Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-I: The Case in
© Rui Zhang et al. 2009
Received: 2 April 2009
Accepted: 31 August 2009
Published: 27 September 2009
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.
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 .
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
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 . 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
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)
Assume that , and hold, then the equilibrium point of (1.2) is globally asymptotically stable.
Assume that holds, then the equilibrium point of (1.2) is globally asymptotically stable.
- (i)Define the Lyapunov function(2.9)
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).
on . In particular, if , then for all .
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.
Now we show the local and global stability of constant equilibrium solutions for (1.3), respectively.
Assume that (2.1) holds, then the equilibrium point of (1.3) is locally asymptotically stable.
Assume that , , and hold, then the equilibrium point of (1.3) is locally asymptotically stable.
Assume that holds, then the equilibrium point of (1.3) is locally asymptotically stable.
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 .
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.
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)
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.
- (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 .
The following proof is similar to (i).
- (i)Assume that ,(3.35)
Assume that hold, then the equilibrium point of (1.3) is globally asymptotically stable.
Assume that and hold, then the equilibrium point of (1.3) is globally asymptotically stable.
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)
then . If, in addition, , then
In this section, we consider the existence and the convergence of global solutions to the system (1.4).
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.
From the maximum principle for parabolic equations, it is not hard to verify that and is bounded.
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 , 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.
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.
where , , , is symbol. It follows from Lemma 4.6 that , .
Proof of Theorem 4.2.
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).
provided that is large enough.
This implies that . So the proof of Theorem 4.8 is completed.
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.
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