Journal of Applied Mathematics

Volume 2012 (2012), Article ID 627419, 30 pages

doi:10.1155/2012/627419

Research Article

Uniqueness and Multiplicity of a Prey-Predator Model with Predator Saturation and Competition

Xiaozhou Feng1,2 and Lifeng Li²

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

²College of Science, Xi'an University of Posts and Telecommunications, Xi'an, Shaanxi 710062, China

Received 15 September 2012; Revised 26 November 2012; Accepted 10 December 2012

Academic Editor: Junjie Wei

Copyright © 2012 Xiaozhou Feng and Lifeng Li. 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.

We investigate positive solutions of a prey-predator model with predator saturation and competition under homogeneous Dirichlet boundary conditions. First, the existence of positive solutions and some sufficient and necessary conditions is established by using the standard fixed point index theory in cones. Second, the changes of solution branches, multiplicity, uniqueness, and stability of positive solutions are obtained by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory. Finally, the exact number and type of positive solutions are proved when or converges to infinity.

1. Introduction

Considering the destabilizing force of predator saturation and the stabilizing force of competition for prey, Bazykin [1] proposed the function response in the prey-predator model instead of the classical Holling-type II functional response. For this functional response, the prey-predator model is taken as the following form:

In this paper, we are concerned with the positive solution of the boundary value problem of the following elliptic system corresponding to the system (1.1):

where is a bounded domain in with smooth boundary ;   are positive constants; is constant; and are nonnegative constants.

If , then (1.2) is reduced to the classical Lotka-Volterra prey-predator model which has received extensive study in the last decade, see [2–8]. In particular, the existence of positive solutions for this case was completely understood, see Dancer [8]. It has been conjectured that there is at most one positive solution, but this was shown only for the case the space dimension is one, see [9]. For space dimension greater than one, this is still an open problem; we also refer to [10, 11] for some partial results on uniqueness. The stability of positive solutions was studied in [10, 11], but the results are still far from being complete.

The case when and was first studied by Blat and Brown [12]. In this case, the term is known as the Holling-Tanner interaction term, and we refer to [5, 12–17] for more discussion on this model. In [12], Blat and Brown studied the existence of positive solutions to (1.2) by making use of both local and global bifurcation theories. The case when goes to infinity was extensively studied by Du and Lou in [13, 14, 18]. They gave a good understanding of the existence, stability, and number of positive solutions for large .

However, the case when and was first studied by Bazykin in the paper [1], more detailed background on this case, we can refer to [1]. And more works can refer to [19], Wang studies the existence, multiplicity, and stability of positive solutions of (1.2). However, Our work is more specific and meticulous than theirs. In particular. Firstly, the changes of solution branches, uniqueness, and stability of positive solutions are obtained by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory. Secondly, the exact number and type of positive solutions are proved when or is large.

This paper is organized as follows: in Section 2, we give sufficient and necessary conditions for the existence of coexistence states of (1.2) by using index theory. In Section 3, by using as a main bifurcation parameter, the multiplicity of coexistence stats to (1.2) is investigated in the gap between the sufficient and necessary conditions for the existence of coexistence states which are found in Section 2. In Section 4, the multiplicity, uniqueness, and stability of coexistence states of (1.2) are investigated when or converges to infinity.

2. Existence and Nonexistence of Coexistence States

In this section, we will obtain existence and nonexistence of coexistence states. Firstly, we present some basic results which will be used in this paper.

Let be all eigenvalues of the following problem:

where . It is easy to see that is simple and is strictly increasing in the sense that and imply . When , we denote by . Moreover, we denote by the eigenfunction corresponding to with normalization and positive in .

Define . It is well known that for any , the problem

has a unique positive solution which we denote by . It is well known that the mapping is strictly increasing, continuously differentiable from to and that uniformly on as . Moreover, we have in . It follows that (1.2) has two semitrivial solutions and if .

Next, we give an a priori estimate based on maximum principle. Its proof will be omitted.

