Stability Analysis of a Delayed Paranthrene tabaniformis (Rott.) Control Model for Poplar Forests in China

Abstract

Forest pests and diseases can diminish forest biodiversity, damage forest ecosystem functions, and have an impact on water conservation. Therefore, it is necessary to analyze the interaction mechanism between plants and pests. In this paper, the prevention and control of a specific pest—namely the larva of Paranthrene tabaniformis (Rott.) (hereinafter referred to as larva)—are studied. Based on the invasion mechanism of the larva in poplar, we establish a delayed differential equation and analyze the existence and stability of equilibria. Next, we assess the existence of a Hopf bifurcation to determine the range of parameters that ensures that the equilibria are stable. Then, we select a set of parameters to verify the results of the stability analysis. Finally, we provide biological explanations and effective theoretical control methods for poplar pests and diseases.

1. Introduction

Forest pests and diseases, including pathogens, fungi, bacteria, insects, and various other agents, represent biological threats to the productivity of forests. Their rampant emergence not only depletes forest resources, disrupts ecological equilibria, and diminishes biodiversity, but also hinders the economic progress of human society and facilitates the spread of viruses [1,2,3,4]. Furthermore, these issues undermine the structure of forest carbon sinks, posing challenges in achieving global goals regarding peak carbon dioxide emissions and carbon neutrality [5,6,7]. The pervasive outbreaks of diverse pests in forests worldwide have had a profound impact on all forest types, strengthening their influence on human economic development [8,9,10,11].

Due to its limited forest resources, China annually imports a substantial volume of wood to meet its demands for economic development. Both domestic and international experiences underscore the significance of establishing timber forests and cultivating fast-growing trees through high-yield methods as crucial strategies to swiftly address the national wood shortage. China, with its recent focus on green development and ecological construction, has increasingly recognized the importance of sustainable practices. The government has consistently emphasized ecological prioritization for green and low-carbon development, including encouraging farmers to actively engage in forest pest control. This commitment is further evidenced by the 1989 enactment of the Law on the Prevention and Control of Forest Pests and Diseases. Concurrently, the government has intensified its efforts regarding monitoring and early warning systems for forest pests and diseases, thus ensuring timely detection and prevention. This holistic approach is in alignment with the nation’s commitment to sustainable development and ecological well-being.

In the realm of forest pest and disease research, scholars have conducted extensive studies across diverse fields, including ecology, biology, and agricultural biotechnology [12,13,14]. Notably, various mathematical models, such as logistic, pulse, and continuous models, have been established and used to simulate the occurrence of pests and diseases, offering scientific insights for forestry and related management [15,16,17,18,19,20,21,22,23]. For example, Abbott et al. [22] introduced a model incorporating diffusion and spatially dependent randomness, thus enhancing our understanding of the synchronization of forest insect populations. Each of these models presents unique advantages and proposes measures to scientifically and reasonably address forest pests and diseases, contributing significantly to the nation’s ecosystem, economy, and environmental protection from an applied mathematics perspective.

The present study is primarily focused on the prevention and control of poplar pests and diseases, driven by several key factors. First, China boasts a poplar planting area exceeding 120 million mu, ranking second nationally and occupying 27% of its artificial forest land. Poplar, a crucial raw material in various wood products, such as packaging, yields substantial economic returns through scientific cultivation. With a harvesting period of 7 to 12 years, poplar provides the opportunity for rapid investment recovery and significant benefits. Besides its economic advantages, poplar cultivation contributes to societal and ecological well-being, given its rich cultural heritage and robust ecological functions (e.g., wind-proofing and sand-fixing capabilities).

A primary threat to poplar growth is the larva of Paranthrene tabaniformis (Rott.), as depicted in Figure 1. These are typically around 30 mm in length and milky white, with slightly upturned dark brown spines on the hip joint and two horizontal belts on the toe hooks of the abdomen and buttocks. These larvae overwinter in damaged poplar branches, initiating movement in mid-April of the following year, pupating in early May, and transforming into imago in late May. In early June, newly hatched larvae invade branches directly or migrate from leaf axils, causing damage to new poplar branches in the form of tumor-like galls. This damage makes the branches susceptible to breakage by wind, leading to seedling loss and even deforestation in newly planted forests.

In the early 1970s, several Chinese cities suffered from severe pest and disease outbreaks, with insect infestation rates soaring to 90% in some nurseries, leading to extensive casualties in newly planted poplar forests (available online at https://xuewen.cnki.net/CJFD-JSLY200004015.html (accessed on 1 January 2024)). The insidious Paranthrene tabaniformis (Rott.) was designated as a supplementary object of forest plant quarantine in China. Since 2000, this pest has posed ecological threats in nurseries, farmland shelterbelts, and green passage engineering woodlands, exhibiting a worrisome pattern of year-on-year expansion and intensification [24]. Targeting this pest requires employing pesticides, closely monitoring tree growth, and promptly eliminating infected areas [25]. Although numerous studies have been conducted on forest pests and diseases in various fields, limited attention has been paid to the use of delayed differential equations to describe the effects of Paranthrene tabaniformis (Rott.) on poplar. Recognizing a time delay in the hatching process from imago to larva, which undergoes continuous change over time, we constructed a delayed differential equation to discuss their dynamic properties, leveraging mathematical modeling techniques.

