Global Existence and Convergence of Solutions to a Cross-Diffusion Cubic Predator-Prey System with Stage Structure for the Prey

  • Huaihuo Cao1Email author and

    Affiliated with

    • Shengmao Fu2

      Affiliated with

      Boundary Value Problems20102010:285961

      DOI: 10.1155/2010/285961

      Received: 3 December 2009

      Accepted: 30 March 2010

      Published: 18 May 2010

      Abstract

      We study a cubic predator-prey system with stage structure for the prey. This system is a generalization of the two-species Lotka-Volterra predator-prey model. Firstly, we consider the asymptotical stability of equilibrium points to the system of ordinary differential equations type. Then, the global existence of solutions and the stability of equilibrium points to the system of weakly coupled reaction-diffusion type are discussed. Finally, the existence of nonnegative classical global solutions to the system of strongly coupled reaction-diffusion type is investigated when the space dimension is less than 6, and the global asymptotic stability of unique positive equilibrium point of the system is proved by constructing Lyapunov functions.

      1. Introduction and Mathematical Model

      The predator-prey model as, which follows, the ordinary differential equation system
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ1_HTML.gif
      (1.1)

      is said to be the general Lotka-Volterra predator-prey model in [13], and to be cubic predator-prey system in [4], where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq1_HTML.gif are the population densities of prey and predator species at time http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq2_HTML.gif , respectively. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq3_HTML.gif are positive constants, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq4_HTML.gif is nonnegative as the intrinsic growth rate of prey population, and the sign of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq5_HTML.gif is undetermined. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq6_HTML.gif is the net mortality rate of predator population, and the survival of predator species is dependent on the survival state of prey species, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq7_HTML.gif are the respective density restriction terms of prey and predator species. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq8_HTML.gif is the predation rate of the predator, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq9_HTML.gif is the conversion rate of the predator. In [4], three questions about system (1.1) are discussed: the stability of nonnegative equilibrium points, and the existence, as well as numbers of limit cycle.

      Referring to [5], we establish cubic predator-prey system with stage structure for the prey as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq11_HTML.gif are the population densities of the immature and mature prey species, respectively, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq12_HTML.gif denotes the density of the predator species. The predators live only on the immature prey species, as well as the survival of the predator species is dependent on the survival state of the immature prey species. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq13_HTML.gif are positive constants, and the sign of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq14_HTML.gif is undetermined. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq15_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq16_HTML.gif are the birth rate and the mortality rate of the immature prey species, respectively. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq17_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq18_HTML.gif are the net mortality rate of the mature prey population and the predator population, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq19_HTML.gif is the conversion rate of the immature prey to the mature prey species. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq21_HTML.gif are the respective density restriction terms of the immature prey species and predator species. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq22_HTML.gif is the predation rate of the predator to the immature prey population, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq23_HTML.gif is the conversion rate of the predator.

      Using the scaling
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ3_HTML.gif
      (1.3)
      and redenoting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq24_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq25_HTML.gif , system (1.2) reduces to
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ4_HTML.gif
      (1.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq27_HTML.gif are positive constants, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq28_HTML.gif is undetermined to the sign.

      To take into account the inhomogeneous distribution of the predators and prey in different spatial locations within a fixed bounded domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq29_HTML.gif at any given time, and the natural tendency of each species to diffuse to areas of smaller population concentration, we derive the following PDE system of reaction-diffusion type:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ5_HTML.gif
      (1.5)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq31_HTML.gif is the unit outward normal vector of the boundary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq32_HTML.gif which we will assume to be smooth. The homogeneous Neumann boundary condition indicates that the above system is self-contained with zero population flux across the boundary. The positive constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq34_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq35_HTML.gif are said to be the diffusion coefficients, and the initial values http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq36_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq37_HTML.gif ) are nonnegative smooth functions.

