Dynamical Behaviors of an Environmental Protection Expenses Model in Protected Areas with Two Delays

Abstract

This paper investigates an environmental protection expenses model, which considers the relations between the visitors to the protected areas V, the quality of the environmental resource E, and the capital stock K. In this model, the total tourism income is used partly to increase the capital stock or as the environmental protection expenses. Two time delays are introduced into the number of visitors, since the visitors need time to respond the changes of the environment, and the environment will take time to respond to the input of money. Stability crossing curves in the plane of delays (${\tau }_1,$ ${\tau }_2$) are used to obtain the stable region of equilibrium. Numerical simulations represent the mutual transformation of the supercritical bifurcation and the subcritical bifurcation. Our model shows that under some parameter conditions, the share of tourism income $\eta$ is related closely to the delay ${\tau }_1$, while the capital stock and the environmental quality can be maintained persistently if the delay ${\tau }_1$ is not too large.

1. Introduction

Protected areas are now expected to achieve an increasing conservation for social and economic objectives. The protection of Protected Areas (PAs) is always a project for researchers [1,2,3]. Recently, the models of environmental protection expenditures have attracted new attention because of the deterioration of environment in some PAs [4,5,6,7,8]. Russu [4] studied a three-variable model among visitors V, quality of environmental resource E and the capital stock K in the PAs:

$$\left\{\begin{array}{c}\dot{V}\left(t\right)=m_{1}E\left(t\right)+m_{2}K\left(t\right)-a{V}^2\left(t\right)\hfill \\ \dot{E}\left(t\right)=r(\overline{P}-E\left(t\right))-(b-c\eta )V(t-\tau )\hfill \\ \dot{K}\left(t\right)=(1-\eta )V(t-\tau )-\delta K\left(t\right),\hfill \end{array}\right.$$

(1)

where $a,r,\overline{P},b,c,\delta ,m_1,$ and $m_2$ are strictly positive constants. $a>0$ represents the crowding influence; this means that the PA becomes less attractive when the number of tourists visiting the PA increases. $\overline{P}$ is the pollution stock of maximum tolerance, while b means the waste generated by every visitor, $0<r<1$ is a constant proportion of the pollution, c determines how much the environmental expenses increase the quality of the environment. Visitors impact negatively on the environmental resource, but environment and infrastructures are attractive for visitors; therefore, the manager of PA uses a part $\eta$ of total tourism income to protect the environmental resource and the remaining part $1-\eta$ to increase the capital stock, where $0\le \eta <1.$ The depreciate rate of capital stock is $\delta$. $m_1,m_2$ mean that the number of visitor is proportional to E and K, $\tau >0$ means the delay from visitors for increasing capital stock and the quality of the environment. The parameters involved in this topic can also see [5,6,7].

In Reference [8], Caraballo et al. suggested a modified version of Russu’s model as:

$$\left\{\begin{array}{c}\dot{V}\left(t\right)=m_{1}E\left(t\right)+m_{2}K\left(t\right)-a{V}^2\left(t\right)\hfill \\ \dot{E}\left(t\right)=r(\overline{P}-E\left(t\right))-(b-c\eta )V(t-\tau )E\left(t\right)\hfill \\ \dot{K}\left(t\right)=(1-\eta )V(t-\tau )-\delta K\left(t\right).\hfill \end{array}\right.$$

(2)

The authors gave some remarks for Russu’s model; they pointed out that there was something wrong in Russu [4].

In many subjects such as biology, epidemiology, ecology, chemistry, and physics, numerous engineering problems delays always occur. The models that have multiple delays are of great interest mathematically and scientifically [9,10,11,12,13,14,15,16].

The stability crossing curve is an effective tool to understand the stable region for a system with multiple delays and to comprehend the bifurcation behaviors. For instance, Hale and Huang [17] investigated the stable region for the two delay differential equations

$$\dot{x}\left(t\right)+ax\left(t\right)+bx(t-r)+cx(t-\sigma )=0.$$

(3)

The authors described the stable region on the ($r,\sigma$) plane and pointed out that the stable region could be unbounded. Gu et al. [18] studied the stability crossing curve carefully for a special case of characteristic equation. Lin and Wang [19] used a different approach to extend the results of Gu et al. result to a general case. An et al. [20] studied the stability switching properties of a model with delay dependent parameters. Matsumto and Szidarovszky [21] considered a delayed Lotka–Volterra competition model with two delays and some symmetries; the stability crossing curves on which stability is switched to instability were investigated.

In this paper, we suggest a modified model of Caraballo et al. Considering that the public praise will affect the amount of visitors, but with a delay, $r_{1}V(t-{\tau }_1)$ is introduced to the model. The crowding effect is considered as $a{V}^2(t-{\tau }_1)$. $m_{1}E\left(t\right)$ in the first equation of (2) is changed into $bV(t-{\tau }_1)E\left(t\right)$ by considering the visitor’s effect on this term, and the influence of the capital stock $m_{2}K\left(t\right)$ to the visitors is not adopted in our model. In the second equation of (2), we consider that the environment resource has self-purification ability, so a term $r_{2}E\left(t\right)$ is added. We hope that the pollution is not tolerable, so we let $\overline{P}=0$. The term $\eta V(t-\tau )E\left(t\right)$ is changed into $\eta V(t-\tau )K\left(t\right)$, since we think in here the capital stock is more important to the change of the environment. So, we obtain the following model:

$$\left\{\begin{array}{c}\dot{V}\left(t\right)=r_{1}V(t-{\tau }_1)-a{V}^2(t-{\tau }_1)+bV(t-{\tau }_1)E\left(t\right)\hfill \\ \dot{E}\left(t\right)=r_{2}E\left(t\right)-cV(t-{\tau }_1)E\left(t\right)+\eta V(t-{\tau }_2)K\left(t\right)\hfill \\ \dot{K}\left(t\right)=(1-\eta )V(t-{\tau }_2)-\delta K\left(t\right),\hfill \end{array}\right.$$

(4)

where $r_1$ is the rate of effect of the public praise; visitors will increase if $r_1$ increases. $r_2$ is the self-purification ability of the environment. $a>0$ represents the crowding effect, b is the rate of the visitors affected by environment resource, c means the waste generated by every visitor which is affected by environment resource, and $0\le \eta <1$ means a share $\eta$ of total revenues is used to protect the environment. Capital stock is depreciated at the rate $\delta$. The increment of the visitors relies on the visitors at the time $t-{\tau }_1$ for their spread of public praise and the crowding effect, while the dynamic evolution of the environment and the capital stock rely on the contribution of visitors at the time $t-{\tau }_2$, where ${\tau }_1\ge 0$ and ${\tau }_2\ge 0$. We name ${\tau }_1$ as the spread delay and ${\tau }_2$ as the protecting delay.

