Spatially Nonhomogeneous Periodic Solutions in a Delayed Predator-Prey Model with Diffusion Effects
This paper is concerned with a delayed predator-prey diffusion model with Neumann boundary conditions. We study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. In particular, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that spatially nonhomogeneous periodic solutions bifurcate from the positive constant steady-state solution when the system parameters are all spatially homogeneous. Meanwhile, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of partial functional differential equations (PFDEs).
Functional differential equations have merited a great deal of attention due to its theoretical and practical significance; they are often used in population dynamics, epidemiology, and other important areas of science; see [1–6]. In particular, Lu and Liu  proposed the following modified Holling-Tanner delayed predator-prey model: where and denote the densities of prey species and predator species, respectively. The first equation states that the prey grows logistically with carrying capacity and intrinsic growth rate in absence of predation. The second equation shows that predators grow logistically with intrinsic growth rate and carrying capacity proportional to the prey populations size . The parameter is the number of prey required to support one predator at equilibrium, when equals . The term of this equation is called the Leslie-Gower term. This interesting formulation for the predator dynamics has been discussed by Leslie and Gower in [8, 9]. is incorporated in the negative feedback of the predator density. is Beddington-DeAngelis functional response. It is known that the Beddington-DeAngelis form of functional response has desirable qualitative features of ratio-dependent form but takes care of their controversial behaviors at low densities . For more details on the background of this functional response, we refer to [10–12].
For convenience, a nondimensional form of system (1.1) will be useful. By defining , , , and dropping the tildes for the sake of simplicity, model (1.1) becomes the following model: where , , , , . Lu and Liu  proved the system (1.2) is permanent under some appropriate conditions and investigated the local and global stability of the equilibria.
In the earlier literature, most population models are often formulated by ordinary differential equations with or without time delays [1, 2, 13–18]. It is well known that the distribution of species is generally heterogeneous spatially, and therefore the species will migrate towards regions of lower population density to add the possibility of survival. Thus, partial differential equations with delay became the subject of a considerable interest in recent years. For a detailed theory and applications of delay equations with diffusion arising in biological and ecological problems, we refer to [19–23]. Therefore, time delays and spatial diffusion should be considered simultaneously in modeling biological interactions. Thus, the growth dynamics of two species corresponding to system (1.2) should be described by the following diffusion system with delay: where and can be interpreted as the densities of prey and predator populations at time and space , respectively; , denote the diffusion coefficients of prey and predator two species, respectively; is the Laplacian operator; Neumann boundary conditions in (1.3) imply that two species have zero flux across the domain boundary. , and defined by with the inner product .
The organization of this paper is as follows. In Section 2, we consider the stability of the positive constant steady-state solutions and the existences of Hopf bifurcations of surrounding the positive constant steady-state solutions. In particular, we show the existence of spatially nonhomogeneous periodic solutions while the system parameters are all spatially homogeneous. In Section 3, we present that the emergence of these spatially nonhomogeneous periodic solutions is clearly due to the effect of the small diffusivity. Finally, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of PFDEs.
2. Stability and Hopf Bifurcations
It is easy to see that the solutions of system (1.2) have a unique boundary equilibrium and a unique positive equilibrium , where Obviously, and are also the spatially homogeneous steady-state solutions of system (1.3). From the point of view of biology, we should consider system (1.3) in the closed first quadrant in the plane, that is, the positive constant steady-state solutions of system (1.3).
Let ; , for convenience, we use and to replace and , respectively; then system (1.3) can be transformed into where Therefore, the positive constant stationary solution of system (1.3) can be transformed into the origin of system (2.2).
Let ,,; therefore, system (2.2) can be rewritten as an abstract form in the phase space : where ,,, , and are given, respectively, by for , , .
Linearizing (2.4) at gives the linear equation whose characteristic equation is where and .
It is well known that the linear operator on with homogeneous Neumann boundary conditions has the eigenvalues , and the corresponding eigenfunctions are Notice that construct an orthogonal basis of the Banach space . Therefore , and thus any element in can be expanded a Fourier series in the form
In addition, some easy computations can show that for .
From (2.9) and (2.11), (2.7) is equivalent to Thus is a characteristic root of (2.7) if and only if for , satisfies where When , (2.13) reduces to the following quadratic equation with respect to : If , then , and since .
Therefore, it is obvious that all roots of equations (2.15) have negative real parts, and we can conclude that the positive constant steady state of system (2.2) is locally asymptotically stable in the absence of delay when . Thus, we can have the following conclusions.
Theorem 2.1. Suppose that the condition is satisfied. Then (i)all roots of each equation in (2.15) have negative real parts for any wave number ,(ii)for any wave number , the positive constant steady-state solution of system (1.3) is locally asymptotically stable in the absence of delay.
In the following, we discuss the effects of delay on the stability of the trivial solution of (2.2). Notice that is a root of (2.13) if and only if for a certain , satisfies the following equation: Thus
Proof. Assuming , is a solution of (2.13) with . From (2.17), (2.18), and (2.19), we get
so we have
Clearly, if and , there are no such that (2.13) with has purely imaginary roots .