Lemma 2.1. Any coexistence state of (1.2) has an a priori boundary, that is,

In the following, we set up the fixed point index theory for later use. Let be a Banach space. is called a wedge if is a closed convex set and for all . For , we define , we always assume that . Let be a compact linear operator on . We say that has property on if there exists and , such that .

For any and , we denote . Assume that is a compact operator and is an isolated fixed point of . If is Fréchet differentiable at , then the derivative has the property that . We denote by the fixed point index of at relative to .

We state a general result of Dancer [20] on the fixed point index with respect to the positive cone (see also [6]).

Lemma 2.2. Suppose that is invertible on .

(i) If has property on , then .(ii) If does not have property on , then , where is the sum of algebra multiplicities of the eigenvalues of which are greater than .

We introduce some notations as follows:

 , where , , where , , .

Define by

where and . It follows from Maximum Principle that is a compact positive operator, is complete continuous and Fréchet differentiable. Denote , observe that (1.2) has a positive solution in if and only if has a positive fixed point in .

If and , then , , and are the only nonnegative fixed points of . Then , , and are well defined. We calculate the Fréchet operator of as follows:

We can obtain the following lemmas by similar methods to those in the proofs of Lemmas 1 and 2 in [19].

Lemma 2.3. Suppose that , one has

(i), where is the degree of in relative to (ii) if , then ;(iii) if , then ;(iv) if , then .

Lemma 2.4. Suppose that , one has

(i) if , then ;(ii) if , then .

Next, we will show some results of existence and nonexistence of positive solutions of (1.2).

Theorem 2.5. If , then (1.2) has no positive solution; if and , then (1.2) has no nonnegative nonzero solution.

If and (1.2) has a positive solution, then .

If and (1.2) has a positive solution, then .

Proof. (i) First assume that is a positive solution of (1.2), then satisfies

and so by the eigenvalue problem. Due to the comparison principle for eigenvalues, we have , a contradiction. Next, assume that is a nonnegative nonzero solution of (1.2). If and , then by the previous proof. We can also similarly derive when and , which is a contradiction again.

(ii) Assume that is a positive solution of (1.2). Then by (i), and so the positive semitrivial solution exists. Since

is a lower solution of (1.2). By the uniqueness of , . Furthermore, since satisfies the equation

one has , which implies the result.

(iii) Let be a positive solution of (1.2); then exists with as in (ii). Similarly, the given assumption implies the existence of positive solution of (1.2) with . Similar to the proof of (i), we have . This follows since the function has a minimum at and (for and ).

Theorem 2.6. If and . Then (1.2) has at least a positive solution.

Suppose that . Then (1.2) has positive solution if and only if and .

Proof. (i) By Lemmas 2.3 and 2.4, we have

So (1.2) has at least one positive solution.

(ii) We first prove the sufficiency. Since , (1.2) has no solution taking the form with . If and , note that ; from Lemma 2.4, we have

Hence (1.2) has at least one positive solution.

Conversely, suppose that is a positive solution of (1.2). Then , and . Since satisfies

It follows that .

Theorem 2.7. If one of the following conditions holds, then (1.2) has no positive solutions:

(i) and ,(ii) and .

Proof. Since the proof of Theorem 2.7 is similar to the proof of Theorem 3 of [19], we omit it.

3. Global Bifurcation and Stability of Positive Solution

In this section, we consider a positive solution bifurcates from the semitrivial nonnegative branch by taking as a bifurcation parameter and fixing . Furthermore, we show that the existence of global bifurcation of (1.2) with respect to parameter and its stability. Moreover, the multiplicity, uniqueness, and stability of positive solutions are obtained by means of perturbation theory of eigenvalues and the fixed point theory.

Let be the principal eigenvalue of the following problem:

and is the corresponding eigenfunction with .

Let ; then , and satisfies

where

Clearly, is continuous, , and the Fréchet derivative . Let be the inverse of with Dirichlet boundary condition. Then, we have

Define the operator as follows:

then is a compact operator on . Let ; then is continuous, and . with if and only if is a nonnegative solution of (1.2).

Lemma 3.1. Assume that . Then is a bifurcation point of (3.2), and there exist positive solutions of (3.2) in the neighborhood of , where .