The impetus behind this research stems from two primary considerations. First, given China’s extensive forests, preventing forest pests and diseases is imperative. The protection of poplar, as the second-largest tree species, is not only a key measure to maintain the trajectory of human economic development but also an important strategy for the protection of China’s forest resources. These efforts lay a robust foundation for the strengthening of carbon absorption and the expeditious achievement of the double-carbon goal. Second, considering the biological characteristics of poplar and its pests, we propose a delayed differential equation that intricately combines the present state of poplar and larvae with their historical states. This creative approach establishes a predictive model for the spread of poplar pests and diseases over time in a forest. The incorporation of a time delay from eggs to larvae of Paranthrene tabaniformis (Rott.) provides a more nuanced reflection of the dynamic situation surrounding poplar diseases and insect pests. Finally, by analyzing the stability of equilibria, we provide a range for each parameter that allows this pest to be controlled. We select a set of parameters to verify the results of this theoretical analysis and observe corresponding phenomena and give biological explanations with regard to the numerical simulation. By leveraging the delayed differential equation, we forecast the states of both poplar and larvae. If the selected parameters deviate from our expectations, we can judiciously adjust them within a reasonable range. Consequently, we can propose practical measures to combat poplar pests and diseases, offering comprehensive suggestions for poplar protection.

The subsequent sections are organized as follows. Section 2 focuses on establishing the mathematical model of poplar pests and diseases under human intervention. In Section 3, we discuss the existence and stability of equilibria under this model. Following this, Section 4 presents the numerical simulations conducted to validate the accuracy of our analysis. Finally, we summarize our findings in Section 5.

2. Mathematical Modeling

Inspired by the infectious disease model, we propose a model that focuses on poplar, diseased poplar, and the larvae of Paranthrene tabaniformis (Rott.) (referred to as larvae). We provide a flow diagram of the invasion mechanism of larvae to poplar and establish a delayed differential equation to analyze their dynamic interaction mechanism according to the flow diagram (see Figure 1).

The associated equation is presented below:

$$\left\{\begin{array}{c}\frac{dS}{dt}=B-d_{1}S-{\phi }_{1}XS,\hfill \\ \frac{dI}{dt}={\phi }_{1}XS-d_{2}I-{\phi }_{2}XI,\hfill \\ \frac{dX}{dt}=(r-d)X(t-\tau )-d_{3}X-\phi {\phi }_{2}XI,\hfill \end{array}\right.$$

(1)

where B, $\phi$, ${\phi }_1$, ${\phi }_2$, $d_1$, $d_2$, $d_3$, $\tau$, r, and d are positive constants. Specific details are provided in

Table 1. Descriptions of the variables and parameters used in system (1).

Symbol	Description	Unit
S	Planting area of poplar	100 K hec
I	Planting area of infected poplar	100 K hec
X	Amount of larva	M pcs
B	Daily planting rate of poplar	-
$d_1$	Daily saturated lethal rate of poplar	-
$d_2$	Daily saturated lethal rate of infected poplar	-
$d_3$	Mortality of larva	-
${\phi }_1$	Influence coefficient of larva on poplar	-
${\phi }_2$	Influence coefficient of larva on infected poplar	-
$\phi$	Non-transfer coefficient on larva	-
r	Spawning rate of eggs	-
d	Mortality of eggs	-
$\tau$	Time delay from egg laying egg to larva hatching	day

.

It is worth noting that the competition among fast-growing poplar trees is considered in the system (1). Specifically, as the number of poplar seedlings increases in a given area, each tree has access to fewer growth resources, resulting in higher mortality rates. Consequently, saturation terms for poplar and diseased trees are incorporated into system (1). Intuitively, more larvae on poplar trees increase the risk of disease, accelerating the transition from diseased poplars to dead poplars. Hence, damage terms for larvae on poplars and diseased poplars are integrated into system (1).

This paper assumes that the survival rate of eggs remains constant. Additionally, we introduce a time delay from egg laying to larva hatching, denoted as $\tau$. X(t-$\tau$) denotes the number of eggs that develop to larvae. Leveraging this delay, we consider the current imago population as the larval population after $\tau$ time and incorporate the successful hatching of the larvae into the model. Regarding hatched larvae, their competition intensifies under specific resource conditions. We also consider natural factors such as extreme weather and natural enemies, which can lead to the natural elimination of larvae. Consequently, we introduce an elimination term for larvae. Moreover, we assume that the death of a diseased poplar might cause the death of its internal larvae; ${\varphi }_{2}XI$ means that larvae have a lethal rate for infected poplars. Therefore, $\varphi {\varphi }_{2}XI$ indicates the rate of death due to the non-transfer of larvae.

Denoting ${\phi }_3=\phi {\phi }_2$ and $c=r-d$, (1) can be written as follows:

$$\left\{\begin{array}{c}\frac{dS}{dt}=B-d_{1}S-{\phi }_{1}XS,\hfill \\ \frac{dI}{dt}={\phi }_{1}XS-d_{2}I-{\phi }_{2}XI,\hfill \\ \frac{dX}{dt}=cX(t-\tau )-d_{3}X-{\phi }_{3}XI.\hfill \end{array}\right.$$

(2)

In the following, we study the dynamic properties of system (2).

3. Existence and Stability of Equilibria

3.1. Existence of Equilibria in System (2)

This section is devoted to discussing the existence and stability of equilibria for system (2).

