1. Introduction
The dynamical behavior of the predator–prey system is one of the main research topics in mathematical ecology and theoretical biology [
1,
2,
3,
4,
5,
6,
7,
8,
9]. Functional response is the core component of the community and food web model, and their mathematical form strongly affects the dynamics and stability of the ecosystem [
10,
11,
12,
13,
14]. Beddington [
15] and DeAngelis [
16] originally proposed the predator–prey model as follows:
where
is the prey density,
is the predator density,
a is the intrinsic growth rate of the prey,
b represents the intensity of intraspecific competition of the prey, and
d represents the predator’s death rate. The Beddington–DeAngelis functional response is similar to the well-known Holling II type [
17,
18] functional response, but it has an extra term
in the denominator modeling mutual interference among predators.
The functional response of consumers is a function of resource density [
17,
19,
20]. However, it has been shown that other species and other predators can alter the predation process directly or indirectly [
21]. In addition, Kratina [
21] showed that predator dependence is very important not only when the predator density on per capita predation rate is very high, but also when the predator density is low. Therefore, we need to consider the realistic level of predator density when we study the predator–prey system.
Moreover, in nature, many species undergo two stages [
22,
23]: immature and mature, and species at these two stages may have different behaviors. The model of single-species stage-structured dynamics [
22] was described as
where
and
represent the immature and mature populations densities, respectively, and
represents a constant time to maturity. Therefore, in order to be in accord with the natural phenomenon, the system with the stage structure recently has been extensively studied [
24,
25,
26,
27,
28,
29,
30]. Liu and Beretta [
27] studied time delay
in the response term
of (
1) in the predator equation, that is,
where
(units: 1/time) and
(units: 1/prey) stand the effects of capture rate and handling time, respectively, on the feeding rate;
n is the birth rate of the predator; and
(units: 1/predator) stands the magnitude of interference among predators. Liu and Beretta [
27] pointed out the difference between Beddington–DeAngelis functional response and Holling type II, and the effect of
(describing mutual interference by predators) on the dynamic of the system (
3).
On the basis of the system (
1) and the system (
2), She and Li [
29] investigated a predator–prey system with density-dependence for predator and stage structure for prey
where
p stands for predator density-dependent mortality rate, the predator consumes prey with functional response of Beddington-DeAngelis type
and contributes to its growth with rate
. Note that compared with the system (
1), the system (
4) contains not only
(which stands for intraspecific competition of prey species), but also
(which stands for intraspecific competition of predator species). That is, they consider both the prey density dependence and the predator density dependence in the predator–prey model (
4). She and Li [
29] studied the dynamics of the system (
4) and pointed out the impact of the predator density-dependent mortality rate
p on the global attraction and permanence of the system (
4).
The development of biological resources, the management of renewable resources, and the harvest of populations are universal human purposes for realizing the economic benefits of fishery, forestry and wildlife management [
31,
32]. Many researchers [
28,
30,
33,
34] have extensively studied the predator–prey model with harvesting and the role of harvesting in renewable resource management. Brauer [
35] introduced the predator–prey system with constant-rate prey harvesting
where prey is harvested at a constant time rate F. May [
36] put forward two types of harvesting regimes: constant-yield harvesting (representing harvested biomass independent of the size of the population) and constant-effort harvesting (representing harvested biomass proportional to the size of the population).
Based on the system (
3), we construct a density-dependent and constant-effort harvesting predator–prey model:
where
denotes the immature or juvenile predator density, juveniles suffer a mortality rate
and take
units of time to be mature and
is the surviving rate of each immature predator to reach maturity.
and
denote the harvesting effort of the mature population of prey and predator, respectively. Further, all the parameters
a,
,
b,
c,
d,
f,
,
,
,
p,
,
, and
are positive.
In the system (
6), as
does not intervene in the dynamics of
and
, system (
6) is equal to the following system:
The initial conditions of the system (
7) is
where
,
, and
is the modulus in
. Normally, we use notation
.
This paper mainly investigates the local and global stability of positive equilibrium in the system (
7) on parameter
, which is organized as follows. In
Section 2, we study local stability of origin equilibrium and boundary equilibrium. In
Section 3, we derive local stability of the positive equilibrium for
and
, respectively. In
Section 4, we obtain the global asymptotical stability of the positive equilibrium for
and
, respectively. Last, we conclude the paper.