      Note that, in recent years, there has been considerable interest to investigate the global behavior of a system of interacting populations by taking into account the effect of self as well as cross-diffusion. According to the ideas in [613], especially to [8, 9], the cross-diffusion term will be only included in the third equation, that is, the following cross-diffusion system:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ6_HTML.gif
      (1.6)

      In the above, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq39_HTML.gif are positive constants. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq40_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq41_HTML.gif are the diffusion rates of the three species, respectively. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq42_HTML.gif are referred to as self-diffusion pressures. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq44_HTML.gif are cross-diffusion pressures. 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. Generally, 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 [9].

      The main purpose of this paper is to study the asymptotic behavior of the solutions of the reaction-diffusion system (1.5) and the global existence of the solution of the cross-diffusion system (1.6). But it is necessary to denonstrate that the conclusion for the existence of global solution of system (1.6) in this paper is the generalization of the work to Lotka-Volterra competition model with cross-diffusion [11] and that the convergence of solution investigated in this paper which is not discussed in [11].

      The paper will be organized as follows. In Section 2, we analyze the asymptotical stability of equilibrium points for the ODE system (1.4) via linearization and the Lyapunov method. In Section 3, we prove the global existence of solutions and the stability of the equilibrium points to the diffusion system (1.5). In Section 4, we investigate the existence of nonnegative classical global solutions by assuming http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq46_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq49_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq50_HTML.gif to be positive constants only for the simplicity of calculation, and the global asymptotic stability of unique positive equilibrium point to the cross-diffusion system (1.6).

      2. Equilibrium Solution of the ODE System

      In this section we discuss the stability of unique positive equilibrium point for system (1.4). The following theorem shows that the solution of system (1.4) is bounded.

      Theorem 2.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq51_HTML.gif be the solution of system (1.4) with initial values http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq52_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq53_HTML.gif be the maximal existence interval of the solution. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq54_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ7_HTML.gif
      (2.1)

      The above http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq55_HTML.gif is a positive constant depending only on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq56_HTML.gif , and further http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq57_HTML.gif .

      Proof.

      It is easy to see that (1.4) has a unique positive local solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq58_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq59_HTML.gif be the maximal existence time of the solution, and combin http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq61_HTML.gif linearly, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq62_HTML.gif , it follows from (1.4) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ8_HTML.gif
      (2.2)
      Using Young inequality, we can check that there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq63_HTML.gif depending only on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq65_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ9_HTML.gif
      (2.3)
      It follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ10_HTML.gif
      (2.4)

      which implies that there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq67_HTML.gif referring to (2.1) such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq68_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq69_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq70_HTML.gif .

      Finally, we note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq71_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq72_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ11_HTML.gif
      (2.5)

      From the comparison inequality for the ODE, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq73_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq74_HTML.gif .

      Thus the solutions for system (1.4) are bounded. Further, from the extension theorem of solutions, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq75_HTML.gif .

      By the simple calculation, the sufficient conditions for system (1.4) having a unique positive equilibrium point as follows:
      1. (i)

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq76_HTML.gif ; (ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq77_HTML.gif , where the left equal sign holds if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq78_HTML.gif ; (iii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq79_HTML.gif ; (iv) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq80_HTML.gif ; (v) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq81_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq82_HTML.gif , where the second equal sign holds if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq83_HTML.gif ; (vi) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq85_HTML.gif .

         
      If one of the above conditions holds, then system (1.4) has the unique positive equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq86_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ12_HTML.gif
      (2.6)

      Theorem 2.2.

      System (1.4) has the unique positive equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq87_HTML.gif when one of the above conditions (i), (ii), (iii), (iv), (v), and (vi) holds. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq88_HTML.gif holds, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq89_HTML.gif is locally asymptotically stable.

      Theorem 2.2 is easy to be obtained by using linearization; therefore, we omit its proof. The objective of this section is to prove the following result.

      Theorem 2.3.

      System (1.4) has the unique positive equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq90_HTML.gif when one of the above conditions (i), (ii), (iii), (iv), (v), and (vi) holds. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq91_HTML.gif holds, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq92_HTML.gif is globally asymptotically stable.

      Proof.