It was found that system (2) has three equilibria:

$$E_1=(\frac{B}{d_1},0,0),E_2=(S_2,I_2,X_2),E_3=(S_3,I_3,X_3),$$

where

$$\left\{\begin{array}{c}{I}_2=I_3=\frac{c-d_3}{{\phi }_3},\hfill \\ X_2=\frac{(B{\phi }_1-{\phi }_1{d}_2{I}_2-{\phi }_2{d}_1{I}_2)-\sqrt{{({\phi }_1{d}_2{I}_2+{\phi }_2{d}_1{I}_2-B{\phi }_1)}^2-4{\phi }_1{\phi }_2{d}_1{d}_2{I}_2^2}}{2{\phi }_1{\phi }_2{I}_2},\hfill \\ X_3=\frac{(B{\phi }_1-{\phi }_1{d}_2{I}_3-{\phi }_2{d}_1{I}_3)+\sqrt{{({\phi }_1{d}_2{I}_3+{\phi }_2{d}_1{I}_3-B{\phi }_1)}^2-4{\phi }_1{\phi }_2{d}_1{d}_2{I}_3^2}}{2{\phi }_1{\phi }_2{I}_3},\hfill \\ S_k=\frac{B}{d_1+{\phi }_1{X}_k},(k=2,3).\hfill \end{array}\right.$$

We consider the following assumption:

$$\left({\mathbf{H}}_{\mathbf{1}}\right)\left\{\begin{array}{c}c-d_3>0,\hfill \\ B{\phi }_1-{\phi }_1{d}_2{I}_k-{\phi }_2{d}_1{I}_k>0,\hfill \\ {({\phi }_1{d}_2{I}_k+{\phi }_2{d}_1{I}_k-B{\phi }_1)}^2-4{\phi }_1{\phi }_2{d}_1{d}_2{I}_k^2>0,(k=2,3)\hfill \end{array}\right.$$

As $E_1$ is a semi-equilibrium, we only consider the equilibria $E_k$$(k=2,3)$. When the parameters meet condition $\left({\mathbf{H}}_{\mathbf{1}}\right)$, the equilibria $E_k$$(k=2,3)$ of the system (2) exist and are positive. When the parameters do not meet condition $\left({\mathbf{H}}_{\mathbf{1}}\right)$, the equilibria do not exist or are not positive; in this case, the equilibria do not have any practical meaning.

3.2. Stability of Equilibria in System (2)

Next, we obtain the characteristic equation of system (2) at $E_k(k=2,3)$ as follows:

$${\lambda }^3+K_k{\lambda }^2+L_k\lambda +R_k-c{\mathrm{e}}^{-\lambda \tau }({\lambda }^2+A_k\lambda +T_k)=0,$$

(3)

where

$K_k=d_1+d_2+{\phi }_1{X}_k+{\phi }_2{X}_k+c,$

$L_k=(d_1+{\phi }_1{X}_k+c)(d_2+{\phi }_2{X}_k)+c(d_1+{\phi }_1{X}_k)+d_2(c-d_3),$

$R_k=c(d_1+{\phi }_1{X}_k)(d_2+{\phi }_2{X}_k)+(c-d_3)(d_1{d}_2-{\phi }_1{\phi }_2{X}_k^2),$

$A_k=d_1+{\phi }_1{X}_k+d_2+{\phi }_2{X}_k,$

$T_k=(d_1+{\phi }_1{X}_k)(d_2+{\phi }_2{X}_k).$

When $\tau$ = 0, the characteristic Equation (3) becomes

$${\lambda }^3+(K_k-c){\lambda }^2+(L_k-c{A}_k)\lambda +R_k-c{T}_k=0.$$

(4)

We consider the following assumption:

$$\left({\mathbf{H}}_{\mathbf{2}}\right)d_1{d}_2>{\phi }_1{\phi }_2{X}_k^2,(k=2,3).$$

When $\left({\mathbf{H}}_{\mathbf{1}}\right)$ and $\left({\mathbf{H}}_{\mathbf{2}}\right)$ hold, we can prove that

$$\begin{array}{c}{K}_k-c=d_1+d_2+{\phi }_1{X}_k+{\phi }_2{X}_k>0,\hfill \\ (K_k-c)(L_k-c{A}_k)-(R_k-c{T}_k)={(d_1+{\phi }_1{X}_k)}^2(d_2+{\phi }_2{X}_k)+{(d_2+{\phi }_2{X}_k)}^2(d_1+{\phi }_1{X}_k)+\hfill \\ {\phi }_3{I}_k[d_2(d_2+{\phi }_2{X}_k+{\phi }_1{X}_k)+{\phi }_1{\phi }_2{X}_k^2]>0,\hfill \\ R_k-c{T}_k={\phi }_3{I}_k(d_1{d}_2-{\phi }_1{\phi }_2{X}_k^2)>0.\hfill \end{array}$$

