-{\tau }_c{\theta }_2{h}_1{\mathtt{e}}^{-\mathtt{i}\omega {\tau }_c},k_4=-1+{\tau }_c{\theta }_2{h}_1{\mathtt{e}}^{-\mathtt{i}\omega {\tau }_c},a=\frac{p_3}{-\mathtt{i}\omega -\beta }.\hfill \end{array}$$

(18)

The influence of the disturbance term $\mu$ on the normal form is in the higher-order part, so it can be ignored. Therefore, we ignore the part containing $\mu$ in the higher order. We suppose the solutions of Equation (14) are given as follows:

$$\begin{array}{cc}\hfill & X_2=g_1{\mathtt{e}}^{2\mathtt{i}\omega {\tau }_c{T}_0}+\overline{g_1}{\mathtt{e}}^{-2\mathtt{i}\omega {\tau }_c{T}_0}+l_1,\hfill \\ \hfill & Y_2=g_2{\mathtt{e}}^{2\mathtt{i}\omega {\tau }_c{T}_0}+\overline{g_2}{\mathtt{e}}^{-2\mathtt{i}\omega {\tau }_c{T}_0}+l_2,\hfill \\ \hfill & W_2=g_3{\mathtt{e}}^{2\mathtt{i}\omega {\tau }_c{T}_0}+\overline{g_3}{\mathtt{e}}^{-2\mathtt{i}\omega {\tau }_c{T}_0}+l_3,\hfill \\ \hfill & Z_2=g_4{\mathtt{e}}^{2\mathtt{i}\omega {\tau }_c{T}_0}+\overline{g_4}{\mathtt{e}}^{-2\mathtt{i}\omega {\tau }_c{T}_0}+l_4.\hfill \end{array}$$

(19)

Then, substituting solution (19) into Equation (14), we can obtain

$$\begin{array}{cc}\hfill & g_1=\frac{b_3{G}^2}{c_1+b_1{c}_2},g_2=\frac{{\theta }_2{\mathtt{e}}^{-2\mathtt{i}\omega {\tau }_c}}{2\mathtt{i}\omega +\beta }{g}_1,g_3=-\frac{{\theta }_3{G}^2{h}_1{h}_3+{\theta }_4{G}^2{h}_3+d_1}{2\mathtt{i}\omega +d_3+d_2},g_4=\frac{-2\mathtt{i}\omega }{2\mathtt{i}\omega +d}{g}_3,\hfill \\ \hfill & l_1=\frac{\beta }{{\theta }_1}{l}_2,l_2=\frac{-{\theta }_{1}G\overline{G}}{N(b_1+b_2)},l_3=\frac{G\overline{G}[{\theta }_4\left(h_1\overline{h_3}\right)+{\theta }_4(h_3+\overline{h_3})]+{\theta }_2{W}^{*}{l}_1+{\theta }_4{W}^{*}{l}_2}{-d_3},l_4=0,\hfill \end{array}$$

with $h_1$, $h_3$ given in (9), and

$$\begin{array}{cc}\hfill & b_1=-{\theta }_1(1-\frac{2Y^{*}}{N}),b_2=\frac{(\alpha +{\theta }_2)\beta }{{\theta }_1},b_3=\frac{-{\theta }_1}{N},\hfill \\ \hfill & c_1=2\mathtt{i}\omega +\alpha +{\theta }_2{\mathtt{e}}^{-2\mathtt{i}\omega {\tau }_c},c_2=\frac{{\theta }_2{\mathtt{e}}^{-2\mathtt{i}\omega {\tau }_c}}{2\mathtt{i}\omega +\alpha },\hfill \\ \hfill & d_1=W^{*}({\theta }_3{g}_1+{\theta }_4{g}_2),d_2=\frac{2P\mathtt{i}\omega }{2\mathtt{i}\omega +d},d_3={\theta }_3{X}^{*}+{\theta }_4{Y}^{*}.\hfill \end{array}$$

(20)

