Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-I: The Case in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq1_HTML.gif

  • Rui Zhang1,

    Affiliated with

    • Ling Guo2 and

      Affiliated with

      • Shengmao Fu2Email author

        Affiliated with

        Boundary Value Problems20092009:378763

        DOI: 10.1155/2009/378763

        Received: 2 April 2009

        Accepted: 31 August 2009

        Published: 27 September 2009

        Abstract

        This paper deals with a Holling type III diffusive predator-prey model with stage structure and nonlinear density restriction in the space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq2_HTML.gif . We first consider the asymptotical stability of equilibrium points for the model of ODE type. Then, the existence and uniform boundedness of global solutions and stability of the equilibrium points for the model of weakly coupled reaction-diffusion type are discussed. Finally, the global existence and the convergence of solutions for the model of cross-diffusion type are investigated when the space dimension is less than 6.

        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 [19]. 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:

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq4_HTML.gif denote the density of the immature and mature population of the prey, respectively, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq5_HTML.gif is the density of the predator. For the prey, the immature population is nonlinear density restriction. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq6_HTML.gif is assumed to consume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq7_HTML.gif with Holling type III functional response http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq8_HTML.gif and contributes to its growth with rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq9_HTML.gif . For more details on the backgrounds of this model see references [11, 12].

        Using the scaling http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq10_HTML.gif and redenoting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq11_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq12_HTML.gif , we can reduce the system (1.1) to

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ2_HTML.gif
        (1.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq13_HTML.gif

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

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ3_HTML.gif
        (1.3)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq14_HTML.gif is a bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq15_HTML.gif with smooth boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq16_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq17_HTML.gif is the outward unit normal vector on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq18_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq19_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq20_HTML.gif are nonnegative smooth functions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq21_HTML.gif . The diffusion coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq22_HTML.gif 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 [1317]).

        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 [1826]. In this paper we are led to the following cross-diffusion system:

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ4_HTML.gif
        (1.4)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq23_HTML.gif are the diffusion rates of the three species, respectively. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq24_HTML.gif are referred as self-diffusion pressures, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq25_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq26_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq27_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq28_HTML.gif , then (1.2) has semitrivial equilibria http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq29_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq30_HTML.gif . To discuss the existence of the positive equilibrium point of (1.2), we give the following assumptions:

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ5_HTML.gif
        (2.1)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq31_HTML.gif . Let one curve http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq32_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq33_HTML.gif , and the other curve http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq34_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq35_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq36_HTML.gif passes the point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq37_HTML.gif . Noting that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq38_HTML.gif attains its maximum at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq39_HTML.gif , thus when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq41_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq42_HTML.gif has the asymptote http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq43_HTML.gif and passes the point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq44_HTML.gif . In this case, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq45_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq46_HTML.gif have unique intersection http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq47_HTML.gif , as shown in Figure 1. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq48_HTML.gif is the unique positive equilibrium point of (1.2), where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq49_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq51_HTML.gif . In addition, the restriction of the existence of the positive equilibrium can be removed, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq52_HTML.gif .
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Fig1_HTML.jpg

        Figure 1

        The Jacobian matrix of the equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq53_HTML.gif is

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ6_HTML.gif
        (2.2)

        The characteristic equation of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq54_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq55_HTML.gif ) is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq56_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq57_HTML.gif is a saddle for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq58_HTML.gif . In addition, the dimensions of the local unstable and stable manifold of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq59_HTML.gif are 1 and 2, respectively. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq60_HTML.gif is locally asymptotically stable for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq61_HTML.gif .

        The Jacobian matrix of the equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq62_HTML.gif is

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ7_HTML.gif
        (2.3)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq63_HTML.gif . The characteristic equation of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq64_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq65_HTML.gif ) is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq66_HTML.gif , where

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ8_HTML.gif
        (2.4)

        According to Routh-Hurwitz criterion, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq67_HTML.gif is locally asymptotically stable for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq69_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq71_HTML.gif .