By the Routh–Hurwitz criterion, we can prove that all the roots of Equation (4) have negative real parts and, thus, the equilibria $E_k$ of system (2) are locally asymptotically stable when $\tau =0$. In this case, we consider that this pest can be controlled. When $\left({\mathbf{H}}_{\mathbf{2}}\right)$ does not hold, the equilibria $E_k$ of system (2) are not stable when $\tau =0$, so we consider that the pest will proliferate. When $\tau >0$, we discuss the existence of a Hopf bifurcation for system (2) in the following. We assume that $\lambda =\mathrm{i}\omega$ $(\omega >0)$ is a pure imaginary root of Equation (3). Substituting it into Equation (3) and separating the real and imaginary parts, we obtain

$$\left\{\begin{array}{c}-{\omega }^3+L_k\omega =c{A}_k\omega cos\left(\omega \tau \right)-c(-{\omega }^2+T_k)sin\left(\omega \tau \right),\hfill \\ -K_k{\omega }^2+R_k\omega =c(-{\omega }^2+T_k)cos\left(\omega \tau \right)+c{A}_k\omega sin\left(\omega \tau \right).\hfill \end{array}\right.$$

(5)

Adding the square of the two equations in Equation (5), and letting $z={\omega }^2$, we obtain

$$h\left(z_k\right)={z_k}^3+{e_k}_2{z_k}^2+{e_k}_1{z}_k+{e_k}_0,$$

(6)

where

$$\begin{array}{cc}\hfill e_{k0}& =R_k^2-c^2{T}_k^2=(R_k+c{T}_k)(R_k-c{T}_k),\hfill \\ \hfill e_{k1}& =L_k^2-2K_k{R}_k-c^2{A}_k^2+2c^2{T}_k\hfill \\ \hfill & ={(d_1+{\phi }_1{X}_k)}^2{(d_2+{\phi }_2{X}_k)}^2+{\left({\phi }_3{d}_2{I}_k\right)}^2+2{\phi }_3{d}_2{I}_k\left[\right(d_1+{\phi }_1{X}_k)(d_2+{\phi }_2{X}_k)+\hfill \\ & c(d_1+{\phi }_1{X}_k)+c(d_2+{\phi }_2{X}_k)]+2{\phi }_3{I}_k(c+d_1+{\phi }_1{X}_k+d_2+{\phi }_2{X}_k)(d_1{d}_2-{\phi }_1{\phi }_2{X}_k^2),\hfill \\ \hfill e_{k2}& =-2L_k+K_k^2-c^2\hfill \\ \hfill & ={(d_1+{\phi }_1{X}_k)}^2+{(d_2+{\phi }_2{X}_k)}^2-2d_2(c-d_3).\hfill \end{array}$$

To facilitate the discussion below, we consider the following assumption:

$$\left({\mathbf{H}}_{\mathbf{3}}\right)\left\{\begin{array}{c}{(d_1+{\phi }_1{X}_k)}^2+{(d_2+{\phi }_2{X}_k)}^2<2d_2(c-d_3),\hfill \\ {(e_{k2}{e}_{k1}-9e_{k0})}^2<4(e_{k2}^2-3e_{k1})(e_{k1}^2-3e_{k2}{e}_{k0}),(k=2,3).\hfill \end{array}\right.$$

Lemma 1.

When the parameters of system (2) meet assumptions $\left({\mathbf{H}}_{\mathbf{1}}\right)$ and $\left({\mathbf{H}}_{\mathbf{2}}\right)$, we have that $e_{k0}>0,e_{k1}>0,$. Suppose that Equation (6) has three real roots $(setas{z}_{k1},z_{k2},z_{k3})$. According to the Vieta theorem, the positive and negative situations of these three roots can be divided into the following two situations:

$$\left({\mathbf{V}}_{\mathbf{1}}\right)\left\{\begin{array}{c}0>z_{k3}⩾z_{k2}⩾z_{k1},\hfill \\ z_{k1}{z}_{k2}{z}_{k3}=-e_{k0}<0,\hfill \\ z_{k1}{z}_{k2}+z_{k2}{z}_{k3}+z_{k1}{z}_{k3}=e_{k1}>0.\hfill \end{array}\right.$$

$$\left({\mathbf{V}}_{\mathbf{2}}\right)\left\{\begin{array}{c}{z}_{k2}⩾z_{k1}>0>z_{k3},\hfill \\ z_{k1}{z}_{k2}{z}_{k3}=-e_{k0}<0,\hfill \\ z_{k1}{z}_{k2}+z_{k2}{z}_{k3}+z_{k1}{z}_{k3}=e_{k1}>0.\hfill \end{array}\right.$$

When the parameters of system (2) meet the three assumptions $\left({\mathbf{H}}_{\mathbf{1}}\right),\left({\mathbf{H}}_{\mathbf{2}}\right),$ and $\left({\mathbf{H}}_{\mathbf{3}}\right)$, Equation (6) has periodic solutions, which means that this pest will emerge periodically. When the parameters of system (2) meet the above three assumptions $\left({\mathbf{H}}_{\mathbf{1}}\right),\left({\mathbf{H}}_{\mathbf{2}}\right),$ and $\left({\mathbf{H}}_{\mathbf{3}}\right)$, we also find that Equation (6) has three real roots. We can prove that the coefficient of the quadratic term in Equation (6) is less than 0. According to the Vieta theorem, the distribution of the roots of Equation (6) belongs to $\left({\mathbf{V}}_{\mathbf{2}}\right)$. In other words, Equation (6) has two positive roots, denoted as ${z_k}_2>{z_k}_1$.

The detailed mathematical proof of Lemma 1 is given in Appendix A.1.