We give the initial conditions of system (4) as:

$$V\left(\theta \right)={\upsilon }_1\left(\theta \right),E\left(\theta \right)={\upsilon }_2\left(\theta \right),K\left(\theta \right)={\upsilon }_3\left(\theta \right),{\upsilon }_i\left(\theta \right)\ge 0,i=1,2,3,-\tau \le \theta \le 0,$$

where $({\upsilon }_1\left(\theta \right),{\upsilon }_2\left(\theta \right),{\upsilon }_3\left(\theta \right))\in C([-\tau ,0],{\mathbf{R}}_{+}^3)$, $\tau =max[{\tau }_1,{\tau }_2]$. Then, according to the theorem on functional differential equations [22], system (4) has one and only one solution $\left(V\right(t),E(t),K(t\left)\right)$ that satisfies the initial conditions. In this paper, we provide the theory of the stability crossing curves and apply it to model (4). By means of numerical simulations, we obtain the stable region of the equilibrium in the ${\tau }_1-{\tau }_2$ plane. Bifurcation directions of the periodic solutions are determined by using the normal form and the center manifold theorem. Numerical simulations show how the equilibrium changes from stable to unstable and how the bifurcation direction changes from supercritical to subcritical and vice versa. Through the research of model (4), we find that the spread delay ${\tau }_1$ we introduced to the model (4) is more important than ${\tau }_2$ which is $\tau$ in [4,8] for the stability of equilibrium, since ${\tau }_1$ needs to be on the left of the stability crossing curves. We find that the share $\eta$ of the tourism user fees and the spread delay ${\tau }_1$ are very important parameters in our discussion.

2. Equilibria and Stability Crossing Curves

By straightforward computation, system (4) has equilibrium $S_0=(0,0,0)$, which is unstable, since there are always positive eigenvalues ${\lambda }_1=r_1$ and ${\lambda }_2=r_2$, while the characteristic equation of (4) at $S_0$ has no relation with delays ${\tau }_1,{\tau }_2$.

If

Hypothesis 1 (H1).

$b\eta (1-\eta )-a\delta c>0,r_2>\frac{r_{1}b\eta (1-\eta )}{(1-a^2)\delta },$

system $\left(4\right)$ has $S_{*}=(V_{*},E_{*},K_{*})$ as a positive equilibrium:

$$V_{*}=\frac{-\delta (a{r}_2+c{r}_1)+\sqrt{{\delta }^2{(a{r}_2+c{r}_1)}^2+4\delta r_1{r}_2(b\eta (1-\eta )-ac\delta )}}{2\left(b\eta \right(1-\eta )-ac\delta )},$$

$$E_{*}=\frac{a{V}_{*}-r_1}{b},K_{*}=\frac{1-\eta }{\delta }{V}_{*}.$$

If

Hypothesis 2 (H2).

$b\eta (1-\eta )-a\delta c<0,{\delta }^2{(a{r}_2+c{r}_1)}^2+4\delta r_1{r}_2(b\eta (1-\eta )-ac\delta )>0,$

system $\left(4\right)$ has $S_{*}=(V_{*},E_{*},K_{*})$ as a positive equilibrium:

$$V_{*}=\frac{-\delta (a{r}_2+c{r}_1)-\sqrt{{\delta }^2{(a{r}_2+c{r}_1)}^2+4\delta r_1{r}_2(b\eta (1-\eta )-ac\delta )}}{2\left(b\eta \right(1-\eta )-ac\delta )},$$

$$E_{*}=\frac{a{V}_{*}-r_1}{b},K_{*}=\frac{1-\eta }{\delta }{V}_{*}.$$

If $b\eta (1-\eta )-a\delta c=0$, there is no positive equilibrium.

Let $u_1\left(t\right)=V\left(t\right)-V_{*},u_2\left(t\right)=E\left(t\right)-E_{*},u_3\left(t\right)=K\left(t\right)-K_{*}$, then (4) becomes

$$\left\{\begin{array}{cc}\hfill {\dot{u}}_1\left(t\right)=& -a{V}_{*}{u}_1(t-{\tau }_1)+b{V}_{*}{u}_2\left(t\right)-a{u}_1^2(t-{\tau }_1)+b{u}_1(t-{\tau }_1)u_2\left(t\right)\hfill \\ \hfill {\dot{u}}_2\left(t\right)=& -c{E}_{*}{u}_1(t-{\tau }_1)+\eta K_{*}{u}_1(t-{\tau }_2)+(r_2-c{V}_{*})u_2\left(t\right)+\eta V_{*}{u}_3\left(t\right)\hfill \\ \hfill & -c{u}_1(t-{\tau }_1)u_2\left(t\right)+\eta u_1(t-{\tau }_2)u_3\left(t\right)\hfill \\ \hfill {\dot{u}}_3\left(t\right)=& (1-\eta )u_1(t-{\tau }_2)-\delta u_3\left(t\right).\hfill \end{array}\right.$$

(5)

The corresponding characteristic equation of system (5) can be written by

$${\lambda }^3+{\alpha }_1{\lambda }^2+{\alpha }_2\lambda +({\alpha }_3{\lambda }^2+{\alpha }_4\lambda +{\alpha }_5)e^{-\lambda {\tau }_1}+({\alpha }_6\lambda +{\alpha }_7)e^{-\lambda {\tau }_2}=0,$$

(6)

where

$$\begin{array}{c}{\alpha }_1=c{V}_{*}-r_2+\delta ,\hfill \\ {\alpha }_2=(c{V}_{*}-r_2)\delta ,\hfill \\ {\alpha }_3=a{V}_{*},\hfill \\ {\alpha }_4=a{V}_{*}(c{V}_{*}-r_2)+a{V}_{*}\delta +bc{V}_{*}{E}_{*},\hfill \\ {\alpha }_5=a\delta V_{*}(c{V}_{*}-r_2)+bc\delta V_{*}{E}_{*}\hfill \\ {\alpha }_6=-b\eta V_{*}{K}_{*},\hfill \\ {\alpha }_7=-b\eta \delta V_{*}{K}_{*}-b\eta (1-\eta )V_{*}^2.\hfill \end{array}$$

(7)

When ${\tau }_1={\tau }_2=0$, by the criterion of Routh–Hurwitz, we have

Theorem 1.

Assume ${\tau }_1={\tau }_2=0$, (H1) (or (H2)) hold. If

Hypothesis 3 (H3).

${\alpha }_1+{\alpha }_3>0,{\alpha }_5+{\alpha }_7>0,({\alpha }_1+{\alpha }_3)({\alpha }_2+{\alpha }_4+{\alpha }_6)>{\alpha }_5+{\alpha }_7$

is satisfied, then $S_{*}$ is asymptotically locally stable.

