# Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-I: The Case in

- Rui Zhang
^{1}, - Ling Guo
^{2}and - Shengmao Fu
^{2}Email author

**2009**:378763

**DOI: **10.1155/2009/378763

© Rui Zhang et al. 2009

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

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

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

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

where

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

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

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

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

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

## 2. Global Stability for System (1.2)

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

The Jacobian matrix of the equilibrium is

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

The Jacobian matrix of the equilibrium is

where . The characteristic equation of ( ) is , where

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

The Jacobian matrix of the equilibrium is

where

The characteristic equation of ( ) is , where

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

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

- (i)Assume that (2.1),(2.8)

- (ii)
Assume that , and hold, then the equilibrium point of (1.2) is globally asymptotically stable.

- (iii)
Assume that holds, then the equilibrium point of (1.2) is globally asymptotically stable.

- (i)Define the Lyapunov function(2.9)

- (ii)Let(2.11)

- (iii)Let(2.13)

Thus, for . This completes the proof of Theorem 2.1.

## 3. Global Behavior of System (1.3)

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

Theorem 3.1.

on . In particular, if , then for all .

Proof.

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

Remark 3.2.

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

- (i)
Assume that (2.1) holds, then the equilibrium point of (1.3) is locally asymptotically stable.

- (ii)
Assume that , , and hold, then the equilibrium point of (1.3) is locally asymptotically stable.

- (iii)
Assume that holds, then the equilibrium point of (1.3) is locally asymptotically stable.

Proof.

From Routh-Hurwitz criterion, we can see that three eigenvalues (denoted by , , ) all have negative real parts if and only if . Noting that , we must have . It is easy to check that if (see Section 2).

- (ii)The linearization of (1.4) is at , where , and(3.11)

- (iii)The linearization of (1.3) is at , where , and(3.13)

Similar to (i), is locally asymptotically stable, when .

Remark 3.4.

When denote . If , then (1.3) has the semitrivial equilibrium point , where . If , then (1.3) has a unique positive equilibrium point . Similar as Theorem 3.3, we have the following.

(i)If , , and (namely, , , ), then is locally asymptotically stable.

(ii)If and , then is locally asymptotically stable.

(iii)If , then is locally asymptotically stable.

Before discussing the global stability, we give an important lemma which has been proved in [31, Lemma ] or in [32, Lemma ].

Lemma 3.5.

Let be positive constants. Assume that , , and is bounded from below. If and for some positive constant , then

- (i)Assume that (2.1),(3.14)

- (ii)
Assume that , and hold, then the equilibrium point of system (1.3) is globally asymptotically stable.

(iii)Assume that and hold, then the equilibrium point of system (1.3) is globally asymptotically stable.

Proof.

- (i)Define the Lyapunov function(3.16)

By Theorem 3.1, is defined well for all solutions of (1.3) with the initial functions . It is easily see that and if and only if .

- (ii)Let(3.30)

- (iii)Define(3.32)

The following proof is similar to (i).

Remark 3.7.

- (i)Assume that ,(3.35)

- (ii)
Assume that hold, then the equilibrium point of (1.3) is globally asymptotically stable.

- (iii)
Assume that and hold, then the equilibrium point of (1.3) is globally asymptotically stable.

Example 3.8.

Using the software Matlab, one can obtain , . 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 [34–36], we have the following result.

Theorem 4.1.

then . If, in addition, , then

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

Theorem 4.2.

Let and the space dimension . Suppose that are nonnegative functions and satisfy zero Neumann boundary conditions. Then (1.4) has a unique nonnegative solution

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

Lemma 4.3.

where .

Proof.

From the maximum principle for parabolic equations, it is not hard to verify that and is bounded.

Lemma 4.4.

Furthermore and

Proof.

where are functions of and so are bounded because of Lemma 4.3.

and . From a disposal similar to the proof of Lemma 2.2 in [23], we have . Using a standard embedding result, we obtain

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

for any with for all .

To obtain -estimates of , we establish -estimates of in the following lemma.

Lemma 4.6.

Proof.

where is arbitrary and .

where . Therefore is bounded from (4.30).

From (4.29), we have . Namely, , . Combining (4.28), we have , where .

Setting in (4.20) (it is easily checked that , i.e., ), we have .