Remark 3.2. The proof of Lemma 3.1 is similar to the proof of Theorem 9 in [19]. The proof of Lemma 3.1 shows that there exist and continuous curve such that , and satisfies , where . Hence is a bifurcation solution of (3.2), where , .

If we take , then the nontrivial nonnegative solutions of (1.2) close to are either on the branch or the branch .

Let be a compact continuously differentiable operator such that . Suppose that we can write as , where is a linear compact operator and the Fréchet derivative . If is an isolated fixed point of , then we can define the index if at as , where is a ball with center at such that is the only fixed point of in . If is invertible, then is an isolated fixed point of and . If , then it is well known that the Leray-Schauder degree , where is equal to the algebraic multiplicities of the eigenvalue of which is greater than one.

Next, we will extend the local bifurcation solution given by Lemma 3.1 to the global bifurcation.

Let .

Theorem 3.3. Suppose that ; then the global bifurcation of bifurcating branch of positive solutions of (1.2) becomes unbounded by going to infinity in .

Proof. Let

Suppose that is an eigenvalue of . Then we have

Clearly , otherwise, , since all eigenvalue of the operator is greater than 0, so , a contradiction. Therefore, for some such that is the eigenvalue of the following problem:

It is well known that is increasing with respect to on and can be ordered as

On the other hand, if , then all eigenvalues of are greater than 0; furthermore, . Thus, is the eigenvalue of if and only if there exists some , such that .

Suppose that . Then for any . Hence, has no eigenvalue greater than 1, and as .

Suppose that . Then for any . Since , , and is increasing with respect to . Hence, there exists a unique , such that . So , where is the principal eigenvalue of the following problem:

where .

In the following, we will prove that . In fact, if the assertion is false, we may assume that . Then there exists , such that , that is,

Multiplying the equation by , integrating over , and using Green's formula, we obtain

which leads to , a contradiction. This proves the assertion, so it is verified that the multiplicity of is one and for . According to global bifurcation theory [12], there exists a continuum of zeros of in bifurcating from , and all zeros of close to lie on the curve whose existence was proved by Lemma 3.1. Let be the maximal continuum defined by . Then, consists of the curve in the neighborhood of the bifurcation point . Let . Then is the solution branch of (1.2) which bifurcates from and remains positive in a small neighborhood of and . Thus the continuum must satisfy one of the following three alternatives.

(i) contains in its closure points and , where is not invertible, and .(ii) is joining up from to in .(iii) is containing points of the form and , where .

Next, we prove that . Assume that . Then there exists and sequence such that when . It is easy to get that or . Suppose , then . Hence, we find either such that or such that . Since satisfies

It follows from the maximum principle that . Similarly, we can show that for .

Thus we only need the following three cases:

Suppose that . Then when . Let ; then satisfies

Thanks to estimates and Sobolev embedding theorem, there exists a convergent subsequence of , which we still denote by , such that in as , and because of . So taking the limit in (3.15) as , we get

It follows from the maximum principle that , which implies . This contradicts .

Suppose that . Then as . Let ; then satisfies

Similarly, By estimates and and Sobolev embedding theorem, there exists a convergent subsequence of , which we still denote by , such that in as , and because of . So taking limit in (3.17) as , we obtain

It follows from the maximum principle that . Hence . A contradiction with .

Suppose that . Similar to the previously mentioned, we can get contradiction.

Thus . By Lemma 2.1, we have . Thanks to estimates and and Sobolev embedding theorem, then there exists a constant such that ,. Hence the global bifurcation of positive solutions of (1.2) bifurcating at contains points with is arbitrarily large in .

In the following, we will study the stability of the bifurcation solution. Let , where is the inclusion mapping. Since is the linearized operator at for (1.2). By the proof of Lemma 3.1, we have span, Codim, and . Since , so it follows from [21] that is an -simple eigenvalue of .

Lemma 3.4. is the eigenvalue of with the largest real part, and all the other eigenvalue of lie in the left half complex plane.