Next, we investigate the distribution of the roots of (6). From Rouche theorem [23], as $({\tau }_1,{\tau }_2)$ vary continuously in ${\mathbf{R}}_{+}^2$, the roots of Equation (6) vary continuously, and the roots (counting multiplicity) can change their symbols of real parts if and only if they cross the imaginary axis [24].

We consider two situations: (I) ${\tau }_1={\tau }_2=\tau >0$, or (II) ${\tau }_1>0,{\tau }_2>0,{\tau }_1\ne {\tau }_2$.

Case (I) ${\tau }_1={\tau }_2=\tau >0$.

Suppose that on the imaginary axis, system (6) has a root $i\omega (\omega >0)$. Substituting it into (6), separating the imaginary part and the real parts, we have

$$\left\{\begin{array}{c}{\alpha }_1{\omega }^2={\alpha }_4\omega sin\omega \tau +(-{\alpha }_3{\omega }^2+{\alpha }_5)cos\omega \tau ,\\ {\alpha }_2\omega -{\omega }^3=(-{\alpha }_3{\omega }^2+{\alpha }_5)sin\omega \tau -{\alpha }_4\omega cos\omega \tau .\end{array}\right.$$

(8)

Then, we obtain

$${\omega }^6+({\alpha }_1^2-2{\alpha }_2-{\alpha }_3^2){\omega }^4+({\alpha }_2^2+2{\alpha }_3{\alpha }_5-{\alpha }_4^2){\omega }^2-{\alpha }_5^2=0.$$

(9)

Let $z={\omega }^2,p={\alpha }_1^2-2{\alpha }_2-{\alpha }_3^2,q={\alpha }_2^2+2{\alpha }_3{\alpha }_5-{\alpha }_4^2,r=-{\alpha }_5^2;$ then, (9) becomes

$$h\left(z\right)=z^3+p{z}^2+qz+r=0.$$

(10)

We see from (10) that since $r<0$, there is at least one positive root. We assume that there are three positive roots for generality, defined by $z_1,z_2$ and $z_3$. According to (8), we have

$${\tau }_k^{\left(j\right)}=\frac{1}{{\omega }_k}\{{cos}^{-1}\left[\frac{({\alpha }_1{\omega }_k^2({\alpha }_5-{\alpha }_3{\omega }_k^2)+{\alpha }_4{\omega }_k^2({\omega }_k^2-{\alpha }_2)}{{({\alpha }_5-{\alpha }_3{\omega }_k)}^2+{\alpha }_4^2{\omega }_k^2}\right]+2j\pi \},$$

(11)

where $j=0,1,\cdots$ and $k=1,2,3$. Denote

$${\tau }_0={\tau }_{k_0}^{\left(0\right)}=\underset{k\in \{1,2,3\}}{min}\left\{{\tau }_k^{\left(0\right)}\right\},{\omega }_0={\omega }_{k_0}.$$

(12)

Then, we know

Lemma 1.

When (H1) (or (H2)), (H3) hold, all roots of (6) have strictly negative real parts when $\tau \in [0,{\tau }_0)$. (6) has simple purely imaginary roots when $\tau ={\tau }_0$.

Near $\tau ={\tau }_k^{\left(j\right)}$, consider $\lambda \left(\tau \right)=\sigma \left(\tau \right)+i\omega \left(\tau \right)$ as the root of (6) where $\sigma \left({\tau }_k^{\left(j\right)}\right)=0,\omega \left({\tau }_k^{\left(j\right)}\right)={\omega }_k,j=0,1,2\cdots ,k=1,2,3.$ For the transversality, we know that

Lemma 2.

Suppose $z_k={\omega }_k^2,k=1,2,3$ and $h^{\prime }\left(z_k\right)\ne 0,$ then ${\left[\frac{d\mathrm{Re}\left(\lambda \right)}{d\tau }\right]}_{\tau ={\tau }_k^{\left(j\right)}}\ne 0$.

According to Lemma 1 and Lemma 2, we have

Theorem 2.

When ${\tau }_1={\tau }_2=\tau$, let ${\tau }_0$ be denoted by (12), and assume that $\left(H1\right)\left(or\right(H2\left)\right)$, $\left(H3\right)$ are satisfied,

(i)

If $\tau \in [0,{\tau }_0)$, then system (4) has a locally asymptotically stable positive equilibrium $S_{*}$.

(ii)

If $\tau >{\tau }_0$ and $h^{\prime }\left(z_k\right)\ne 0,$ then for system (4), Hopf bifurcation will occur at $S_{*}$ when $\tau ={\tau }_0$.

Case (II) ${\tau }_1>0,{\tau }_2>0,{\tau }_1\ne {\tau }_2$

Characteristic Equation (6) can be rewritten as

$$P(\lambda ,{\tau }_1,{\tau }_2)\equiv P_0\left(\lambda \right)+P_1\left(\lambda \right)e^{-\lambda {\tau }_1}+P_2\left(\lambda \right)e^{-\lambda {\tau }_2}=0,$$

(13)

where

$$P_0\left(\lambda \right)={\lambda }^3+{\alpha }_1{\lambda }^2+{\alpha }_2\lambda ,$$

$$P_1\left(\lambda \right)={\alpha }_3{\lambda }^2+{\alpha }_4\lambda +{\alpha }_5,$$

$$P_2\left(\lambda \right)={\alpha }_6\lambda +{\alpha }_7.$$

We can easily confirm that Equation (13) satisfies:

(I)

$deg\left(P_0\left(\lambda \right)\right)\ge max\{deg\left(P_1\left(\lambda \right)\right),deg\left(P_2\left(\lambda \right)\right)\};$

(II)

$P_0\left(0\right)+P_1\left(0\right)+P_2\left(0\right)={\alpha }_5+{\alpha }_7\ne 0,if\left(H3\right)holds;$

(III)

$P_0\left(\lambda \right),P_1\left(\lambda \right)$, $P_2\left(\lambda \right)$ have no common zeros;

(IV)

${lim}_{\lambda \to \infty }\left(\right|P_1\left(\lambda \right)/P_0\left(\lambda \right)|+|P_2\left(\lambda \right)/P_0\left(\lambda \right)\left|\right)<1.$

Lemma 3.

For each $\omega >0,P_0\left(i\omega \right)\ne 0$, $P(\lambda ,{\tau }_1,{\tau }_2)=0$ has $\lambda =i\omega$ as its root for some $({\tau }_1,{\tau }_2)\in R_{+}^2$ if and only if

Hypothesis 4 (H4).

$$|P_0\left(i\omega \right)|\le |P_1\left(i\omega \right)|+|P_2\left(i\omega \right)|,$$

(14)

and

$$-|P_0\left(i\omega \right)|\le |P_1\left(i\omega \right)|-|P_2\left(i\omega \right)|\le |P_0\left(i\omega \right)|.)$$

(15)

The proof of Lemma 3 can be found in [18].

