1. Introduction

Population models with stage structure have been investigated by many researchers, and various methods and techniques have been used to study the existence and qualitative properties of solutions [1–9]. However, most of the discussions in these works are devoted to either systems of ODE or weakly coupled systems of reaction-diffusion equations. In this paper we investigate the global existence and convergence of solutions for a strongly coupled cross-diffusion predator-prey model with stage structure and nonlinear density restriction. Nonlinear problems of this kind are quite difficult to deal with since the usual idea to apply maximum principle arguments to get priori estimates cannot be used here [10].

Consider the following predator-prey model with stage-structure:

where , denote the density of the immature and mature population of the prey, respectively, is the density of the predator. For the prey, the immature population is nonlinear density restriction. is assumed to consume with Holling type III functional response and contributes to its growth with rate . For more details on the backgrounds of this model see references [11, 12].

Using the scaling and redenoting by , we can reduce the system (1.1) to

where

To take into account the natural tendency of each species to diffuse, we are led to the following PDE system of reaction-diffusion type:

where is a bounded domain in with smooth boundary , is the outward unit normal vector on , and . are nonnegative smooth functions on . The diffusion coefficients are positive constants. The homogeneous Neumann boundary condition indicates that system (1.3) is self-contained with zero population flux across the boundary. The knowledge for system (1.3) is limited (see [13–17]).

In the recent years there has been considerable interest to investigate the global behavior for models of interacting populations with linear density restriction by taking into account the effect of self-as well as cross-diffusion [18–26]. In this paper we are led to the following cross-diffusion system:

where are the diffusion rates of the three species, respectively. are referred as self-diffusion pressures, and is cross-diffusion pressure. The term self-diffusion implies the movement of individuals from a higher to a lower concentration region. Cross-diffusion expresses the population fluxes of one species due to the presence of the other species. The value of the cross-diffusion coefficient may be positive, negative, or zero. The term positive cross-diffusion coefficient denotes the movement of the species in the direction of lower concentration of another species and negative cross-diffusion coefficient denotes that one species tends to diffuse in the direction of higher concentration of another species [27]. For , problem (1.4) becomes strongly coupled with a full diffusion matrix. As far as the authors are aware, very few results are known for cross-diffusion systems with stage-structure.

The main purpose of this paper is to study the asymptotic behavior of the solutions for the reaction-diffusion system (1.3), the global existence, and the convergence of solutions for the cross-diffusion system (1.4). The paper will be organized as follows. In Section 2 a linear stability analysis of equilibrium points for the ODE system (1.2) is given. In Section 3 the uniform bound of the solution and stability of the equilibrium points to the weakly coupled system (1.3) are proved. Section 4 deals with the existence and the convergence of global solutions for the strongly coupled system (1.4).

2. Global Stability for System (1.2)

Let . If , then (1.2) has semitrivial equilibria , where . To discuss the existence of the positive equilibrium point of (1.2), we give the following assumptions:

where . Let one curve : , and the other curve : . Obviously, passes the point . Noting that attains its maximum at , thus when , . has the asymptote and passes the point . In this case, and have unique intersection , as shown in Figure 1. is the unique positive equilibrium point of (1.2), where , , . In addition, the restriction of the existence of the positive equilibrium can be removed, if .

Figure 1

The Jacobian matrix of the equilibrium is

The characteristic equation of () is . is a saddle for . In addition, the dimensions of the local unstable and stable manifold of are 1 and 2, respectively. is locally asymptotically stable for .

The Jacobian matrix of the equilibrium is

where . The characteristic equation of () is , where

According to Routh-Hurwitz criterion, is locally asymptotically stable for and , that is, and .

The Jacobian matrix of the equilibrium is

where

The characteristic equation of () is , where

According to Routh-Hurwitz criterion, is locally asymptotically stable for . Obviously, can be checked by (2.1).

Now we discuss the global stability of equilibrium points for (1.2).

Theorem 2.1. (i) Assume that (2.1), hold, then the equilibrium point of (1.2) is globally asymptotically stable.
(ii) Assume that , and hold, then the equilibrium point of (1.2) is globally asymptotically stable.
(iii) Assume that holds, then the equilibrium point of (1.2) is globally asymptotically stable.

Proof. (i) Define the Lyapunov function Calculating the derivative of along the positive solution of (1.2), we have When , the minimum of and is and 0, respectively; the maximum of is are and , respectively. Thus, when (2.8) hold, According to the Lyapunov-LaSalle invariance principle [28], is globally asymptotically stable if (2.1)–(2.3) hold.
(ii) Let Then Noting that the maximum of is , and , we find . Therefore,
(iii) Let then Thus, for . This completes the proof of Theorem 2.1.

3. Global Behavior of System (1.3)

In this section we discuss the existence, uniform boundedness of global solutions, and the stability of constant equilibrium solutions for the weakly coupled reaction-diffusion system (1.3). In particular, the unstability results in Section 2 also hold for system (1.3) because solutions of (1.2) are also solutions of (1.3).

Theorem 3.1. Let be nonnegative smooth functions on . Then system (1.3) has a unique nonnegative solution , and on . In particular, if , then for all .

Proof. It is easily seen that is sufficiently smooth in and possesses a mixed quasimonotone property in . In addition, and are a pair of lower-upper solutions of problem (1.3) (cf. in (3.1)). From [29, Theorem  5.3.4], we conclude that (1.3) exists a unique classical solution satisfying (3.1). According to strong maximum principle, it follows that . So the proof of the Theorem is completed.

Remark 3.2. When (namely ), system (1.3) reduces to a system in which the immature population of the prey is linear density restriction. Similar to the proof of Theorem 3.1, we have

Now we show the local and global stability of constant equilibrium solutions for (1.3), respectively.

Theorem 3.3. (i) Assume that (2.1) holds, then the equilibrium point of (1.3) is locally asymptotically stable.
(ii) Assume that , , and hold, then the equilibrium point of (1.3) is locally asymptotically stable.
(iii) Assume that holds, then the equilibrium point of (1.3) is locally asymptotically stable.

Proof. Let be the eigenvalues of the operator on with Neumann boundary condition, and let be the eigenspace corresponding to in . Let where is an orthonormal basis of , then
Let , , where
The linearization of (1.3) is at . For each , is invariant under the operator L, and is an eigenvalue of L on , if and only if is an eigenvalue of the matrix . The characteristic equation is , where From Routh-Hurwitz criterion, we can see that three eigenvalues (denoted by , , ) all have negative real parts if and only if . Noting that , we must have . It is easy to check that if (see Section 2).
We can conclude that there exists a positive constant , such that In fact, let , then Since as , it follows that Clearly, has the three roots . Let . By continuity, there exists such that the three roots of satisfy Let , then . Let , then (3.7) holds. According to [30, Theorem  5.1.1], we have the locally asymptotically stability of .
(ii) The linearization of (1.4) is at , where , and