Proof. We assume that is the eigenvalue of with the largest real part , and is the corresponding eigenfunction; then , equivalently,

Suppose that . Then is an eigenvalue of the operator , and then and , a contradiction. Thus . It follows that is an eigenvalue of the operator . Since is the principal eigenvalue of the operator ; furthermore, . This contradicts the assumption, so assumption does not hold. Which proves our conclusion.

We will use the linearized stability theory from [22]. Let , and be the linearized operators of (1.2) at , and , respectively. It follows that from Lemma 7, Corollary 1.13, and Theorem 1.16 [22, 23] that Lemma 3.5 holds.

Lemma 3.5. There exists -function: , and , defined from the neighborhood of and into , respectively, such that and

where . Moreover , whence and have the same sign for . Where is the derivative of with respect to at , and is the derivative of with respect to .

Lemma 3.6. The derivative of with respect to at is positive.

Proof. It follows from , that is,

Since , so . Clearly, , otherwise, , then , a contradiction. Hence is an eigenvalue of the operator . We consider , then , as . So is the principal eigenvalue of the operator , and is increasing with respect to as . Moreover, . Hence .

Lemma 3.7. The derivative of with respect to at satisfies

Proof. By substituting into (1.2), differentiating with respect to , and then setting , we find that

where is the derivative of with respect to at .

Taking the inner product with , using Green's formula, and noting the definition of , we have

It follows from Lemmas 3.4–3.7 that we obtain the following Theorem.

Theorem 3.8. Let . If , then bifurcation solution is stable; If , then bifurcation solution is unstable.

In Section 2, from Theorems 2.5–2.6, we can obtain the sufficient condition and necessary condition on the existence of positive solution and find that there exists a gap between and when . Next, we will consider the multiplicity, stability, and uniqueness of positive solutions in the gap.

Theorem 3.9. Assume and . Then there exist such that bifurcation positive solution is nondegenerate and unstable for and . Moreover, Problem (1.2) has at least two positive solutions.

Proof. We first prove that bifurcation positive solution is nondegenerate and unstable. To this end, it suffices to show that there exists a sufficiently small such that for , any positive solution of (1.2) is nondegenerate and the linearized eigenvalue problem:

has a unique eigenvalue and with algebra multiplicity one.

Let and be sequences which approach as . Due to , we can set sequences and such that and as . It follows that is a solution of (1.2). Then the corresponding linearized problem (3.25) can become the following form:

where and

Observe that as ,   converges to

It is easy to get that is a simple eigenvalue of the operator with corresponding eigenfunction . Moreover, all the other eigenvalues of are positive and stand apart from . Therefore, using perturbation theory [24], we get that for large has a unique eigenvalue which is close to zero. In addition, all the other eigenvalues of have positive real parts and stand apart from 0. Note that is simple real eigenvalue which converges to zero, and we can take the corresponding eigenfunction such that as . If we show that for large , then the result follows. By multiplying to the first equation of and integrating over , we obtain

Multiplying the first equation of (1.2) with by and integrating, we have

Due to , the previous equation becomes

Using (3.29) and (3.31), we have

Recall that , here is defined in Remark 3.2, and so dividing the previous equation by and taking the limit, we have

which implies that for large . This proved our claim.

Next, we apply the method in [25] to show the remaining part of Theorem 3.9. A contradiction argument will be used; we assume that (1.2) has a unique positive solution , then this solution must be bifurcated from . Since there exists a positive solution near by the local bifurcation theory. So is nondegenerate, and the corresponding linearized eigenvalue problem has a unique eigenvalue with algebra multiplicity one such that . Due to these facts, it is easy to show that is invertible and does not have property on ; it follows that by Lemma 2.2 (ii). Finally, Using Lemmas 2.3–2.4 and the additivity property of the index, we obtain

which derives a contradiction. Hence the Proof is complete.

Remark 3.10. In Theorem 3.9, the multiplicity can be shown easily when . Note that for a sufficiently small , since , and so . Since there is no positive solution of (1.2) if by Theorem 2.5 (i) when and . Therefore we easily see that there must be at least two positive solutions for and some .

