Stability and Optimal Control of Tree-Insect Model under Forest Fire Disturbance

Abstract

In this article, we propose a mathematical model for insect outbreaks coupled with wildfire disturbances and an optimization model for finding suitable wildfire frequencies. We use a refined Holling II function as a model for the nonlinear response of fire frequency against trees and insects. The results show that for the tree–insect–wildfire model, there is a coexistence equilibrium in the system. Sensitivity analysis is performed to determine the effect of wildfire on trees in the optimization model. The results show that forest fires have a significant impact on the equilibrium mechanism of tree–insect coexistence. Numerical simulations suggest that in some areas of high fire intensity, there may be positive feedback between disturbances from wildfires and insect outbreaks. The result is consistent with the present theory in this field.

1. Introduction

Disturbance interactions are receiving increasing attention in today’s ecological studies [1,2]. Ecologists now have a much better understanding of individual disturbances, recognizing that natural ecosystems are affected by many types of natural disturbances, and studies have found significant interactions between these disturbances [3]. These disturbances are divided into natural and anthropogenic. Natural disturbances include wildfires, hurricanes, insect pests, diseases, floods, droughts, etc. Anthropogenic disturbances include artificial logging, water pollution, air pollution, etc. Among them, wildfires and pests are the two main natural disruptions of forest grassland. The synergistic relationship between wildfires and insects has been clearly described in previous studies. The study concluded that the massive tree mortality caused by insect outbreaks was an important cause of subsequent wildfires [4]. Correspondingly, the damage and death caused by the fire will produce focus trees that attract more beetles [5]. In separate studies, McHugh [6] and Cunningham [7] followed tree mortality and beetle infestation in a forest for three years following a wildfire and found that fire damage resulted in a higher probability of beetle attacks on trees. Similar results have been found in other studies of wildfires and pests (Bradley and Tueller [8]). However, some other studies have shown that beetles do not prioritize attacking trees with fire-damaged trunks, but that focal trees have a higher success rate of attack when beetle numbers are low (Elkin and Reid [9]). Furthermore, Sanchez-Martinez and Wagner [10] found that bark beetle numbers are at low levels regardless of the number of trees in the forest, including a large number of trees destroyed by wildfire. These results confirm the complexity of the link between insects and fire when assessed on a long-term and large-scale basis. Fleming et al. [11] took a statistical approach to examine the interaction between spruce budworm damage and forest fire risk. In their paper, they use a GIS overlay of the fire and spruce budworm histories in Ontario in order to define polygons with unique histories of forest fire and insect damage. McHugh et al. [12] measured insect abundance in several burned stands in Northern Arizona and found that trees targeted by insects had more canopy damage from fires compared to trees that were not targeted. Santoro et al. [13] found increased post-fire populations of the pine engraver (Ips pini) despite also measuring increased insects. Bebi et al. [14] found that forests burned in 1879 during the spruce beetle explosion were less affected by the infestation than old-growth forests that had not been burned in the 1940s. Historically, insect pests and wildfires have had a predominant impact on evolutionary progression in the forests of the American West. In the last 100 years, fire suppression has led to larger and more serious wildfire and insect outbreaks. In the year 2002, more than 50,000 acres of land in three northeastern provinces of China were damaged by wildfires and about 800,000 acres were damaged by insects such as bark beetles and pine wood nematodes. The relationship between fires and insect outbreaks is often described as a mutual and synergistic one. Although there are underlying feedbacks between these two natural disturbances, consensus on responses is lacking in the accepted papers, with some published studies recording substantial impacts on fires and burning from trees killed by beetles, while others report none or diminished impacts. The temporal dynamics of the B. Chen Charpentier model is consistent with the assumed two-stage model of beetle demography (also known as the binary theory) [15] and is able to fit the existing data. In particular, the model consists of the thoughts of the classical model described and developed by Safranyik [16]. The optimal control model is sufficiently accurate to allow an analytical characterization of the dynamic behavior. At the same time, this model extends the tree–beetle system model to include average wildfire disturbance and its interaction dynamics with insect outbreaks. Fire can simultaneously slay beetles and trees and can also diminish the defenses of surviving trees against beetle invasion.