      We make use of the general Lyapunov function
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ13_HTML.gif
      (2.7)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq93_HTML.gif are positive constants. It holds that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq94_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq95_HTML.gif . Calculating the derivative along each solution of system (1.4), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ14_HTML.gif
      (2.8)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq97_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ15_HTML.gif
      (2.9)
      We observe that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ16_HTML.gif
      (2.10)
      is a sufficient condition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq98_HTML.gif . So, when condition (2.10) holds, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ17_HTML.gif
      (2.11)

      Set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq99_HTML.gif . According to the Lyapunov-LaSalle invariance principle [14], http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq100_HTML.gif is global asymptotic stability if inequality (2.10) and all conditions of Theorem 2.2 are satisfied. Theorem 2.3 is, thus, proved.

      3. Stability of the PDE System without Cross-Diffusion

      In this section, we first prove the global existence and uniform boundedness of solutions, then discuss the stability of unique positive equilibrium solution for the weakly coupled reaction-diffusion system (1.5).

      Denote that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq101_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq102_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq103_HTML.gif . It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq104_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq105_HTML.gif . The standard PDE theory [15] shows that (1.5) has the unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq106_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq107_HTML.gif is the maximal existence time. The following theorem shows that the solution of (1.5) is uniformly bounded, and thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq108_HTML.gif .

      Theorem 3.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq109_HTML.gif be the solution of system (1.5) with initial values http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq110_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq111_HTML.gif be the maximal existence time. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq112_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq113_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq114_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq115_HTML.gif is a positive constant depending only on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq116_HTML.gif and all coefficients of (1.5) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq117_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq118_HTML.gif . Furthermore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq119_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq120_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq121_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq122_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq123_HTML.gif .

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq124_HTML.gif be the solution of (1.5) with initial values http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq125_HTML.gif . From the maximum principle for parabolic equations [16], it is not hard to verify that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq126_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq127_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq128_HTML.gif is the maximal existence time of the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq129_HTML.gif . Furthermore, we know by the strong maximum principle that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq130_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq131_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq132_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq133_HTML.gif . Next we prove that the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq134_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq135_HTML.gif .

      Integrating the first two equations of (1.5) over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq136_HTML.gif and adding the results linearly, we have that, by Young inequality,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ18_HTML.gif
      (3.1)

      for some positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq137_HTML.gif depending only on the coefficients of (1.5). Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq138_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq139_HTML.gif . Using [17, Exercise 5 of Section http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq140_HTML.gif ], we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq141_HTML.gif is also bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq142_HTML.gif . Now note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq143_HTML.gif The maximum principle gives http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq144_HTML.gif . The proof of Theorem 3.1 is completed.

      In order to prove the global stability of unique positive equilibrium solution for system (1.5), we first recall the following lemma which can be found in [7, 17].

      Lemma 3.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq146_HTML.gif be positive constants. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq149_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq150_HTML.gif is bounded from below. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq151_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq152_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq153_HTML.gif for some constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq154_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq155_HTML.gif

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq156_HTML.gif be the eigenvalues of the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq157_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq158_HTML.gif with the homogeneous Neumann boundary condition, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq159_HTML.gif be the eigenspace corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq160_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq161_HTML.gif . Denote that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq162_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq163_HTML.gif is an orthonormal basis of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq164_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq165_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ19_HTML.gif
      (3.2)

      Next we present the clear proof of the the global stability by two steps:

      Step 1 (Local Stability).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq166_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq167_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ20_HTML.gif
      (3.3)
      The linearization of (1.5) at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq168_HTML.gif is
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ21_HTML.gif
      (3.4)

      For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq169_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq170_HTML.gif is invariant under the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq171_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq172_HTML.gif is an eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq173_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq174_HTML.gif if and only if it is an eigenvalue of the matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq175_HTML.gif .