By computing, we have
In addition, according to , we have . It is clear that − when (). Furthermore, if , we can get when . Therefore, (2.13) with has no purely imaginary roots when the conditions and hold. Thus the proof of Theorem 2.2 is accomplished.
Let be a root of (2.13) with near satisfying , ,.
Then the following result holds.
Lemma 2.3. The following transversality conditions hold:
From the previous discussions, we have the following theorem on the stability of positive steady-state solution of system (1.3) and the existence of Hopf bifurcation near .
Theorem 2.4. Assume that the conditions and hold. Then (i)if , the positive constant steady state of (1.3) is asymptotically stable;(ii)if , the positive constant steady state of (1.3) is unstable;(iii) are Hopf bifurcation values of system (1.3), and these Hopf bifurcations are all spatially homogeneous.
3. Effect of Small Diffusivity
In the previous section, we have obtained the conditions under which spatially homogeneous Hopf bifurcations bifurcate from the positive steady-state solutions of system (1.3) when the parameter crosses through the critical value . In this sense, we say that the diffusion terms do not have effect on the Hopf bifurcations. In this section, we discuss the effect of small diffusivity on Hopf bifurcations for system (1.3) when the condition is not satisfied. For the simplicity of discussion which follows, throughout this section, we always suppose that the condition : holds.
Assume is a solution of (2.13) with . From the discussion in Section 2, we have If the condition () is not satisfied, and (), then (2.13) with has roots , where From the discussion in Section 2, we know that there exists , such that (2.13) with has only characteristic roots with negative real parts when .
In addition, from (2.17), we have Thus
Suppose the condition is not satisfied, that is, , assume further satisfy. Then (2.13) with has a pair of purely imaginary eigenvalues , and all other zeros have negative real parts, where is given by Therefore, we can obtain the following.
Lemma 3.1. Suppose that and . If the condition holds, then (2.13) with has a simple pair of purely imaginary roots and all other roots except have strictly negative real parts, where is defined by (3.6).
For system (1.3), by the similar discussion to that of Theorem 2.2, when crosses through the critical values , where it can give rise to Hopf bifurcation at the positive constant steady state . By the results in , bifurcating periodic solutions of (1.3) at are spatially nonhomogeneous.
Therefore, we have the following conclusion.
In general, we have the following.
Theorem 3.3. Suppose that , if there exist , such that and holds, then (2.13) with has purely imaginary roots and system (1.3) has a family of spatially nonhomogeneous periodic solutions bifurcating from the spatially homogeneous steady state , when crosses through the critical values , where and are defined by (3.2) and (3.4) with , , respectively.
From Theorems 2.2 and 3.3, we can know that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that the system bifurcates spatially nonhomogeneous periodic solutions at the positive constant steady state under which the system parameters are all spatially homogeneous. These exhibit that the emergence of these spatially nonhomogeneous periodic solutions is clearly due to the effect of the small diffusivity.
4. Properties of Hopf Bifurcation
In Theorem 3.2, we have obtained the conditions under which a family of spatially nonhomogeneous periodic solutions bifurcates from the spatially homogeneous steady-state solutions of system (1.3) when the parameter crosses through the critical value . In this section, we redefine an inner product to study the properties of the spatially nonhomogeneous Hopf bifurcation applying normal form theory of PFDEs by developed [22, 25].
Normalizing the delay in system (2.2) by the time-scaling , (2.2) is transformed into where , are defined by (2.2). Letting , then, (4.1) can be written in abstract form in as where , , are given by for .
Linearizing (4.2) at leads to the following linear equation:
Let ; consider the following FDE on : that is,
Obviously, is a continuous linear function mapping into . According to the Riesz representation theorem, there exists a matrix function . , whose elements are of bounded variation such that Thus, we can choose then (4.7) is satisfied.
Letting denote the infinitesimal generator of strongly continuous semigroup, according to , then, where .
For , define and a bilinear inner product of the Sobolev space : where and are the formal adjoint of .
It is easy to see from Section 2 that has a pair of simple purely imaginary eigenvalues and they are also eigenvalues of since and are adjoint operators. Let and be the center spaces, that is, the generalized eigenspaces, of and associated with , respectively. Then is the adjoint space of and .
Let and , where
From the above expression, we can easily see that , .
Let , be defined by for and for . Then the center space of linear equation (4.4) is given by , where and ; here denotes the complementary subspace of in .
Let be defined by where : is given by
Then is the infinitesimal generator induced by the solution of (4.4) and (4.2) and can be rewritten as the following operator differential equation: Using the decomposition and (4.13), the solution of (4.16) can be written as where , and with .
Thus, we describe the flow on the center manifold for (4.2) as where .
Letting and , when , then satisfies where
Noting that , therefore, Since and for () appear in , we still need them.
Thus, by using the chain rule
From (4.23) and (4.30), we can obtain Noticing that has only two eigenvalues , therefore, (4.33) has the unique solution in and Note that for , So, for , By the definition of , for , we have Note that ; hence