The structure of this paper is as follows: In Section 2, based on the work of reference [17], a tree–beetle–fire model is considered. The threshold value of boundary equilibrium point stability is obtained, and the internal equilibrium point global stability is proved. In Section 3, the optimal control strategy of wildfire frequency is considered by optimizing and rewriting the model to find the appropriate frequency of wildfire occurrence to minimize the damage to trees. Using the optimization model to simulate the reported forest fire data from State Forestry Administration 2004 to 2017 and predict the development trend of a forest area after the occurrence of forest fires. The sensitivity analysis is used to identify the intensity of forest fires in Section 4, and the results show that the frequency of forest fires had a great influence on the tree–insect coexistence mechanism.

2. Deterministic Model for Tree–Beetle–Fire System

In this section, based on the work of reference [17], we describe the formulation of the deterministic model for the tree–beetle–fire system introduced. Positive feedback may occur between forest fires and beetle outbreaks, which will enhance the frequencies and severities of these two types of naturally occurring disturbances so that it includes the mortality of trees caused by fire. By considering an ordinary differential equation model, we propose the following

$$\left\{\begin{array}{c}\frac{dV}{dt}=r_{v}V(1-\frac{V}{K_v}-f_k\frac{B}{r+B})-P\frac{V}{K_v}{M}_{v}V,\hfill \\ \frac{dB}{dt}=r_{b}B(1-\frac{B}{K_e})-\frac{\alpha B}{1+\beta B}-M_{b}P\frac{V}{K_v}{M}_{v}V.\hfill \end{array}\right.$$

(1)

The variables are V and B, indicating a number of vulnerable trees and mountain pine beetles in each tree. The constant $r_v$ is the inherent percentage of growth of vulnerable trees (1/time) and the constant $r_b$ is the inherent percentage of growth of vulnerable beetles (1/time). $K_v$ represents the carrying capacity of the system. $K_e$ is the carrying capacity of the beetles per tree. $\alpha$ represents the defense rate of the pine tree, and $\beta$ represents the reciprocal of beetle density. When the pine tree is saturated with defense, $f_k$ is the fraction of trees killed by successful attacks and r represents the threshold value of the number of beetles successfully attacked (beetles per tree). The special relationship models the reality that the increased death rate of trees due to beetle existence is linearly related to B when the beetle population is small, and it reaches saturation at $f_k$ when the beetle population is extremely high. As described above, the equation is an extension of the logistic type of growth, which includes the host pine’s defense against the beetle, described by the Holling II functional response. The constant $M_v$ represents the percentage of trees lost/damaged in the fire. The constant $M_b$ represents the effect of forest fires on beetles, and c is the defense parameter of fire weakening pine against beetles. The term $0\le P\left(\frac{V}{K_v}\right)\le 1$ is the frequency of fire or probability of fire, and we assume that it is proportional to the population of trees within the forest. The constant P is a measure of the average intensity of vegetation feedback to the fire for different thermogenic properties and $0<f_k<1,0<M_v<1,0<c<1,0<M_b<1,t>0$. We define

$$v=\frac{V}{K_v},b=\frac{B}{K_v},r_1=\frac{r_b}{r_v},{\beta }_1=\beta K_e^2,K=f_k{K}_e,m_v=\frac{M_{v}P}{K_v},m_b=\frac{M_b{M}_{v}P}{K_v},{\alpha }_f=\frac{\alpha (1-c{M}_v)K_e}{r_v},$$

(2)

which then can be rewritten as $(1+m_v)<m_b,m_b>m_v$

$$\left\{\begin{array}{c}\frac{dv}{dt}=v(1-v-\frac{Kb}{r+K_{e}b}-m_{v}v),\hfill \\ \frac{db}{dt}=b[r_1(1-b)-\frac{{\alpha }_f}{1+{\beta }_{1}b}-m_{b}v].\hfill \end{array}\right.$$

(3)

The parameters of the above two systems are all positive.

Existence and Stability of Equilibrium

In this subsection, we demonstrate the existence of the equilibria. According to the biological meaning, we need all equilibria to be nonnegative. By using system (3), the equilibrium satisfies the following equations:

$$\left\{\begin{array}{c}v(1-v-\frac{Kb}{r+K_{e}b}-m_{v}v)=0,\hfill \\ b[r_1(1-b)-\frac{{\alpha }_f}{1+{\beta }_{1}b}-m_{b}v]=0.\hfill \end{array}\right.$$

(4)

Consequently, we can immediately calculate that system (4) has only one boundary equilibrium $E^0=(\frac{1}{m_v+1},0)$.