      The characteristic polynomial of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq176_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ22_HTML.gif
      (3.5)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ23_HTML.gif
      (3.6)
      Thus
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ24_HTML.gif
      (3.7)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq178_HTML.gif are given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ25_HTML.gif
      (3.8)
      According to the Routh-Hurwitz criterion [18], for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq179_HTML.gif , the three roots http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq180_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq181_HTML.gif all have negative real parts if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq182_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq183_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq184_HTML.gif . Noting that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq185_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq186_HTML.gif , the three roots have negative real parts if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq187_HTML.gif . A direct calculation shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq188_HTML.gif is negative if
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ26_HTML.gif
      (3.9)
      Now we can conclude that there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq189_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ27_HTML.gif
      (3.10)
      In fact, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq190_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ28_HTML.gif
      (3.11)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq191_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq192_HTML.gif , it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ29_HTML.gif
      (3.12)
      It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq193_HTML.gif are the three roots of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq194_HTML.gif . Thus, there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq195_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ30_HTML.gif
      (3.13)
      By continuity, we see that there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq196_HTML.gif such that the three roots http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq197_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq198_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ31_HTML.gif
      (3.14)
      So
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ32_HTML.gif
      (3.15)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ33_HTML.gif
      (3.16)

      then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq199_HTML.gif , and (3.10) holds for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq200_HTML.gif .

      Consequently, the spectrum of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq201_HTML.gif , consisting only of eigenvalues, lies in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq202_HTML.gif if (3.9) holds, and the local stability of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq203_HTML.gif follows [19, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq204_HTML.gif ].

      Step 2 (Global Stability).

      In the following, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq205_HTML.gif denotes a generic positive constant which does not depend on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq206_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq207_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq208_HTML.gif be the unique positive solution. Then it follows from Theorem 3.1 that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq209_HTML.gif is bounded uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq210_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq211_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq212_HTML.gif . By [20, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq213_HTML.gif ],
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ34_HTML.gif
      (3.17)
      Define the Lyapunov function
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ35_HTML.gif
      (3.18)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq214_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq215_HTML.gif . Using (1.5) and integrating by parts, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ36_HTML.gif
      (3.19)
      Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq216_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq217_HTML.gif , we have that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ37_HTML.gif
      (3.20)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq218_HTML.gif holds for
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ38_HTML.gif
      (3.21)
      From Theorem 3.1 the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq219_HTML.gif of (1.5) is bounded, and so are the derivatives of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq220_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq221_HTML.gif by equations in (1.5). Applying Lemma 3.2, we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ39_HTML.gif
      (3.22)
      As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq222_HTML.gif , it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ40_HTML.gif
      (3.23)
      Using inequality (3.17) and system (1.5), the derivative of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq223_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq224_HTML.gif . From Lemma 3.2, we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq225_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq226_HTML.gif . Therefore
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ41_HTML.gif
      (3.24)
      Using the Poincaré inequality yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ42_HTML.gif
      (3.25)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq227_HTML.gif Thus, it follows from (3.22) and (3.25) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ43_HTML.gif
      (3.26)
      as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq228_HTML.gif . So we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq229_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq230_HTML.gif . Similarly, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq231_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq232_HTML.gif . Therefore, there exists a sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq233_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq234_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq235_HTML.gif . As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq236_HTML.gif is bounded, there exists a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq237_HTML.gif , still denoted by the same notation, and nonnegative constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq238_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ44_HTML.gif
      (3.27)
      At http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq239_HTML.gif , from the first equation of (1.5), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ45_HTML.gif
      (3.28)
      In view of (3.22) and (3.27), it follows from (3.28) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq240_HTML.gif , thus
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ46_HTML.gif
      (3.29)
      According to (3.17), there exists a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq241_HTML.gif , denoted still by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq242_HTML.gif , and nonnegative functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq243_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ47_HTML.gif
      (3.30)
      In view of (3.29) and noting that in fact http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq244_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq245_HTML.gif , we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq246_HTML.gif . Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ48_HTML.gif
      (3.31)

      The global asymptotic stability of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq247_HTML.gif follows from (3.31) and the local stability of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq248_HTML.gif .

      Theorem 3.3.

      System (1.5) has the unique positive equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq249_HTML.gif when one of the conditions (i), (ii), (iii), (iv), (v), and (vi) in Section 2 holds. If (3.9) and (3.21) hold, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq250_HTML.gif is globally asymptotically stable.

      4. Global Existence of Classical Solutions and Convergence

      In this section, we discuss the existence of nonnegative classical global solutions and the global asymptotic stability of unique positive equilibrium point of system (1.6).

