Journal of Applied Mathematics

Volume 2013 (2013), Article ID 851028, 8 pages

http://dx.doi.org/10.1155/2013/851028

Research Article

Positive Steady States of a Strongly Coupled Predator-Prey System with Holling-() Functional Response

Xiao-zhou Feng1,2 and Zhi-guo Wang²

¹College of Science, Xi'an Technological University, Xi'an 710032, China

²Institute of Mathematics, Shaanxi Normal University, Xi'an 710062, China

Received 25 December 2012; Accepted 2 March 2013

Academic Editor: Junjie Wei

Copyright © 2013 Xiao-zhou Feng and Zhi-guo Wang. 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.

Abstract.

This paper discusses a predator-prey system with Holling-() functional response and the fractional type nonlinear diffusion term in a bounded domain under homogeneous Neumann boundary condition. The existence and nonexistence results concerning nonconstant positive steady states of the system were obtained. In particular, we prove that the positive constant solution is asymptotically stable when the parameter k satisfies some conditions.

1. Introduction

In this paper, we are interested in the positive steady states of the strongly coupled predator-prey system with Holling- functional response. The specific system is as follows:

where ; is a bounded domain in with smooth boundary ; is the outward directional derivative normal to ; and stand for the densities of the prey and predator; the given coefficients , , , , and are positive constants. The term is named Holling- functional response [1, 2]. In the second equation, the fractional type nonlinear diffusion term models a situation in which the population pressure of the predator species weakens in high-density areas of the prey species. For more precise details, we can refer to [3, 4]. Paper [3] discusses a strongly coupled predator-prey system with nonmonotonic functional response, the existence and nonexistence results concerning nonconstant positive steady states of the system were proved by degree theory. Paper [4] considers the positive steady states for a prey-predator model with some nonlinear diffusion terms, and the sufficient conditions for the existence of positive steady state solutions were obtained by bifurcation theory.

In recent years, there has been considerable interest in the dynamics of strongly coupled reaction-diffusion systems with cross-diffusion. We point out that most efforts have concentrated on the Lotka-Volterra competition system which was proposed first by Shigesada et al. [5]. Since their pioneering work, many authors have studied population models with cross-diffusion terms from various mathematical viewpoints, for example, the global existence of time-depending solutions [6–11], the stability analysis for steady states [12–14], and the steady state problems [15–21]. In this paper, we mainly consider the existence of solutions of (1). The research method refers to [3, 22, 23].

For convenience of the research, we write (1) as the following form:

where

The main work of this paper is to study the effects of the fractional type nonlinear diffusion pressures on the existence of nonconstant positive steady states of (1). Here, a positive solution means a smooth solution with both and being positive. We will demonstrate that the cross-diffusion pressure may help forming more patterns. Obviously, for system (1), one notes that when , there holds , so that the only nonnegative solutions to (1) are and . Consequently, (1) does not have any positive solution. On the other hand, when , the unique positive constant solution to (1) is ; that is,

The organization of our paper is as follows. In Section 2, we establish a priori upper and lower bounds for positive solutions of (1). In Section 3, we use a degree theory to develop a general result to enable one to conclude the existence or nonexistence of nonconstant steady-state solutions or patterns as the index of positive constant steady states changes. In Section 4, we establish the existence of nonconstant positive solutions to (1) for a large range of diffusion and cross-diffusion coefficients. Meanwhile, we prove that the positive constant solution is asymptotically stable for different ranges of parameters.

2. Upper and Lower Bounds for Positive Solutions

The main purpose of this section is to give a priori upper and lower positive bounds for positive solutions of (1). Firstly, we cite two known results.

Harnack Inequality (see [24]). Let and be a positive classical solution to in subject to the homogeneous Neumann boundary condition. Then, there exists a positive constant such that .

Maximum Principle (see [25]). Suppose that . If satisfies

and , then .

Theorem 1. Let and , be fixed constants. Assume that , and there exists a positive constant , such that any positive solution of (1) satisfies

Proof. Let and be defined as in (3), and denote that

Then, (1) becomes

For the first equation of (9), by the Maximum Principle, we have

The function satisfies

where . It is easy to verify that the norm

then, is bounded by a constant depending only on and . By the Harnack inequality, we have

where . It follows that

Integrating the equations of (1) over by parts and making use of the boundary conditions, we have

For any , we denote that . Then, from (15) and (10), we have

Along with (14), we have

Similarly, consider the equation of as follows:

where . Then, is bounded by a constant depending only on and . By the Harnack Inequality, there exists a positive constant such that . Then,

Thus, along with (10), (14), and (17), we can complete the proof.