Let

$$G1\left(\omega \right)=-|P_0\left(i\omega \right)|+|P_1\left(i\omega \right)|-|P_2\left(i\omega \right)|,$$

$$G2\left(\omega \right)=-|P_0\left(i\omega \right)|-|P_1\left(i\omega \right)|+|P_2\left(i\omega \right)|,$$

$$G3\left(\omega \right)=|P_0\left(i\omega \right)|-|P_1\left(i\omega \right)|-|P_2\left(i\omega \right)|,$$

then we know that $P(\lambda ,{\tau }_1,{\tau }_2)=0$ has $\lambda =i\omega$ as its solution if and only if $G1\left(\omega \right)\le 0,G2\left(\omega \right)\le 0,G3\left(\omega \right)\le 0$ simultaneously.

Denote the crossing set $\mathsf{\Omega }$ of all $\omega$, which satisfy (14) and (15). For given $\omega \in \mathsf{\Omega },P_k\left(i\omega \right)\ne 0,k=0,1,2.$ From (14) and (15), we can find all of $({\tau }_1\left(\omega \right),{\tau }_2\left(\omega \right))$:

$${\tau }_1={\tau }_1^{m\pm }\left(\omega \right)=(arg\frac{P_1\left(i\omega \right)}{P_0\left(i\omega \right)}+(2m-1)\pi \pm {\psi }_1)/\omega ,m=m_0^{\pm },m_0^{\pm }+1,\cdots ,$$

(16)

$${\tau }_2={\tau }_2^{n\pm }\left(\omega \right)=(arg\frac{P_2\left(i\omega \right)}{P_0\left(i\omega \right)}+(2n-1)\pi \mp {\psi }_2)/\omega ,n=n_0^{\pm },n_0^{\pm }+1,\cdots ,$$

(17)

where ${\psi }_1,{\psi }_2\in [0,\pi ]$ can be calculated as

$${\psi }_1={cos}^{-1}\left\{\frac{|P_0{\left(i\omega \right)|}^2+|P_1{\left(i\omega \right)|}^2-{\left|P_2\left(i\omega \right)\right|}^2}{2|P_0\left(i\omega \right)\left|\right|P_1\left(i\omega \right)|}\right\},$$

(18)

$${\psi }_2={cos}^{-1}\left\{\frac{|P_0{\left(i\omega \right)|}^2-|P_1{\left(i\omega \right)|}^2+{\left|P_2\left(i\omega \right)\right|}^2}{2|P_0\left(i\omega \right)\left|\right|P_2\left(i\omega \right)|}\right\},$$

(19)

$m_0^{+},m_0^{-},n_0^{+}$ and $n_0^{-}$ are the smallest integers such that the corresponding ${\tau }_1^{m_0^{+}+},{\tau }_1^{m_0^{-}-},{\tau }_2^{n_0^{+}+}$, and ${\tau }_2^{n_0^{-}-}$ are non-negative.

Let ${\omega }_0\in \mathsf{\Omega }$, we can obtain ${\tau }_{10}={\tau }_{10}\left({\omega }_0\right),{\tau }_{20}={\tau }_{20}\left({\omega }_0\right)$ from (16) and (17).

Next, we discuss the transversality. Choose ${\tau }_2$ to be the bifurcating parameter, and take the derivative of $\lambda \left({\tau }_2\right)$ in (6); then, we obtain

$$\begin{array}{c}\{3{\lambda }^2+2{\alpha }_1\lambda +{\alpha }_2+[2{\alpha }_3\lambda +{\alpha }_4-({\alpha }_3{\lambda }^2+{\alpha }_4\lambda +{\alpha }_5){\tau }_1]e^{-\lambda {\tau }_1}\hfill \\ +[{\alpha }_6-({\alpha }_6\lambda +{\alpha }_7){\tau }_2]e^{-\lambda {\tau }_2}\}\frac{d\lambda }{d{\tau }_2}\hfill \\ =\lambda ({\alpha }_6\lambda +{\alpha }_7)e^{-\lambda {\tau }_2},\hfill \end{array}$$

Therefore,

$$\begin{array}{c}{\left[\frac{d\lambda }{d{\tau }_2}\right]}^{-1}=\frac{(3{\lambda }^2+2{\alpha }_1\lambda +{\alpha }_2)e^{\lambda {\tau }_2}}{\lambda ({\alpha }_6\lambda +{\alpha }_7)}+\frac{[2{\alpha }_3\lambda +{\alpha }_4-({\alpha }_3{\lambda }^2+{\alpha }_4\lambda +{\alpha }_5){\tau }_1]e^{\lambda ({\tau }_2-{\tau }_1)}}{\lambda ({\alpha }_6\lambda +{\alpha }_7)}\hfill \\ +\frac{{\alpha }_6}{\lambda ({\alpha }_6\lambda +{\alpha }_7)}-\frac{{\tau }_2}{\lambda }.\hfill \end{array}$$

(20)

When ${\tau }_2={\tau }_{20}$, $\lambda =i{\omega }_0$, then

$$\begin{array}{c}{\left((3{\lambda }^2+2{\alpha }_1\lambda +{\alpha }_2)e^{\lambda {\tau }_2}\right)}_{{\tau }_2={\tau }_{20}}=(({\alpha }_2-3{\omega }_0^2)cos\left({\omega }_0{\tau }_{20}\right)-2{\alpha }_1{\omega }_{0}sin\left({\omega }_0{\tau }_{20}\right))\hfill \\ +i(2{\alpha }_1{\omega }_{0}cos\left({\omega }_0{\tau }_{20}\right)+({\alpha }_2-3{\omega }_0^2)sin\left({\omega }_0{\tau }_{20}\right)),\hfill \\ {\left([2{\alpha }_3\lambda +{\alpha }_4-({\alpha }_3{\lambda }^2+{\alpha }_4\lambda +{\alpha }_5){\tau }_1]e^{\lambda ({\tau }_2-{\tau }_1)}\right)}_{{\tau }_2={\tau }_{20}}\hfill \\ =({\alpha }_4+{\tau }_{10}{\alpha }_3{\omega }_0^2-{\tau }_{10}{\alpha }_5)\cos \left(\omega ({\tau }_{20}-{\tau }_{10})\right)-(2{\alpha }_3-{\tau }_{10}{\alpha }_4){\omega }_0\sin \left({\omega }_0({\tau }_{20}-{\tau }_{10})\right)\hfill \\ +i[(2{\alpha }_3-{\tau }_{10}{\alpha }_4){\omega }_0\cos \left({\omega }_0({\tau }_{20}-{\tau }_{10})\right)+({\alpha }_4+{\tau }_{10}{\alpha }_3{\omega }_0^2-{\tau }_{10}{\alpha }_5)\sin \left({\omega }_0({\tau }_{20}-{\tau }_{10})\right)].\hfill \\ \lambda {({\alpha }_6\lambda +{\alpha }_7)}_{{\tau }_2={\tau }_{20}}=-{\alpha }_6{\omega }_0^2+i{\alpha }_7{\omega }_0.\hfill \end{array}$$