Substituting ${{\omega }_k}_l=\sqrt{{z_k}_l}$$(k=2,3;l=1,2)$ into Equation (5), we obtain the following expression for $\tau$:

$${{{\tau }_k}_l}^{\left(j\right)}=\left\{\begin{array}{c}\frac{1}{{{\omega }_k}_l}[arccos\left({P_k}_l\right)+2j\pi ],Q_{kl}>0,\hfill \\ \frac{1}{{{\omega }_k}_l}[2\pi -arccos\left({P_k}_l\right)+2j\pi ],Q_{kl}<0,k=1,2,3,l=1,2,j=0,1,2,\cdots ,\hfill \end{array}\right.$$

(7)

where $Q_{kl}$ and $P_{kl}$, obtained from Equation (5), are given as follows:

$$\left\{\begin{array}{c}{Q_k}_l=sin\left({{\omega }_k}_l{{\tau }_k}_l\right)=\frac{c{A}_k{\omega }_{kl}(-K_k{\omega }_{kl}^2+R_k)-c(-{\omega }_{kl}^2+T_k)(-{\omega }_{kl}^3+L_k{\omega }_{kl})}{{\left(c{A}_k{\omega }_{kl}\right)}^2+{\left[c(-{\omega }_{kl}^2+T_k)\right]}^2},\hfill \\ {P_k}_l=cos\left({{\omega }_k}_l{{\tau }_k}_l\right)=\frac{c{A}_k{\omega }_{kl}(-{\omega }_{kl}^3+L_k{\omega }_{kl})+c(-{\omega }_{kl}^2+T_k)(-K_k{\omega }_{kl}^2+R_k)}{{\left(c{A}_k{\omega }_{kl}\right)}^2+{\left[c(-{\omega }_{kl}^2+T_k)\right]}^2}.\hfill \end{array}\right.$$

(8)

Summarizing the above process, we obtain the following lemma.

Lemma 2.

If $\left(\mathbf{H}\mathbf{1}\right)$ holds, if $\tau ={{\tau }_k}_l^{\left(j\right)}$ ($k=2,3;l=1,2;j=0,1,2,\cdots )$, then Equation (3) has a pair of purely imaginary roots ${\pm \mathrm{i}\omega }_k$, and all the other roots of Equation (3) have non-zero real parts.

Furthermore, let $\lambda \left(\tau \right)=\alpha \left(\tau \right)+\mathrm{i}\omega \left(\tau \right)$ be the root of Equation (3) satisfying $\alpha \left({{{\tau }_k}_l}^{\left(j\right)}\right)=0$, $\omega \left({{{\tau }_k}_l}^{\left(j\right)}\right)={{\omega }_k}_l,(k=2,3;l=1,2;j=0,1,2,\cdots ).$ Then, we have the following transversality condition (the detailed mathematical proof of the following lemma is given in Appendix A.2).

Lemma 3.

If $\left(\mathbf{H}\mathbf{1}\right)$ holds and ${z_k}_l={\omega }_{kl}^2$, $h^{\prime }\left({z_k}_l\right)\ne 0$, where $h^{\prime }\left(z\right)$ is the derivative of $h\left(z\right)$ with respect to z, we have the following transversality condition:

$$\begin{array}{cc}\hfill \mathrm{Re}\left(\frac{d\lambda }{d\tau }\right){{|}_{\tau ={{{\tau }_k}_l}^{\left(j\right)}}=\mathrm{Re}{\left(\frac{d\tau }{d\lambda }\right)}^{-1}|}_{\tau ={{{\tau }_k}_l}^{\left(j\right)}}=& \frac{h^{\prime }\left({z_k}_l\right)}{{\left(c{A}_k{\omega }_{kl}\right)}^2+{(c{T}_k-c{\omega }_{kl}^2)}^2}\ne 0,\hfill \end{array}$$

(9)

where $k=2,3;l=1,2;j=0,1,2,\cdots$.

Summarizing the above Lemmas 1–3, we obtain the following theorem (the detailed mathematical proof of the following theorem is given in Appendix A.3).

Theorem 1.

If $\left(\mathbf{H}\mathbf{1}\right)$ and $\left(\mathbf{H}\mathbf{2}\right)$ hold, we can draw the following conclusions for system (2).

• $\left(1\right)$ When the distribution of the roots of Equation (6) belongs to $\left({\mathbf{V}}_{\mathbf{1}}\right)$—that is, if Equation (6) has no positive root—the equilibria $E_k$$(k=2,3)$ for system (2) are locally asymptotically stable for any $\tau ⩾0$.
• $\left(2\right)$ When the distribution of the roots of Equation (6) belongs to $\left({\mathbf{V}}_{\mathbf{2}}\right)$—that is, if Equation (6) has two positive roots—and ${z_k}_1<{z_k}_2$, then $h^{\prime }\left({z_k}_1\right)<0,h^{\prime }\left({z_k}_2\right)>0$. Note that ${{\tau }_k}_1^{\left(0\right)}>{{\tau }_k}_2^{\left(0\right)}$. Then, ∃$m\in \mathbb{N}$ such that $0<{{{\tau }_k}_2}^{\left(0\right)}<{{{\tau }_k}_1}^{\left(0\right)}<{{{\tau }_k}_2}^{\left(1\right)}<{{{\tau }_k}_1}^{\left(1\right)}<\dots <{{{\tau }_k}_1}^{(m-1)}<{{{\tau }_k}_2}^{\left(m\right)}<{{{\tau }_k}_2}^{(m+1)}$. When $\tau \in [0,{{{\tau }_k}_2}^{\left(0\right)})\cup {\displaystyle \bigcup _{l=1}^m}({{{\tau }_k}_1}^{(l-1)},{{{\tau }_k}_2}^{\left(l\right)})$, the equilibria $E_k$ of system (2) are locally asymptotically stable and, when $\tau \in {\displaystyle \bigcup _{l=0}^{m-1}}({{{\tau }_k}_2}^{\left(l\right)},{{{\tau }_k}_1}^{\left(l\right)})\cup ({{{\tau }_k}_2}^{\left(m\right)},+\infty )$, the equilibria $E_k$ of system (2) are unstable.

