1. Introduction
Since the classical Lotka–Volterra model proposed by Lotka [
1] and Volterra [
2], predator–prey models have become one of the best-developed areas in ecology systems and have been widely studied. To better reflect the realistic phenomena and features of the ecosystem, the Lotka–Volterra model has been modified and improved in many forms [
3,
4,
5,
6]. In order to improve the random process, many researchers added spatial variables to the system, and the spatiotemporal dynamics of the diffusion predator–prey system have received widespread attention [
7,
8,
9]. In fact, the spatiotemporal dynamics of the reaction–diffusion predator–prey system were affected not only by the random diffusion of predators and the prey, but also by the spatiotemporal variations of the predator’s velocity, which is often influenced by prey taxis. After the prey-taxis phenomenon was first observed by Kareiva and Odell [
10] in the experiment, numerous diffusion systems with prey taxis have been studied to interpret the phenomenon of predator aggregation in high-prey-density areas [
11,
12,
13]. It may be noted that the prey taxis yields complex spatiotemporal dynamics, such as periodic solutions, quasi-periodic solutions, or sphere-like surfaces of solutions [
14,
15,
16].
It should be noted that many existing discussions [
17,
18,
19] are focused on the Gause–Kolmogorov-type predator–prey model, there is little work on a general Leslie type predator–prey system with prey taxis. Therefore, we shall consider the following system with prey taxis
where
is a bounded domain in
with a smooth boundary
and outward unit normal vector
;
and
represent the densities of the prey and predator at time
and position
, respectively; the Laplacian operator
denotes the random movement of two species;
and
are the diffusion coefficients for the prey and predator; the natural per capita growth function
describes the specific growth rate of the prey in the absence of predation; the function
represents the functional response of predators to the prey, and the function
denotes the per capita growth function of the predator with prey-dependent carrying capacity.
represents the tendency of predators to move along the prey gradient direction,
is the taxis coefficient of the predator, and the prey taxis is called attractive (repulsive) if
. For more biological background information, readers can refer to [
20,
21].
For some specific forms of system (
1), many excellent results have been obtained through qualitative or numerical analysis. For the case
, the system (
1) has been considered by many authors [
22,
23,
24]. Lindstrom [
25] showed that the system may appear to have multiple equilibria and multiple limit cycles, as the half saturation of predators is low. By using Liapunov function and LaSalle’s invariance principle, the global asymptotic stability of the unique interior equilibrium was proved in [
26]. Freedman and Mathsen [
27] derived the criteria for persistence by transforming variables. Lan and Zhu [
28] studied the phase portraits and Hopf bifurcation, and they proved that there is a unique stable limit cycle of the system by computing the Lyapunov number.
For the case
, the system (
1) also has more interesting dynamical behavior, such as the existence and boundedness of positive solutions, the global stability of the interior equilibrium, bifurcation, and the stability of spatially nonhomogeneous steady-state solutions [
29,
30,
31]. Du and Hsu [
32] obtained the existence conditions of positive steady-state solutions, and proved that positive steady states with certain spatial patterns can appear for a suitable heterogeneous environment. The Turing instability of the equilibrium and the Hopf bifurcation were discussed in [
33], and they also researched the stability of the bifurcating spatially homogeneous periodic solution. Qi and Zhu [
34] obtained an improved condition for the global asymptotic stability of the unique positive equilibrium by using a novel comparison argument. For a general diffusive system, Zou and Guo [
35] investigated the existence and boundedness conditions of positive steady-state solutions and also carried out the Hopf and steady-state bifurcation analyses.
Recently, when
in system (
1), Wang et al. [
36] considered the linearized stability of the positive equilibrium, and investigated the local existence and stability of nonconstant positive steady states through rigorous local bifurcation analysis in the 1D case. Zhang and Fu [
37] discussed the structure of the set of the nonconstant steady states in detail for the Holling–Tanner predator–prey model. They found that attractive prey taxis can stabilize the homogeneous equilibrium even if diffusion-driven instability has occurred. By the asymptotic analysis and bifurcation theory, Qiu and Guo [
38] obtained the local and global structure of nonconstant positive steady states of a modified Leslie–Gower model.
The main purpose of this paper is to find the nonconstant steady states of (
1) by solving the following system
In order to explore the steady-state bifurcation of system (
2), we always assume that:
, , there exists such that and for any ;
, and for any ;
, , there exists such that and for any .
The remainder of this paper is organized as follows. In
Section 2, we discuss the local linear stability of the unique positive equilibrium of system (
1) by analyzing the characteristic equation. And, we also show that the pattern formation may arise only when
. In
Section 3, the existence of nonconstant positive solutions is proved by an abstract bifurcation theorem. The linear stability of these bifurcating solutions for system (
2) is considered in
Section 4. In
Section 5, the paper ends with a conclusion.
2. Stability of the Positive Equilibrium
According to the assumptions of
, it is easy to see that system (
1) has a boundary equilibrium
and a unique positive equilibrium
, where
is the unique one that satisfies
.