(21)

We have

$$\begin{array}{c}Re{\left[\frac{d\lambda }{d{\tau }_2}\right]}_{{\tau }_2={\tau }_{20}}^{-1}=\frac{M_1{N}_1+M_2{N}_2}{{\alpha }_6^2{\omega }_0^3+{\alpha }_7^2{\omega }_0},\hfill \end{array}$$

(22)

where

$$\begin{array}{cc}\hfill M_1=& -{\alpha }_6{\omega }_0,M_2={\alpha }_7,\hfill \\ \hfill N_1=& ({\alpha }_2-3{\omega }_0^2)cos\left({\omega }_0{\tau }_{20}\right)-2{\alpha }_1{\omega }_{0}sin\left({\omega }_0{\tau }_{20}\right)+{\alpha }_6+({\alpha }_4+{\tau }_{10}{\alpha }_3{\omega }_0^2-{\tau }_{10}{\alpha }_5)cos\left({\omega }_0({\tau }_{20}-{\tau }_{10})\right)\hfill \\ \hfill & -(2{\alpha }_3-{\tau }_{10}{\alpha }_4){\omega }_{0}sin\left({\omega }_0({\tau }_{20}-{\tau }_{10})\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill N_2=& 2{\alpha }_1{\omega }_{0}cos\left({\omega }_0{\tau }_{20}\right)+({\alpha }_2-3{\omega }_0^2)sin\left({\omega }_0{\tau }_{20}\right)+(2{\alpha }_3-{\tau }_{10}{\alpha }_4){\omega }_{0}cos\left({\omega }_0({\tau }_{20}-{\tau }_{10})\right)\hfill \\ \hfill & +({\alpha }_4+{\tau }_{10}{\alpha }_3{\omega }_0^2-{\tau }_{10}{\alpha }_5)sin\left({\omega }_0({\tau }_{20}-{\tau }_{10})\right).\hfill \end{array}$$

Let

Hypothesis 5 (H5).

$M_1{N}_1+M_2{N}_2\ne 0.$

Denote ${\mathit{T}}^{\mathit{j}}=\{({\tau }_{10}^j\left(\omega \right),{\tau }_{20}^j\left(\omega \right)),\omega \in \mathsf{\Omega }\},j=1,2,\cdots ,p$ are p sections of continuous curves defined on $\mathsf{\Omega }$, ${\mathit{T}}^{\circ }$ is the internal region surrounded by $\mathit{T}={\bigcup }_{j=1}^p{\mathit{T}}^{\mathit{j}}$ with coordinate axis ${\tau }_1=0$ and ${\tau }_2=0$. Then, we can obtain the following:

Theorem 3.

Assume that $\left(H1\right)$ (or $\left(H2\right)$), $\left(H3\right),\left(H4\right)$ hold,

(I)

If $({\tau }_1,{\tau }_2)\in {\mathit{T}}^{\circ }$, then system (4) has a positive equilibrium $S_{*}$ which is locally asymptotically stable.

(II)

If $({\tau }_1,{\tau }_2)$ crossing $\mathit{T}$ and $\left(H5\right)$ holds, then $({\tau }_1,{\tau }_2)=({\tau }_{10},{\tau }_{20})\in \mathit{T}$ is a critical point, system (4) has Hopf bifurcation at $S_{*}$

We denote continuous curve ${\mathit{T}}^{\mathit{j}},j=1,2,\cdots ,p$ as stability crossing curves.

3. Hopf Bifurcation Direction and the Stability of Periodic Solution

In this section, we suppose $0<{\tau }_1<{\tau }_2$. If $0<{\tau }_2<{\tau }_1$; we can discuss it using the same method.

Let ${\tau }_1={\tau }_{10},{\tau }_2={\tau }_{20}+\mu$, then system $\left(5\right)$ has $\mu =0$ as its bifurcation value. Let $t=\tau \overline{t},u_i\left(\tau \overline{t}\right)=\overline{u_i}\left(\overline{t}\right),i=1,2,3$ and omit $‘‘{-}^{\prime \prime }$ above, $\left(5\right)$ can be expressed as

$$\dot{u}\left(t\right)=L_{\mu }{u}_t+f(\mu ,u_t),$$

(23)

where $u\left(t\right)={(u_1\left(t\right),u_2\left(t\right),u_3\left(t\right))}^{\top }\in {\mathbf{R}}^3$, $u_t\left(\theta \right)=u(t+\theta ),\theta \in [-1,0]$.

$$L_{\mu }\varphi =({\tau }_{20}+\mu )(A\varphi \left(0\right)+B\varphi (-\frac{{\tau }_1}{{\tau }_2})+C\varphi (-1)),\varphi \left(\theta \right)={({\varphi }_1\left(\theta \right),{\varphi }_2\left(\theta \right),{\varphi }_3\left(\theta \right))}^{\top },$$

(24)

$$\begin{array}{c}f(\mu ,\varphi )=({\tau }_{20}+\mu ){(f_1,f_2,f_3)}^{\top }\hfill \\ =({\tau }_{20}+\mu ){(-a{\varphi }_1^2(-\frac{{\tau }_1}{{\tau }_2})+b{\varphi }_1(-\frac{{\tau }_1}{{\tau }_2}){\varphi }_2\left(0\right),-c{\varphi }_1(-\frac{{\tau }_1}{{\tau }_2}){\varphi }_2\left(0\right)+\eta {\varphi }_1(-1){\varphi }_3\left(0\right),0)}^{\top },\hfill \end{array}$$

where

$$A=\left(\begin{array}{ccc}0& b{V}_{*}& 0\\ 0& r_2-c{V}_{*}& \eta V_{*}\\ 0& 0& -\delta \end{array}\right),B=\left(\begin{array}{ccc}-a{V}_{*}& 0& 0\\ -c{E}_{*}& 0& 0\\ 0& 0& 0\end{array}\right),C=\left(\begin{array}{ccc}0& 0& 0\\ \eta K_{*}& 0& 0\\ 1-\eta & 0& 0\end{array}\right).$$

By the Riesz representation theorem, for $\theta \in [-1,0],$ we write

$$L_{\mu }\varphi ={\int }_{-1}^{0}d\rho (\theta ,\mu )\varphi \left(\theta \right),$$

(25)

where

$$\rho (\theta ,\mu )=({\tau }_{20}+\mu )(A\delta \left(\theta \right)+B\delta (\theta +\frac{{\tau }_1}{{\tau }_2})+C\delta (\theta +1)).$$

(26)

$\delta \left(\theta \right)$ satisfies $\delta \left(\theta \right)=0$ if $\theta \ne 0$, and $\delta \left(0\right)=1$.