Next, substituting solutions (16) and (19) into (15), and the coefficient vector of ${\mathtt{e}}^{\mathtt{i}\omega {\tau }_c{T}_0}$ denoted as $m_2$, under the solvability condition, $<h^{*},m_2>=0$. We note that $\mu$ is the perturbation parameter and ${\mu }^2$ has little effect on the small unfolding parameter, so we can ignore the ${\mu }^{2}G$ term. Then $\frac{\partial G}{\partial T_2}$ can be solved in the following form:

$$\frac{\partial G}{\partial T_2}=H{G}^2\overline{G},$$

(21)

where $H=\frac{2{\theta }_1{\tau }_c(f_1+f_2{f}_3)}{N(-h_1+J-a{h}_2-aJ)}$. $h_1$, $h_2$ are given in (9), a is given in (18), $b_1$, $b_2$, $c_1$, $c_2$ are given in (20), and $f_1=-\frac{{\theta }_1}{N{b}_1{b}_2}$, $f_2=\frac{{\theta }_2{\mathtt{e}}^{-2\mathtt{i}\omega {\tau }_c}}{2\mathtt{i}\omega +\beta }$, $f_3=\frac{-{\theta }_1}{N(c_1+b_1{c}_2)}$, $J={\tau }_c{\theta }_2{h}_1{\mathtt{e}}^{-\mathtt{i}\omega {\tau }_c}$.

Let $G\to G/\epsilon$. We obtain the normal form of Hopf bifurcation of Equation (8) truncated at the cubic-order terms:

$$\dot{G}=K\mu G+H{G}^2\overline{G},$$

(22)

where K is given in (17) and H is given in (21).

We substitute the polar $G=r{\mathtt{e}}^{\mathtt{i}\theta }$ into Equation (22) to obtain the normal form of Hopf branch in polar coordinates:

$$\left\{\begin{array}{c}\dot{r}=\mathtt{Re}\left(K\right)\mu r+\mathtt{Re}\left(H\right)r^3,\hfill \\ \dot{\theta }=\mathtt{Im}\left(K\right)\mu +\mathtt{Im}\left(H\right)r^2.\hfill \end{array}\right.$$

(23)

According to the normal form of Hopf bifurcation in polar coordinates, we only need to consider the first equation in system (23). Thus, the following theorem holds:

Theorem 2.

For the first equation of system (23), when $\frac{\mathtt{Re}\left(K\right)\mu }{\mathtt{Re}\left(H\right)}<0$, there is an equilibrium $r^{*}=\sqrt{-\frac{\mathtt{Re}\left(K\right)\mu }{\mathtt{Re}\left(H\right)}}$, and system (1) exhibits periodic solutions.

(1) 

If $\mathtt{Re}\left(K\right)\mu <0$, then the periodic solution reduced on the center manifold is unstable.

(2) 

If $\mathtt{Re}\left(K\right)\mu >0$, then the periodic solution reduced on the center manifold is stable.

5. Numerical Simulations

In this section, we carry out reasonable analysis and speculation on some parameter values based on official data. On the basis of these parameters, we verify our theoretical analysis by numerical simulations, and give some suggestions for controlling Saperda populnea.

5.1. Determination of Parameter Values

According to the official statistics of the National Bureau of Statistics, we obtain data on the prevention rate of forest pests and the total area of artificial afforestation in different regions of China. In order to ensure that the data can reflect the average value, we retain representative data and eliminate outliers. Finally, we screen the forest control rate of 31 regions and the total afforestation area of 26 regions in China, and based on these data, we obtain bar charts, which are presented in Figure 1 and Figure 2.

(1) Recovery rate: P

We use the average value of forest pest prevention and control rate in different regions of the country as the prevention and control rate of Saperda populnea, that is, the recovery rate of infected poplars after human intervention is P. Figure 1 shows that most of the prevention and control rates of forest pests and diseases in the whole country are in the range of 0.5 to 0.8, so we calculate the mean value 0.78 and choose it as the value of P, that is, P $= 0.78$.

(2) Number of poplar trees in artificial afforestation: R

According to Figure 2, we can calculate that the average area of artificial afforestation in each region in 2020 is 114,166.7 hectares, and about 2000 to 3000 trees can be planted per hectare. Since there are other species of trees that need to be planted, we assume that 227 poplars are planted per hectare, and that the average number of poplars planted per district per week in 2020 $R=\frac{\mathrm{114,166.7} \times 227}{52}\approx$ 500,000, that is, $R=0.05$.

(3) Mortality: $\alpha$, $\beta$, d and infection rate: ${\theta }_3$, ${\theta }_4$

In Ref. [24], larval mortality of Bruchus pisorum of the same order as Saperda populnea on pods of several F-2:3 families ranged from 36 to 52.9%, and total weevil mortality on pods and seeds of eight F-2:3 families was estimated to be 50–70%. Therefore, we consider the average mortality rate of Bruchus pisorum larvae to be 40% and use it as the larval mortality rate of Saperda populnea and take the adult mortality rate of Saperda populnea to be 25%, that is, $\alpha =0.4$, $\beta =0.25$.