4. Multiplicity, Uniqueness and Stability as or Is Large

In this section, taking or as a parameter, we investigate the multiplicity, stability, and uniqueness of positive solutions of (1.2) as or is large. In the following, we will always assume that , and let be fixed, unless otherwise specified.

Firstly, we consider the case that is large and is bounded away from . Hence the upper solution for and the lower solution for do not depend on when and .

Lemma 4.1. For any small , there exists such that for , (1.2) has at least one positive solution which satisfies

Proof. Since the proof of Lemma 4.1 is similar to the proof of Lemma 3 in [19], we omit it.

Lemma 4.2. If , then any positive solution of (1.2) approaches .

There exists a large such that any positive solution of (1.2) is nondegenerate and linearly stable when .

Proof. (i) Let , we show that the compact operator converges to , where

It follows that any positive solution of (1.2) converges to the fixed point of in this case. It is easy to see that is a unique fixed point of , so the positive solutions of (1.2) could not converge to semitrivial solutions when . Thus the conclusion is complete.

(ii) We use a contradiction method. Assume that there exists , with and with such that

where is a positive solution of (1.2) with . By computing, we have

where are the complex conjugates of . From Lemma 2.1, we know that and . It follows that is bounded and is bounded from the following. Thus is bounded as we assume . Hence we can suppose that and . Thanks to estimate, we obtain and are bounded. Hence we may assume that and in strongly, here . Setting in (3.1) and (3.2), we know that satisfy the following two single equations weakly (then strongly):

Clearly . If , then . However, . Hence . Similarly, we have , a contradiction; hence the proof is complete.

Theorem 4.3. Assume that . Then (1.2) has no positive solution when is sufficiently large.

Assume that and is sufficiently large. Then (1.2) has a unique positive solution, and it is asymptotically stable.

Proof. (i) Assume that there exists a positive solution of (1.2) for sufficiently large . It is easy to show that by Lemma 4.2. By Lemmas 2.3–2.4 and the additivity property of the index, we obtain

which gives a contradiction.

(ii) From Theorem 2.6, the existence is trivial. Since and , for sufficiently large, the nondegenerate positive solutions may not converge to semitrivial solutions by the proof of Lemma 4.2 (i). We need only to show the uniqueness. It follows from compactness and nondegeneracy that has at most finitely many positive fixed points in the region . We denote them by for . From the proof of (i), we obtain . Applying Lemmas 2.3–2.4 and the additivity property of the index again, we have

Hence the uniqueness is obtained. The stability has been given in Lemma 4.2.

In the following, we investigate the case when is large. This part is motivated by the work of Du and Lou in [13, 14, 18], and many of our methods used nextly come from their work.

Theorem 4.4. For any to be small, there that exists is large such that for ,

(i) if , then (1.2) has at least two positive solutions;(ii) if , then (1.2) has a unique positive solution and it is asymptotically stable.

Theorem 4.4 is the main result we will prove that when is large. The cases and will be treated separately. First, we deal with the multiplicity in Theorem 4.4. Similar to the method of [19], if , and is sufficiently large, then we can get the following result on (1.2).

Lemma 4.5. For any given small , there exists that such that if and , (1.2) has a positive solution , which satisfy

Lemma 4.6. For any to be small and any , there exists that is large such that if and , then any positive solution which satisfies (4.8) is nondegenerate and linearly stable.

Proof. Assume that and is large, then we can easily get that (1.2) is a regular perturbation of

It is well know that (4.9) has a unique positive solution which is linearly stable. So the positive solutions cannot bifurcate from semitrivial ones. Hence, Lemma 4.6 is proved by a standard regular perturbation argument. We omit the details.

Proof of (i) of Theorem 4.4. For any to be small, let

where and are defined in Lemmas 4.5 and 4.6, respectively. Assume that for some and some , Thus the unique positive solution must be the one found in Lemma 4.5. It follows from Lemma 4.5 that is invertible in and has no eigenvalue greater than one. Hence . Applying Lemmas 2.3–2.4 and the additivity property of the index, we obtain

a contradiction. Hence the proof if complete.

Part (i) in Theorem 4.4 implies that when and is large, (1.2) has at least two positive solutions. In the following, we will show that (1.2) has only two types of positive solutions in this case, one of which is close to and asymptotically stable and the other is close to which is not stable.

