Open Access

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

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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq2_HTML.gif , respectively. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq3_HTML.gif are positive constants, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq5_HTML.gif is undetermined. https://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 https://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. https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq10_HTML.gif and https://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 https://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. https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq14_HTML.gif is undetermined. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq15_HTML.gif and https://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. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq17_HTML.gif and https://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 https://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. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq20_HTML.gif and https://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. https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ3_HTML.gif
(1.3)
and redenoting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq24_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq25_HTML.gif , system (1.2) reduces to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq27_HTML.gif are positive constants, and https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq30_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq34_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq36_HTML.gif ( https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ6_HTML.gif
(1.6)

In the above, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq39_HTML.gif are positive constants. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq40_HTML.gif and https://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. https://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. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq43_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq46_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq47_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq49_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq52_HTML.gif , and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq54_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ7_HTML.gif
(2.1)

The above https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq56_HTML.gif , and further https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq58_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq61_HTML.gif linearly, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq62_HTML.gif , it follows from (1.4) that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq63_HTML.gif depending only on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq65_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ9_HTML.gif
(2.3)
It follows that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq67_HTML.gif referring to (2.1) such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq68_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq69_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq70_HTML.gif .

Finally, we note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq71_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq72_HTML.gif , then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq73_HTML.gif , https://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 https://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)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq76_HTML.gif ; (ii) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq78_HTML.gif ; (iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq79_HTML.gif ; (iv) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq80_HTML.gif ; (v) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq81_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq83_HTML.gif ; (vi) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq84_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq86_HTML.gif , where
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq88_HTML.gif holds, then https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq91_HTML.gif holds, then https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ13_HTML.gif
(2.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq93_HTML.gif are positive constants. It holds that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq94_HTML.gif for any https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ14_HTML.gif
(2.8)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq97_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ15_HTML.gif
(2.9)
We observe that
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ17_HTML.gif
(2.11)

Set https://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], https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq101_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq102_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq103_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq104_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq106_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq108_HTML.gif .

Theorem 3.1.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq110_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq111_HTML.gif be the maximal existence time. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq112_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq113_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq114_HTML.gif , where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq117_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq118_HTML.gif . Furthermore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq119_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq120_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq121_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq122_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq123_HTML.gif .

Proof.

Let https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq126_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq127_HTML.gif , where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq130_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq131_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq132_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq133_HTML.gif . Next we prove that the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq134_HTML.gif is bounded on https://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 https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ18_HTML.gif
(3.1)

for some positive constant https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq138_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq139_HTML.gif . Using [17, Exercise 5 of Section https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq140_HTML.gif ], we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq141_HTML.gif is also bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq142_HTML.gif . Now note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq143_HTML.gif The maximum principle gives https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq146_HTML.gif be positive constants. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq147_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq148_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq150_HTML.gif is bounded from below. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq151_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq152_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq153_HTML.gif for some constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq154_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq155_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq156_HTML.gif be the eigenvalues of the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq157_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq159_HTML.gif be the eigenspace corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq160_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq161_HTML.gif . Denote that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq162_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq163_HTML.gif is an orthonormal basis of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq164_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq165_HTML.gif . Then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq166_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq167_HTML.gif , where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq168_HTML.gif is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ21_HTML.gif
(3.4)

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

The characteristic polynomial of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq176_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ22_HTML.gif
(3.5)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ23_HTML.gif
(3.6)
Thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ24_HTML.gif
(3.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq178_HTML.gif are given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq179_HTML.gif , the three roots https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq180_HTML.gif of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq182_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq184_HTML.gif . Noting that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq185_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq187_HTML.gif . A direct calculation shows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq188_HTML.gif is negative if
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq189_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ27_HTML.gif
(3.10)
In fact, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq190_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ28_HTML.gif
(3.11)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq191_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq192_HTML.gif , it follows that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq193_HTML.gif are the three roots of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq194_HTML.gif . Thus, there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq195_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq196_HTML.gif such that the three roots https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq197_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq198_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ31_HTML.gif
(3.14)
So
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ32_HTML.gif
(3.15)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ33_HTML.gif
(3.16)

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

Consequently, the spectrum of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq201_HTML.gif , consisting only of eigenvalues, lies in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq203_HTML.gif follows [19, Theorem https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq207_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq209_HTML.gif is bounded uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq210_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq211_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq212_HTML.gif . By [20, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq213_HTML.gif ],
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ34_HTML.gif
(3.17)
Define the Lyapunov function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ35_HTML.gif
(3.18)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq214_HTML.gif for all https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ36_HTML.gif
(3.19)
Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq217_HTML.gif , we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ37_HTML.gif
(3.20)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq218_HTML.gif holds for
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq220_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ39_HTML.gif
(3.22)
As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq222_HTML.gif , it follows that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq223_HTML.gif is bounded in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq225_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq226_HTML.gif . Therefore
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ42_HTML.gif
(3.25)
where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ43_HTML.gif
(3.26)
as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq228_HTML.gif . So we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq229_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq230_HTML.gif . Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq231_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq232_HTML.gif . Therefore, there exists a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq233_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq234_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq235_HTML.gif . As https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq238_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ44_HTML.gif
(3.27)
At https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq240_HTML.gif , thus
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq241_HTML.gif , denoted still by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq242_HTML.gif , and nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq243_HTML.gif , such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq244_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq245_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq246_HTML.gif . Therefore,
https://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 https://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 https://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 https://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 https://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: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq251_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq252_HTML.gif means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq253_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq254_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq255_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq256_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq257_HTML.gif means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq258_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq259_HTML.gif are in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq260_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq261_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq262_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq263_HTML.gif .