We next consider the coexistence equilibrium $E^{*}=(v_{*},b_{*})$ such that $v_{*}>0,b_{*}>0$. By using the first equation of system (4), we obtain

$$v=\frac{r+(K_e-K)b}{(1+m_v)(r+K_{e}b)}.$$

(5)

Substituting (5) into the second equation of system (4), we obtain

$$H\left(b\right)=c_1{b}^3+c_2{b}^2+c_{3}b+c_4=0,$$

(6)

where

$$\begin{array}{cc}\hfill & c_1=-{\beta }_1{K}_e{r}_1(1+m_v),\hfill \\ \hfill & c_2=[K{m}_b-r{r}_1(1+m_v)]{\beta }_1+K_e[(1+m_v)(-1+{\beta }_1)r_1-m_b{\beta }_1],\hfill \\ \hfill & c_3=(1+m_v)[-K_e{\alpha }_f+r_1[K_e+r(-1+{\beta }_1)]+m_b(K-K_e-r{\beta }_1)],\hfill \\ \hfill & c_4=r[(r_1-{\alpha }_f)(1+m_v)-m_b].\hfill \end{array}$$

Suppose $(r_1-{\alpha }_f)(1+m_v)>m_b$, then $H\left(0\right)>0$. Since ${\beta }_1,K_e,r_1,m_v>0$, then $c_1<0$ and $H(+\infty ))=-\infty$. From the real continuation method, there exist $b_{*}>0$ and $H\left(b_{*}\right)=0$. Since $0<f_k<1$ and $K_e{f}_k=K$, then $K_e-K>0$, hence

$$v=\frac{r+(K_e-K)b_{*}}{(1+m_v)(r+K_e{b}_{*})}>0.$$

Summarizing the above discussions, when $(r_1-{\alpha }_f)(1+m_v)>m_b$, the system (3) has at least one coexistence equilibrium $E^{*}=(v_{*},b_{*})$.

In the following, we consider the stability of the equilibria. To determine the local stability of these equilibria, we consider the Jacobian matrix of system (3):

$$J=\left(\begin{array}{cc}1-2v-\frac{bK}{r+K_{e}b}-2m_{v}v& \frac{K_{e}Kvb}{{(r+K_{e}b)}^2}-\frac{Kv}{r+K_{e}b}\\ -m_{b}b& r_1(1-2b)-m_{b}v+\frac{b{\alpha }_f{\beta }_1}{{(1+b{\beta }_1)}^2}-\frac{{\alpha }_f}{1+b{\beta }_1}\end{array}\right).$$

(7)

The Jacobian matrix of system (3) at $E^0=(\frac{1}{m_v+1},0)$ is

$$J^0=\left(\begin{array}{cc}{m}_1& m_2\\ n_1& n_2\end{array}\right).$$

where $m_1=-1<0,m_2=-\frac{K}{r(1+m_v)},n_1=0,n_2=r_1-{\alpha }_f-\frac{m_b}{1+m_v}$. The eigenvalues of $J^0:|\lambda E-J^0|={\lambda }^2-T{R}_1+DE{T}_1=0$, can be determined by

$$\left\{\begin{array}{c}T{R}_1=n_2+m_1,\hfill \\ DE{T}_1=-m_2{n}_1+m_1{n}_2=m_1{n}_2.\hfill \end{array}\right.$$

Summing up the above discussions, we obtain the following Theorem 1.

Theorem 1.

Let $r_1,m_b,{\alpha }_f,m_v,K,r$ be positive parameters. Then we have

 (1) 

If $(r_1-{\alpha }_f)(1+m_v)<m_b$, boundary equilibrium $E^0$ of system (3) is local asymptotically stable.

 (2) 

If $(r_1-{\alpha }_f)(1+m_v)>m_b$, boundary equilibrium $E^0$ of system (3) is unstable.

Remark 1.

Let $r_1>{\alpha }_f$. Defining threshold $R^{*}=\frac{(r_1-{\alpha }_f)(1+m_v)}{m_b}$. Then if $R^{*}<1$ boundary equilibrium $E^0$ is asymptotically stable; if $R^{*}>1$, and boundary equilibrium $E^0$ is unstable, coexistence equilibrium $E^{*}$ exists.