      Some notations throughout this section are as follows: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq251_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq252_HTML.gif means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq253_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq254_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq255_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq256_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq257_HTML.gif means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq258_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq259_HTML.gif are in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq260_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq261_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq262_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq263_HTML.gif .

      To obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq264_HTML.gif normal estimates of the solution for (1.6), we present a series of lemmas in the following.

      Lemma 4.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq265_HTML.gif be the solution of (1.6). Then there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq266_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq267_HTML.gif 1) such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ49_HTML.gif
      (4.1)

      Proof.

      By applying the comparison principle [20] to system (1.6), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq268_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq269_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq270_HTML.gif . To prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq271_HTML.gif in the following, we consider the auxiliary problem
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ50_HTML.gif
      (4.2)
      Notice that the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq272_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq273_HTML.gif are sufficiently smooth in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq274_HTML.gif , and are quasimonotone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq275_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq276_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq277_HTML.gif be a pair of upper-lower solutions for (4.2), where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq278_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq279_HTML.gif are positive constants. Direct calculation with inequalities
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ51_HTML.gif
      (4.3)

      yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq280_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq281_HTML.gif . It follows that there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq282_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq283_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq284_HTML.gif is a big enough positive constant such that (4.1) holds.

      Lemma 4.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq285_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq286_HTML.gif for the solution to following equation:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ52_HTML.gif
      (4.4)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq287_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq288_HTML.gif are positive constants and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq289_HTML.gif . Then there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq290_HTML.gif , depending on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq291_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq292_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ53_HTML.gif
      (4.5)
      Furthermore,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ54_HTML.gif
      (4.6)

      Proof.

      It is easy to check, from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq293_HTML.gif , that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ55_HTML.gif
      (4.7)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq294_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq295_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq296_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq297_HTML.gif are bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq298_HTML.gif from (4.1). Multiplying (4.7) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq299_HTML.gif , and integrating by parts over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq300_HTML.gif , yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ56_HTML.gif
      (4.8)
      Using Hölder inequality and Young inequality to estimate the right side of (4.8), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ57_HTML.gif
      (4.9)
      with some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq301_HTML.gif . Substituting (4.9) into (4.8) yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ58_HTML.gif
      (4.10)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq302_HTML.gif depends on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq303_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq304_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq305_HTML.gif , the elliptic regularity estimate [10, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq306_HTML.gif ] yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ59_HTML.gif
      (4.11)

      From (4.7), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq307_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq308_HTML.gif . Moreover, the Sobolev embedding theorem shows that (4.6) holds.

      Lemma 4.3 (Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq309_HTML.gif can be presented by combining Lemmas http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq310_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq311_HTML.gif in [11]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq312_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq313_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ60_HTML.gif
      (4.12)
      and there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq314_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq315_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq316_HTML.gif . Then there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq317_HTML.gif independent of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq318_HTML.gif but possibly depending on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq319_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq320_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq321_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq322_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq323_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ61_HTML.gif
      (4.13)

      Finally, one proposes some standard embedding results which are important to obtain the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq324_HTML.gif normal estimates of the solution for (1.6).

      Lemma 4.4.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq325_HTML.gif be a fixed bounded domain and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq326_HTML.gif . Then for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq327_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq328_HTML.gif , one has

      (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq329_HTML.gif

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq330_HTML.gif

      (3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq331_HTML.gif

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq332_HTML.gif is a positive constant dependent on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq333_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq334_HTML.gif .

      The main result about the global existence of nonnegative classical solution for the cross-diffusion system (1.6) is given as follows.

      Theorem 4.5.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq335_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq336_HTML.gif satisfy homogeneous Neumann boundary conditions and belong to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq337_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq338_HTML.gif . Then system (1.6) has a unique nonnegative solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq339_HTML.gif when the space dimension is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq340_HTML.gif .

      Proof.

      Step 1.