Theorem 2. Assume that . Let , be fixed constants. There exists a constant , such that any positive solution of (1) satisfies

provided that and .

Proof. Since , there exists , such that ; that is,

It follows that . Consequently, by in Theorem 1, we have

where and .

In the following, we mainly prove that as . Now, we suppose that claim is not true; then, there exists a sequence with and . And the positive solution of (1) corresponding to is such that

By (14), we have , and then, ; furthermore,   uniformly holds as . Then, the first equation of (1) becomes the following:

According to (7), there exists such that ,  , where is a positive constant which does not depend on . For each given in problem (24), it follows from estimate that , where . Let ; then, by Sobolev Imbedding Theorems, we get

where . We choose such that Imbedding is compact, and along with elliptic equation regularity theory, there exists the subsequence of , which is still denoted by , and exists such that uniformly holds in . On the other hand, , and when , the limit of (24) becomes the following problem:

Applying Maximum Principle to problem (26) and noting that , we have .

However, when , the limit of (24) becomes the following problem:

which implies that for some nonnegative constant . Since , by letting and noting that ,  , and , we also have .

By a similar argument as that in (24), for the second equation of (1), we can prove that there exists a subsequence in , such that . Since , then . Dividing the second equation of (1) by , and integrating over , we have

Let , and note that and ; then, . This contradiction to the assumption completes the proof.

3. A Result on Degree Theory

In this section, we obtain nonexistence of nonconstant positive solutions to (1) as . Meanwhile, by degree theory, a general result to establish the existence of nonconstant positive solutions to (1) in the next section is proved.

Denote and . We will fix and take as bifurcation parameters, the dependence of will often be suppressed. Define

Since

and is positive for all nonnegative , exists. Hence, is a positive solution to (1) if and only if

where is the inverse of in . As is a compact perturbation of the identity operator , the Leray-Schauder is well defined if for all .

By Theorems 1 and 2, there exists a positive constant such that if in , then . Note that can be taken as a continuous function for . By the invariance property of the Leray-Schauder degree, we then conclude that does not depend on if , and it also does not depend on , if changes continuously in without touching the surface .

For the case , by degree invariance, we need only consider a special ; say with large . For this we can use the following nonexistence result. To compare the existence regions of (1) with and without cross-diffusion, we give a nonexistence result stronger than what is needed here.

Theorem 3. Let be given. Then, there exists a positive constant such that when and , (1) does not have any nonconstant positive solution.

Proof. First, by (7), there exists a positive constant such that a classical solution to (1) satisfies , provided that . Now, we write as average of over , where

Multiplying the equation for of (1) by and integrating over by parts, we have

for some positive constant and , where and lie between and , and , respectively. Similarly, we have

for some positive constant and , where and are the same as in (33). Adding (33) and (34), we obtain

It follows from the Poincaré Inequality that

Since we can choose such that , we may also choose and sufficiently small such that . Consequently, by (36),

which implies that constant, and, hence, constant if . Thus, we complete the proof of the theorem.

In the following, we only calculate when all solutions to are positive constant solutions in .

Let be the eigenvalues of the operator in with zero flux boundary condition, be the eigenspace corresponding to in ,  ,  , an orthonormal basis of , and . Then, , where . This decomposed method is similar to that of [22].

Let be a positive root to . We can calculate

where is a linear mapping from to itself.

Denote that .

Lemma 4. Let be a positive root to , and assume that for all . Then,

Proof. If is invertible, then the index of at is defined as index, where is the number of eigenvalues of with negative real parts. The is then equal to summation of the indexes over all solutions to in , provided that on . Since for each integer is invariant under , and is an eigenvalue of on if and only if is an eigenvalue of following matrix:

Since ,   is invertible. Therefore, the number of eigenvalues with negative real parts of on is odd if and only if , and therefore

Theorem 5. Assume that with . Then, for every , there exists a constant such that for every ,

Proof. Since (1) does not have any positive solution in , when , we then conclude that

Next, to complete the proof of Theorem 5. We need only to calculate the degree for the case . In this case, by Theorem 3, all positive solutions to are the unique positive constant solution to , denoted by , when so that ,  . This implies that when is sufficiently large, for all , and thus . It follows from Lemma 4 that . This completes the proof.