Define

$$\mathcal{A}\left(\mu \right)\varphi =\left\{\begin{array}{c}\frac{d\varphi \left(\theta \right)}{d\theta },\theta \in [-1,0),\hfill \\ {\int }_{-1}^{0}d\rho (s,\mu )\varphi \left(s\right),\theta =0,\hfill \end{array}\right.$$

and

$$R\left(\mu \right)\varphi =\left\{\begin{array}{c}0,\theta \in [-1,0),\hfill \\ f(\mu ,\varphi ),\theta =0.\hfill \end{array}\right.$$

Then, (23) can be expressed as

$${\dot{u}}_t=\mathcal{A}\left(\mu \right)u_t+R\left(\mu \right)u_t,$$

(27)

it is a differential equation in functional space $C^1([-1,0],R^3)$.

Denote $\mathcal{A}=\mathcal{A}\left(0\right)$,

$${\mathcal{A}}^{*}\psi \left(s\right)=\left\{\begin{array}{cc}-\frac{d\psi \left(s\right)}{ds},& s\in (0,1],\\ {\int }_{-1}^{0}d{\rho }^{\top }(t,0)\psi (-t),& s=0,\end{array}\right.$$

and

$$\langle \psi \left(s\right),\varphi \left(\theta \right)\rangle =\overline{\psi }\left(0\right)\varphi \left(0\right)-{\int }_{-1}^0{\int }_{\xi =0}^{\theta }\overline{\psi }(\xi -\theta )d\rho (\theta ,0)\varphi \left(\xi \right)d\xi ,$$

(28)

where $\psi \in C^{1*}([0,1],{\left(R^3\right)}^{*})$. Then, ${\mathcal{A}}^{*}$ is an adjoint operator of $\mathcal{A}$.

Suppose $q\left(\theta \right)={(1,\beta ,\gamma )}^{\top }{e}^{i{\omega }_0{\tau }_{20}\theta }$ satisfies $\mathcal{A}q\left(\theta \right)=i{\omega }_0{\tau }_{20}q\left(\theta \right),$ which means $q\left(\theta \right)$ is an eigenvector of $\mathcal{A}$. We obtain that

$$\beta =\frac{(i{\omega }_0+\delta )(\eta K_{*}{e}^{-i{\omega }_0{\tau }_{20}}-c{E}_{*}{e}^{-i{\omega }_0{\tau }_{10}})+\eta (1-\eta )V_{*}{e}^{-i{\omega }_0{\tau }_{20}}}{(i{\omega }_0+\delta )(i{\omega }_0+c{V}_{*}-r_2)},\gamma =\frac{(1-\eta )e^{-i{\omega }_0{\tau }_{20}}}{i{\omega }_0+\delta }.$$

Let $-i{\omega }_0{\tau }_{20}$ be an eigenvalue of ${\mathcal{A}}^{*}$, and $q^{*}\left(s\right)=D(1,{\beta }^{*},{\gamma }^{*})e^{i{\omega }_0{\tau }_{20}s}$ is an eigenvector; then, we have

$${\beta }^{*}=\frac{b{V}_{*}}{-i{\omega }_0+c{V}_{*}-r_2},{\gamma }^{*}=\frac{b\eta V_{*}^2}{(-i{\omega }_0+\delta )(-i{\omega }_0+c{V}_{*}-r_2)}.$$

By (28),

$$\begin{array}{c}〈q^{*}\left(s\right),q\left(\theta \right)〉=\overline{D}(1+\beta \overline{{\beta }^{*}}+\gamma \overline{{\gamma }^{*}}-{\int }_{-1}^0(1,\overline{{\beta }^{*}},\overline{{\gamma }^{*}})\theta e^{i\theta {\omega }_0{\tau }_{20}}d\eta (\theta ,0){(1,\beta ,\gamma )}^{\top })\hfill \\ =\overline{D}(1+\beta \overline{{\beta }^{*}}+\gamma \overline{{\gamma }^{*}}+{\tau }_{20}{e}^{-i{\omega }_0{\tau }_{20}}(\eta \overline{{\beta }^{*}}{K}_{*}+(1-\eta )\overline{{\gamma }^{*}})+{\tau }_{10}{e}^{-i{\omega }_0{\tau }_{10}}(-a{V}_{*}-c\overline{{\beta }^{*}}{E}_{*})).\hfill \end{array}$$

For $\langle q^{*}\left(s\right),q\left(\theta \right)\rangle =1$, $\langle q^{*}\left(s\right),\overline{q}\left(\theta \right)\rangle =0$, we choose

$$\overline{D}=\frac{1}{1+\beta \overline{{\beta }^{*}}+\gamma \overline{{\gamma }^{*}}+{\tau }_{20}{e}^{-i{\omega }_0{\tau }_{20}}(\eta \overline{{\beta }^{*}}{K}_{*}+(1-\eta )\overline{{\gamma }^{*}})+{\tau }_{10}{e}^{-i{\omega }_0{\tau }_{10}}(-a{V}_{*}-c\overline{{\beta }^{*}}{E}_{*})}$$

(29)

Define

$$z\left(t\right)=\langle q^{*},u_t\rangle ,u_t\left(\theta \right)=W(t,\theta )+2Re\left\{z\left(t\right)q\left(\theta \right)\right\}$$

(30)

when $\mu =0$. Then, (27) becomes

$$\left\{\begin{array}{c}\dot{z}=i{\omega }_{0}z+\frac{g_{20}}{2}{z}^2+g_{11}z\overline{z}+\frac{g_{02}}{2}{\overline{z}}^2+g_{21}{z}^2\overline{z}+\cdots ,\hfill \\ \dot{W}=\mathcal{A}W+H(z,\overline{z},0).\hfill \end{array}\right.$$

(31)