4. Numerical Simulations

In this section, we determine some parameters for system (2) by leveraging official data. Subsequently, we conduct numerical simulations on the model to validate the precision of our analysis.

Given that artificial afforestation constitutes the primary method for poplar cultivation in China, we extracted annual data for artificial afforestation areas and forest pest control rates from 2005 to 2021, as reported in the China Statistical Yearbook (refer to

Table 2. Data on artificial afforestation area and prevention rates for forest pests in China.

Year	Artificial Afforestation Area (100 K Hectares)	Prevention Rate for Forest Pests and Diseases (%)
2005	32.32	66.7
2006	24.46	66.8
2007	27.39	66.2
2008	36.85	68.7
2009	41.56	71.8
2010	38.73	69.8
2011	40.66	62.4
2012	38.21	66.5
2013	42.10	62.7
2014	40.53	65.3
2015	43.63	72.0
2016	38.24	68.8
2017	42.96	76.8
2018	36.78	77.8
2019	34.83	82.1
2020	30	78.9

). This data set allowed for the selection of reasonable intervals for the daily planting rate and larval mortality rate. Moreover, considering that poplar planting predominantly occurs in March, we make the assumption that March encompasses the entire period of plantation activities.

The approach used to calculate the daily planting rate is as follows. We record the annual afforestation area as $S^{\left(i\right)}$ and utilize a quadratic function to simulate the poplar planting curve for each March, starting with $(0,0)$ and $(30,S^{\left(i\right)})$ as the vertex of the quadratic function. As it was the year in which China reached its largest artificial afforestation area, we choose the 2015 data for calculation, resulting in the creation of Figure 2.

Next, we construct a graph of the derivative function, where the points in the graph represent the afforestation area for each day. By calculating the average afforestation area for the month, we identify the intersection point with the derivative function, representing the desired point. By following this method, Figure 2 is generated.

Remark 1.

The black curve in Figure 2 illustrates the artificial afforestation situation in March 2015, highlighting that the artificial afforestation area reached its maximum after 31 March 2015. Thus, the slope of the curve at this point is zero.

We deduce that the daily slope of the curve in this figure represents the daily afforestation area. Considering that the inclined straight line is the derivative curve of the afforestation area in 2015, we use the red points to simulate the daily afforestation area in March 2015 (as shown in Figure 3). Subsequently, we mark the daily afforestation area in March 2015. Moreover, we calculate the average of the daily afforestation area and identify the closest two points to the average. The daily planting rate for this year is then deduced as the rate calculated from these two points.

We know that a stable equilibrium indicates that a pest is under control. Therefore, we aim to prevent the spread of pests through controlling the parameters. After computation, we determine that the reasonable interval for the daily planting rate B is $[0.05,0.2]$. As observed in Table 2, the highest prevention rate for forest pests and diseases was 82.1%, achieved in 2019, while the lowest was 62.4%, in 2011. Consequently, we establish the reasonable interval for the larval mortality rate as $[0.6,0.85]$.

We identified a certain set of parameters, denoted as $\left(\mathbf{P}\right)$, as follows:

$$\left(\mathbf{P}\right):B=0.2;{\phi }_1=0.4;{\phi }_2=0.6;{\phi }_3=0.3;c=0.9;d_1=0.005;d_2=0.008;d_3=0.7.$$

Utilizing MATLAB for computation, we determine the three equilibria of system (2), given as follows:

$$E_1=(40,0,0),E_2=(38.91,0.67,0.000352),E_3=(1.03,0.37,0.47).$$

The parameter set $\left(\mathbf{P}\right)$ satisfies condition $\left({\mathbf{H}}_{\mathbf{1}}\right)$ for each equilibrium—that is, the equilibria of system (2) are all positive. Notably, equilibrium $E_2$ complies with $\left({\mathbf{H}}_{\mathbf{2}}\right)$, while $E_3$ does not. Considering that $E_1$ serves as the boundary equilibrium of system (2), we only simulate the stability of the equilibrium point $E_2$ in this section.

When $\tau =0$, the time delay from egg hatching to larva has no impact on the poplar. We choose initial values of $[30,0.2,0.0001]$ and simulate the stability of these three variables with the parameter group $\left(\mathbf{P}\right)$ (see Figure 4).

Remark 2.

From Figure 4, it can be observed that the number of poplars tends to stabilize after 800 days, and the number of diseased poplars and larvae tends to stabilize after approximately 1000 days. The values converge to around 0, indicating the absence of a large-scale pest outbreak.