Remark 2.

$$\begin{array}{cc}\hfill R^{*}& =\frac{(r_1-{\alpha }_f)(1+m_v)}{m_b}\hfill \\ \hfill & =\frac{1}{m_b}[(r_1-\frac{\alpha K_e}{m_v})+(r_1-\frac{\alpha K_e}{r_v}+\frac{\alpha rc{K}_e^2}{r_{v}P})m_v+\frac{\alpha rc{K}_e^2}{r_{v}P}{m}_v^2].\hfill \end{array}$$

Defining $m_b=(r_1-\frac{\alpha K_e}{m_v})+(r_1-\frac{\alpha K_e}{r_v}+\frac{\alpha rc{K}_e^2}{r_{v}P})m_v+\frac{\alpha rc{K}_e^2}{r_{v}P}{m}_v^2$. The properties of $R^{*}$ can be seen in Figure 1.

In the following, we consider the properties of $E^{*}=(v_{*},b_{*})$. The Jacobian matrix of system (3) at $E^{*}=(v_{*},b_{*})$ is

$$J^1=\left(\begin{array}{cc}{p}_1& p_2\\ s_1& s_2\end{array}\right).$$

where $p_1=-(1+m_v)v_{*}<0,p_2=-\frac{Kr{v}_{*}}{{(r+K_e{b}_{*})}^2}<0,s_1=-m_b{b}_{*}<0,s_2=r_1(1-2b_{*})-\frac{{\alpha }_f}{{(1+{\beta }_1{b}_{*})}^2}-m_b{v}_{*}$. Take the eigenvalues of $J^1$ can be determined by $|\lambda E-J^1|={\lambda }^2-T{R}_2+DE{T}_2=0$, where

$$\left\{\begin{array}{c}T{R}_2=s_2+p_1,\hfill \\ DE{T}_2=-p_2{s}_1+p_1{s}_2.\hfill \end{array}\right.$$

Summarizing the above discussions, we obtain the following Theorem 2.

Theorem 2.

Let $r_1,m_b,{\alpha }_f,m_v,K,r,K_e$ be positive parameters and $R^{*}>1$.

 (1) 

If $s_2<\frac{p_2{s}_1}{p_1}$, coexistence equilibrium $E^{*}$ of system (3) is local asymptotically stable.

 (2) 

If $\frac{p_2{s}_1}{p_1}<s_2<0$, coexistence equilibrium $E^{*}$ of system (3) is unstable.

In order to verify the asymptotic stability of the forest fire-forest beetle outbreak model system, numerical simulation is performed on the model (3). Using the parameter value in [16]: $m_b=366.75,m_v=75,{\beta }_1=156.9398,{\alpha }_f=699.3189,r_1=33.75,r=9.1,K=1467$, and $c=0.5,r_v=0.08,K_e=1956,K_v=100,P=0.1,\alpha =0.04086$, we obtain positive equilibrium points $(v_{*},b_{*})\approx (0.00334654,0.799878)$. These parameter values satisfy the conditions of the theorem analyzed above. The above analysis shows that the system is asymptotically stable, as shown in Figure 2. The figure on the left shows the tree’s endemic state for fire, and the figure on the right shows the beetle’s endemic state for fire.

Next we derive some basic properties of solutions to system (3), such as the nonexistence of limit cycles. According to the biological setting of the model described above, we define

$$\mathsf{\Omega }=\left\{\right(v,b):v>0,b>0\}.$$

(8)

Then, we have

Theorem 3.

Let Ω be defined as in system (3), suppose that $r_1>{\alpha }_f{\beta }_1$, system (3) has no periodic orbits in Ω.

Proof of Theorem 3. 

We define Ω $\frac{1}{vb}>0$, then we have

$$\left\{\begin{array}{c}QF\triangleq \frac{1}{vb}\left[v(1-v-\frac{Kb}{r+K_{e}b}-m_{v}v)\right],\hfill \\ QG\triangleq \frac{1}{v}[r_1(1-b)-\frac{{\alpha }_f}{1+{\beta }_{1}b}-m_{b}v].\hfill \end{array}\right.$$

(9)

an easy computation yields that

$$\frac{\partial QF}{\partial v}=\frac{1}{b}(-1-m_v)<0,b\in \mathsf{\Omega }.$$

and

$$\frac{\partial QG}{\partial b}=\frac{1}{v}(-r_1+\frac{{\alpha }_f{\beta }_1}{{(1+{\beta }_1)}^2})<\frac{1}{v}(-r_1+{\alpha }_f{\beta }_1).$$