        The Jacobian matrix of the equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq72_HTML.gif is

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ9_HTML.gif
        (2.5)

        where

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ10_HTML.gif
        (2.6)

        The characteristic equation of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq73_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq74_HTML.gif ) is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq75_HTML.gif , where

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ11_HTML.gif
        (2.7)

        According to Routh-Hurwitz criterion, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq76_HTML.gif is locally asymptotically stable for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq77_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq78_HTML.gif can be checked by (2.1).

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

        Theorem 2.1.
        1. (i)
          Assume that (2.1),
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ12_HTML.gif
          (2.8)
           
        hold, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq79_HTML.gif of (1.2) is globally asymptotically stable.
        1. (ii)

          Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq80_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq81_HTML.gif hold, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq82_HTML.gif of (1.2) is globally asymptotically stable.

           
        2. (iii)

          Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq83_HTML.gif holds, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq84_HTML.gif of (1.2) is globally asymptotically stable.

           
        Proof.
        1. (i)
          Define the Lyapunov function
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ13_HTML.gif
          (2.9)
           
        Calculating the derivative of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq85_HTML.gif along the positive solution of (1.2), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ14_HTML.gif
        (2.10)
        When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq86_HTML.gif , the minimum of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq88_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq89_HTML.gif and 0, respectively; the maximum of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq90_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq91_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq93_HTML.gif , respectively. Thus, when (2.8) hold, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq94_HTML.gif According to the Lyapunov-LaSalle invariance principle [28], http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq95_HTML.gif is globally asymptotically stable if (2.1)–(2.3) hold.
        1. (ii)
          Let
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ15_HTML.gif
          (2.11)
           
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ16_HTML.gif
        (2.12)
        Noting that the maximum of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq96_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq97_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq98_HTML.gif , we find http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq99_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq100_HTML.gif
        1. (iii)
          Let
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ17_HTML.gif
          (2.13)
           
        then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ18_HTML.gif
        (2.14)

        Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq101_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq102_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq103_HTML.gif be nonnegative smooth functions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq104_HTML.gif . Then system (1.3) has a unique nonnegative solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq105_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ19_HTML.gif
        (3.1)

        on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq106_HTML.gif . In particular, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq107_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq108_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq109_HTML.gif .

        Proof.

        It is easily seen that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq110_HTML.gif is sufficiently smooth in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq111_HTML.gif and possesses a mixed quasimonotone property in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq112_HTML.gif . In addition, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq113_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq114_HTML.gif are a pair of lower-upper solutions of problem (1.3) (cf. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq115_HTML.gif in (3.1)). From [29, Theorem  5.3.4], we conclude that (1.3) exists a unique classical solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq116_HTML.gif satisfying (3.1). According to strong maximum principle, it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq117_HTML.gif . So the proof of the Theorem is completed.

        Remark 3.2.

        When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq118_HTML.gif (namely http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq119_HTML.gif ), 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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ20_HTML.gif
        (3.2)

        Now we show the local and global stability of constant equilibrium solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq120_HTML.gif for (1.3), respectively.

        Theorem 3.3.
        1. (i)

          Assume that (2.1) holds, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq121_HTML.gif of (1.3) is locally asymptotically stable.

           
        2. (ii)

          Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq123_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq124_HTML.gif hold, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq125_HTML.gif of (1.3) is locally asymptotically stable.

           
        3. (iii)

          Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq126_HTML.gif holds, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq127_HTML.gif of (1.3) is locally asymptotically stable.

           

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq128_HTML.gif be the eigenvalues of the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq129_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq130_HTML.gif with Neumann boundary condition, and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq131_HTML.gif be the eigenspace corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq132_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq133_HTML.gif . Let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ21_HTML.gif
        (3.3)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq134_HTML.gif is an orthonormal basis of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq135_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ22_HTML.gif
        (3.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq136_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq137_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq138_HTML.gif , where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ23_HTML.gif
        (3.5)
        The linearization of (1.3) is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq139_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq140_HTML.gif . For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq141_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq142_HTML.gif is invariant under the operator L, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq143_HTML.gif is an eigenvalue of L on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq144_HTML.gif , if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq145_HTML.gif is an eigenvalue of the matrix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq146_HTML.gif . The characteristic equation is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq147_HTML.gif , where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ24_HTML.gif
        (3.6)

        From Routh-Hurwitz criterion, we can see that three eigenvalues (denoted by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq149_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq150_HTML.gif ) all have negative real parts if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq151_HTML.gif . Noting that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq152_HTML.gif , we must have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq153_HTML.gif . It is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq154_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq155_HTML.gif (see Section 2).