Remark 6. The change of degree when passes the borderline is due to the appearance (disappearance) of a positive constant steady-state bifurcating from .

4. Existence of Nonconstant Positive Solutions

In this section, we establish the existence of positive nonconstant solutions for (1). In particular, we show that for certain ranges of parameters where (1) does not have any positive nonconstant steady state, our model can still produce patterns. The idea is as follows. First we calculate the index of at positive constant steady states. Suppose that the sum of all these indices is not equal to the degree stated in Theorem 5. Then, in for must have a nonconstant positive solution, which also solves (1).

In the following, we always assume that , and (1) has a unique positive constant solution , when is a positive solution to . We can get the following results at by simply calculating

To calculate the roots of , we will restrict our attention to large . Note that

where

The sign of the trace is determined by and the sign of the determinant .

Hence, we will discuss separately the following cases:

(i) , obviously , then ;(ii) :

(iia)if , then ;(iib)if , then .

4.1. The Case   

         In this subsection, we consider local stability of the constant steady state for evolution dynamics

where is a positive constant solution to by (5).

Theorem 7. Let   . Then the positive constant solution is asymptotically stable with respect to the dynamics (47). Consequently, in a small neighborhood of  , (1) does not have any nonconstant positive solution.

Proof. The linearization of (47) at takes the form

Denote that the corresponding linear operator . For each eigen-pair    of on with no flux boundary condition, is a solution to (48) if and only if is an eigen-pair of the matrix . is invariant under the operator . From (44), we have

Then

Hence, we conclude that the two eigenvalues and of have negative real parts. More specifically, we have the following:

(i) For , , if , then ; if , then

(ii) For , as is increasing with respect to and as , it follows that if , then

if , since and , then

for some positive constant that does not depend on . The previous arguments show that there exists a positive constant , which does not depend on , such that , . Consequently, the spectrum of , which consists of eigenvalues, lies in . It then follows from Theorem of [26, page 98] that the constant steady-state is asymptotically stable to (47).

4.2. The Case

4.2.1.

Theorem 8. Let . Then the positive constant solution is asymptotically stable with respect to the dynamics (47).

Proof. Similar to the proof of Theorem 7.

4.2.2.

In this case, is the only positive constant solution to . By fixing the diffusion coefficients (for prey) and using the diffusion coefficients and (for predator) as bifurcation parameters, we will show that (1) can create nonconstant positive solutions. We want to emphasize that it is caused by the presence of cross-diffusion which has a more complex role than that of the diffusion coefficients and .

Theorem 9 (existence with suitable and ). Assume that and , define and as (46).

(i) Suppose that and are given such that for some positive even integer . There exists a positive constant such that if , then (1) has at least one nonconstant positive solution.

(ii) Suppose that is given such that for some positive even integer . Then, for any given , there exists a positive constant such that if , (1) has at least one nonconstant positive solution.

Proof. Denote by , with , the two roots to , and then

From (45), we see that

Suppose that for some positive even integer ; then, there exists a positive constant such that , and we have

Hence, if , is equivalent to , since is even. It follows from Lemma 4 that

Consequently, has at least one nonconstant positive solution that is different from the constant function . Otherwise, the degree of in would be for all large enough , which would contradict Theorem 5. This proves, the first assertion of the theorem, and the second assertion is similarly proved.

Remark 10. For to be positive, it is necessary and sufficient to have

When this inequality holds, is also positive provided that is large, and we then can adjust to and make the assumptions in (i) or (ii) of theorem hold.

Remark 11. In fact, if , then . This case is similar to case (i). Furthermore, the positive constant solution is also asymptotically stable with respect to the dynamics (47).

Acknowledgments

This project is supported by the NSF of China under Grant 11001160, the Scientific Research Plan Projects of Shaanxi Education Department (no. 12JK0865), and the President Fund of Xi’an Technological University (XAGDXJJ1136).

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