Suppose that $r_1>{\alpha }_f{\beta }_1$, then $\frac{\partial QG}{\partial b}<0$. For $(v,b)\in \mathsf{\Omega }$, we have

$$\frac{\partial QF}{\partial v}+\frac{\partial QG}{\partial b}<0.$$

Therefore, the Dulac criterion can be applied to system (3) in $\mathsf{\Omega }$ and there exist no periodic orbits in $\mathsf{\Omega }$. According to the above analysis, $E^{*}$ is globally approaching stability. □

According to Theorems 2 and 3, we have the following Theorem 4:

Theorem 4.

Suppose $R^{*}>1$ and $r_1,m_b,{\alpha }_f,m_v,K,r,K_e$ are positive parameters. If $r_1>{\alpha }_f{\beta }_1$, then the internal positive equilibrium point is globally asymptotically stable.

3. Optimal Control Strategy

There are potential interactions between insect outbreaks and wildfire disturbances in forests. Wildfires are a potential threat to tree growth, and they also kill beetles. Therefore, it is an optimization problem to find the suitable frequency of wildfire so that there is little damage to trees and the beetles can be destroyed. Optimization has always played a critical part in the management of forests and pests in the design and operation. In the paragraphs below, we discuss optimal control strategies for wildfire frequency. The $m_v$ in system (3) is set to be time-dependent $m_v\left(t\right)$, describing the time-varying wildfire intensity, and there exists $\overline{m_v}$ such that $0\le m_v\left(t\right)\le \overline{m_v}$. Within the fixed time $[0,T]$ with $T>0$, the constraint set reads

$$U=\left\{m_v\left(t\right)\right|0\le m_v\left(t\right)\le m_v,0\le t\le T,m_v\left(t\right)areLebesguemeasurable\}.$$

(10)

The optimal objectives are to minimize the number of insects and the wildfire frequency. We rewrite system (3) as

$$\left\{\begin{array}{c}\frac{dv}{dt}=v(1-v-\frac{Kb}{r+K_{e}b}-m_{v}v),\hfill \\ \frac{db}{dt}=b[r_1(1-b)-\frac{{\alpha }_1}{1+{\beta }_{1}b}+\frac{{\alpha }_2{m}_v}{1+{\beta }_{1}b}-{\alpha }_3{m}_{v}v].\hfill \end{array}\right.$$

(11)

The quadratic optimal objective function reads:

$$J\left(m_v\right)={\int }_0^T(\frac{1}{2}{b}^2\left(t\right)+\frac{1}{2}{m}_v^2\left(t\right))dt$$

with $b\left(0\right)=b_0,v\left(0\right)=v_0$. The optimal control problem rewrites

$$J^{*}\left(m_v^{**}(·)\left(t\right)\right)=\underset{m_v\left(t\right)\in U}{min}J\left(m_v\left(t\right)\right).$$

(12)

Using the method in [18], we check the existence of optimal control $m_v\left(t\right)$ by satisfied H1–H4 the following:

Hypothesis 1

(H1). The set of state and control variables are nonempty;

Hypothesis 2

(H2). The set U of the control variables is closed and convex;

Hypothesis 3

(H3). The right side of each equation in control problem (11) is continuous with a bounded sum of controls and states above, which can be written as a linear function of U with coefficients depending on time and state [19];

Hypothesis 4

(H4). There exists constants ${\alpha }_1,{\alpha }_2,{\alpha }_3>0$ such that the integrand $L\left(m_v\left(t\right)\right)$ of the objective functional J is convex and satisfied.

The Hypothesis H4 can be obtained since $L\left(m_v\left(t\right)\right)=\frac{1}{2}(b^2\left(t\right)+m_v^2\left(t\right))$. Using the Hypothesis (H1–H4), we have

Theorem 5.

For the optimal control problem (10)–(12), there exists an optimal control $m_v^{**}$ such that $J^{*}\left(m_v^{**}(·)\left(t\right)\right)=$$\underset{m_v\left(t\right)\in U}{min}$$J\left(m_v\left(t\right)\right)$.