Theorem 4.7. For any to be small, there exists that is large such that if and , one obtains either or , where is any positive solution of (1.2). In particular, if occurs, by choosing suitably larger, one gets , where is a positive solution of the following equation:

Proof. Suppose that the conclusion does not hold. Then there exist and a positive solution of (1.2) with such that is bounded away from and . We may assume that and with . Since , we have as . Thanks to estimate and Sobolev embedding theorems, we may assume that and satisfies

If , then , which contradicts our assumption that is bounded away from . If from maximum principle, we have in . Hence and , which also contradicts our assumption. Thus, the first part of the proof is complete.

To complete the proof, it suffices to prove that if and , then approaches some positive solution of (4.12) with in the norm. It is easy to see that (4.12) has a positive solution when . First we claim that is uniformly bounded. If this is not true, we may assume that . Let . Then we have

Thanks to standard elliptic regularity theory, we may assume that , and . By taking the limit in (4.12), we know that satisfies the following equation weakly:

By Harnack's inequality, we have in . Since and , then in any compact subset of . Hence and , which contradicts the assumption that . Therefore is uniformly bounded.

Let . Then satisfies

Since is bounded, due to standard elliptic regularity theory and Sobolev embedding theorems, we may assume that . By letting in (4.16), we know that is a nonnegative solution of (4.12). There are two possibilities as follows.

(i). For this case, . Since any positive solution of (4.12) with approaches zero when , is certainly close to positive solutions of (4.12) with .(ii). In this case, we will prove that is a positive solution of (4.12). If not, by Harnack's inequality, we have . Let . Then we have

Hence we may assume that . By taking the limit in (4.17), we find

Since , we must have , which contradicts . The proof is complete.

The proof of (ii) in Theorem 4.4 is more difficult. To this end, we need several lemmas. First we show that there is no positive solution of (1.2) with small component if and is large.

Lemma 4.8. There exists a large such that if , then for all , any positive solution of (1.2) satisfies , where .

Proof. Suppose that our conclusion is not true, then there exist , and a positive solution sequence of (1.2) with such that does not hold.

First, let . Since , for large , we obtain

where is small such that , which is possible when . Due to the super- and subsolution method, we obtain , where is a unique positive solution of

Hence for large , we get

Applying the super- and subsolution method again, we obtain , which contradicts the assumption.

Secondly, we consider the case that . Thanks to standard elliptic regularity theory, we may assume that in and weakly converge to in with . Hence satisfies the following equation weakly:

If by Harnack's inequality, we have in . Thus and . Since , we have for large . This contradicts our assumption at the beginning of the proof. If , by letting , then we know that satisfies

Thanks to standard elliptic regularity theory, we may assume that with and in . Since , we have . By taking the weak limit in (4.23), we have

Let be the positive solution to

Multiplying (4.24) by and integrating, we obtain

Since , we must have and . Investigate the following equation for :

Multiplying (4.27) by and using (4.25), we obtain

After some rearrangements, we obtain

Set . By the definition of and , we have

Multiplying (4.30) by and integrating, we obtain

Since , and are bounded, we know that is bounded by Hölder inequality. Therefore from estimate and Sobolev embedding theorem, it is easy to see that is bounded. Dividing (4.29) by , we show that

Since weakly in and are uniformly bounded, we obtain

as . Hence if is large enough, which contradicts our assumption that for all . Thus, the proof is complete.

By Lemma 4.8 and a simple variant of the proof of Lemma 4.6, we immediately get the following result.

Lemma 4.9. For any given , there exists that is large such that if and ; then any positive solution of (1.2) is nondegenerate and linearly stable.

Next we investigate the case that is large.

Lemma 4.10. For any , there exists that is large such that if and , then any positive solution of (1.2) is nondegenerate and linearly stable.

