1. Introduction
Interactions between predator and prey can generate rich dynamics and have engaged numerous investigators’ attention. In the existing literature, the following homogeneous diffusive predator-prey model has been extensively considered:
where
is an open bounded domain in
with
;
is the outer unit normal to the boundary
, which is assumed to be sufficiently smooth;
and
are the population densities of the prey and the predator at time
s and position
, respectively;
and
are the diffusion coefficients of
U and
V, respectively;
are all of the positive constants;
A is the intrinsic growth rate;
N is the carrying capacity;
B and
E are the strength of the relative effect on the two species in the interaction;
is the functional response of the predator to the prey density;
C is the “saturation” effect; and
F is the death rate of
V.
Then, by a non-dimensionalized change of variables (see also [
1]):
we can reduce the system (
1) to the simplified dimensionless form as follows:
where
u and
v are the scaled densities of the prey and predator, respectively;
is the growth rate of
u in the absence of the predator;
is the death rate of the predator;
is the functional response determining the predator’s consumption of the prey’s abundance;
k is the fraction of the prey’s biomass, which can be transformed into the predator’s biomass; and
and
are the diffusion coefficients of
u and
v, respectively.
System (
2) and the like have been studied extensively in the existing literature. For example, for the corresponding ODE system of (
2), Hsu [
2] showed that the local stability of the positive equilibrium solution can also indicate its global asymptotic stability. In [
3], Hsu and Shi studied the relaxation oscillations of (
2), while in [
4], Cheng observed that the periodic solution of the ODEs in system (
2) is unique and stable. For the reaction-diffusion system of system (
2), in [
5], Ko and Ryu not only studied the existence of non-constant positive equilibrium solutions but also investigated the local existence of periodic solutions. In [
1], Yi, Wei, and Shi performed steady-state bifurcation and Hopf bifurcation analysis of the system. In [
6], Peng and Shi considered global steady-state bifurcations of the system, and their results proved that the global bifurcation of steady-state solutions comprises bounded loops.
In this paper, we mainly consider the following reaction-diffusion predator-prey system with dormancy:
where
,
,
,
,
,
, and
;
w is the predator’ density with a dormant state or resting eggs;
and
denote the proportion of reproduction effects on predators between active and dormant states, respectively; and
stands for the hatching of dormant predators or the average dormancy period.
In [
7], Kuwamura showed that the hatching of resting eggs can keep the population dynamics stable when the switching between non-resting and resting eggs is sharp. In [
8], Kuwamura, Nakazawa, and Ogawa studied the stationary and oscillatory diffusion-induced instabilities of the constant equilibrium solutions.
For system (
3), we are mainly interested in the influence of the dormancy of the predators on the dynamics of the system. In particular, we focus on the diffusion-induced instability of the Hopf bifurcating periodic solutions of the system, which is less understood for this particular model in the existing literature [
9,
10,
11,
12,
13,
14,
15,
16,
17,
18]. We shall prove that from suitable conditions on the diffusion rates
, the spatially homogeneous periodic solution can undergo diffusion-induced instability and can induce the new spatiotemporal patterns emerging consequently. We would like to remark that for the system without the dormancy of predators (e.g., system (
2)), Yi, Wei, and Shi proposed that under suitable conditions, once the periodic solution is stable with respect to the ODEs, it is still stable with respect to the PDEs; thus, there is no diffusion-induced instability of the periodic solutions. Based on this, we shall present a quite interesting difference between the system with the dormancy of predators and the system without the dormancy of predators.
The rest of this paper is organized in the following way. In
Section 2, we consider the dynamics of the ODEs system; in
Section 3, we consider the diffusion-induced instability of the periodic solutions bifurcating from Hopf bifurcations; in
Section 4, we present some numerical simulations to illustrate our theoretical analysis; and in
Section 5, we draw some conclusions.
2. The Dynamical Behaviors of the Kinetic System
In this section, we consider the following kinetic system:
2.1. The Auxiliary System: The Predator-Prey System without Dormancy of Predators
To begin with, we consider the following kinetic system of system (
2):
System (
5) has a trivial solution
, a semi-trivial solution
, and a unique positive equilibrium solution under certain conditions stated below.
We now state the following results on system (
5) due to Hsu [
2] (see also [
1]):
Lemma 1 ([
1,
2]).
The following conclusions hold true:- 1.
Suppose that either or holds. Then, system (5) has no positive equilibrium solutions; in this case, is unstable, while is globally asymptotically stable; - 2.
Suppose that holds. Then, system (5) has a unique positive equilibrium solution , where In this case, both and are unstable, is globally asymptotically stable if either and or and holds, while is unstable if and . In particular, the loss of the stability of leads to a Hopf bifurcation at .