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq341_HTML.gif - http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq342_HTML.gif -Estimates and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq343_HTML.gif -Estimates of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq344_HTML.gif . Firstly, integrating the third equation of (1.6) over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq345_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ62_HTML.gif
      (4.14)
      Thus
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ63_HTML.gif
      (4.15)
      Furthermore
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ64_HTML.gif
      (4.16)
      Integrating (4.14) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq346_HTML.gif and moving terms yield
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ65_HTML.gif
      (4.17)
      Secondly, multiplying the third equation of (1.6) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq347_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq348_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ66_HTML.gif
      (4.18)
      Integrating the above expression in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq349_HTML.gif yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ67_HTML.gif
      (4.19)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq350_HTML.gif from Lemma 4.2, and using Hölder inequality and Young inequality, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ68_HTML.gif
      (4.20)
      From (4.1) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq351_HTML.gif , it holds that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ69_HTML.gif
      (4.21)
      Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq352_HTML.gif and selecting a proper http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq353_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq354_HTML.gif , then applying (4.20) and (4.21) to (4.19) yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ70_HTML.gif
      (4.22)
      Denote that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq355_HTML.gif . Then it follows from (4.22) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ71_HTML.gif
      (4.23)
      It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq356_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq357_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq358_HTML.gif ; hence
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ72_HTML.gif
      (4.24)
      Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq359_HTML.gif . Then it follows from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq360_HTML.gif -estimates of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq361_HTML.gif namely (4.15), that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ73_HTML.gif
      (4.25)
      It follows from Lemma 4.3 and (4.24) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ74_HTML.gif
      (4.26)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq362_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq363_HTML.gif is bounded by contrary proof. It follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq364_HTML.gif is bounded, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq365_HTML.gif . It is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq366_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq367_HTML.gif still denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq368_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq369_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ75_HTML.gif
      (4.27)
      Finally, we observe that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq370_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq371_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq372_HTML.gif . So take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq373_HTML.gif for (4.17) and (4.19). Then there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq374_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ76_HTML.gif
      (4.28)

      Step 2.

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq375_HTML.gif -Estimates of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq376_HTML.gif . We rewrite the third equation of (1.6) as a linear parabolic equation
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ77_HTML.gif
      (4.29)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq377_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq378_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq379_HTML.gif are Kronecker symbols.

      To apply the maximum principle [15, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq380_HTML.gif , page 181] to (4.15) to obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq381_HTML.gif , we need to verify that the following conditions hold: (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq382_HTML.gif is bounded; (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq383_HTML.gif (3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq384_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq385_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq386_HTML.gif are positive constants, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq387_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq388_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ78_HTML.gif
      (4.30)
      Next we verify conditions (1)–(3) in turn. From (4.28), condition (1) is true for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq389_HTML.gif . One can choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq390_HTML.gif such that condition (2) holds. To verify condition (3), the first equation of (1.6) is written in the divergence form
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ79_HTML.gif
      (4.31)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq391_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq392_HTML.gif by Lemma 4.1, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq393_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq394_HTML.gif from (4.27). Application of the Hölder continuity result [15, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq395_HTML.gif , page 204] to (4.19) yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ80_HTML.gif
      (4.32)
      Returning to (4.7), since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq396_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq397_HTML.gif by (4.1) and (4.27), and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq398_HTML.gif by (4.32), then by applying the parabolic regularity theorem [15, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq399_HTML.gif , pages 341-342] to (4.7) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ81_HTML.gif
      (4.33)
      Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq400_HTML.gif from Lemma 4.4, which shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq401_HTML.gif . Similarly, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq402_HTML.gif by the second equation of (1.6). Now we can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq403_HTML.gif , which imply that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq404_HTML.gif . In addition, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq405_HTML.gif obviously belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq406_HTML.gif . It follows that one can select http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq407_HTML.gif . Now the above three conditions are satisfied, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq408_HTML.gif from [15, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq409_HTML.gif , page 181]. Recalling Lemma 4.1, thus there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq410_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq411_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ82_HTML.gif
      (4.34)

      Step 3.