In 1998–2002, the mortality rate of heavily and moderately mistletoe-infected trees was 44.44%, and the mortality rate of lightly infected trees was 9.5% [25]. Therefore, the mortality rate of mistletoe-infected trees is about 55%, so we take 55% as the mortality rate of poplars infected by Saperda populnea, that is, $d=0.55$.

In 2015, scholars found that the prevalence of aspen infected by an invasive fungal ulcer disease at six study sites distributed across 88 ecoregions in Alaska ranged from 1% to 69% [26]. Therefore, we take the intermediate values of 30% and 40% as the infection rates of Saperda populnea larvae and adults on poplars, respectively, that is, ${\theta }_3=0.3$, ${\theta }_4=0.4$.

Based on the above consideration and values, we choose a group of parameter as follows:

${\theta }_1=0.4$, ${\theta }_2=0.7$, ${\theta }_3=0.3$, ${\theta }_4=0.4$, $\alpha =0.4$, $\beta =0.25$, $P=0.78$, $N=6$, $R=0.05$, $d=0.55$.

5.2. Numerical Simulations and Interpretations of System (1)

For the above parameters, it is clear that condition (A1) holds. We calculate the equilibrium $E=(0.04,0.1,2.2,0.09)$ of system (1). According to Equation (5), we find that ${\alpha }^2+{\beta }^2-{\theta }_2^2<0$, $-\frac{{({\alpha }^2+{\beta }^2-{\theta }_2^2)}^2}{4}<M<0$, so Equation (5) has two positive roots $z_1=0.0037$, $z_2=0.2638$, and we can calculate ${\omega }_1=0.0608$, ${\omega }_2=0.5136$. Substituting the parameter into Equation (6), we obtain ${\tau }_1^{\left(0\right)}=90.0195$, ${\tau }_2^{\left(0\right)}=5.7315$, ${\tau }_2^{\left(1\right)}=17.9646$, so $0<{\tau }_2^{\left(0\right)}<{\tau }_2^{\left(1\right)}<{\tau }_1^{\left(0\right)}<\cdots$. According to Theorem 1, the trivial equilibrium E is local asymptotically stable for $\tau \in [0,{\tau }_2^{\left(0\right)})=[0,5.7315)$ and unstable for $\tau >{\tau }_2^{\left(0\right)}$.

We first choose $\tau =4.5\in [0,5.7315)$. Clearly, the equilibrium E is locally asymptotically stable, as shown in Figure 3.

From Figure 3, we find that the equilibrium E is locally asymptotically stable. The fluctuation is obvious in the first 150 weeks, but it gradually stabilized after the 150th week, and the pests can also be controlled. However, when the number of susceptible poplars and the number of infected poplars tended to be stable, the number of adults of Saperda populnea is larger. Therefore, physical control, chemical control, and other means must be used to control Saperda populnea.

W represents the number of susceptible poplars, and Z represents the number of infected poplars. We set $\delta =W-Z$ as the difference between the number of susceptible poplars and the number of infected poplars, and then we discuss the change in $\delta$ with time t when the time delay $\tau$ is 4.5, 5, 5.5, as shown in Figure 4.

From Figure 4, it can be seen that under different time delays $\tau$, the number of susceptible poplars W almost always reaches the maximum at the 100th week, which means that within a certain time range, even if the growth delay $\tau$ from larva to adult becomes larger and larger, and the number of susceptible poplars always reaches the maximum at the 100th week. However, with the increase in time delay $\tau$, the difference $\delta$ between susceptible and infected poplars becomes smaller and smaller, and it is difficult to reach a stable value. Therefore, in order to control Saperda populnea as soon as possible, chemical agents can be sprayed on poplars when the larvae are ready to pupate into adults. On the one hand, it can kill Saperda populnea, and on the other hand, it can reduce the growth delay $\tau$ of the larvae and provide convenient conditions for subsequent control of pests and diseases.

According to Theorem 1, we obtain that when $\tau >{\tau }_2^{\left(0\right)}=5.7315$, the equilibrium E is unstable. From Equations (17) and (21), we have $K=0.1097-0.2234\mathtt{i},H=-0.0693-0.2248\mathtt{i}$. Obviously, $\frac{\mathtt{Re}\left(K\right)\mu }{\mathtt{Re}\left(H\right)}<0$ and $\mathtt{Re}\left(K\right)\mu >0$; therefore, according to Theorem 2, system (1) has a stable periodic solution near the bifurcation critical value ${\tau }_2^{\left(0\right)}$ when $\mu >0$. We choose $\tau =5.8>5.7315$, and the numerical simulation is consistent with the theoretical analysis, as shown in Figure 5.