2.2. The Predator-Prey Model with Dormancy of Predators
In this subsection, we study the predator-prey system with dormancy, which is system (
4). Clearly, system (
4) has
and
as its equilibrium solutions. We have the following results:
Theorem 1. The following conclusions hold true:
- 1.
is always unstable in (4). - 2.
is locally asymptotically stable in (4) when , while it is unstable when .
Proof. The Jacobian matrix of system (
4) evaluated at
is given by
which has three eigenvalues:
. Thus,
is unstable with respect to (
4).
The Jacobian matrix of system (
4) evaluated at
is given by
The characteristic equation of
is given by
If
, for
, all of the eigenvalues of (
7) have negative real parts. Thus,
is stable.
If
, then (
7) has a positive eigenvalue. Thus,
is unstable. □
Clearly, if
is a positive equilibrium solution of (
5), then
is a positive equilibrium solution of (
4), where
Then, by Lemma 1, we have the following results on the existence of positive equilibrium solution of system (
4).
Theorem 2. Suppose that holds. Then, system (4) has a unique positive equilibrium solution , which is defined by (8). Next, we study the stability of
in system (
4).
We choose
as the bifurcation parameter. Linearizing system (
4) at
, we obtain its Jacobian matrix:
where
The characteristic equation of
is
where
To study the stability of
, by Appendix of [
19] (see also Lemma 2 below), we need to know the signs of
,
,
and
.
We make the following assumptions:
- (H)
Suppose that either (1):
and
or (2):
and
holds so that
is stable in system (
5).
Under assumption (H), we have and . Thus, , , and for all and .
Thus, to study the stability of
, it remains to study the sign of
, which takes the following form:
where
Clearly, under assumption (H), we have . We now consider the sign of .
We can check that
which has a unique positive root, denoted by
, which is given by
It can be directly checked that
Clearly, for , while for .
Then, for any
,
and
,
. Therefore, for any
,
. By Appendix of [
19],
is locally asymptotically stable in system (
4).
In what follows, we study the case when
so that
, where
If we regard A and B as the functions of , we can check that is increasing and is decreasing in . Moreover, for .
One can check that for any
,
(resp.,
) is equivalent to
When
,
,
, then
as
; When
, since
,
, we have
as
. Then, for any
, we have
. Since
, a unique
exists such that
If
, then for any
, it always holds true that
, which means that
. Thus,
for any
,
is locally stable in system (
4).
If
, then
. Therefore, for any
, a unique
exists, satisfying
or equivalently
Then, for any
,
is locally stable in system (
4).
Next, we assume the case of
, in which
. Regarding
as the quadratic function
, we can obtain its discriminant
Assume that for some
, we have
. Then, for any
, we have
. Hence, for any
,
is locally stable in system (
4).
Assume that for some
, we have
. Then,
has two distinct positive solutions given by
Thus, for any
, we have
. Then,
is locally stable in system (
4). While if
, we have
. Then,
is unstable in system (
4).
We are now in the position to state the stability results of :
Theorem 3. Suppose that either but or but holds so that is stable in system (5). Let , , , and be defined in (16), (17), (20) and (21), respectively. Then, we have . In particular, - 1.
Suppose that holds. Let be defined by (22). - (a)
If, additionally, , then for any , is locally asymptotically stable in system (4); - (b)
If, additionally, , then for any , is locally asymptotically stable in (4), while for , is unstable in system (4).
- 2.
Suppose that holds. Then, for any , is locally asymptotically stable in system (4).
Remark 1.
- 1.
We would like to remark that it is analytically demanding to analyze the sign of . Indeed, we need to resort to numerical simulations to determine when or . It is found from numerical simulations that for some θ, we have , while for the other θ, ;
- 2.
We assume that either but or but holds so that is stable in system (5). However, for case of 2(b), when , is unstable. From this, we can see a difference between the system without dormancy and the system with dormancy.
Theorem 4. Suppose that either but or but holds, so that is stable in system (5). Let and so that and are well-defined. Then, at , the Hopf bifurcation around occurs. Moreover, at , the Hopf bifurcating periodic solution is stable and the bifurcation direction is forward if , while the Hopf bifurcating periodic solution is unstable and the bifurcation direction is backward if , where or , and is defined by (26). Proof. 1. The proof of the existence of Hopf bifurcations at
and
. By the aforementioned analysis, at
and
, we have
. Thus, at
and
, the eigenvalue problem has a pair of purely imaginary roots and a negative root. Furthermore, according to Theorem 3, we have
Therefore, by the Hopf bifurcation theorem, at and , the Hopf bifurcation around occurs.
2. Now, we derive conditions to determine the bifurcation direction and the stability of the periodic solutions.
By Theorem A.1 of [
19] (or see also Lemma 2 below), the bifurcation direction (forward or backward) and the stability/instability of the periodic solutions can be determined by the sign of
, where
or
, and
is defined in (
14).