      The Proof of the Classical Solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq412_HTML.gif of (1.6) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq413_HTML.gif for Any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq414_HTML.gif . Because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq415_HTML.gif , we have from (4.34) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq416_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq417_HTML.gif . So http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq418_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq419_HTML.gif . It follows from [15, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq420_HTML.gif , page 80] that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq421_HTML.gif . And direct calculation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq422_HTML.gif yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq423_HTML.gif . So we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ83_HTML.gif
      (4.35)
      The third equation of (1.6) can be written as
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ84_HTML.gif
      (4.36)
      Summarizing the above conclusions that are proved, we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq424_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq425_HTML.gif are all bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq426_HTML.gif . It follows from [15, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq427_HTML.gif page 204] that there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq428_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ85_HTML.gif
      (4.37)
      The proof of Lemma 4.2 is similar. Then we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq429_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq430_HTML.gif . Applying the [13, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq431_HTML.gif page 204] to the second equation (1.6), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq432_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ86_HTML.gif
      (4.38)
      Furthermore, applying Schauder estimate [15, page 320-321] yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq433_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq434_HTML.gif . Selecting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq435_HTML.gif and using Sobolev embedding theorem, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq436_HTML.gif . Still applying Schauder estimate, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ87_HTML.gif
      (4.39)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq437_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq438_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ88_HTML.gif
      (4.40)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq439_HTML.gif . By (4.35)–(4.38), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq440_HTML.gif . So applying Schauder estimate to (4.40) yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq441_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq442_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ89_HTML.gif
      (4.41)
      The first equation of (1.6) can be written as
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ90_HTML.gif
      (4.42)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq443_HTML.gif . By (4.35), (4.39), and (4.41), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq444_HTML.gif . So applying Schauder estimate to (4.42) yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ91_HTML.gif
      (4.43)

      In particular, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq445_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq446_HTML.gif ; in other words, Theorem 4.5 is proved. For the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq447_HTML.gif , from Sobolev embedding theorem, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq448_HTML.gif . Repeating the above bootstrap and Shauder estimate arguments, this completes the proof of Theorem 4.5. About space dimension http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq449_HTML.gif , see [21].

      Theorem 4.6.

      System (1.6) has the unique positive equilibrium point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq450_HTML.gif when one of the conditions (i), (ii), (iii), (iv), (v), and (vi) in Section 2 holds. Let the space dimension be http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq451_HTML.gif , and let the initial values http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq452_HTML.gif be nonnegative smooth functions and satisfy the homogenous Neumann boundary conditions. If the following condition (4.44) holds, then the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq453_HTML.gif of (1.6) converges to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq454_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq455_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ92_HTML.gif
      (4.44)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq456_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq457_HTML.gif .

      Proof.

      Define the Lyapunov function
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ93_HTML.gif
      (4.45)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq458_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq459_HTML.gif have been given in Theorem 4.6. Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq460_HTML.gif is nonnegative, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq461_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq462_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq463_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq464_HTML.gif is a positive solution of system (1.6), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq465_HTML.gif is well posed for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq466_HTML.gif from Theorem 4.5. According to system (1.6), the time derivative of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq467_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ94_HTML.gif
      (4.46)
      It is easy to check that the final three integrands on the right side of the above expression are positive definite because of the electing of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq468_HTML.gif , and the sufficient and necessary conditions of the first integrand being positive definite are the following:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ95_HTML.gif
      (4.47)
      Noticing that (4.44) is the sufficient conditions of (4.47), so there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq469_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ96_HTML.gif
      (4.48)
      Similar to the tedious calculations of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq470_HTML.gif , using integration by parts, Hölder inequality, and (4.34), one can verify that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq471_HTML.gif is bounded from above. Thus we have from (4.48) and Lemma 3.2 in Section 3 that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ97_HTML.gif
      (4.49)

      In addition, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq472_HTML.gif is decreasing for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq473_HTML.gif , so we can conclude that the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq474_HTML.gif is globally asymptotically stable. The proof of Theorem 4.6 is completed.

      Declarations

      Acknowledgments

      The work of this author was partially supported by the Natural Science Foundation of Anhui Province Education Department (KJ2009B101) and the NSF of Chizhou College (XK0833) (caguhh@yahoo.com.cn). The work of this author was partially supported by the China National Natural Science Foundation (10871160), the NSF of Gansu Province (096RJZA118), and NWNU-KJCXGC-03-47, 61 Foundations (fusm@nwnu.edu.cn).

