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:
where
and
are strictly positive constants.
represents the crowding influence; this means that the PA becomes less attractive when the number of tourists visiting the PA increases.
is the pollution stock of maximum tolerance, while
b means the waste generated by every visitor,
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
of total tourism income to protect the environmental resource and the remaining part
to increase the capital stock, where
The depreciate rate of capital stock is
.
mean that the number of visitor is proportional to
E and
K,
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:
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
The authors described the stable region on the (
) 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,
is introduced to the model. The crowding effect is considered as
.
in the first equation of (2) is changed into
by considering the visitor’s effect on this term, and the influence of the capital stock
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
is added. We hope that the pollution is not tolerable, so we let
. The term
is changed into
, since we think in here the capital stock is more important to the change of the environment. So, we obtain the following model:
where
is the rate of effect of the public praise; visitors will increase if
increases.
is the self-purification ability of the environment.
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
means a share
of total revenues is used to protect the environment. Capital stock is depreciated at the rate
. The increment of the visitors relies on the visitors at the time
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
, where
and
. We name
as the spread delay and
as the protecting delay.
We give the initial conditions of system (4) as:
where
,
. Then, according to the theorem on functional differential equations [
22], system (4) has one and only one solution
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
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
we introduced to the model (4) is more important than
which is
in [
4,
8] for the stability of equilibrium, since
needs to be on the left of the stability crossing curves. We find that the share
of the tourism user fees and the spread delay
are very important parameters in our discussion.
2. Equilibria and Stability Crossing Curves
By straightforward computation, system (4) has equilibrium , which is unstable, since there are always positive eigenvalues and , while the characteristic equation of (4) at has no relation with delays .
If
Hypothesis 1 (H1). system
has
as a positive equilibrium:
If
Hypothesis 2 (H2).
system
has
as a positive equilibrium:
If , there is no positive equilibrium.
Let
, then (4) becomes
The corresponding characteristic equation of system (5) can be written by
where
When , by the criterion of Routh–Hurwitz, we have
Theorem 1. Assume , (H1) (or (H2)) hold. If
Hypothesis 3 (H3). is satisfied, then is asymptotically locally stable.
Next, we investigate the distribution of the roots of (6). From Rouche theorem [
23], as
vary continuously in
, 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) , or (II) .
Case (I) .
Suppose that on the imaginary axis, system (6) has a root
. Substituting it into (6), separating the imaginary part and the real parts, we have
Let
then, (9) becomes
We see from (10) that since
, there is at least one positive root. We assume that there are three positive roots for generality, defined by
and
. According to (8), we have
where
and
. Denote
Then, we know
Lemma 1. When (H1) (or (H2)), (H3) hold, all roots of (6) have strictly negative real parts when . (6) has simple purely imaginary roots when .
Near , consider as the root of (6) where For the transversality, we know that
Lemma 2. Suppose and then .
According to Lemma 1 and Lemma 2, we have
Theorem 2. When , let be denoted by (12), and assume that , are satisfied,
- (i)
If , then system (4) has a locally asymptotically stable positive equilibrium .
- (ii)
If and then for system (4), Hopf bifurcation will occur at when .
Case (II)
Characteristic Equation (
6) can be rewritten as
where
We can easily confirm that Equation (
13) satisfies:
- (I)
- (II)
- (III)
, have no common zeros;
- (IV)
Lemma 3. For each , has as its root for some if and only if
The proof of Lemma 3 can be found in [
18].
Let
then we know that
has
as its solution if and only if
simultaneously.
Denote the crossing set
of all
, which satisfy (14) and (15). For given
From (14) and (15), we can find all of
:
where
can be calculated as
and are the smallest integers such that the corresponding , and are non-negative.
Let , we can obtain from (16) and (17).
Next, we discuss the transversality. Choose
to be the bifurcating parameter, and take the derivative of
in (6); then, we obtain
When
,
, then
Let
Hypothesis 5 (H5).
Denote are p sections of continuous curves defined on , is the internal region surrounded by with coordinate axis and . Then, we can obtain the following:
Theorem 3. Assume that (or ), hold,
- (I)
If , then system (4) has a positive equilibrium which is locally asymptotically stable.
- (II)
If crossing and holds, then is a critical point, system (4) has Hopf bifurcation at
We denote continuous curve as stability crossing curves.
4. Numerical Simulations of the System
We consider some numerical results with different values of . Let the parameters of system be , then condition holds. The unique positive equilibrium is .
The corresponding characteristic equation of system
at
is
When
, the roots of Equation (
33) are
. Thus,
is asymptotically stable.
Next, we just consider
. From Lemma 3, we obtain that if and only if
,
has a solution
(see
Figure 1).
From Theorem 3, we know that if
, which is surrounded by
, coordinate axis
,
and
(see
Figure 2, here,
can be larger); thus,
is asymptotically stable, where
from (16) and (17).
When crossing and holds, there are periodic solutions bifurcating from . We choose some points to illustrate the result.
We know from
Figure 2 that if
, then
is always stable. Let
; we see from
Figure 2 that there are three critical points on the stability crossing curves,
,
,
For each point, we calculate
.
When delay
remains unchanged,
, since
is in the left of
, we know that
is asymptotically stable (see
Figure 3).
Let the delays
increase and pass through the critical point
, where
. We obtain that at
,
,
,
, and
. Therefore, the bifurcation is supercritical; the bifurcation periodic solution is in the direction of
and is stable (see
Figure 4).
Let the delays
increase further to arrive at the critical point
, where
. We obtain
,
,
, and
. Therefore, the bifurcation is subcritical, and the bifurcation periodic solution is stable, which is on the side less than
. When
increases crossing
, the delay
enters the area of
, and
becomes stable again (see
Figure 5).
Let the delays
increase further to arrive at the critical point
, where
. We obtain
,
,
, and
. Therefore, the bifurcation is supercritical; when
increases crossing
,
becomes unstable, and a stable periodic solution bifurcates from
(see
Figure 6).
In this example, we can see that if the Hopf bifurcation is supercritical for one point
on the stability crossing curve, then for another point
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
is closely related to the delay
, with a smaller
, a smaller
is needed for the equilibrium’s stability.