To obtain https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq266_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq267_HTML.gif 1) such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq268_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq269_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq270_HTML.gif . To prove that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ50_HTML.gif
(4.2)
Notice that the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq272_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq273_HTML.gif are sufficiently smooth in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq274_HTML.gif , and are quasimonotone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq275_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq276_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq278_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ51_HTML.gif
(4.3)

yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq280_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq281_HTML.gif . It follows that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq282_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq283_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq285_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq286_HTML.gif for the solution to following equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ52_HTML.gif
(4.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq287_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq288_HTML.gif are positive constants and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq289_HTML.gif . Then there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq290_HTML.gif , depending on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq291_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq292_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ53_HTML.gif
(4.5)
Furthermore,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq293_HTML.gif , that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ55_HTML.gif
(4.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq294_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq295_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq296_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq297_HTML.gif are bounded in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq299_HTML.gif , and integrating by parts over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq300_HTML.gif , yields
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ57_HTML.gif
(4.9)
with some https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ58_HTML.gif
(4.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq302_HTML.gif depends on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq303_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq304_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq305_HTML.gif , the elliptic regularity estimate [10, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq306_HTML.gif ] yields
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq307_HTML.gif . Hence, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq309_HTML.gif can be presented by combining Lemmas https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq310_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq311_HTML.gif in [11]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq312_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq313_HTML.gif satisfy
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq314_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq315_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq316_HTML.gif . Then there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq317_HTML.gif independent of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq318_HTML.gif but possibly depending on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq319_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq320_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq321_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq322_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq323_HTML.gif such that
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq325_HTML.gif be a fixed bounded domain and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq326_HTML.gif . Then for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq327_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq328_HTML.gif , one has

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

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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq332_HTML.gif is a positive constant dependent on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq333_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq335_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq337_HTML.gif for some https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq339_HTML.gif when the space dimension is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq340_HTML.gif .

Proof.

Step 1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq341_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq342_HTML.gif -Estimates and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq343_HTML.gif -Estimates of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq345_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ62_HTML.gif
(4.14)
Thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ63_HTML.gif
(4.15)
Furthermore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ64_HTML.gif
(4.16)
Integrating (4.14) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq346_HTML.gif and moving terms yield
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq347_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq348_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq349_HTML.gif yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ67_HTML.gif
(4.19)
Since https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ68_HTML.gif
(4.20)
From (4.1) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq351_HTML.gif , it holds that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ69_HTML.gif
(4.21)
Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq352_HTML.gif and selecting a proper https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq353_HTML.gif such that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ70_HTML.gif
(4.22)
Denote that https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq356_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq357_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq358_HTML.gif ; hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ72_HTML.gif
(4.24)
Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq359_HTML.gif . Then it follows from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq360_HTML.gif -estimates of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq361_HTML.gif namely (4.15), that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ74_HTML.gif
(4.26)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq362_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq364_HTML.gif is bounded, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq365_HTML.gif . It is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq366_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq367_HTML.gif still denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq368_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq369_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ75_HTML.gif
(4.27)
Finally, we observe that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq370_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq371_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq372_HTML.gif . So take https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq374_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ76_HTML.gif
(4.28)

Step 2.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq375_HTML.gif -Estimates of https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ77_HTML.gif
(4.29)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq377_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq378_HTML.gif , https://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 https://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 https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq382_HTML.gif is bounded; (2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq383_HTML.gif (3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq384_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq385_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq386_HTML.gif are positive constants, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq387_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq388_HTML.gif satisfy
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq389_HTML.gif . One can choose https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ79_HTML.gif
(4.31)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq391_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq392_HTML.gif by Lemma 4.1, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq393_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq395_HTML.gif , page 204] to (4.19) yields
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq396_HTML.gif for any https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ81_HTML.gif
(4.33)
Hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq401_HTML.gif . Similarly, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq403_HTML.gif , which imply that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq404_HTML.gif . In addition, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq405_HTML.gif obviously belongs to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq406_HTML.gif . It follows that one can select https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq408_HTML.gif from [15, Theorem https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq410_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq411_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq412_HTML.gif of (1.6) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq413_HTML.gif for Any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq414_HTML.gif . Because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq415_HTML.gif , we have from (4.34) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq416_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq417_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq418_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq419_HTML.gif . It follows from [15, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq420_HTML.gif , page 80] that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq421_HTML.gif . And direct calculation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq422_HTML.gif yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq423_HTML.gif . So we have
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq424_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq425_HTML.gif are all bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq426_HTML.gif . It follows from [15, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq427_HTML.gif page 204] that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq428_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq429_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq430_HTML.gif . Applying the [13, Theorem https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq432_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq433_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq434_HTML.gif . Selecting https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq436_HTML.gif . Still applying Schauder estimate, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ87_HTML.gif
(4.39)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq437_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq438_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ88_HTML.gif
(4.40)
where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq441_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq442_HTML.gif , we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ90_HTML.gif
(4.42)
where https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ91_HTML.gif
(4.43)

In particular, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq445_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq447_HTML.gif , from Sobolev embedding theorem, we have https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq451_HTML.gif , and let the initial values https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq453_HTML.gif of (1.6) converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq454_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq455_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ92_HTML.gif
(4.44)

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

Proof.

Define the Lyapunov function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ93_HTML.gif
(4.45)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq458_HTML.gif and https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq460_HTML.gif is nonnegative, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq461_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq462_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq463_HTML.gif . When https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq465_HTML.gif is well posed for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq467_HTML.gif satisfies
https://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 https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq469_HTML.gif such that
https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_Equ97_HTML.gif
(4.49)

In addition, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F285961/MediaObjects/13661_2009_Article_913_IEq472_HTML.gif is decreasing for https://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 https://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

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Copyright

© H. Cao and S. Fu. 2010

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