By using the framework of Theorem A.1 of [
19], we need to calculate the term
. To that end, we define the matrix
P in the following way:
where
Then, we can calculate
where the determinant of
P denotes as
and
and
denote the transformation from the variables
. Then, by (A.17) in Appendix of [
19], we have
where
,
, and
are defined in (
25).
By Theorem A.1 of [
19] (see also Lemma 2 below), we can draw the following conclusions: at
, the bifurcating periodic solution is unstable and the bifurcation occurs for
for sufficiently small
if
, and the bifurcating periodic solution is stable and the bifurcation occurs for
for sufficiently small
if
holds. On the other hand, at
, the bifurcating periodic solution is unstable and the bifurcation occurs for
for sufficiently small
if
, and the bifurcating periodic solution is stable and the bifurcation occurs for
for sufficiently small
if
holds. □
Remark 2.
- 1.
It is analytically demanding to obtain a explicit expression of , and we shall resort to numerical tools to calculate it in the part of numerical simulations;
- 2.
For simplicity, we denote , and P by the Hopf bifurcating periodic solution and its minimum period.
4. Numerical Examples
In this section, we present some numerical examples. We divided our numerical simulations into two parts: “larger” ( is very close to 1) and “smaller” ( is very close to 0).
Case 1 (“larger”
). We set
,
,
, and
. In this case, we have
By Theorem 4, at and , the supercritical Hopf bifurcation occurs around . That is, the bifurcating periodic solution, denoted by , is stable in the ODEs system.
At
,
which indicates that Hopf bifurcation direction is backward. Set
,
,
,
,
.
Firstly, we set
. Numerical simulation shows that
remains stable in the diffusive system. No diffusion-induced instability of
occurs (see
Figure 1).
Secondly, we set
,
,
;
;
,
,
. By Theorem 5,
becomes diffusion-induced unstable in diffusive system (
3). This is demonstrated by
Figure 2.
At
, we have
which implies that the bifurcating direction is forward. Then, we choose
,
,
,
,
. In this case, there is a stable periodic solution in the system (
4) and denoted by
.
First, we set
.
,
,
,
. In this case,
remains stable in system (
3). No diffusion-induced instability of
occurs. This is demonstrated by
Figure 3.
Secondly, we set
,
,
,
,
,
,
. In this case, by theorem 5,
becomes diffusion-induced unstable in system (
3). This is demonstrated by
Figure 4.
Case 2 (“smaller”
). We set
,
,
and
. In this case, we have
According to Theorem 4, at or , the supercritical Hopf bifurcation occurs around . That is, the bifurcating periodic solution, denoted by , is asymptotically stable in the ODEs system.
At
,
which indicates that the Hopf bifurcation direction is backward. Set
,
,
,
, and
. System (
4) has a stable periodic solution, denoted by
.
First, we set
.
. No diffusion-induced instability of
occurs. This is demonstrated by
Figure 5.
Secondly, we set
,
,
.
, and
,
,
. By Theorem 5,
becomes diffusion-induced unstable and the emerging spatiotemporal patterns can be found. This is demonstrated by
Figure 6.
At
, we have
which confirms that the Hopf bifurcation is forward. We set
,
,
,
,
. Then, the kinetic system (
4) possesses a periodic solution
, which is stable.
First, we set
.
. The diffusion-induced instability of
cannot be found. This is demonstrated by
Figure 7.
Secondly, let
,
,
.
, and
,
,
. By Theorem 5,
becomes diffusion-induced unstable and the emerging spatiotemporal patterns can be observed. This is demonstrated by
Figure 8.
5. Concluding Remarks
In this paper, a homogeneous diffusive predator-prey system with the dormancy of predators is mainly considered. It concentrates on the diffusion-induced instability of the Hopf bifurcating periodic solutions.
Without regard to the dormancy effect, the predator-prey system is a system with two components. Motivated by [
2,
3], we choose the first component
of the positive equilibrium solution
as the bifurcation parameter. We assume that the unique positive equilibrium solution of the system (the 2-component predator-prey system) is stable with respect to the corresponding ODEs system, say
holds so that
is stable in system (
5). By [
2],
is globally asymptotically stable in system (
5).
In the presence of the dormancy effect, the predator-prey system becomes a system with 3-components. Our results indicated that for some , if , then for suitable and (say, , and ), the ODEs predator-prey system might exhibit temporal oscillations. This suggests that the dormancy effects can favor the emergence of temporal oscillatory patterns. Precisely, the smaller (the modeling the strengthen of the dormancy effect) is, the larger stability range of is. At and , Hopf bifurcations around occur. By calculating the first Lyapunov coefficients, we can derive conditions to determine the stability of the periodic solutions.
When diffusions are introduced into the predator-prey system with dormancy, we can deduce the reaction-diffusion equations with the 3-components system. Referring to the abstract results in [
19], we are able to expound some precise conditions on the diffusion coefficients to determine the diffusion-induced instability of the periodic solutions.