Subsequently, we analyze the stability of equilibrium $E_2$ for the system when $\tau >0$ and with parameter set $\left(\mathbf{P}\right)$. In this case, Equation (6) has no real roots. According to Theorem 1, the equilibrium $E_2$ is locally asymptotically stable for any $\tau >0$.

Given that it takes time for eggs to hatch into larvae, the influence of the time delay on poplar growth was determined by leveraging official information from the official website (available online at www.caf.ac.cn/info/1301/30164.htm (accessed on 1 January 2024)). We obtain approximate developmental stage durations from this official information and illustrate their developmental stages in Figure 5.

Remark 3.

From Figure 5, it can be observed that the larvae develop over approximately 20 days in early April, transform into pupae by late April, and emerge as adults in early May. Subsequently, after oviposition, they undergo another developmental cycle, reaching adulthood again by the end of July. Therefore, it is assumed that, without human intervention, the process of development from eggs to larvae takes approximately 75 days. Hence, in this paper, τ is set as 75.

Consequently, we select $\tau =75$ for the simulation, demonstrating that system (2) is locally asymptotically stable at equilibrium $E_2$, as shown in Figure 6.

Remark 4.

Figure 6 illustrates the stabilization of the number of poplars and larvae after approximately 2000 days, and it also illustrates the stabilization of the number of diseased poplars after 2500 days. Notably, no widespread pest disaster was observed. The duration of stability was prolonged due to the 75-day period required for eggs to hatch into larvae.

Considering the variable time required for eggs to hatch into larvae, as influenced by natural and human factors, we study the stability of the equilibrium point $E_2$ under varying time delays ranging from 10 to 90 days, with increments of 20 days (see Figure 7).

Remark 5.

As can be observed from Figure 7, an extended time delay correlates with a prolonged stabilization duration for equilibrium $E_2$. Notably, an increased delay results in fewer diseased poplars and an elongated time to reach the peak. This phenomenon is deemed reasonable and can be attributed to the effect of the presence of pests in the form of eggs when the time required for adults to hatch larvae is extended. The slowed consumption rate of poplars subsequently decelerates the variation in the number of diseased poplars.

In the context of poplar pest infestation, a large number of Paranthrene tabaniformis (Rott.) larvae can lead to the internal decay of poplar trees, nutrient losses, and reduced wood quality, and it can even contribute to disease transmission and ecological imbalances. Therefore, we study the impact of changes in the parameter c in system (2) on the equilibrium states of system (2). We depict the asymptotic stability of system (2) at equilibrium point $E_2$ when the parameter c varies from 0.6 to 0.9, as shown in Figure 8.

In system (2), we utilize $c=r-d$, where r is the spawning rate of eggs and d is the survival rate of eggs. In this case, c is referred to as the success rate of larval hatching. A low success rate of larval hatching implies that the preventive measures are continuously improving. This indicates efforts to reduce the pest population through various means or to disrupt their life cycle before they enter the destructive stage.

Remark 6.

From Figure 8, we can observe that as the parameter c (representing the success rate of larval hatching) increases, the time to reach a balanced state becomes longer for each component at equilibrium point $E_2$. Moreover, with a larger value of c, the quantity of poplars decreases, while the number of diseased poplars increases. Additionally, the pest population suddenly increases at certain times; for instance, when $c=0.9$, the pest population reaches peak values at around 20 and 200 days, at approximately 30,000 and 10,000 individuals, respectively. Furthermore, larger c values lead to earlier occurrences of these extreme values.

These phenomena are justified for two main reasons: first, an increased success rate of larval hatching results in a larger number of larvae, which require more food for survival. This intensifies the extent of pest damage to poplars, leading to an increased disease rate, a decrease in the normal population of poplars, and an increase in the number of diseased poplars. Second, under natural conditions favorable for the survival of a large number of larvae, the larval population exhibits exponential growth, with a larger hatching success rate resulting in a larger maximum larval quantity. For example, when $c=0.9$, the larval population reaches approximately 30,000 individuals. However, as the pest detection and control techniques improve, biological measures are taken to reduce the pest population rapidly. The more pests there are, the more effective these measures become, resulting in the earlier occurrence of negative larval growth.

This necessitates the continuous improvement of detection technologies for foreign organisms, the enhancement of the sensitivity of pest detection technologies, and the early detection of pest disasters. However, the development of pest detection and preventive control technologies requires substantial investments in terms of funds and personnel, making it difficult to achieve rapid advancements in detection technologies within a short period.

As of 2022, China’s forest planting area totals 231 million hectares and the unused land in China totals approximately 348.7 million hectares, accounting for about 36.3% of the total land area. Therefore, increasing the planting rate and improving the planting techniques to expand the forest area is crucial. This not only provides timber products and improves quality of life, but is also a key force in regulating the global climate, mitigating greenhouse effects, and promoting China’s early achievement of carbon peaking and carbon neutrality. Based on the development of detection technology, we propose measures to increase the forest planting rate and expand the forest area.

Next, we study the stability of system (2) at equilibrium point $E_2$ when the parameters B and c change—that is, when the daily planting rate and the success rate of larval hatching both change. Specifically, by varying B from 0.1 to 0.2 (in increments of 0.05) and c from 0.9 to 0.6 (in increments of 0.15), we obtain Figure 9 using MATLAB R2021b.

Remark 7.