        We can conclude that there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq156_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ25_HTML.gif
        (3.7)
        In fact, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq157_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ26_HTML.gif
        (3.8)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq158_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq159_HTML.gif , it follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ27_HTML.gif
        (3.9)
        Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq160_HTML.gif has the three roots http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq161_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq162_HTML.gif . By continuity, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq163_HTML.gif such that the three roots http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq164_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq165_HTML.gif satisfy
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ28_HTML.gif
        (3.10)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq166_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq167_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq168_HTML.gif , then (3.7) holds. According to [30, Theorem  5.1.1], we have the locally asymptotically stability of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq169_HTML.gif .
        1. (ii)
          The linearization of (1.4) is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq170_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq171_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq172_HTML.gif , and
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ29_HTML.gif
          (3.11)
           
        The characteristic equation of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq173_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq174_HTML.gif , where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ30_HTML.gif
        (3.12)
        The three roots of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq175_HTML.gif all have negative real parts for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq176_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq177_HTML.gif . Namely, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq178_HTML.gif is the locally asymptotically stable, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq179_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq180_HTML.gif .
        1. (iii)
          The linearization of (1.3) is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq181_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq182_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq183_HTML.gif , and
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ31_HTML.gif
          (3.13)
           

        Similar to (i), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq184_HTML.gif is locally asymptotically stable, when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq185_HTML.gif .

        Remark 3.4.

        When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq186_HTML.gif denote http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq187_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq188_HTML.gif , then (1.3) has the semitrivial equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq189_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq190_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq191_HTML.gif , then (1.3) has a unique positive equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq192_HTML.gif . Similar as Theorem 3.3, we have the following.

        (i)If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq193_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq194_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq195_HTML.gif (namely, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq196_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq197_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq198_HTML.gif ), then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq199_HTML.gif is locally asymptotically stable.

        (ii)If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq200_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq201_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq202_HTML.gif is locally asymptotically stable.

        (iii)If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq203_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq204_HTML.gif is locally asymptotically stable.

        Before discussing the global stability, we give an important lemma which has been proved in [31, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq205_HTML.gif ] or in [32, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq206_HTML.gif ].

        Lemma 3.5.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq207_HTML.gif be positive constants. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq208_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq209_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq210_HTML.gif is bounded from below. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq211_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq212_HTML.gif for some positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq213_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq214_HTML.gif

        Theorem 3.6.
        1. (i)
          Assume that (2.1),
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ32_HTML.gif
          (3.14)
           
        hold, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq215_HTML.gif of system (1.3) is globally asymptotically stable.
        1. (ii)

          Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq216_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq217_HTML.gif hold, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq218_HTML.gif of system (1.3) is globally asymptotically stable.

           

        (iii)Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq219_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq220_HTML.gif hold, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq221_HTML.gif of system (1.3) is globally asymptotically stable.

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq222_HTML.gif be the unique positive solution of (1.3). By Theorem 3.1, there exists a positive constant C which is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq223_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq224_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq225_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq226_HTML.gif . By [33, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq227_HTML.gif ],
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ33_HTML.gif
        (3.15)
        1. (i)
          Define the Lyapunov function
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ34_HTML.gif
          (3.16)
           

        By Theorem 3.1, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq228_HTML.gif is defined well for all solutions of (1.3) with the initial functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq229_HTML.gif . It is easily see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq230_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq231_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq232_HTML.gif .