We give the Hamiltonian function to obtain the minimum value of (12).

$$H=\frac{1}{2}(b^2+m_v^2)+{\lambda }_1\left[v(1-v-\frac{Kb}{r+K_{e}b}-m_{v}v)\right]+{\lambda }_2\left[b(r_1(1-b)-\frac{{\alpha }_1}{1+{\beta }_{1}b}+\frac{{\alpha }_2{m}_v}{1+{\beta }_{1}b}-{\alpha }_3{m}_{v}v)\right].$$

Using Pontryagin’s Maximum Principle, the optimal solution of (10)–(12) can be obtained as follows

$$\left\{\begin{array}{c}\frac{\partial H}{\partial m_v}{|}_{(m_v^{**},{\lambda }_1,{\lambda }_2,t)}=0,\hfill \\ \frac{\partial {\lambda }_1}{\partial m_v}{|}_{(m_v^{**},{\lambda }_1,{\lambda }_2,t)}=-{\dot{\lambda }}_1,{\lambda }_1=0,\hfill \\ \frac{\partial {\lambda }_2}{\partial m_v}{|}_{(m_v^{**},{\lambda }_1,{\lambda }_2,t)}=-{\dot{\lambda }}_2,{\lambda }_2=0.\hfill \end{array}\right.$$

Theorem 6.

The optimal control of (10)–(12) is given by

$$m_v^{**}=min\{\overline{m_v},max\{{\lambda }_{1}vb+{\lambda }_2{\alpha }_{3}vb-\frac{{\lambda }_2{\alpha }_2{b}^2}{1+{\beta }_{1}b},0\}\}.$$

(13)

where ${\lambda }_1,{\lambda }_2$ are the adjoint variables satisfying (13).

Proof of Theorem 6. 

By Pontryagin’s Maximum Principle, finding the optimal control of (10)–(12) is equivalent to minimizing the following Hamiltonian function H above. The optimal control

$\frac{\partial H}{\partial m_v}{|}_{(m_v^{**},{\lambda }_1,{\lambda }_2,t)}=0$ gives $m_v^{**}-{\lambda }_{1}vb-{\lambda }_2{\alpha }_{3}vb+\frac{{\lambda }_2{\alpha }_2{b}^2}{1+{\beta }_{1}b}=0$, where the adjoint variables are satisfied. Furthermore,

$\dot{{\lambda }_1}=-\frac{\partial H}{\partial b}{|}_{(m_v^{**},{\lambda }_1,{\lambda }_2,t)}=-b+\frac{{\lambda }_{1}vKr}{{(r+K_{e}b)}^2}-{\lambda }_2[r_1-r_{1}b-\frac{{\alpha }_1}{{(1+{\beta }_{1}b)}^2}+\frac{{\alpha }_2{m}_v}{{(1+{\beta }_{1}b)}^2}-{\alpha }_3{m}_{v}v]$

$\dot{{\lambda }_2}=-\frac{\partial H}{\partial b}{|}_{(m_v^{**},{\lambda }_1,{\lambda }_2,t)}=-{\lambda }_1(1-2v-\frac{Kb}{r+K_{e}b}-m_v)+{\lambda }_2{\alpha }_3{m}_v$ and the transversality conditions ${\lambda }_1\left(T\right)=0,{\lambda }_2\left(T\right)=0$.

Moreover, since $m_v^{**}\in U$, using the lower and upper bounds of $\overline{m_b}\left(t\right)$, the optimal $m_v^{**}$ can be characterized by (13). □

On the basis of model (11), we discuss the influence of different $m_v$ on the model. We define ${\alpha }_1=\frac{\alpha K_e}{r_v},{\alpha }_2=\frac{\alpha c{K}_e}{r_v},{\alpha }_3=M_b$. Using the wildfire and beetle data from some provinces in China (Heilongjiang, Jilin, Inner Mongolia. et al. [20]). With these data, the averages of these parameters from 2004 to 2017 are obtained for 11 provinces, see

Table 1. The parameter and values to be used in the model (11).