Figure 9 illustrates the asymptotic stability phenomena of system (2) at equilibrium $E_2$ under variations in the parameters B and c. In the graph with $B=0.1$ and $c=0.9$, we can observe that the pests reach their maximum populations (approximately 30,000 individuals) at around day 20, leading to a pest outbreak. Subsequently, around day 190, the pest population peaks again, reaching about 15,000 individuals. In the graph with $B=0.15$ and $c=0.75$, the pests achieve a maximum population of approximately 16,000 individuals at around day 21, resulting in a pest outbreak. Later, at around day 190, the pest population peaks again, reaching about 5000 individuals, indicating a smaller-scale pest outbreak.

We observed that as B increased and c decreased, the number of poplars increased, the number of diseased poplars decreased, and the pest population exhibited a more stable pattern. These observations are rational. First, as B increases and c decreases (i.e., with a higher daily planting rate and more refined detection and control measures), poplars receive better care and thrive, leading to an increase in their population and a decrease in the number of diseased poplars. Additionally, with the continuous improvement of preventive measures, the pest population stabilizes and large-scale pest outbreaks become rare.

The continuous advancement of modern biological technologies plays a significant role in the prevention and treatment of forest pests and diseases, providing powerful tools for the more effective and precise understanding, detection, and targeting of these issues. For instance, enhancing the biological activity of natural enemies or improving the formulations of biopesticides can enhance the effectiveness of biological control. Therefore, the ongoing development of modern biological technology reduces the probability of poplar (including diseased poplar) mortality. Consequently, we proceed to analyze the stability of system (2) at equilibrium point $E_2$ under variations in the parameters $d_1$ and $d_2$. Specifically, $d_1$ (the saturation mortality rate of poplars) changes from 0.005 to 0.002 and $d_2$ (the saturation mortality rate of diseased poplars) changes from 0.008 to 0.003 (both with intervals of 0.00025), and we examine the stability of system (2) at equilibrium $E_2$. The results are presented in Figure 10.

Remark 8.

Figure 10 illustrates the asymptotic stability of the equilibrium point under the variation in parameters $d_1$ and $d_2$ in system (2). Local plots of the two variables I and X are also provided. As $d_1$ and $d_2$ decrease, we observed an exponential increase in the quantity of poplar trees, while the numbers of diseased poplar trees and pests remain relatively constant. Additionally, as $d_1$ and $d_2$ decrease, the amplitude of the change in diseased poplar trees increases. Specifically, the maximum and minimum values of diseased poplar trees increase and decrease, respectively. From Figure 10, it is evident that the time taken to reach stability for diseased poplar trees is prolonged. Simultaneously, the fluctuation in the pest population diminishes, with the maximum number decreasing and the time to reach the maximum being extended.

These observations are also considered reasonable. Due to the continuous advancement of polymer biotechnology, techniques such as tissue culture can be employed for poplar tree reproduction. This biotechnological approach can significantly increase the poplar population by regulating plant growth, promoting faster growth, and increasing yields in specific periods. As the population grows, solutions to address diseased poplar trees may involve the introduction of resistance genes. However, the pathogen might evolve to adapt to this resistance, potentially resulting in breakthroughs and increasing the complexity of diseases. Consequently, the rate of change in diseased poplar trees slows down and, while the maximum value increases, this is reasonable given the overall increase in the poplar tree population.

Next, we provide a reasoned explanation for the impact of changes in the parameters $d_1$ and $d_2$ on pest variation. With advancements in biotechnology, the introduction of pest-resistant genes in poplar trees reduces the harm caused by pests, ultimately decreasing pest numbers. Chemical compounds resembling insect hormones can be used to influence the growth, development, and reproduction of pests. Moreover, gene editing techniques can be utilized to modify pest genes, rendering them incapable of harming poplar trees. These measures collectively contribute to a significant reduction in the maximum pest population and prolong the pest’s reproductive cycle, resulting in fewer and smaller-scale occurrences of infestation.

In summary, a combination of various measures is the most effective approach to prevent and control forest pest infestations. Strategies should involve increasing planting rates, improving planting techniques, and expanding forest areas. Simultaneously, it is crucial to enhance the detection technology for foreign organisms, increase the pest detection sensitivity, and promptly identify and eradicate pest outbreaks. Finally, we emphasize the effective prevention and management of forest pest infestations through the use of polymer biotechnology; however, it is crucial to recognize that biotechnology may pose threats to local ecosystems and biodiversity, potentially influencing human health.

5. Conclusions

In this study, we established a delayed differential equation to simulate the relationship between poplars and Paranthrene tabaniformis (Rott.) and analyzed the existence and stability of equilibria in the developed system. Then, we selected a set of parameters for use in numerical simulations, in order to validate our theoretical analysis. Furthermore, we conducted simulations considering various time delays and provided a thoughtful analysis. To comprehensively explore the system’s behavior, we systematically varied the parameters within reasonable ranges for system (2). This investigation yielded insights into the asymptotic stability of the system, revealing distinct patterns based on parameter variations. We recognized the corresponding biological significance of these patterns and proposed relevant practical measures.

It is worth noting that our study emphasizes the significance of increasing the planting rates, expanding the forest areas, improving the detection technologies, and utilizing advanced polymer biotechnologies. These measures collectively serve as robust strategies for the prevention and management of forest pest infestations.