      Authors’ Affiliations

      (1)
      Department of Mathematics and Computer Science, Chizhou College
      (2)
      College of Mathematics and Information Science, Northwest Normal University

      References

      1. Kuno E: Mathematical models for predator-prey interaction. Advances in Ecological Research 1987, 16: 252-265.
      2. Zheng JB, Yu ZX, Sun JT: Existence and uniqueness of limit cycle for prey-predator systems with sparse effect. Journal of Biomathematics 2001,16(2):156-161.MATHMathSciNet
      3. Shen C, Shen BQ: A necessary and sufficient condition of the existence and uniqueness of the limit cycle for a class of prey-predator systems with sparse effect. Journal of Biomathematics 2003,18(2):207-210.MATHMathSciNet
      4. Huang X, Wang Y, Zhu L: One and three limit cycles in a cubic predator-prey system. Mathematical Methods in the Applied Sciences 2007,30(5):501-511. 10.1002/mma.791MATHMathSciNetView Article
      5. Zhang X, Chen L, Neumann AU: The stage-structured predator-prey model and optimal harvesting policy. Mathematical Biosciences 2000,168(2):201-210. 10.1016/S0025-5564(00)00033-XMATHMathSciNetView Article
      6. Shigesada N, Kawasaki K, Teramoto E: Spatial segregation of interacting species. Journal of Theoretical Biology 1979,79(1):83-99. 10.1016/0022-5193(79)90258-3MathSciNetView Article
      7. Pang PYH, Wang M: Strategy and stationary pattern in a three-species predator-prey model. Journal of Differential Equations 2004,200(2):245-273. 10.1016/j.jde.2004.01.004MATHMathSciNetView Article
      8. Kuto K: Stability of steady-state solutions to a prey-predator system with cross-diffusion. Journal of Differential Equations 2004,197(2):293-314. 10.1016/j.jde.2003.10.016MATHMathSciNetView Article
      9. Dubey B, Das B, Hussain J: A predator-prey interaction model with self and cross-diffusion. Ecological Modelling 2001,141(1–3):67-76.View Article
      10. Lou Y, Ni W-M, Wu Y: On the global existence of a cross-diffusion system. Discrete and Continuous Dynamical Systems 1998,4(2):193-203.MATHMathSciNetView Article
      11. Choi YS, Lui R, Yamada Y: Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete and Continuous Dynamical Systems 2004,10(3):719-730.MATHMathSciNetView Article
      12. Zhang R, Guo L, Fu SM:Global behavior for a diffusive predator-prey model with stage-structure and nonlinear density restriction-II: the case in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq475_HTML.gif . Boundary Value Problems 2009, 2009:-19.
      13. Zhang R, Guo L, Fu SM: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%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq476_HTML.gif . Boundary Value Problems 2009, 2009:-26.
      14. Hale JK: Ordinary Differential Equations. 2nd edition. Robert E. Krieger, Malabar, Fla, USA; 1980:xvi+361.MATH
      15. Ladyzenskaja OA, Solonnikov VA, Uralceva NN: Linear and Quasilinear Partial Differential Equations of Parabolic Type, Translations of Mathematical Monographs. Volume 23. American Mathematical Society, Providence, RI, USA; 1968.
      16. Protter MH, Weinberger HF: Maximum Principles in Differential Equations. 2nd edition. Springer, New York, NY, USA; 1984.MATHView Article
      17. Wang MX: Nonlinear Partial Differential Equations of Parabolic Type. Science Press, Beijing, China; 1993.
      18. May RM: Stability and Complexity in Model Ecosystems. Princeton Univesity Press, Princeton, NJ, USA; 1974.
      19. Henry D: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. Volume 840. Springer, Berlin, Germany; 1993.
      20. Protter MH, Weinberger HF: Maximum Principles in Differential Equations. 2nd edition. Springer, New York, NY, USA; 1984:x+261.MATHView Article
      21. Cao HH, Fu SM: Global solutions for a cubic predator-prey cross-diffusion system with stage structure. Mathematics in Practice and Theory 2008,38(21):161-177.MATHMathSciNet

      Copyright

      © H. Cao and S. Fu. 2010

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.