Parameter	${\mathit{M}}_{\mathit{v}}$	${\mathit{r}}_{\mathit{v}}$	${\mathit{m}}_{\mathit{v}}$	${\mathit{\alpha }}_1$	${\mathit{\alpha }}_2$
Beijing	0.764572	0.048204	158.611171	1657.993	828.9964
Tianjin	0.445343	0.021843	203.882824	3658.925	1829.463
Hebei	0.154946	0.023128	66.995922	3455.703	1727.852
Shanxi	0.252996	0.036412	69.4821260	2194.957	1097.479
Inner Mongoria	0.561737	0.045826	122.5812760	1744.046	872.0231
Liaoning	0.337569	0.016873	200.0673550	4736.759	2368.379
Jilin	0.38393	0.006038	635.8630730	13236.66	6618.332
Heilongjiang	0.239552	0.008569	279.546910	9326.556	4663.278
Shandong	0.524125	0.024039	218.031458	3324.691	1662.345
Henan	0.258396	0.046559	55.498005	1716.561	858.2804

. The curves in Figure 3 show the influence of $m_v$ on trees and beetles. Figure 4 magnifies the function image of beetle time to see the change trend of the beetle more clearly.

Sensitivity Analysis and Numerical Simulations

In this section, we first use (11) to simulate the reported forest fire data from 2004 to 2017 and predict the development trend of the forest area after the occurrence of a forest fire. Then, we conduct a sensitivity analysis for some critical parameters, perform the numerical simulation for optimal control, and search for some valid control and preventive measures. The forest area data of forest fires are extracted from the State Forestry Administration (See

Table 2. Annual forest area data of forest fires in each province (unit: ten thousand hectares).

Year	2004	2008	2009	2010	2011	2012	2013	2014	2015	2016	2017
Shanxi	0.18412	0.1013	0.3149	0.0499	0.6372	0.048	0.2145	0.076	0.0814	0.0272	0.0299
Inner Mongoria	0.51265	1.5358	1.7764	0.9129	0.1741	0.4464	0.0826	0.595	0.3847	0.1622	2.4805
Liaoning	0.04303	0.1249	0.1424	0.0404	0.0577	0.0374	0.0143	0.1045	0.1922	0.1706	0.0802
Jilin	0.03272	0.0445	0.0351	0.0084	0.024	0.0178	0.0063	0.0134	0.0541	0.0123	0.045
Heilong jiang	18.55472	1.8402	9.9819	1.3779	0.1741	0.1046	0.0106	0.0301	0.0787	0.064	0.0883
Hebei	0.13467	0.033	0.0443	0.0221	0.2169	0.2112	0.0422	0.1168	0.0471	0.0573	0.1157

).

Based on the data of the forest fire area in Table 2, we calculate v and the parameter $K_v=100$ in the model (11). For numerical simulation, most parameters of (11) were obtained from the literature. The influence trend of forest fires on forest areas in six provinces (Heilongjiang, Jilin, Liaoning, etc.) was predicted by fitting (11) with the data from 2004 to 2017. A numerical simulation shows that the model (11) with reasonable parameter values agrees well with the measured data. From 2004 to 2017, the area of forest fires in the same province decreased year by year. In order to better study the impact of forest fire on forest trees, we conduct a study. The key parameters are analyzed by sensitivity; Figure 5a–f show the forecast figures of Shanxi, Inner Mongolia, Liaoning, Jilin, Heilongjiang and Hebei, respectively.

With the help of model (11), we fit it to the data from 2004 to 2017 to predict the dynamics of trees and insects in the presence of fire (Figure 5). A numerical simulation shows that model (11) and reasonable parameter values can match the reported data well. From 2004 to 2017, the area of forest affected by fires decreased year by year and stabilized. Then, the rationality of the parameters selected by the model (11) is verified.

The accuracy of the original data or the stability of the optimal solution when the system changes is an important step in model optimization [21]. In the process of model optimization, sensitivity analysis is the most commonly used test method. Sensitivity analysis can quickly identify a few key factors from a large number of parameters and input states in the power system, but usually, it is not necessary to calculate the sensitivity coefficient of each parameter, and only those parameters with great uncertainty can be selected for sensitivity analysis. Based on the sensitivity analysis results of the tree–beetle–fire model, it is shown that the intensity of forest fire is the key factor in controlling and predict the mechanism of tree–beetle coexistence. To better understand the impact of fire on forests and insects, we conduct a study by analyzing the impact of key parameters $m_v$ in sensitivity characterization.