        Calculating the derivative of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq233_HTML.gif along positive solution of (1.3) by integration by parts and the Cauchy inequality, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ35_HTML.gif
        (3.17)
        It is not hard to verify that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ36_HTML.gif
        (3.18)
        if (3.14) hold. Applying Lemma 3.5, we can obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ37_HTML.gif
        (3.19)
        Recomputing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq234_HTML.gif , we find
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ38_HTML.gif
        (3.20)
        From (3.15), we can see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq235_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq236_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq237_HTML.gif . It follows from Lemma 3.5 and (3.15) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq238_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq239_HTML.gif . Namely,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ39_HTML.gif
        (3.21)
        Using the Pioncaré inequality, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ40_HTML.gif
        (3.22)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq240_HTML.gif Noting that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ41_HTML.gif
        (3.23)
        according to (3.19) and (3.22), we can see
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ42_HTML.gif
        (3.24)
        Thus, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq241_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq242_HTML.gif . Applying the boundness of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq243_HTML.gif , there exists a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq244_HTML.gif , denoted still by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq245_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq246_HTML.gif On the one hand
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ43_HTML.gif
        (3.25)
        On the other hand
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ44_HTML.gif
        (3.26)
        According to (3.19) to compute the limit of the previous equation and using the uniqueness of the limit, we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq247_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ45_HTML.gif
        (3.27)
        It follows from (3.15) that there exists a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq248_HTML.gif , denoted still by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq249_HTML.gif , and nonnegative functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq250_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ46_HTML.gif
        (3.28)
        Applying (3.19)–(3.27), we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq251_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ47_HTML.gif
        (3.29)
        In view of Theorem 3.3, we can conclude that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq252_HTML.gif is globally asymptotically stable.
        1. (ii)
          Let
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ48_HTML.gif
          (3.30)
           
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ49_HTML.gif
        (3.31)
        Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq253_HTML.gif It follows that the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq254_HTML.gif of (1.3) is globally asymptotically stable.
        1. (iii)
          Define
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ50_HTML.gif
          (3.32)
           
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ51_HTML.gif
        (3.33)
        When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq255_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ52_HTML.gif
        (3.34)

        The following proof is similar to (i).

        Remark 3.7.

        When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq256_HTML.gif , Theorem 3.6 shows the following.
        1. (i)
          Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq257_HTML.gif ,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ53_HTML.gif
          (3.35)
           
        hold, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq258_HTML.gif of (1.3) is globally asymptotically stable.
        1. (ii)

          Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq259_HTML.gif hold, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq260_HTML.gif of (1.3) is globally asymptotically stable.

           
        2. (iii)

          Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq261_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq262_HTML.gif hold, then the equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq263_HTML.gif of (1.3) is globally asymptotically stable.

           

        Example 3.8.

        Consider the following system:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ54_HTML.gif
        (3.36)

        Using the software Matlab, one can obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq264_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq265_HTML.gif . It is easy to see that the previous system satisfies the all conditions of Theorem 3.6(i). So the positive equilibrium point (0.5637,0.5637,0.1199) of the previous system is globally asymptotically stable.

        4. Global Existence and Stability of Solutions for the System (1.4)

        By [3436], we have the following result.

        Theorem 4.1.

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq266_HTML.gif , then (1.4) has a unique nonnegative solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq267_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq268_HTML.gif is the maximal existence time of the solution. If the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq269_HTML.gif satisfies the estimate
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ55_HTML.gif
        (4.1)

        then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq270_HTML.gif . If, in addition, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq271_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq272_HTML.gif

        In this section, we consider the existence and the convergence of global solutions to the system (1.4).

        Theorem 4.2.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq273_HTML.gif and the space dimension http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq274_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq275_HTML.gif are nonnegative functions and satisfy zero Neumann boundary conditions. Then (1.4) has a unique nonnegative solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq276_HTML.gif

        In order to prove Theorem 4.2, some preparations are collected firstly.

        Lemma 4.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq277_HTML.gif be a solution of (1.4). Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ56_HTML.gif
        (4.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq278_HTML.gif .

        Proof.

        From the maximum principle for parabolic equations, it is not hard to verify that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq279_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq280_HTML.gif is bounded.

