Since C.S. Holling proposed several kinds of functional responses (Holling functional responses) to model the phenomena of predation in 1965; the classical Lotka-Volterra predator-prey system was extended and more realistic     in biomathematics. These functional responses describe how predators transform the harvested prey into the growth of itself and were discussed by numbers of researchers  - . When the spatial distributions of the two populations are also of interest, the passive dispersal of the populations can be modeled and simulated by diffusive operators . Complicated diffusive predator-prey systems in the form of partial differential equations (PDEs) with Holling type II functional response have been constructed and analyzed in several previous literatures  - .
For instance, in , a continuous diffusive predator-prey model incorporating Holling type II functional response of the predator and a logistic growth of the prey was shown to exhibit temporal chaos at a fixed point in space. Numerical results demonstrated that low diffusion values drive a periodic system into aperiodic behavior with sensitivity to initial conditions.  considered the case where densities of predator and prey are both spatially inhomogeneous in a bounded domain subject to homogeneous Neumann boundary condition, and they also studied qualitative properties of solutions to this reaction-diffusion system. They showed that even though positive constant steady state is globally asymptotically stable for the ordinary differential equation (ODE) dynamics, non-constant positive steady states can coexist in a PDE system.
With regard to Hopf bifurcation analysis,  carried out Hopf and steady state bifurcation, and the existence of multiple spatially non-homogeneous periodic orbits are showed in particular, while the system parameters are all spatially homogeneous. The global bifurcation theory also suggested the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions. Based on this reference,  considered the possibility of the occurrence of Turing patterns and performed detailed Hopf bifurcation analysis in a diffusive predator-prey system with Holling type III functional response. They showed that the system has multiple oscillatory patterns.
Motivated by the reference , in this paper, we mainly consider a homogeneous reaction-diffusion predator-prey system with Holling type II functional response with density-dependent predator specific death rate and predator mutual interference:
Here functions and are prey and predator densities, respectively. The one dimensional spatial domain is . All above positive constants have practically biological considerations. Parameter is the intrinsic growth rate of prey; represents the carrying capacity of environment; a is the half-saturation constant; is the search efficiency of predator for prey; and are the mortality rate of prey and predator species, respectively; e is the biomass conversion; d is the intra-specific competition coefficient of predator; and are two positive diffusive rates of prey and
predator, respectively. The specific growth term governs the increase of prey in the lack of predator. The coupled term , named Holling
type II functional response, describes the functional response of predator, which also refers to the change in the density of prey attached per unit time per predator as prey density changes. The square term denotes intrinsic decrease and mutual interference of predator. In the absence of diffusion, its corresponding ODEs system is familiar to the Lotka-Volterra system in which populations have the addition of damping terms(or self inhabit) . To describe an environment surrounded by dispersal barriers, we take zero flux at such boundary . The symbol is the outer flux, and no flux boundary condition is imposed, thus the system is closed  and we have above Neumann boundary conditions (1c). Bazykin  once looked at the ODEs version of above system, and it is also researched by some authors (  and , etc.).
For simplicity and convenience in the later, we introduce a new dimensionless change of variables and parameters:
and still denote and as and , . Thus we have following simplified dimensionless system in the form of PDEs:
Our main contribution in this paper is detailed global asymptotic stability proof and Hopf bifurcation analysis of the system (3). The rest of this paper is organized as follows. In Section 2, we will analyze global stability of trivial equilibria , and interior equilibrium by using the comparison principle. In Section 3, we firstly give standard stability analysis to show the nonexistence of Turing patterns of this system, then we conduct the Hopf bifurcation analysis to show the existence of oscillatory patterns. The directions of Hopf bifurcation are also performed analytically. Finally, a short summary and some remarks are in Section 4.
2. Global Asymptotic Stability
In this section, we devote to give priori foundations for our system. Firstly, we discuss non-negative equilibria of the system (3) with their sufficient existence conditions. It is obvious to see that this system has following trivial equilibria:
and , where is defined as . For practical considerations, we omit a singular point . The point is
a desired equilibrium only if . Then we make a special effort to derive the existence conditions of an interior equilibrium which is denoted by or . If , an interior equilibrium exists. Meanwhile, we
have and , where .
Then we will give analysis of global asymptotic stability at the equilibria , and . These conclusions can also be extended to a generalized bounded domain with smooth boundary. Here and positive solutions , where .
2.1. Equilibria E0 and E2
Firstly, we consider global asymptotic stability of the trivial equilibrium by using the comparison principle  or .
Theorem 1 (Global asymptotic stability at E0) If , then the equilibrium is globally asymptotically stable.