2. Local Stability of Origin Equilibrium and Boundary Equilibrium
For any value of all parameters, system (
7) has the equilibria
and
, denoted as the origin and the boundary equilibrium, respectively. In the following, we determine the local stability of two equilibria by the sign of eigenvalue for the corresponding characteristic matrix.
First, for origin equilibrium
, the corresponding characteristic matrix is
and the eigenvalues are
Clearly, is hyperbolic saddle and is unstable.
Next, for boundary equilibrium
, the corresponding characteristic matrix is
We can obtain one eigenvalue
, the second eigenvalue is determined by the following equation:
Let n be the independent variable and g be the dependent variable; then, the straight line and the curve must intersect at a unique point . Further, we can get the following:
- (i)
if , then ;
- (ii)
if , then ;
- (ii)
if , then .
Therefore, we have the following conclusion about local stability of boundary equilibrium.
Theorem 1.
(1) If , boundary equilibrium is locally asymptotically stable;
(2) If , boundary equilibrium is unstable;
(3) If , boundary equilibrium is linearly neutrally stable.
Remark 1. By Theorem 1, when , is locally asymptotically stable; when , is unstable; when , is linearly neutrally stable, where That is, is a threshold for the stability of the boundary equilibrium.
3. Local Stability of the Positive Equilibrium
We denote
as equilibrium other than the origin equilibrium and boundary equilibrium for the system (
7) and satisfying the algebraic equations
From
, we have the curve
by
, we have another curve
By considering the intersection of curves
and
in the first quadrant, we can obtain if the condition
holds,
is a positive equilibrium of system (
7).
By the condition (
10), we can directly obtain the following result.
Remark 2. The positive equilibrium exists for any predation maturation time τ in the interval .
The Jacobian matrix [
37] of differential system (
7) with delay coefficients at
is
Therefore, the characteristic equation is
where
We noticed that if
, then
. Therefore, when
,
cannot be a characteristic root of Equation (
11).
The parameter is the time from immature to mature predator. Time delay , it shows the predator population is divided into immature and mature; time delay , it shows the model does not consider the immature predators. In the following, we discuss the local stability of positive equilibrium for and , respectively.
3.1. Local Stability of the Positive Equilibrium for
When
, characteristic Equation (
11) becomes
Obviously, when
, all roots of characteristic Equation (
12) have negative real part. Therefore,
is locally asymptotically stable.
Further, by
we can obtain that if
, then
. According to the previous analysis, we also have if
,
. Therefore, we get the following conclusion.
Theorem 2. Let is a positive equilibrium of system (7), if the inequality (10) and the following inequalitieshold, the positive equilibrium of system (7) is locally asymptotically stable for . Example 1. Let , , , , , , , , , , , and , the system (7) becomes Clearly, . Therefore, the system (14) exists a positive equilibrium. Moreover, we obtain the positive equilibrium of system (14). As and , the equilibrium is locally asymptotically stable by Theorem 2. 3.2. Local Stability of the Positive Equilibrium for
In this subsection, we further discuss the sign of the real part of the characteristic root for the characteristic Equation (
11) with the change of parameter
, and determine stability switch.
First, assume that characteristic Equation (
11) has a characteristic root with zero real part, and let it be
and
. Putting it into the characteristic Equation (
11), we have
Here,
,
,
and
are abbreviated as
,
,
and
, respectively. From Equation (
15), we can get
Regarding
as an invariant, by the quadratic root formula, we have
We discuss the case of roots for the Equation (
16):
(I) If
and
are negative, which obviously contradicts with
. Therefore, Equation (
16) does not have real roots, that is, characteristic Equation (
11) does not have purely imaginary roots. Moreover, if the conditions (
10) and (
13) hold, all roots of the characteristic Equation (
12) have negative real part for
. Therefore, by Rouche’s theorem, it follows that the roots of the characteristic Equation (
11) also have negative real part.
Equation (
16) has a positive root
. Furthermore, putting
into Equations (
15), we have
Equation (
16) has two positive roots
. Putting
into Equations (
15), we obtain
From the above analysis, we have the following conclusion.
Theorem 3.