        Multiplying the second equation of (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq281_HTML.gif , adding up the first equation of (1.4), and integrating the result over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq282_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ57_HTML.gif
        (4.3)
        Using Young inequality and H http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq283_HTML.gif lder inequality, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ58_HTML.gif
        (4.4)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq284_HTML.gif It follows from (4.3) and (4.4) that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ59_HTML.gif
        (4.5)
        Thus,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ60_HTML.gif
        (4.6)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq285_HTML.gif depends on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq286_HTML.gif and coefficients of (1.4). In addition, there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq287_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ61_HTML.gif
        (4.7)
        Integrating the first equation of (1.4) over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq288_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ62_HTML.gif
        (4.8)
        Integrating (4.8) from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq289_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq290_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ63_HTML.gif
        (4.9)
        According to (4.7), there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq291_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ64_HTML.gif
        (4.10)
        Multiplying the second equation of (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq292_HTML.gif and integrating it over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq293_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ65_HTML.gif
        (4.11)
        Integrating the previous inequation from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq294_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq295_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ66_HTML.gif
        (4.12)

        Lemma 4.4.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq296_HTML.gif be a solution of (1.4), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq297_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq298_HTML.gif . Then there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq299_HTML.gif depending on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq300_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq301_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ67_HTML.gif
        (4.13)

        Furthermore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq302_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq303_HTML.gif

        Proof.

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq304_HTML.gif satisfies the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ68_HTML.gif
        (4.14)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq305_HTML.gif are functions of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq306_HTML.gif and so are bounded because of Lemma 4.3.

        Multiply the second equation of (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq307_HTML.gif and integrate it over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq308_HTML.gif to obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ69_HTML.gif
        (4.15)
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ70_HTML.gif
        (4.16)

        and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq309_HTML.gif . From a disposal similar to the proof of Lemma  2.2 in [23], we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq310_HTML.gif . Using a standard embedding result, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq311_HTML.gif

        Lemma 4.5 (see [23, Lemmas  2.3 and 2.4]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq312_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq313_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq314_HTML.gif be any number which may depend on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq315_HTML.gif . Then there is a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq316_HTML.gif depending on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq317_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq318_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ71_HTML.gif
        (4.17)

        for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq319_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq320_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq321_HTML.gif .

        To obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq322_HTML.gif -estimates of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq323_HTML.gif , we establish http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq324_HTML.gif -estimates of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq325_HTML.gif in the following lemma.

        Lemma 4.6.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq326_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq327_HTML.gif , then there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq328_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq329_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ72_HTML.gif
        (4.18)

        Proof.

        Multiply the first equation of (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq330_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq331_HTML.gif and integrate by parts over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq332_HTML.gif to obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ73_HTML.gif
        (4.19)
        Integrating (4.19) from 0 to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq333_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ74_HTML.gif
        (4.20)
        Then substitution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq334_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq335_HTML.gif into (4.20) leads to
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ75_HTML.gif
        (4.21)
        It follows from H http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq336_HTML.gif lder inequality and Lemma 4.3 that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ76_HTML.gif
        (4.22)
        Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq337_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq338_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq339_HTML.gif . From H http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq340_HTML.gif lder inequality, Young inequality, and Lemma 4.4, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ77_HTML.gif
        (4.23)
        Substitution of (4.22) and (4.23) into (4.21) leads to
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ78_HTML.gif
        (4.24)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq341_HTML.gif is arbitrary and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq342_HTML.gif .

        Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq343_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ79_HTML.gif
        (4.25)
        then it follows from (4.24) that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ80_HTML.gif
        (4.26)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ81_HTML.gif
        (4.27)
        Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq344_HTML.gif for
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ82_HTML.gif
        (4.28)
        According to Lemma 4.5 and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq345_HTML.gif , we can see
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ83_HTML.gif
        (4.29)
        Combining (4.26) and (4.29), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ84_HTML.gif
        (4.30)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq346_HTML.gif . Therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq347_HTML.gif is bounded from (4.30).

        From (4.29), we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq348_HTML.gif . Namely, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq349_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq350_HTML.gif . Combining (4.28), we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq351_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq352_HTML.gif .

        Setting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq353_HTML.gif in (4.20) (it is easily checked that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq354_HTML.gif , i.e., http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq355_HTML.gif ), we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq356_HTML.gif .

        Multiplying the second equation of (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq357_HTML.gif and integrating it over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq358_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ85_HTML.gif
        (4.31)

        The result of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq359_HTML.gif can be obtained from an analogue of the previous proof of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq360_HTML.gif 's.

        Lemma 4.7.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq361_HTML.gif , then there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq362_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ86_HTML.gif
        (4.32)

        Proof.