Proof. Suppose the conclusion is not true. Then there exist some , and with such that

where is a positive solution of (1.2) with . Since , we may assume that . Let . From , we have

Hence . Due to Kato's inequality, we have

Multiplying (4.37) by and integrating by parts, we obtain

By the method of Lemma 2.2 in [18], there exists such that

It follows from (4.38) and (4.39) that . Multiplying (4.34) by and integrating, we have

Multiplying (4.35) by and integrating, we have

Adding the previous two identities, we have

It is easy to show that the imaginary part of the right-hand side of the previous identity is bounded. On the other hand, due to (4.38), (4.39) and the fact that , we have is bounded nextly. Hence is bounded as we assume that . Thus we may assume that with . Thanks to (4.35) and standard elliptic regularity theory, is bounded. Therefore we may assume that in . Since and , we have and in . By letting in (4.35), we know that satisfies the following equation weakly:

with . The self-disjointness of the previous problem implies . Since , we have

Hence , which implies and . Using Kato's inequality again, we have

It follows that satisfies

Multiplying (4.45) by , integrating by parts, and using (4.46), we obtain

Set . Then satisfies

Thanks to standard elliptic regularity theory, we may assume that . Then satisfies

Since , we have . Dividing both sides of (4.47) by , we know that

Since , the right-hand side of (4.50) converges to 0 by Hölder inequality. However, the previous discussion implies that . The contradiction completes the proof.

Proof of (ii) of Theorem 4.4. It suffices to prove the uniqueness. Investigate the following system with :

Set and by

where . Thanks to standard regularity results, we can prove that is a completely continuous operator. Clearly, is a positive solution of (4.51) if and only if it is a positive fixed point of in . For and , we first show that has no fixed point on .

We claim that any positive solution of (4.51) satisfies , where . Suppose that this claim is not true. Then there exist , and a positive solution of (4.51) with such that fails. Since the case that is considered in Lemma 4.8, it remains to discuss the case that . Since , we have

for large , where . Therefore is a supersolution to

Due to the choice of , (4.54) has a unique positive solution . Thus we have for all large . Hence,

By the super- and subsolution method again, we have for large . This is a contradiction.

Our claim implies that has no fixed point on . Hence . In particular,

Since has a unique fixed point in and , we have . From Lemmas 4.8–4.10, we see that for , all fixed points of fall into , and they are nondegenerate and linearly stable. Then by compactness, there are at most finitely many fixed points of , which we denote by . As shown in the proof of part (i), we have . Using the additivity property of the index, we know that

Hence for and , (1.2) has a unique positive solution, and it is stable.

Our final task is to establish the exact multiplicity and stability results for large and close to or . Firstly we consider the elliptic equation (4.12), which acts as a limiting problem of (1.2) when . Applying the similar method to Lemma 2.7 in paper [18], we have the following conclusion.

Lemma 4.11. The problem (4.12) has a positive solution if and only if . Moreover, all positive solutions of (4.12) are unstable. Furthermore, there exists some such that if , then (4.12) has at most one positive solution and it is nondegenerate (if it exists).

Define , where is defined by Lemma 4.11.

Theorem 4.12. For any , one can find that is large such that if , and , then (1.2) has exactly two positive solutions, one asymptotically stable and the other unstable.

To verify Theorem 4.12, we need some intermediate results. Theorem 4.7 has shown that (1.2) has only two types of positive solutions for when large and . In the following lemma, we will prove another result.

Lemma 4.13. There exist that is small and is large; both depend only on , and , such that if , and , then (1.2) has exactly two positive solutions, one asymptotically stable and the other unstable.

Proof. First we prove that for large , (1.2) has a unique asymptotically stable positive solution of type (i) in Theorem 4.7. In fact, if we choose small enough in Theorem 4.7, then any positive solution of (1.2) of type (i) satisfies (4.8). Hence by Lemma 4.6, they are nondegenerate and linearly stable. Now by a simple variant of the proof of part (ii) of Theorem 4.4, we find that there is only one positive solution of (1.2) satisfying type (i), and it is asymptotically stable.

