Global Asymptotic Stability and Hopf Bifurcation in a Homogeneous Diffusive Predator-Prey System with Holling Type II Functional Response ()
1. Introduction
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 [1] [2] [3] [4] in biomathematics. These functional responses describe how predators transform the harvested prey into the growth of itself and were discussed by numbers of researchers [5] - [10]. 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 [11]. 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 [12] - [18].
For instance, in [12], 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. [13] 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, [18] 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, [19] 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 [18], 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:
(1a)
(1b)
(1c)
(1d)
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) [9]. To describe an environment surrounded by dispersal barriers, we take zero flux at such boundary
[12]. The symbol
is the outer flux, and no flux boundary condition is imposed, thus the system is closed [18] and we have above Neumann boundary conditions (1c). Bazykin [20] once looked at the ODEs version of above system, and it is also researched by some authors ( [9] and [21], etc.).
For simplicity and convenience in the later, we introduce a new dimensionless change of variables and parameters:
(2)
and still denote
and
as
and
,
. Thus we have following simplified dimensionless system in the form of PDEs:
(3a)
(3b)
(3c)
(3d)
where
.
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
[21].
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
[22].
2.1. Equilibria E0 and E2
Firstly, we consider global asymptotic stability of the trivial equilibrium
by using the comparison principle [22] or [23].
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
(4)
By using lemmas in [22] or [23], we have
. Thus for any sufficiently small
, there must exist
, such that
,
, we have
(5)
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
(6)
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
(7)
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
(8)
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
(9)
This implies
(10)
and
(11)
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.
(12)
where
. Hence we have a positive lower bound
(13)
Similarly, we have
. This positive lower bound enquires that
(14)
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
(15)
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:
(16)
or
(17)
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
(18)
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 [18] 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:
(19)
and the Jacobian matrix of the system (19) at
reads
(20)
Then we denote some notations
,
and
, where
and
are the trace and determinant of the matrix
, respectively. If the parameter
satisfies
(21)
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
(22)
Suppose now that
is an eigenvector of operator
corresponding to an eigenvalue
, i.e.
or equations
(23)
is a basis and
,
have following expressions [19]:
(24)
Substitute them into above Equation (23), we have following algebraic linear equations
(25)
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
(26)
where
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
(27)
where smooth functions are
Take the operator of (27) near
which determines the eigenvalues of linearized operator
:
(28)
where
then we derive its characteristic equation
or an algebraic equation
(29)
where coefficients are:
From the theorem 5, we know that any potential Hopf bifurcation occurs
when
or
(30)
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.
or
(31)
with
(32)
Hence the transversality condition is satisfied as long as
(if
exist) or
can not holds, where
Let
, we obtain potential Hopf bifurcation points
(33)
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
(34)
Suppose further that the discriminant of above quadratic function in righthand side satisfies following condition
(35)
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 [18]). 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
(36)
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 [24], 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 [24] (Chapter 5), [18] and [19], and only need to calculate
. For convenience, we denote
,
,
,
,
and
, where
(37)
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
(38)
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 [25]):
,
and
, where coefficients are
Notice that the integrals
, it is straightforward to drive relations
(39)
So far, we have
and
. From following inverse operators
(40a)
(40b)
(40c)
(40d)
where
then we have
(41)
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
(42)
Since the integrals of
are
(43)
by some calculations, we obtain
(44)
and
(45)
Finally, from the Formula (2.31) in [18] or page 47 in [24], we have
(46)
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
(47)
and (see the Formula (2.31) in [18])
(48)
From following calculations of inner product:
(49)
we have the real part
(50)
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 [18]. 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 [21], 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 [18] and [19], 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.
Acknowledgements
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.