Notations:
By the assumptions, we know that
and
. In order to discuss the influence of prey taxis on the stability of the positive equilibrium, we assume the condition
always holds to guarantee that the positive equilibrium
is stable for the corresponding ordinary differential system of (
1).
The linearization of the system (
1) at the positive equilibrium
is
where
Denote to be the simple eigenvalues of on subject to the homogeneous Neumann boundary condition and let the corresponding normalized eigenfunction be .
Then, the characteristic equations corresponding to the positive equilibrium
are
where
Clearly,
for arbitrarily
and
by
. Thus, in order to investigate the instability of the positive equilibrium
, we need to find a
such that
. Let
and
Through calculation, we determine that
is decreasing on
and increasing on
, where
, and there exists a constant
such that
. Set
Thereby, there is at least some
such that
when
. This leads to the following theorem.
Theorem 1. Assume holds; then, the positive equilibrium of system (1) is stable for and it is unstable if . Remark 1. It is easy to check that if for all , then and the positive equilibrium is stable for the system (1) without prey taxis. By Theorem 1, this implies that only repulsive prey taxis can lead to Turing instability and induces the emergence of spatial patterns. However, if there exists some such that , then and the positive equilibrium are instable for the system (1) without prey taxis. From Theorem 1, which shows that attractive prey taxis can stabilize positive equilibrium and inhibits the emergence of spatial patterns. These are all different from indirect prey taxis; readers can refer to Theorem in [39]. Example 1. To verify our above analysis, we investigate the diffusive Leslie–Gower predator–prey model with herd behavior and prey taxis in 1D spatial domain . That is, and .
Let us first take ; we have that for all and . From Theorem 1, the positive equilibrium is unstable for and it is stable for ; see Figure 1 and Figure 2. When we choose , we have for all and . From Theorem 1, the positive equilibrium is stable for and it is unstable for ; see Figure 3 and Figure 4. 3. Global Bifurcation
From the ecological viewpoint, a nonconstant positive solution corresponds to the coexistence steady state of prey and predator. Thus, the purpose of this section is to seek the nonconstant positive solutions of system (
2) by an abstract bifurcation theorem. To discuss the bifurcation analysis of system (
2), it is necessary to give a priori bound for any positive solutions of system (
2).
Lemma 1. Let be a positive solution of system (2). Then, for all andwhere C is a positive constant. Proof. By the first equation of system (
2) and the maximum principle, we ensure that any positive solutions of system (
2) satisfy
. And then, due to the Harnack inequality, there exists a positive constant
such that
. Hence, there exists a positive constant
C such that
for all
.
Let
; then, the second equation of system (
2) becomes
Suppose that
is a maximum point of
w; that is,
. In view of the boundedness of
u and the maximum principle, we obtain
Thus,
. Similarly, let
be a point such that
, then
Hence,
.
Next, we shall show that
for all
. Since the first equation of system (
2) and
, we have
for any
by standard elliptic regularity theory. Then, it follows from Sobolev embedding theorem that
. Thus,
for
. Then, we rewrite the second equation of system (
2) as follows
Based on
and
, one obtains
for any
from elliptic regularity theory. And, together with the Sobolev embedding theorem, we gain
for all
. Therefore, the proof is completed. □
In the following, we shall take the prey-taxis sensitivity coefficient
as the bifurcation parameter; a branch of nonconstant stationary solutions is established by the Crandall–Rabinowitz’ss bifurcation theorem [
40] and its user-friendly version given by Shi and Wang in [
41]. The structure of positive solutions in the vicinity of
is presented in detail by the local bifurcation theory, and then we extend the local curves by applying global bifurcation theory. To this end, we define
Let
, and rewrite
back to
; system (
2) becomes
where
Theorem 2. Assume holds. For each, is fixed ifwhere and , then a branch of nonconstant positive solution of system (2) bifurcates from the positive equilibrium , and the bifurcating branch near the bifurcation point can be parameterized aswith and sufficiently small. Moreover, the bifurcating branch is part of a connected component of the set , whereand extends to infinity in χ as . Proof. By direct computation, the Fr
chet derivative of
is given by
where
and
. It is easy to verify that
is continuous and differentiable with respect to
in
.
When defining
, (
5) can be rewritten as
where
Clearly,
and
. Thus, from Corollary 2.11 of [
41],
is the Fredholm operator with zero index.
To find the potential bifurcation point, we should show that the implicit function theorem fails on
under some values of
. This means that there exists a nontrivial solution for
; that is
In fact, any pair of function
can be expanded as follows
If
, then there is at least one of
. Thus, substituting the formula (
7) into system (
6), and multiplying the system (
6) by
and integrating over
, together with
as the normalized orthogonal eigenfunction, we obtain the following matrix equation
Note that
can be easily ruled out. Hence, for
, system (
6) has nontrivial solutions equivalent to
According to
, we have
In addition, in order to apply the bifurcation theory of a simple eigenvalue, we take
, and multiply
to the first equation of system (
6), then add it to the second equation; we obtain
where
To consider the eigenvalues of the matrix
A, we should find a value
such that
After calculations, we obtain
. And, from Equation (
8), it is easy to check that
is one of eigenvalues of
A. So, another eigenvalue of
A is
. Thus, if
, then the matrix
A has two different positive real eigenvalues.