From Figure 5, we find that the bifurcating periodic solution is stable, which verifies our theoretical analysis. In fact, no matter how much the time is, the number of larvae and adults will be greatly affected, so the number of X and Y will fluctuate with the change in time delay.

Next, we discuss the effects of egg-to-larvae survival rate ${\theta }_1$ and larva-to-adult survival rate ${\theta }_2$ on critical delay ${\tau }_2^{\left(0\right)}$, as shown in Figure 6.

From Figure 6, we can obtain the following:

(1)

${\tau }_2^{\left(0\right)}$ increases with the increase in ${\theta }_1$ and decreases with the increase in ${\theta }_2$. When the survival rate of Saperda populnea eggs to larvae ${\theta }_1$ increases, $\tau$ increases, which is not conducive to our control of pests. Therefore, in the prevention and control of Saperda populnea, we can mainly eliminate the larvae to control Saperda populnea, in order to achieve the purpose of protecting trees.

(2)

In fact, after the larvae hatch, they begin to eat the trunk. Even very small larvae dig deep holes in the trunk, which are deadly to trees, and the larvae feed on the trunk in the hole. In addition, the feces of the larvae can also accumulate in the hole, and once the amount is large enough, the tree will die. Therefore, when controlling Saperda populnea, we focus on eliminating the larvae of Saperda populnea, which is consistent with our theoretical analysis.

5.3. Analysis of Simulations

Based on the results of the above numerical simulations, we put forward the following suggestions for the prevention and control of Saperda populnea.

(1)

From Figure 3, we find that even after the infestation has stabilized, there are still more adults and larvae, so we still need to take control measures for Saperda populnea. We can prune branches with insects to prevent the spread of Sarerda populnea. The insect source trees around roadsides, villages, and especially nurseries, should be completely removed. We should formulate appropriate control measures according to the characteristics of Saperda populnea in different periods. The activity ability of Saperda populnea adults is insufficient, which can be controlled by artificial trapping. It can be wrapped in the tree branches with tape during the emergence of adult insects to do a good job of sealing and preventing emergence holes. At the peak of adult development, adults are artificially killed by taking advantage of characteristics such as its “false mortality” or roosting at the base of the trunk.

(2)

The survival rate from egg to larva of Saperda populnea has a greater impact on the critical delay ${\tau }_2^{\left(0\right)}$. When the survival rate from egg to larva of Saperda populnea decreases, that is, when there are fewer larvae, it is more beneficial for Saperda populnea control. Therefore, we can use chemical control and inject the liquid into the bore hole with a syringe to kill the larvae of Saperda populnea. In the damage period of larvae, when there is excrement on the trunk, the bark can be peeled off to dig out the larvae, kill them, or stab the larvae in the wormhole with metal wire.

(3)

In addition, it is necessary to ensure the scientific collocation in the selection of tree species in forestry regions and adhere to the principle of adapting to local conditions. Based on the actual local climate conditions, the basic nature of pests, and the growth conditions of different tree species, it is necessary to reasonably match the tree species with strong immunity and establish a perfect mixed forest structure with strong pest defense ability. When setting up the matching scheme, the tree species with strong disease resistance and vitality should be the main focus. When the disease is completely eliminated, it is necessary to appropriately supplement the tree species with strong immune capacity, so as to achieve the goal of pest control.

6. Conclusions

In this paper, we have established a delay differential equation for the damage of Saperda populnea larvae and adults to poplars. We have studied the stability of the equilibrium of system (1) and the existence of Hopf bifurcation. Then, we use the multiple time scales method to calculate the normal form of Hopf bifurcation and analyze the stability of Hopf bifurcating periodic solutions.

On the basis of official data, we have analyzed and speculated on some parameter values. Then we have carried out numerical simulation to verify the theoretical analysis results. We have found that when the insect pest is stable, the adults and larvae of Saperda populnea are still stable at a large value. Therefore, it is still necessary to take control measures against Saperda populnea. Next, we have found that time delay $\tau$ affects the number of susceptible poplars reaching the maximum. Then, we have studied the effect of the survival rate from egg to larva and the survival rate from larva to adult on the the critical time $\tau$, and have concluded that the main task in the control process is to eliminate the larvae.

After giving the biological suggestions for controlling Saperda populnea, we also have emphasized that in the process of controlling Saperda populnea, we should pay attention to the selection of poplars with strong insect resistance for cultivation, so as to reduce the number of infected poplars.