This paper is concerned with a diffusive predator-prey system with Beddington-DeAngelis
functional response and delay effect. By analyzing the distribution of the eigenvalues, the stability
of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous
periodic solutions are investigated. Also, it is shown that the small diffusion can affect the Hopf
bifurcations. Finally, the direction and stability of Hopf bifurcations are determined by normal form
theory and center manifold reduction for partial functional differential equations.
In this paper, we will study the stability and Hopf bifurcations of a diffusive predator-prey system with Beddington-DeAngelis functional response and delay effect as follows:
where and denote the population densities of prey and predator species at time and space , respectively; the positive constants and represent the diffusion coefficients of prey and predator species, respectively; ( is called the capturing rate) and ( is called the conversion rate) represent the strength of the relative effect of the interaction on the two species; denotes the death rate of predator species; is the Beddington-DeAngelis functional response function with and are positive numbers; denotes the generation time of the prey species; is a bounded domain in ( is any positive integer) with a smooth boundary ; is the Laplacian operator on ; is the outward normal to ; homogeneous Neumann boundary conditions reflect the situation where the population cannot move across the boundary of the domain.
System (1.1) includes the models which have been discussed by many researchers; for examples, when , the models were considered in [1, 2]; if and , it was discussed in ; if , it was discussed in . Moreover, when and , system (1.1) can be transformed into Narcisa Apreutesei’s model (see ).
There has been an increasing interest in the study of diffusive predator-prey system (see [1, 2, 4, 6–14] and references therein) with functional response. As is known to all, the Beddington-DeAngelis functional response, proposed by Beddington  and DeAngelis et al. , is more general than those the above authors considered, and it has been studied extensively in the literature [1–3, 7, 14–16]. However, to the authors’ best knowledge, few researches have been done on the diffusive predator-prey system with Beddington-DeAngelis functional response and time delay.
The aim of this paper is to extend and develop the work in [1, 2]; that is, we will study the stability and Hopf bifurcation of a diffusive predator-prey system with Beddington-DeAngelis functional response and delay. The system we consider here is more general than the system in [1, 2].
The rest of the paper is organized as follows. In Section 2, we analyze the distribution of the roots of the characteristic equation and give various conditions on the stability of a positive constant steady state and the existence of Hopf bifurcation. In Section 3, we discuss the effect of diffusion on the Hopf bifurcation. In Section 4, by applying the normal form theory and the center manifold reduction of partial functional differential equations by Wu , an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is given.
2. Analysis of the Characteristic Equations
In this section, by choosing the delay as the bifurcation parameter and analyzing the associated characteristic equation of (1.1) at the positive constant steady state, we investigate the stability of the positive constant steady state of (1.1) and obtain the conditions under which (1.1) undergoes Hopf bifurcation.
It can be seen that homogeneous Neumann boundary conditions imposed on (1.1) lead to and , always being two boundary equilibria for any feasible parameters, and (1.1) always having a unique positive constant steady state provided that the condition(A1), hold, where
Under (A1), let and drop the bars for simplicity of notations, then (1.1) can be transformed into the following equivalent system:
Let By and (2.2) becomes
where , , , , and
where denoted by and . for shorthand of “higher order terms.”
Denote , and Then (2.3) can be transformed into an abstract differential equation in the phase space ,
The linearization of (2.5) is given by
and its characteristic equation is
where and , .
From the properties of the Laplacian operator defined on the bounded domain, the operator on has the eigenvalues with the relative eigenfunctions
where . Clearly, construct a basis of the phase space and therefore any element in can be expanded as Fourier series in the following form:
Some simple computations show that
From (2.10)-(2.11), (2.8) is equivalent to
Assume that(A2) .
Let , , , , , , then we conclude that the characteristic equation (2.8) is equivalent to the sequence of the characteristic equations:
Obviously, for , is not a root of (2.13).
Equation (2.13) with is equivalent to the following quadratic equations:
Let and be the two roots of (2.14). Then, for ,
Therefore, we have the following Lemma.
Lemma 2.1. Assume that (A1) and (A2) hold. Then the equilibrium of (1.1) with is asymptotically stable.
Assume that (A3);
(A4), or , if .
Theorem 2.2. If (A1)–(A4) hold, then all roots of (2.13) have negative real parts for all . Furthermore, the equilibrium of the system (1.1) is asymptotically stable for all .
Proof. Let be a root of the characteristic equation (2.13). Then satisfies the following equation for some :
Separating the real and imaginary parts of (2.16) leads to
which implies that
Let , then (2.18) can be rewritten into the following form:
By (A3) and (A4), for all , we have
which imply that (2.19) has no positive roots. Hence, the characteristic equation (2.13) has no purely imaginary roots. By Lemma 2.1 and the theorem proved by Ruan and Wei , all roots of (2.13) have negative real parts.
Notice that (2.13) with is the characteristic equation of the linearization of (1.1) corresponding system without diffusion (ordinary differential equations, ODEs) at the positive equilibrium. And it has been considered under the condition:(B1) .
It is easy to get that when
Equation (2.13) with has simple imaginary roots , and , where is the root of (2.13) with satisfying , and
and . Assume that(B2) .
We have the following result.
Theorem 2.3. Assume that (A1), (A2), (B1), and (B2) are satisfied. Then for , (2.13) has a pair of simple imaginary roots , and all roots of (2.13), except , have no zero real parts. Moreover, all the roots of (2.13) with , except , have negative real parts.
Proof. Let be a root of (2.13) with . By the same way in Theorem 2.2, we can obtain
for all . Set , then
Let and be the roots of (2.24) with . We know that if and then (2.13) with has no purely imaginary roots. By (B2), it follows that, for ,
Therefore, (2.13) with have no purely imaginary roots. Summarizing the above results and combining Theorem 2.3, we have the following theorem on the stability of the positive equilibrium of system (1.1) and the existence of Hopf bifurcation at .
