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

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 [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).

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:

where . Let one curve : , and the other curve : . Obviously, passes the point . Noting that attains its maximum at , thus when , . has the asymptote and passes the point . In this case, and have unique intersection , as shown in Figure 1. is the unique positive equilibrium point of (1.2), where , , . In addition, the restriction of the existence of the positive equilibrium can be removed, if .

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).

Theorem 2.1.

(i) Assume that (2.1),

hold, then the equilibrium point of (1.2) is globally asymptotically stable.

(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.

Proof.

(i) Define the Lyapunov function

Calculating the derivative of along the positive solution of (1.2), we have

When , the minimum of and is and 0, respectively; the maximum of is are and , respectively. Thus, when (2.8) hold, According to the Lyapunov-LaSalle invariance principle [28], is globally asymptotically stable if (2.1)–(2.3) hold.

(ii) Let

Then

Noting that the maximum of is , and , we find . Therefore,

(iii) Let

then

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

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 be nonnegative smooth functions on . Then system (1.3) has a unique nonnegative solution , and

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.

When (namely ), 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

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

Theorem 3.3.

(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.

Let be the eigenvalues of the operator on with Neumann boundary condition, and let be the eigenspace corresponding to in . Let

where is an orthonormal basis of , then

Let , , where

The linearization of (1.3) is at . For each , is invariant under the operator L, and is an eigenvalue of L on , if and only if is an eigenvalue of the matrix . The characteristic equation is , where

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).

We can conclude that there exists a positive constant , such that

In fact, let , then

Since as , it follows that

Clearly, has the three roots . Let . By continuity, there exists such that the three roots of satisfy

Let , then . Let , then (3.7) holds. According to [30, Theorem  5.1.1], we have the locally asymptotically stability of .

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

The characteristic equation of is , where

The three roots of all have negative real parts for and . Namely, is the locally asymptotically stable, if and .

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

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

Theorem 3.6.

(i) Assume that (2.1),

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

(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.

Let be the unique positive solution of (1.3). By Theorem 3.1, there exists a positive constant C which is independent of and such that , for . By [33, Theorem ],

(i) Define the Lyapunov function

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 .

Calculating the derivative of along positive solution of (1.3) by integration by parts and the Cauchy inequality, we have

It is not hard to verify that

if (3.14) hold. Applying Lemma 3.5, we can obtain

Recomputing , we find

From (3.15), we can see that is bounded in , . It follows from Lemma 3.5 and (3.15) that as . Namely,

Using the Pioncaré inequality, we have

where Noting that

according to (3.19) and (3.22), we can see

Thus, there exists as . Applying the boundness of , there exists a subsequence of , denoted still by , such that On the one hand

On the other hand

According to (3.19) to compute the limit of the previous equation and using the uniqueness of the limit, we have , and

It follows from (3.15) that there exists a subsequence of , denoted still by , and nonnegative functions , such that

Applying (3.19)–(3.27), we obtain that , and

In view of Theorem 3.3, we can conclude that is globally asymptotically stable.

(ii) Let

Then

Therefore, It follows that the equilibrium point of (1.3) is globally asymptotically stable.

(iii) Define

Then

When ,

The following proof is similar to (i).

Remark 3.7.

When , Theorem 3.6 shows the following.

(i) Assume that ,

hold, then the equilibrium point of (1.3) is globally asymptotically stable.

(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.

Consider the following system:

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.

By [34–36], we have the following result.

Theorem 4.1.

If , then (1.4) has a unique nonnegative solution , where is the maximal existence time of the solution. If the solution satisfies the estimate

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.

Let be a solution of (1.4). Then

where .

Proof.

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

Multiplying the second equation of (1.4) by , adding up the first equation of (1.4), and integrating the result over , we obtain

Using Young inequality and Hlder inequality, we have

where It follows from (4.3) and (4.4) that

Thus,

where depends on and coefficients of (1.4). In addition, there exists a positive constant , such that

Integrating the first equation of (1.4) over , we have

Integrating (4.8) from to , we have

According to (4.7), there exists a positive constant , such that

Multiplying the second equation of (1.4) by and integrating it over , we obtain

Integrating the previous inequation from to , we have

Lemma 4.4.

Let be a solution of (1.4), , and . Then there exists a positive constant depending on and , such that

Furthermore and

Proof.

satisfies the equation

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

Multiply the second equation of (1.4) by and integrate it over to obtain

Then

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]).

Let , , and let be any number which may depend on . Then there is a constant depending on , and such that

for any with for all .

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

Lemma 4.6.

Let , , then there exist positive constants and , such that

Proof.

Multiply the first equation of (1.4) by for and integrate by parts over to obtain

Integrating (4.19) from 0 to , we have

Then substitution of , into (4.20) leads to

It follows from Hlder inequality and Lemma 4.3 that

Note that , and for . From Hlder inequality, Young inequality, and Lemma 4.4, we have

Substitution of (4.22) and (4.23) into (4.21) leads to

where is arbitrary and .

Choose such that

then it follows from (4.24) that

Let

Then for

According to Lemma 4.5 and the definition of , we can see

Combining (4.26) and (4.29), we have

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 .

Multiplying the second equation of (1.4) by and integrating it over , we have

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

Lemma 4.7.

Let , then there exists a positive constant such that

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

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

By the third equation of (1.4), we have

It follows from Lemma 4.3 that is bounded in . Applying Theorem [37, Page 204] to (4.34), we have

Recall that satisfy (4.14) in Lemma 4.4, that is,

where is bounded. Since by (4.35), applying Theorem [37, page 341-342] to (4.36), we have

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 .

Secondly, for , applying Lemma [37, Page 80] to (4.36), we have

Since , we obtain

The first two equations can be written in the divergence form as

where . It follows from Lemmas 4.1, 4.5, and (4.39) that are bounded. Thus applying Theorem [37, Page 204] to (4.40) leads to

We rewrite the third equation of (1.4) as

where . Applying Schauder estimate [29, Theorem , page 114] to (4.42) gives

Let

then

where , . From (4.41), we have . It follows from (4.41) and (4.43) that . Applying Schauder estimate to (4.45) gives

Solving equations (4.44) for , respectively, we have

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.

Assume that the all conditions in Theorem 4.2, (2.1), and

hold. Let be the unique positive equilibrium point of (1.4), and let be a positive solution for (1.4). Then

provided that is large enough.

Proof.

Define the Lyapunov function

Let be a positive solution of (1.4), Then

The first integrand in the right hand of the previous inequality is positive definite if

Therefore, when the all conditions in Theorem 4.8 hold, there exists a positive constant such that

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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