The result of can be obtained from an analogue of the previous proof of 's.

Lemma 4.7.

Proof.

where , , , is symbol. It follows from Lemma 4.6 that , .

It follows from [37, Lemma , page 80] that and so . Recall from Lemma 4.6 that , so that by applying Theorem [37, Page 181] to (4.33).

Proof of Theorem 4.2.

Firstly, Theorem 4.2 can be proved in a similar way as Theorem in [21, 25] when the space dimension .

In particular, to conclude , we need to repeat the above bootstrap technique. Since is arbitrary, so the classical solution of (1.4) exists globally in time.

Now we discuss the global stability of the positive equilibrium (see Section 2) for (1.4).

Theorem 4.8.

provided that is large enough.

Proof.

This implies that . 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

## References

- Aiello WG, Freedman HI: A time-delay model of single-species growth with stage structure.
*Mathematical Biosciences*1990, 101(2):139–153. 10.1016/0025-5564(90)90019-UMATHMathSciNetView Article - Zhang X, Chen L, Neumann AU: The stage-structured predator-prey model and optimal harvesting policy.
*Mathematical Biosciences*2000, 168(2):201–210. 10.1016/S0025-5564(00)00033-XMATHMathSciNetView Article - Liu S, Chen L, Liu Z: Extinction and permanence in nonautonomous competitive system with stage structure.
*Journal of Mathematical Analysis and Applications*2002, 274(2):667–684. 10.1016/S0022-247X(02)00329-3MATHMathSciNetView Article - Lin Z: Time delayed parabolic system in a two-species competitive model with stage structure.
*Journal of Mathematical Analysis and Applications*2006, 315(1):202–215. 10.1016/j.jmaa.2005.06.012MATHMathSciNetView Article - Xu R: A reaction-diffusion predator-prey model with stage structure and nonlocal delay.
*Applied Mathematics and Computation*2006, 175(2):984–1006. 10.1016/j.amc.2005.08.014MATHMathSciNetView Article - Xu R, Chaplain MAJ, Davidson FA: Global convergence of a reaction-diffusion predator-prey model with stage structure for the predator.
*Applied Mathematics and Computation*2006, 176(1):388–401. 10.1016/j.amc.2005.09.028MATHMathSciNetView Article - Xu R, Chaplain MAJ, Davidson FA: Global convergence of a reaction-diffusion predator-prey model with stage structure and nonlocal delays.
*Computers & Mathematics with Applications*2007, 53(5):770–788. 10.1016/j.camwa.2007.02.002MATHMathSciNetView Article - Wang M: Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion.
*Mathematical Biosciences*2008, 212(2):149–160. 10.1016/j.mbs.2007.08.008MATHMathSciNetView Article - Wang Z, Wu J: Qualitative analysis for a ratio-dependent predator-prey model with stage structure and diffusion.
*Nonlinear Analysis: Real World Applications*2008, 9(5):2270–2287. 10.1016/j.nonrwa.2007.08.004MATHMathSciNetView Article - Galiano G, Garzón ML, Jüngel A: Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model.
*Numerische Mathematik*2003, 93(4):655–673. 10.1007/s002110200406MATHMathSciNetView Article - Chen L:
*Mathematical Models and Methods in Ecology*. Science Press, Beijing, China; 1988. - Chen LJ, Sun JH: The uniqueness of a limit cycle for a class of Holling models with functional responses.
*Acta Mathematica Sinica*2002, 45(2):383–388.MATHMathSciNet - Li W-T, Wu S-L: Traveling waves in a diffusive predator-prey model with Holling type-III functional response.
*Chaos, Solitons & Fractals*2008, 37(2):476–486. 10.1016/j.chaos.2006.09.039MATHMathSciNetView Article - Ko W, Ryu K: Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge.
*Journal of Differential Equations*2006, 231(2):534–550. 10.1016/j.jde.2006.08.001MATHMathSciNetView Article - Zhang H, Georgescu P, Chen L: An impulsive predator-prey system with Beddington-DeAngelis functional response and time delay.
*International Journal of Biomathematics*2008, 1(1):1–17. 10.1142/S1793524508000072MATHMathSciNetView Article - Fan Y, Wang L, Wang M: Notes on multiple bifurcations in a delayed predator-prey model with nonmonotonic functional response.
