Open Access

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

Boundary Value Problems20092009:378763

DOI: 10.1155/2009/378763

Received: 2 April 2009

Accepted: 31 August 2009

Published: 27 September 2009

Abstract

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

1. Introduction

Population models with stage structure have been investigated by many researchers, and various methods and techniques have been used to study the existence and qualitative properties of solutions [19]. However, most of the discussions in these works are devoted to either systems of ODE or weakly coupled systems of reaction-diffusion equations. In this paper we investigate the global existence and convergence of solutions for a strongly coupled cross-diffusion predator-prey model with stage structure and nonlinear density restriction. Nonlinear problems of this kind are quite difficult to deal with since the usual idea to apply maximum principle arguments to get priori estimates cannot be used here [10].

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

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

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

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

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

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

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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq14_HTML.gif is a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq15_HTML.gif with smooth boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq17_HTML.gif is the outward unit normal vector on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq18_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq19_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq20_HTML.gif are nonnegative smooth functions on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq21_HTML.gif . The diffusion coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq22_HTML.gif are positive constants. The homogeneous Neumann boundary condition indicates that system (1.3) is self-contained with zero population flux across the boundary. The knowledge for system (1.3) is limited (see [1317]).

In the recent years there has been considerable interest to investigate the global behavior for models of interacting populations with linear density restriction by taking into account the effect of self-as well as cross-diffusion [1826]. In this paper we are led to the following cross-diffusion system:

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq23_HTML.gif are the diffusion rates of the three species, respectively. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq24_HTML.gif are referred as self-diffusion pressures, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq25_HTML.gif is cross-diffusion pressure. The term self-diffusion implies the movement of individuals from a higher to a lower concentration region. Cross-diffusion expresses the population fluxes of one species due to the presence of the other species. The value of the cross-diffusion coefficient may be positive, negative, or zero. The term positive cross-diffusion coefficient denotes the movement of the species in the direction of lower concentration of another species and negative cross-diffusion coefficient denotes that one species tends to diffuse in the direction of higher concentration of another species [27]. For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq26_HTML.gif , problem (1.4) becomes strongly coupled with a full diffusion matrix. As far as the authors are aware, very few results are known for cross-diffusion systems with stage-structure.

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

2. Global Stability for System (1.2)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq27_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq28_HTML.gif , then (1.2) has semitrivial equilibria https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq29_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq30_HTML.gif . To discuss the existence of the positive equilibrium point of (1.2), we give the following assumptions:

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

Figure 1

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

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

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

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

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

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

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

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

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

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

where

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

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

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

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

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

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

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

     
  2. (iii)

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

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

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq101_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq102_HTML.gif . This completes the proof of Theorem 2.1.

3. Global Behavior of System (1.3)

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

Theorem 3.1.

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

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

Proof.

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

Remark 3.2.

When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq118_HTML.gif (namely https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_IEq119_HTML.gif ), system (1.3) reduces to a system in which the immature population of the prey is linear density restriction. Similar to the proof of Theorem 3.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F378763/MediaObjects/13661_2009_Article_843_Equ20_HTML.gif
(3.2)

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

Theorem 3.3.
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     

Proof.

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

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

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

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

Remark 3.4.

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

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

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

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

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

Lemma 3.5.

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

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

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

     

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

Proof.

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

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

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

The following proof is similar to (i).

Remark 3.7.

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

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

     
  2. (iii)

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

     

Example 3.8.

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

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

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

By [3436], we have the following result.

Theorem 4.1.

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

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

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

Theorem 4.2.

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

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

Lemma 4.3.

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

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

Proof.

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

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

Lemma 4.4.

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

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

Proof.

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

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

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

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

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

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

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

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

Lemma 4.6.

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

Proof.

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

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

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

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

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

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

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

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

Lemma 4.7.

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

Proof.

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

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

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

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

Proof of Theorem 4.2.

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

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

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

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

Theorem 4.8.

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

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

Proof.

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

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

Declarations

Acknowledgments

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

Authors’ Affiliations

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

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