        We will prove this lemma by [37, Theorem  7.1, page 181]. At first, we rewrite the first two equations of (1.4) as
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ87_HTML.gif
        (4.33)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq363_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq364_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq365_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq366_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq367_HTML.gif symbol. It follows from Lemma 4.6 that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq368_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq369_HTML.gif .

        By the third equation of (1.4), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ88_HTML.gif
        (4.34)
        It follows from Lemma 4.3 that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq370_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq371_HTML.gif . Applying Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq372_HTML.gif [37, Page 204] to (4.34), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ89_HTML.gif
        (4.35)
        Recall that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq373_HTML.gif satisfy (4.14) in Lemma 4.4, that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ90_HTML.gif
        (4.36)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq374_HTML.gif is bounded. Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq375_HTML.gif by (4.35), applying Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq376_HTML.gif [37, page 341-342] to (4.36), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ91_HTML.gif
        (4.37)

        It follows from [37, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq377_HTML.gif , page 80] that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq378_HTML.gif and so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq379_HTML.gif . Recall from Lemma 4.6 that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq380_HTML.gif , so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq381_HTML.gif by applying Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq382_HTML.gif [37, Page 181] to (4.33).

        Proof of Theorem 4.2.

        Firstly, Theorem 4.2 can be proved in a similar way as Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq383_HTML.gif in [21, 25] when the space dimension http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq384_HTML.gif .

        Secondly, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq385_HTML.gif , applying Lemma http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq386_HTML.gif [37, Page 80] to (4.36), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ92_HTML.gif
        (4.38)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq387_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ93_HTML.gif
        (4.39)
        The first two equations can be written in the divergence form as
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ94_HTML.gif
        (4.40)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq388_HTML.gif . It follows from Lemmas 4.1, 4.5, and (4.39) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq389_HTML.gif are bounded. Thus applying Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq390_HTML.gif [37, Page 204] to (4.40) leads to
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ95_HTML.gif
        (4.41)
        We rewrite the third equation of (1.4) as
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ96_HTML.gif
        (4.42)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq391_HTML.gif . Applying Schauder estimate [29, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq392_HTML.gif , page 114] to (4.42) gives
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ97_HTML.gif
        (4.43)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ98_HTML.gif
        (4.44)
        then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ99_HTML.gif
        (4.45)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq393_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq394_HTML.gif . From (4.41), we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq395_HTML.gif . It follows from (4.41) and (4.43) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq396_HTML.gif . Applying Schauder estimate to (4.45) gives
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ100_HTML.gif
        (4.46)
        Solving equations (4.44) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq397_HTML.gif , respectively, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ101_HTML.gif
        (4.47)

        In particular, to conclude http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq398_HTML.gif , we need to repeat the above bootstrap technique. Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq399_HTML.gif is arbitrary, so the classical solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq400_HTML.gif of (1.4) exists globally in time.

        Now we discuss the global stability of the positive equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq401_HTML.gif (see Section 2) for (1.4).

        Theorem 4.8.

        Assume that the all conditions in Theorem 4.2, (2.1), and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ102_HTML.gif
        (4.48)
        hold. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq402_HTML.gif be the unique positive equilibrium point of (1.4), and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq403_HTML.gif be a positive solution for (1.4). Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ103_HTML.gif
        (4.49)

        provided that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq404_HTML.gif is large enough.

        Proof.

        Define the Lyapunov function
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ104_HTML.gif
        (4.50)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq405_HTML.gif be a positive solution of (1.4), Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ105_HTML.gif
        (4.51)
        The first integrand in the right hand of the previous inequality is positive definite if
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ106_HTML.gif
        (4.52)
        Therefore, when the all conditions in Theorem 4.8 hold, there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq406_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ107_HTML.gif
        (4.53)

        This implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq407_HTML.gif . So the proof of Theorem 4.8 is completed.

        Declarations

        Acknowledgments

        This work has been partially supported by the China National Natural Science Foundation (no. 10871160), the NSF of Gansu Province (no. 096RJZA118), the Scientific Research Fund of Gansu Provincial Education Department, and NWNU-KJCXGC-03-47 Foundation.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Lanzhou Jiaotong University
        (2)
        Department of Mathematics, Northwest Normal University

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