Proof. With respect to the Equation (1a), it is obvious to get an inequality
By using lemmas in  or , we have . Thus for any sufficiently small , there must exist , such that , , we have
This implies . Thus we complete the proof.
With the same technique at hand, we have following theorem about the axis equilibrium .
Theorem 2 (Global asymptotic stability at E2) If and
, then the equilibrium is globally asymptotically stable.
Proof. For the Equation (3a), from the inequality (4) we have
. That is to say, for any sufficiently small , we have
. Thus from the Equation (3b), a similar inequality
is derived. By using the lemmas illustrated above, we have , i.e.
an inequality holds. Substitute it into the Equation (3a), we have a new inequality
This implies and positive solutions u converge uniformly
to in . Thus we complete the proof.
2.2. Equilibrium E*
Here we consider the global stability of the equilibrium . Firstly, we define two functions
and a discriminant
Notice that is a monotonic increasing function but is a monotonic decreasing function.
From the existence conditions of point and inequality (10), for sufficiently small , we have , where . Substitute it into the Equation (3b), we obtain an inequality
For any sufficiently small , suppose now that the numerator in the right hand side of the inequality (11) has two different real roots and , i.e.
where . Hence we have a positive lower bound
Similarly, we have . This positive lower bound enquires that
With the same procedure at hand, we have new bounds , .
Without loss of generality, for these positive lower and upper bound sequences, we have following iteration relations
and comparison relations
It is obvious to see that four limitations , , and
are all exist with the help of mathematical analysis. But we denote them as , , and , respectively. Notice that they satisfy following equations:
thus is equivalent to from above last two equations. If holds or holds, we know that and due to the existence condition of point .
Case 1. If condition holds, substitute the equation into the equation , then we have
Similarly, we have
Let the two equations to subtract each other, we derive a contradiction! Thus we have a theorem.
Theorem 3 (Global asymptotic stability at E*) Suppose , if conditions (12), (14) and hold, then is globally asymptotically stable.
Case 2. Substitute the equation into the equation , then we have
Similarly, we have
Let the above two equations to divide each other, we derive , where is a quadratic function. If there exist an index , such that
for instance, , i.e. or
then the equilibrium is also globally asymptotically stable.
Theorem 4 (Global asymptotic stability at E*) Suppose , if conditions (12) and (18) hold, then is globally asymptotically stable.
3. Hopf Bifurcation
In this section, we concentrate on the Hopf bifurcation analysis. Firstly, we define a real-valued Sobolev space
and the complexification of X  as
Denote the complex-valued inner product on space as , where column vectors , and is the Hermitian conjugate (or adjoint) vector of U, thus we notice that the space equipped with inner product is a Hilbert space. It is easy to verify the “linear” relation .
3.1. Nonexistence of Turing Instability
In this section, we consider the nonexistence of Turing instability of above positive constant steady state . Firstly, we recall the corresponding ODEs system of (3) again:
and the Jacobian matrix of the system (19) at reads
Then we denote some notations , and , where and are the trace and determinant of the matrix , respectively. If the parameter satisfies
conditions and hold, we know that is asymptotically stable in the ODEs system (19).
The linearized operator of the system (3) at steady state is
Suppose now that is an eigenvector of operator corresponding to an eigenvalue , i.e. or equations
is a basis and , have following expressions :
Substitute them into above Equation (23), we have following algebraic linear equations
Set or above linear equations have nonzero solutions for index
, then the determinant of the equation must be zero, i.e. we have a characteristic equation
With the condition (21) at hand, we know , i.e. , then the steady state is asymptotically stable in PDEs system (3). Turing instability will not occur. See following theorem to summarize above analysis.
Theorem 5 (Turing instability) Suppose that conditions and (21) hold, then is asymptotically stable in the PDEs system (3) and Turing instability phenomenon will not occur.
3.2. Existence of Hopf Bifurcation and Spatial Periodic Patterns
In this section, with the help of standard Hopf bifurcation theory, we will prove the existence of spatially homogeneous and non-homogeneous periodic patterns of the system (3). Here we choose as a bifurcation parameter (or equivalently d as a bifurcation parameter). Firstly, take the linear transformation for later use: , , and still denote , as u, v, we have a new system
where smooth functions are
Take the operator of (27) near which determines the eigenvalues of linearized operator :
then we derive its characteristic equation or an algebraic equation
where coefficients are:
From the theorem 5, we know that any potential Hopf bifurcation occurs
We shall identify possible Hopf bifurcation values under this condition accompanied by .
Suppose now that the bifurcation parameter satisfies the condition (30), and are a pair of conjugate eigenvalues of , i.e.