Here,

$$\begin{array}{cc}\hfill g_{20}& =2{\tau }_{20}\overline{D}((-a{e}^{-i{\omega }_0{\tau }_{10}}+(b-c\overline{{\beta }^{*}})\beta e^{-i{\omega }_0{\tau }_{10}}+\eta \gamma \overline{{\beta }^{*}}{e}^{-i{\omega }_0{\tau }_{20}}),\hfill \\ \hfill g_{11}& ={\tau }_{20}\overline{D}(-2a+(b-c\overline{{\beta }^{*}})\overline{\beta }{e}^{-i{\omega }_0{\tau }_{10}}+(b-c\overline{{\beta }^{*}})\beta e^{i{\omega }_0{\tau }_{10}}+\eta \overline{\gamma }\overline{{\beta }^{*}}{e}^{-i{\omega }_0{\tau }_{20}}+\eta \gamma \overline{{\beta }^{*}}{e}^{i{\omega }_0{\tau }_{20}}),\hfill \\ \hfill g_{02}& =2{\tau }_{20}\overline{D}((-a{e}^{i{\omega }_0{\tau }_{10}}+(b-c\overline{{\beta }^{*}})\overline{\beta })e^{i{\omega }_0{\tau }_{10}}+\eta \overline{\gamma }\overline{{\beta }^{*}}{e}^{i{\omega }_0{\tau }_{20}}),\hfill \\ \hfill g_{21}& ={\tau }_{20}\overline{D}\{\frac{1}{2}(-2a{e}^{i{\omega }_0{\tau }_{10}}+(b-c\overline{{\beta }^{*}})\overline{\beta })W_{20}^1(-\frac{{\tau }_{10}}{{\tau }_{20}})+\frac{1}{2}(b-c\overline{{\beta }^{*}})e^{i{\omega }_0{\tau }_{10}}{W}_{20}^2\left(0\right)\hfill \\ \hfill & +\frac{1}{2}\eta \overline{{\beta }^{*}}{e}^{i{\omega }_0{\tau }_{20}}{W}_{20}^3\left(0\right)+\frac{1}{2}\eta \overline{{\beta }^{*}}\overline{\gamma }{W}_{20}^1(-1)+(-2a{e}^{-i{\omega }_0{\tau }_{10}}+(b-c\overline{{\beta }^{*}})\beta )W_{11}^1(-\frac{{\tau }_{10}}{{\tau }_{20}})\hfill \\ \hfill & +(b-c\overline{{\beta }^{*}})e^{-i{\omega }_0{\tau }_{10}}{W}_{11}^2\left(0\right)+\eta \overline{{\beta }^{*}}{e}^{-i{\omega }_0{\tau }_{20}}{W}_{11}^3\left(0\right)+\eta \gamma \overline{{\beta }^{*}}{W}_{11}^1(-1)\}.\hfill \end{array}$$

$$W_{20}\left(\theta \right)=\frac{i{g}_{20}}{{\omega }_0{\tau }_{20}}q\left(0\right)e^{i\theta {\omega }_0{\tau }_{20}}+\frac{i{\overline{g}}_{02}}{3{\omega }_0{\tau }_{20}}\overline{q}\left(0\right)e^{-i\theta {\omega }_0{\tau }_{20}}+C_1{e}^{2i\theta {\omega }_0{\tau }_{20}},$$

$$W_{11}\left(\theta \right)=-\frac{i{g}_{11}}{{\omega }_0{\tau }_{20}}q\left(0\right)e^{i\theta {\omega }_0{\tau }_{20}}+\frac{i{\overline{g}}_{11}}{{\omega }_0{\tau }_{20}}\overline{q}\left(0\right)e^{-i\theta {\omega }_0{\tau }_{20}}+C_2.$$

Here, $C_1,C_2\in R^3$ satisfy

$$A_1{C}_1=f_0\left(20\right),$$

$$A_1=\left(\begin{array}{ccc}2i{\omega }_0+a{V}_{*}{e}^{-2i{\omega }_0{\tau }_{10}}& -b{V}_{*}& 0\\ c{E}_{*}{e}^{-2i{\omega }_0{\tau }_{10}}-\eta K_{*}{e}^{-2i{\omega }_0{\tau }_{20}}& 2i{\omega }_0-r_2+c{V}_{*}& -\eta V_{*}\\ -(1-\eta )e^{-2i{\omega }_0{\tau }_{20}}& 0& 2i{\omega }_0+\delta \end{array}\right),$$

$$f_0\left(20\right)=2\left(\begin{array}{c}(-a{e}^{-i{\omega }_0{\tau }_{10}}+b\beta )e^{-i{\omega }_0{\tau }_{10}}\\ -c\beta e^{-i{\omega }_0{\tau }_{10}}+\eta \gamma e^{-i{\omega }_0{\tau }_{20}}\\ 0\end{array}\right),$$

and

$$A_2{C}_2=f_0\left(11\right),$$

$$A_2=\left(\begin{array}{ccc}-a{V}_{*}& b{V}_{*}& 0\\ -c{E}_{*}+\eta K_{*}& r_2-c{V}_{*}& \eta V_{*}\\ 1-\eta & 0& -\delta \end{array}\right),$$

$$f_0\left(11\right)=2\left(\begin{array}{c}-a+b{R}_e\left(\beta e^{i{\omega }_0{\tau }_0}\right)\\ -c{R}_e\left(\beta e^{i{\omega }_0{\tau }_0}\right)+\eta R_e\left(\gamma e^{i{\omega }_0{\tau }_0}\right)\\ 0\end{array}\right).$$

From Hassard’s method [25], we have

$$l_1\left(0\right)=\frac{i}{2{\omega }_0{\tau }_{20}}(g_{20}{g}_{11}-2{\left|g_{11}\right|}^2-\frac{1}{3}{\left|g_{02}\right|}^2)+g_{21}.$$

(32)

${\mu }_2=-\frac{Re\left\{l_1\left(0\right)\right\}}{Re\left\{{\lambda }^{\prime }\left({\tau }_{20}\right)\right\}}$ confirms the Hopf bifurcation direction: if $Re\left\{{\lambda }^{\prime }\left({\tau }_{20}\right)\right\}>0$, ${\mu }_2>0$ (or if ${\mu }_2<0)$, the Hopf bifurcation periodic solution exists for ${\tau }_2>{\tau }_{20}$ (or ${\tau }_2<{\tau }_{20}$), the bifurcation is supercritical ( or subcritical). If $Re\left\{{\lambda }^{\prime }\left({\tau }_{20}\right)\right\}<0$, the bifurcation is on the opposite direction.

${\nu }_2=2Re\left\{l_1\left(0\right)\right\}$ confirms the stability of the Hopf bifurcation periodic solutions: the solution is stable if ${\nu }_2<0$ or it is unstable if ${\nu }_2>0$.

4. Numerical Simulations of the System

We consider some numerical results with different values of ${\tau }_1,{\tau }_2$. Let the parameters of system $\left(4\right)$ be $r_1=1,r_2=0.1,a=0.09,b=0.2,c=4,\eta =0.8,\delta =0.5$, then condition $\left(H2\right)$ holds. The unique positive equilibrium is $S_{*}=(13.5189,1.0835,5.4076)$.

The corresponding characteristic equation of system $\left(4\right)$ at $S_{*}$ is

$${\lambda }^3+54.4757{\lambda }^2+26.9879\lambda +(1.2167{\lambda }^2+77.9992\lambda +38.6954)e^{-\lambda {\tau }_1}+(-11.6967\lambda -11.6967)e^{-\lambda {\tau }_2}=0.$$

(33)

When ${\tau }_1=0,{\tau }_2=0$, the roots of Equation (33) are $-53.9732,-1.3481,-0.3710$. Thus, $S_{*}$ is asymptotically stable.