(i) If conditions (10) and (19) hold and , then characteristic Equation (11) has a pair of purely imaginary roots . (ii) If conditions (10) and (21) hold and (), then characteristic Equation (11) has a pair of purely imaginary roots (). Second, we have that when
,
, and
, there are roots with positive real part in the characteristic Equation (
11). Let these characteristic roots with positive real part be
and
where
According to the characteristic Equation (
11), we get
Obviously,
. Therefore, in order to judge the sign of
, we only need to judge the sign of
. According to (
23), we can get
Last, we can easily prove that
and
satisfy the following transversality conditions:
It follows that
and
are bifurcation values. Thus, we have the following theorem about the distribution of the characteristic roots of Equation (
11).
Theorem 4.
(i) If conditions (10), (13), and (18) hold, then all roots of Equation (11) have negative real parts for all . Therefore, the positive equilibrium of system (7) is locally asymptotically stable. (ii) If conditions (10) and (19) hold, when , all roots of Equation (11) have negative real parts, and the positive equilibrium of system (7) is locally asymptotically stable; when , Equation (11) has a pair of purely imaginary roots , and when , Equation (11) has at least one root with positive real part, and the positive equilibrium of system (7) is unstable, that is, when τ keeps increasing and passes , the positive equilibrium bifurcates into two periodic solutions with small amplitudes. (iii) If conditions (10) and (21) hold, there is a positive integer k such that the positive equilibrium of system (7) has k stability switches from stability to instability to stability. That is, whenall roots of Equation (11) have negative real parts, and the positive equilibrium of system (7) is locally asymptotically stable. When Equation (11) has at least one root with positive real part, the positive equilibrium of system (7) is unstable. Furthermore, it shows when time delay τ passes the critical value , the positive equilibrium of the system (7) loses its stability and Hopf bifurcation occurs. 5. Conclusions
In this paper, we investigate dynamics of a stage-structured density-dependent predator–prey system with Beddington–DeAngelis functional response and harvesting.
First, according to the sign for real part of eigenvalue, the local stability of the origin equilibrium and the boundary equilibrium is determined.
In addition, we discuss local stability of positive equilibrium
for the case of
and
, respectively. When
, the positive equilibrium
of system (
7) is locally asymptotically stable. However, when
, the local stability of positive equilibrium
of the system (
7) includes the following three cases:
(i) If (
10), (
13), and (
18) hold, the positive equilibrium
of system (
7) is locally asymptotically stable for any
;
(ii) If (
10) and (
19) hold, when
increases and passes
, the stability of positive equilibrium
for system (
7) switches from local asymptotical stability to unstability and bifurcates into two periodic solutions with small amplitudes;
(iii) If (
10) and (
21) hold, there is a positive integer
k such that the positive equilibrium
of system (
7) has
k switches from stability to instability to stability, which shows when the delay
passes the critical value
, the positive equilibrium
of system (
7) loses its stability and Hopf bifurcation occurs.
Last, we consider the global asymptotical stability of the positive equilibrium
for system (
7). When
, stability theory of the periodic solution is used to prove that the positive equilibrium
is globally asymptotically stable by contradiction; when
, by constructing Lyapunov functions we prove the positive equilibrium
for system (
7) is globally asymptotically stable. Moreover, examples are given to illustrate the obtained results.
The predator–prey model studied in this paper, where not only the prey density dependence but also the predator density dependence are considered such that the system conforms to real biological environment. In Remarks 1, 2, and 4, we explain the effect of time delay on the stability of boundary equilibrium, the existence of the positive equilibrium, and the global asymptotic stability of the positive equilibrium, respectively. Moreover, we study the effect of harvesting on the global asymptotic stability of the positive equilibrium in Remark 3. Simultaneously, regarding the predator density-dependent mortality rate
p, by Theorem 1 and the condition
10, the parameter
p does not affect the stability of boundary equilibrium, the existence of the positive equilibrium, respectively. As
all contain parameter
p, the predator density-dependent mortality rate
p not only affects the local asymptotic stability of the positive equilibrium by Theorems 2 and 4, but also the global asymptotic stability of the positive equilibrium by Theorems 5 and 6. However, we cannot describe exactly how the predator density-dependent mortality rate affects the stability of the positive equilibrium. In our future work, we will further study the influence of parameter
p on the predator–prey system.