*International Journal of Biomathematics*2009, 2(2):129–138. 10.1142/S1793524509000583MathSciNetView Article - Wang F, An Y: Existence of nontrivial solution for a nonlocal elliptic equation with nonlinear boundary condition.
*Boundary Value Problems*2009, 2009:-8. - Lou Y, Ni W-M: Diffusion, self-diffusion and cross-diffusion.
*Journal of Differential Equations*1996, 131(1):79–131. 10.1006/jdeq.1996.0157MATHMathSciNetView Article - Lou Y, Ni W-M, Wu Y: On the global existence of a cross-diffusion system.
*Discrete and Continuous Dynamical Systems*1998, 4(2):193–203.MATHMathSciNetView Article - Shim S-A: Uniform boundedness and convergence of solutions to cross-diffusion systems.
*Journal of Differential Equations*2002, 185(1):281–305. 10.1006/jdeq.2002.4169MATHMathSciNetView Article - Shim S-A: Uniform boundedness and convergence of solutions to the systems with cross-diffusions dominated by self-diffusions.
*Nonlinear Analysis: Real World Applications*2003, 4(1):65–86. 10.1016/S1468-1218(02)00014-7MATHMathSciNetView Article - Choi YS, Lui R, Yamada Y: Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion.
*Discrete and Continuous Dynamical Systems*2003, 9(5):1193–1200.MATHMathSciNetView Article - Choi YS, Lui R, Yamada Y: Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion.
*Discrete and Continuous Dynamical Systems*2004, 10(3):719–730.MATHMathSciNetView Article - Pang PYH, Wang MX: Existence of global solutions for a three-species predator-prey model with cross-diffusion.
*Mathematische Nachrichten*2008, 281(4):555–560. 10.1002/mana.200510624MATHMathSciNetView Article - Fu S, Wen Z, Cui S: Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model.
*Nonlinear Analysis: Real World Applications*2008, 9(2):272–289. 10.1016/j.nonrwa.2006.10.003MATHMathSciNetView Article - Yang F, Fu S: Global solutions for a tritrophic food chain model with diffusion.
*The Rocky Mountain Journal of Mathematics*2008, 38(5):1785–1812. 10.1216/RMJ-2008-38-5-1785MATHMathSciNetView Article - Dubey B, Das B, Hussain J: A predator-prey interaction model with self and cross-diffusion.
*Ecological Modelling*2001, 141(1–3):67–76.View Article - Hale JK:
*Ordinary Differential Equations*. Krieger, Malabar, Fla, USA; 1980.MATH - Ye Q, Li Z:
*Introduction to Reaction-Diffusion Equations*. Science Press, Beijing, China; 1999. - Henry D:
*Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics*.*Volume 840*. Springer, Berlin, Germany; 1993. - Lin Z, Pedersen M: Stability in a diffusive food-chain model with Michaelis-Menten functional response.
*Nonlinear Analysis: Theory, Methods & Applications*2004, 57(3):421–433. 10.1016/j.na.2004.02.022MATHMathSciNetView Article - Wang M:
*Nonliear Parabolic Equation of Parabolic Type*. Science Press, Beijing, China; 1993. - Brown KJ, Dunne PC, Gardner RA: A semilinear parabolic system arising in the theory of superconductivity.
*Journal of Differential Equations*1981, 40(2):232–252. 10.1016/0022-0396(81)90020-6MATHMathSciNetView Article - Amann H: Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations.
*Nonlinear Analysis: Theory, Methods & Applications*1988, 12(9):895–919. 10.1016/0362-546X(88)90073-9MATHMathSciNetView Article - Amann H: Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems.
*Differential and Integral Equations*1990, 3(1):13–75.MATHMathSciNet - Amann H: Dynamic theory of quasilinear parabolic systems. III. Global existence.
*Mathematische Zeitschrift*1989, 202(2):219–250. 10.1007/BF01215256MATHMathSciNetView Article - Ladyženskaja OA, Solonnikov VA, Ural'ceva NN:
*Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs*.*Volume 23*. American Mathematical Society, Providence, RI, USA; 1967:xi+648.

## Copyright

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