The specific steps of the sensitivity coefficient algorithm are as follows. We consider function $\dot{y\left(t\right)}=f(t,y,p)$, the absolute sensitivity of variable $y_i$ to parameter P:

$$S_i\left(t\right)=\frac{\partial y_i(t.P)}{\partial P},i=v,b.$$

We denote the relative sensitivity of a variable to a parameter:

$$s_i\left(t\right)=\frac{\partial y_i(t.P)}{\partial P}\frac{P}{y_i},i=v,b,$$

the absolute sensitivity equation of parameter $S_i$:

$$\dot{S_i}=\frac{\partial f}{\partial y}{S}_i+\frac{\partial f}{\partial p},i=v,b.$$

Next, the sensitivity of forest fire frequency and the impact of forest fires on trees to various variables in the system is discussed through sensitivity analysis and relative sensitivity analysis. The sensitivity equation of the system contains four equations for parameter $m_v$.

$$\left\{\begin{array}{c}\dot{v}=v(1-v-\frac{Kb}{r+K_{e}b}-m_{v}v),\hfill \\ \dot{b}=b[r_1(1-b)-\frac{{\alpha }_1}{1+{\beta }_{1}b}+\frac{{\alpha }_2{m}_v}{1+{\beta }_{1}b}-{\alpha }_3{m}_{v}v],\hfill \\ \dot{S_v}=S_v(1-2v-\frac{Kb}{r+K_{e}b}-2m_{v}v)-S_b\frac{Krv}{r+K_{e}b}-v^2,\hfill \\ \dot{S_b}=S_b(r_1-2r_{1}b-\frac{{\alpha }_1}{{(1+{\beta }_{1}b)}^2}+\frac{{\alpha }_2{m}_v}{{(1+{\beta }_{1}b)}^2}-{\alpha }_3{m}_{v}v)-S_v{\alpha }_3{m}_{v}b+\frac{{\alpha }_{2}b}{1+{\beta }_{1}b}-{\alpha }_{3}vb.\hfill \end{array}\right.$$

(14)

The Rk-4 method was used to calculate the system, and the sensitivity and relative sensitivity analysis of forest fire intensity and the impact of forest fires on trees were obtained. The conclusions obtained can reflect the impact intensity of each variable in the system, as shown in the figure.

As shown in Figure 6 and Figure 7, both the sensitivity and relative sensitivity of $m_v$ to $v\left(t\right)$ show a trend less than 0, indicating that the increase in $m_v$ will lead to the decrease in $v\left(t\right)$, and when $t=10$, $S_v=-1.109<{10}^{-6}<0$, $\frac{m_v{S}_v}{v}=-0.02485<0$. That means that when $m_v$ increases by $10\%$, trees will decrease by $0.2485\%$. The sensitivity and relative sensitivity of $m_v$ to $b\left(t\right)$ show a trend greater than 0, indicating that the increase in $m_v$ will lead to the decrease in $b\left(t\right)$. When $b\left(t\right)$, $S_b=0.001797>0$ and $\frac{m_v{S}_b}{v}=0.2593>0$. That means that when $m_v$ increases by $10\%$, beetles will increase by $2.593\%$. Ecologically speaking, we assume that the beetle–tree system has been exposed to sequential fires of constant intensity. This means that a sufficiently intense fire will result in a large number of trees being burned and weaken the defenses of living trees, thus allowing beetle epidemics (beetle outbreaks) to become established in the forest. With our results, there may be positive feedback between disturbances between wildfires and insect outbreaks, which would enhance the frequency and severity of forest damage from these two natural disturbances. This situation is consistent with current theories in the field.

4. Discussion

We present mathematical models with generalizations that provide a basis for exploring the disturbance effects of wildfires and insect outbreaks on forests. Current research reports show that there is a lack of proof of the global stability of the model, as well as analysis and fitting of actual data; see [17]. In Section 2, we prove the coexistence equilibrium is globally stable under certain conditions. In this paper, the tree–insect–fire model is optimized and rewritten. The optimal control strategy for the frequency of wildfire occurrence is discussed. In order to find the appropriate wildfire occurrence frequency and reduce the loss of trees. Based on the forest fire data from 2004 to 2017 provided by the State Forestry Administration, the development trend of a forest area after a forest fire is predicted. The sensitivity analysis is used to identify the frequency and intensity of forest fires, and the results show that the frequency of forest fires greatly influenced the tree–insect coexistence mechanism, or to be more precise, when the beetle–tree system is exposed to continuous fires of constant intensity. Intense fires will result in the burning of large amounts of forest and, in the process, also weaken the defense systems of living trees, thus allowing beetle epidemics to become established in the forest, which is consistent with the subsequent research theory in Reference [17].