On the other hand, by reversible transformation, system (
9) is equivalent to the following system
where
and
Hence, if
is not an eigenvalue of
on
subject to the homogeneous Neumann boundary condition, i.e.,
we have
and
, where
c is a constant. Since the transformation is reversible, we obtain
and
, where
and
are constants. By computing, we have
Therefore,
. Because we have proved that
is the Fredholm operator of index 0, it follows that
.
Finally, we also need to prove the transversality condition
holds, where
By (
5) we have
is continuous, which yields
Suppose that
then there exist a nontrivial solution
, such that
that is
Note that
. Multiplying the first two equations in system (
12) by
and then integrating them over
by parts, we have
However, (
13) is impossible because the determinant of the coefficient matrix on the left hand of (
13) is zero by (
8). So we obtain the contradiction, and it follows that
,
Thus, the local structure of nonconstant positive solutions near
can be obtained by Crandall and Rabinowitz in Theorem 1.7 in [
40]. Then, we extend the local solution branch to be a global one. According to the abstract global bifurcation theory of Shi and Wang [
41], we know that
must fulfill one of the following:
is unbounded in ;
contains a point with ;
contains a point where .
Note that positive solutions of system (
2) bifurcate from
as
. Hence,
is invalid.
By calculating (
5) at
with
, we have
Since
and
, which means that the boundary equilibrium
is nondegenerate as
. Hence,
can be excluded.
It follows from Lemma 1 that any positive solution of system (
2) is bounded and satisfies
for all
. By the Sobolev imbedding theorem, we determine that any positive solution of system (
2) is bounded in the norm of
. Therefore,
extends to infinity in
. This completes the proof. □
4. Stability of the Bifurcating Branches
By Theorem 2, we know that system (
2) has a branch of nonconstant positive solutions
, which bifurcate from the homogeneous patterns
. In this section, we shall employ linearized stability and perturbation of simple eigenvalues as in [
42,
43] and spectrum theory to study the local stability of the nonconstant positive solutions.
Obviously, the stability of the nonconstant positive solutions of system (
2) is consistent with that of system (5), and the nonconstant positive solutions of the system (5) are stable as all eigenvalues of
have negative real parts, where
Hence, we will investigate all of the eigenvalues of
for each
. To this end, we will first present the relation between eigenvalues of
and
.
Definition 1 ([
42]).
Suppose that are bounded linear maps of to . Then, is a simple eigenvalue of T if and . By the proof of Theorem 2, we know that 0 is a
-simple eigenvalue of
. Then, from Corollary
and Theorem
in [
42], we can draw the following results.
Lemma 2. If and are open intervals with , , and , , , are continuously differentiable functions, thenMoreover,where is defined in (10). Lemma 3. When the conditions of Lemma 2 are satisfied, then . Moreover, when , the functions and are both zeros or have the same sign whenever near . More exactly, From Lemma 3, we determine that for with sufficiently small. Thus, if , then for on one side of zero with sufficiently small.
In fact, by (
6) and Lemma 2, we know that
where
and
are defined as (
3). Together with
and
, we have
For , we present the following theorem.
Proposition 1. Let the conditions of Theorem 2 hold. For each, is fixed if , then , where Proof. Substituting
into Equation (
2) and differentiating it with respect to
twice, and then setting
. Note that
and
, it is deduced that
Multiplying (
14) by
and integrating over
by parts, we have
Then, by multiplying (
15) by
on the left, it is derived that
Thus,
as
. The proof is completed. □
Remark 2. In particular, if and , then . Thus, we have . This implies that each bifurcation curve around is of pitchfork type.
We now proceed to find the largest eigenvalue of at .
Proposition 2. Assume that the conditions of Proposition 1 are satisfied. For each, is fixed if , where is arbitrary and , then is the largest eigenvalue of at .
Proof. For each
, the eigenvalue problem of
is
Multiplying the first two equations in system (
16) by
and integrating over
, one can have
where
is an eigenvalue of
on
subject to the homogeneous Neumann boundary condition.
According to the formula of
in (9), we have
. Hence, the corresponding eigenvalues of (
16) are
and
as
. Furthermore, we can factor
as
Thus, the other root of
is
Hence, if
, then
for all
. Note that if
, we can determine that all eigenvalues
of
arising from eigenvalues
will be negative. This implies that
is the largest eigenvalue of
. The proof is complete. □
From Lemmas 2 and 3, and Propositions 1 and 2, we have the following result about the stability of the bifurcating solutions.
Proposition 3. Assume that all the conditions of Proposition 2 are valid; then, the spatially inhomogeneous patterns of system (2), which bifurcate from the positive equilibrium , are locally stable, as is ϵ on one side of zero with small enough.