Hence the transversality condition is satisfied as long as (if exist) or can not holds, where
Let , we obtain potential Hopf bifurcation points
with a discriminant
Note that the Hopf bifurcation at occurs without any restriction on l and is always non-positive. Now we only need to verify that , for instance, holds forever.
Recall the condition (30) again, we have , ; , ; , and , . If there is a positive lower bound (or a local positive minimum) such that , for instance, , then
Suppose further that the discriminant of above quadratic function in righthand side satisfies following condition
then since the quadratic function is always positive. Summarizing discussions above, we obtain following theorem.
Theorem 6 (Hopf bifurcation) Suppose that and points exist with , the parameters satisfy and (35), then the system (3) undergoes a Hopf bifurcation at , and the bifurcating periodic solutions can be parameterized (see the Formula (2.32) in ). Furthermore, we have:
(1) The bifurcation periodic solutions from are spatially homogeneous, which coincides with the periodic solutions of the corresponding ODEs system;
(2) The bifurcation periodic solutions from are spatially non-homogeneous.
Example From the existence condition of point and the condition (30), we have
then the positive lower bound is
Here we let , , , , , and , from some complicated calculations, the Hopf bifurcation values are and .
3.3. Direction of Hopf Bifurcation
Under the given conditions in above subsection, by the center manifold theorem and the normal form theory , the system (3) has a series of periodic solutions. In this subsection, we consider the direction and stability of spatially non-homogeneous periodic solutions at corresponding to and , respectively. Here we obey the framework in references  (Chapter 5),  and , and only need to calculate . For convenience, we denote , , , , and , where
To summarize above discussions, we firstly give following Hopf bifurcation theorem for our diffusive predator-prey system.
Theorem 7 (Direction of Hopf bifurcation) For the diffusive system (3), suppose that the theorem 6 holds, then Hopf bifurcation at point is supercritical (subcritical) if following number
Moreover, we have:
(1) The bifurcating (spatially homogeneous) periodic solutions are stable (unstable) at if ;
(2) The bifurcating periodic solutions are all unstable at .
3.3.1. The General Case:
For the operators , we take an eigenvector and a “conjugate” vector , such that and , where and
For later use, from functions and , we obtain partial derivatives evaluated at as follows:
and vectors in the form of symmetric B, C functions (see ):
, and , where coefficients are
Notice that the integrals , it is straightforward to drive relations
So far, we have and . From following inverse operators
then we have
These calculations of inverse operators in and are restricted to the subspaces spanned by the eigen-modes 1 and . Precisely, we have
and (41) yield
Since the integrals of are
by some calculations, we obtain
Finally, from the Formula (2.31) in  or page 47 in , we have
where coefficients , , , are
and decompositions of real part and imaginary part are: , , , , and , where
3.3.2. The Special Case:
In this subsection, we consider the special case: . Similarly, we take two
vectors and , where
Suppose that , and , where
it is straightforward to see and , thus we have
and (see the Formula (2.31) in )
From following calculations of inner product:
we have the real part
where some coefficients are
and coefficients unlisted here are zero.
4. Summary and Remarks
In summary, with the framework of homogeneous reaction-diffusion systems, we have considered global asymptotic stability and Hopf bifurcation in a homogeneous diffusive predator-prey system with Holling type II functional response subject to Neumann boundary conditions, which is also an extended version of the predator-prey system in . Some sufficient results were obtained to ensure that the equilibria of this system were globally asymptotically stable and Hopf bifurcation could occur. In the Example is negative while is positive due to the non-existence of , . That is to say, the bifurcation directions are subcritical at ; the bifurcating periodic solutions are stable at .
In Section 2, we induced global asymptotic stability theorems but neglected critical cases due to the used lemmas, which need to be considered further. In Subsections 3.1 and 3.2, combing the phenomenon that Turing instability will not occur, more sufficient conditions could be used to ensure asymptotic stability and existence of Hopf bifurcation, such as the conditions of Theorem 7 in reference , but the condition (36) is well-done for the Hopf bifurcation analysis. In Subsection 3.3, similar to the references listed above, we derived a useful algorithm for determining direction of Hopf bifurcation and stability of bifurcating periodic solutions. Furthermore, in  and , interior equilibria are all analytically and easily solvable, but the interior equilibrium in our system can not be solved easily. The methods in this paper are forward guidance for other complicated reaction-diffusion consumer-resource (predator-prey) systems, even some general reaction-diffusion systems in other fields. Finally, in some extent, it is our expectancy that these conclusions can provide theoretical support for more complex problems in biomathematics.
This work was supported by the National Natural Science Foundation of China (Grant No.31570364) and the National Key Research and Development Program of China (Grant No.2018YFE0103700). We thank the Editor and the referee for their works.
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