Next, we just consider ${\tau }_1\ne {\tau }_2$. From Lemma 3, we obtain that if and only if $\omega \in [1.173,1.676]$, $P(\lambda ,{\tau }_1,{\tau }_2)=0$ has a solution $\lambda =i\omega$ (see Figure 1).

From Theorem 3, we know that if $({\tau }_1,{\tau }_2)\in {\mathit{T}}^{\circ }$, which is surrounded by $\mathit{T}={\bigcup }_{j=1}^5{\mathit{T}}^{\mathit{j}}$, coordinate axis ${\tau }_1=0$, ${\tau }_2=0$ and ${\tau }_2=7$ (see Figure 2, here, ${\tau }_2$ can be larger); thus, $S_{*}$ is asymptotically stable, where ${\mathit{T}}^{\mathit{1}}=(({\tau }_{10}^{1-}\left(\omega \right),{\tau }_{20}^{2-}\left(\omega \right)),{\mathit{T}}^{\mathit{2}}=(({\tau }_{10}^{1-}\left(\omega \right),{\tau }_{20}^{2+}\left(\omega \right)),{\mathit{T}}^{\mathit{3}}=(({\tau }_{10}^{1-}\left(\omega \right),{\tau }_{20}^{1-}\left(\omega \right)),$${\mathit{T}}^{\mathit{4}}=(({\tau }_{10}^{1-}\left(\omega \right),{\tau }_{20}^{1+}\left(\omega \right)),{\mathit{T}}^{\mathit{5}}=(({\tau }_{10}^{1-}\left(\omega \right),{\tau }_{20}^{0-}\left(\omega \right))$ from (16) and (17).

When $({\tau }_1,{\tau }_2)$ crossing $\mathit{T}$ and $\left(H5\right)$ holds, there are periodic solutions bifurcating from $S_{*}$. We choose some points $({\tau }_1,{\tau }_2)$ to illustrate the result.

We know from Figure 2 that if ${\tau }_1<0.9034$, then $S_{*}$ is always stable. Let ${\tau }_1=1.0003$; we see from Figure 2 that there are three critical points on the stability crossing curves, $({\tau }_{10}^4,{\tau }_{20}^4)\doteq (1.0003,1.9807)\in {\mathit{T}}^{\mathit{4}}$, $({\tau }_{10}^3,{\tau }_{20}^3)\doteq (1.0003,4.3577)\in {\mathit{T}}^{\mathit{3}}$, $({\tau }_{10}^2,{\tau }_{20}^2)\doteq (1.0003,6.4977)\in {\mathit{T}}^{\mathit{2}}.$ For each point, we calculate $Re\left\{{\lambda }^{\prime }\left({\tau }_{20}\right)\right\},{\mu }_2,{\nu }_2$.

When delay ${\tau }_1=1.0003$ remains unchanged, ${\tau }_{20}<1.9807$, since $({\tau }_1,{\tau }_2)$ is in the left of $\mathit{T}$, we know that $S_{*}$ is asymptotically stable (see Figure 3).

Let the delays $({\tau }_1,{\tau }_2)$ increase and pass through the critical point $({\tau }_{10},{\tau }_{20})\doteq (1.0003,1.9807)$, where $\omega =1.372$. We obtain that at $({\tau }_{10},{\tau }_{20})=(1.0003,1.9807)$, $Re\left\{{\lambda }^{\prime }\left({\tau }_{20}\right)\right\}=0.2762>0$, $Re\left\{l_1\left(0\right)\right\}=-0.0565<0$, ${\mu }_2=0.2047>0$, and ${\nu }_2=-0.1131<0$. Therefore, the bifurcation is supercritical; the bifurcation periodic solution is in the direction of ${\tau }_{20}>1.9807$ and is stable (see Figure 4).

Let the delays $({\tau }_1,{\tau }_2)$ increase further to arrive at the critical point $({\tau }_{10},{\tau }_{20})\doteq (1.0003,4.3577)$, where $\omega =1.372$. We obtain $Re\left\{{\lambda }^{\prime }\left({\tau }_{20}\right)\right\}=-0.0935<0$, $Re\left\{l_1\left(0\right)\right\}=-0.0580<0$, ${\mu }_2=-0.6206<0$, and ${\nu }_2=-0.1160<0$. Therefore, the bifurcation is subcritical, and the bifurcation periodic solution is stable, which is on the side less than ${\tau }_{20}=4.3577$. When ${\tau }_{20}$ increases crossing $4.3577$, the delay $({\tau }_1,{\tau }_2)$ enters the area of ${\mathit{T}}^{\circ }$, and $S_{*}$ becomes stable again (see Figure 5).

Let the delays $({\tau }_1,{\tau }_2)$ increase further to arrive at the critical point $({\tau }_{10},{\tau }_{20})\doteq (1.0003,6.4977)$, where $\omega =1.372$. We obtain $Re\left\{{\lambda }^{\prime }\left({\tau }_{20}\right)\right\}=0.2771>0$, $Re\left\{l_1\left(0\right)\right\}=-0.0538<0$, ${\mu }_2=0.1941>0$, and ${\nu }_2=-0.1076<0$. Therefore, the bifurcation is supercritical; when ${\tau }_{20}$ increases crossing $6.4977$, $S_{*}$ becomes unstable, and a stable periodic solution bifurcates from $S_{*}$ (see Figure 6).

In this example, we can see that if the Hopf bifurcation is supercritical for one point $({\tau }_{10}^{m_1},{\tau }_{20}^{n_1})$ on the stability crossing curve, then for another point $({\tau }_{10}^{m_2},{\tau }_{20}^{n_2})$ on the adjacent stability crossing curve, the Hopf bifurcation is subcritical and vice versa. The Hopf bifurcation will be supercritical and subcritical alternately. From Figure 7, we understand that $\eta$ is closely related to the delay ${\tau }_1$, with a smaller $\eta$, a smaller ${\tau }_1$ is needed for the equilibrium’s stability.

5. Conclusions

In this paper, we proposed a delayed environmental protection expenditures model with two delays. We discussed the existence of equilibrium and bifurcation using delays ${\tau }_1$ and ${\tau }_2$ as the bifurcation parameters. We depicted the stability crossing curves and obtained the stability of the equilibrium $S_{*}$; the direction of bifurcation was also considered.

Since the financial support for the protection of the environment is only depending on the share $\eta$ of the tourism user fees in this model, the stability of the equilibrium relies closely on $\eta$ with delay ${\tau }_1$, the lower the share $\eta$, the smaller the spread delay ${\tau }_1$ needs to be. The external capital support for sustaining the PAs has not been considered in the model; it will be an interesting problem in the management of PAs. We leave this subject for future work.