Theorem 2.4. Assume that (A1), (A2), (B1), and (B2) hold. For system (1.1), the following statements are true:(I)If , then the equilibrium point is asymptotically stable; (II)If , then the equilibrium point is unstable; (III) are Hopf bifurcation values of system (1.1), and these Hopf bifurcations are all spatially homogeneous.
By the same way in Theorem 2.2, let be a root of the characteristic equation (2.13), then satisfies the following equation:
Now, we make the following assumptions. For a certain ,(C1) ;
(C3) , .
Under the assumptions (C1) and (C2), (2.26) with has only a positive solution ,
Set , then (2.26) can be transformed into the following equation:
Let and be the roots of (2.28). If the assumptions (C1)–(C3) hold, we have
Therefore, (2.13) with has no solutions with zero real parts.
In addition, similar to the proof of Theorem 2.2, we have
which implies that
From the above analysis, we have the following Theorem.
Theorem 2.5. Assume that (A1), (A2), and (C1)–(C3) hold. Then for , (2.13) has a pair of simple imaginary roots , and all roots of (2.13), except , have no zero real parts. Moreover, all the roots of (2.13) with , except , have negative real parts.
Let be the root of (2.13) near satisfying
where and are given by (2.27) and (2.32), respectively. Then we have the following transversality condition.
Lemma 2.6. Assume that (A1), (A2), and (C1)–(C3) hold. Then
Proof. Differentiating the two sides of (2.13) with respect to yields
From (2.30), we have
according to (2.27), and then
Applying Lemma 2.6 and Theorem 2.5, we draw the following conclusions.
Theorem 2.7. Assume that (A1), (A2), and (C1)–(C3) hold. For system (1.1), the following statements are true: (I)If , then the equilibrium point is asymptotically stable; (II)If , then the equilibrium point is unstable; (III) are Hopf bifurcation values of system (1.1), and these Hopf bifurcations are all spatially inhomogeneous.
3. The Effect of Diffusion on Hopf Bifurcations
In the previous section, we have studied the Hopf bifurcations from the positive constant steady-state of (1.1) when crosses through the critical value and have the following conclusions.(I) If (B2) holds, then system (1.1) and the corresponding system without diffusion (ODEs) have the same Hopf bifurcations, containing the existence and properties of Hopf bifurcations. In this case, the diffusion has no effect on the Hopf bifurcations of ODEs. (II)If (B2) does not hold, then system (1.1) and ODEs have the different Hopf bifurcations. In this case, the diffusion has the effect on the Hopf bifurcations of ODEs.
According to Theorems 2.4 and 2.7, system (1.1) undergoes Hopf bifurcations under the different conditions. Comparing the conditions of Theorems 2.4 and 2.7, we have the following conclusions.(I)When system (1.1) undergoes spatially homogeneous Hopf bifurcation, diffusion coefficients satisfy the condition:
and in this case, system (1.1) and ODEs have the same properties of Hopf bifurcation. (II)When system (1.1) undergoes spatially inhomogeneous Hopf bifurcation, diffusion coefficients satisfy the condition
and in this case, system (1.1) and ODEs have the different properties of Hopf bifurcation.
Summarizing the above results, we can obtain the conclusion. The big diffusion has no effect on the Hopf bifurcation of system (1.1), the small diffusion can make system (1.1) undergo the spatially inhomogeneous Hopf bifurcation.
4. Direction of Hopf Bifurcation and Stability of the Bifurcating Periodic Orbits
In this section, we will study the directions, stability, and the period of bifurcating periodic solutions by using normal formal theory and center manifold theorem of partial functional differential equations presented in . For fixed , we denote by . Let
and drop the tilde for the sake of simplicity. Then system (1.1) can be written as
where , and are given, respectively, by
for , where . denotes high order terms.
Consider the linear equation
From the discussion of Theorems 2.3 and 2.5 in Section 2, we know that the origin (0,0) is an equilibrium of (4.2), and for , the characteristic equation of (4.5) has a pair of simple purely imaginary eigenvalues .
Consider the ordinary functional differential equation
By the Riesz representation theorem, there exists a matrix function , whose entry is of bounded variation such that
for . In fact, we can choose
Let denote the infinitesimal generators of the semigroup induced by the solutions of (4.6) and be the formal adjoint of under the bilinear pairing
for . Then and are a pair of adjoint operators. From the discussion in Section 2, we know that has a pair of simple purely imaginary eigenvalues , and they are also eigenvalues of . Let and be the center subspaces, that is, the generalized eigenspace of and associated with , respectively. Then is the adjoint space of and , see .
Direct computations give the following results.
Lemma 4.1. Let
is a basis of with , and
is a basis of with .
Let and with
for , and
then construct a new basis for and , see . In addition, , where
Let be defined by .
Then the center subspace of linear equation (4.5) is given by , where
and , and denotes the complement subspace of in .
Let be the infinitesimal generator induced by the solution of (4.5). Then (4.2) can be rewritten as the abstract form
Using the decomposition and (4.17), the solution of (4.2) can be written as
where , and . In particular, the solution of (4.2) on the center manifold is given by
Let and , and notice that , then
Equation (4.21) can be transformed into
where . Moreover, by , satisfies
From (4.4) and (4.23), it follows that
Notice that , for .
Let . Then we can obtain the following quantities:
Since and for appear in , we need to compute them. It follows from (4.26) that
In addition, by , satisfies
Thus, from (4.24), (4.27)–(4.33), we can obtain that
Noticing that has only two eigenvalues ; therefore, (4.34) has a unique solution in given by
From (4.33), we know that for ,