Next we show that (1.2) has a unique unstable positive solution of type (ii). If we can prove this, then by Theorem 4.7, our proof of Lemma 4.13 is complete. Due to Theorem 4.7 and Lemmas 4.11, if any solution of (1.1) is close to , then must be close to , where is the unique positive solution of (4.12). Hence to prove uniqueness, it suffices to show that for , and , there is a unique pair , being a positive solution of (1.2), close to for certain and . Set , , and discuss

Clearly solves (1.2) if and only if solves (4.58) with . Thus it suffices to prove uniqueness for (4.58). For fixed , regarding as a parameter, we know that is a simple bifurcation point of (4.58). Due to a variant of Theorem 1 in [21], there exist and curves

such that, if , then all positive solutions of (4.58) are close to

Hence we need only to prove that these curves uniformly cover , for suitably chosen , and for fixed and cover the range only once. It is easy to obtain (see Theorem 3.9)

Hence

By the continuity of , there exist such that

Therefore, if , then for any , . This implies that for , covers the . Moreover, since for , each curve covers the range only once. By choosing , we get that for and , (1.2) has exactly one positive solution of type (ii) in Theorem 4.7.

It remains to show that the positive solution of (1.2) close to is unstable. In fact, when is sufficiently large, applying the method of the proof in Theorem 3.9, we can show that the positive solution of (1.2) close to is unstable. we omit the proof procedure.

Proof of Theorem 4.12. By Lemma 4.13, it suffices to establish the exact multiplicity and stability when for any given , where is defined in Lemma 4.13

From Theorem 4.7 we know that the solutions of (1.2) for and large are of two types, that is, types (i) or (ii). As in the proof of Lemma 4.13, we can prove that there is a unique asymptotically stable positive solution of type (i). Thus to end the proof, we need only to show that there is a unique unstable positive solution of (1.2) close to if and is large. Again by Lemma 4.11, it suffices to prove that there is a unique unstable positive solution of (1.2) such that is close to , where is the unique positive solution of (4.12) as shown in Lemma 4.11. In this connection, we investigate (4.58) with and small. Let . Since the unique solution of (4.12) with is nondegenerate, then is a nondegenerate solution of (4.58) with . Clearly, (4.58) with small is a regular perturbation of (4.58) with , and the perturbation is uniform for in the compact set . Thus it follows from the implicit function theorem that there exist small such that for any , (4.58) possesses a unique positive solution which satisfies

Set , where is defined in Lemma 4.11. It is easy to see that for any , there exists such that if and , then (1.2) has a unique positive solution of type (ii).

It remains to prove the instability for the unique positive solution of (1.2) of type (ii). Define and by

It is easy to show that, as , in the operator norm uniformly for approaches with close to and . Since belongs to the resolvent set of and

is an eigenvalue of . Due to standard perturbation theory, it is easy to get that also belongs to the resolvent set of and that has an eigenvalue close to . In particular, . This shows that for all large , the positive solution of (1.2) close to is nondegenerate and unstable. Thus, the proof of Theorem 4.12 is complete.

Acknowledgments

The authors would like to thank the referees for their valuable suggestions. The project supported by the Scientific Research Plan Projects of Shaanxi Education Department (no. 12JK0865) and the President Fund of Xi’an Technological University (XAGDXJJ1136).

References

A. D. Bazykin, Nonlinear Dynamics of interacting Populations, World Scientific, Singapore, 1998.

A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, Md, USA, 1925.

V. Volterra, “Variations and fluctuations of the number of individuals in animal species living together,” in Animal Ecology, R. N. Chapman, Ed., pp. 409–448, 1931.

H. Nie and J. Wu, “Multiplicity and stability of a predator-prey model with non-monotonic conversion rate,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 10, no. 1, pp. 154–171, 2009.

J. Wu and G. Wei, “Coexistence states for cooperative model with diffusion,” Computers & Mathematics with Applications, vol. 43, no. 10-11, pp. 1277–1290, 2002.

L. Li, “Coexistence theorems of steady states for predator-prey interacting systems,” Transactions of the American Mathematical Society, vol. 305, no. 1, pp. 143–166, 1988.

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.

E. N. Dancer, “On positive solutions of some pairs of differential equations. II,” Journal of Differential Equations, vol. 60, no. 2, pp. 236–258, 1985.

J. López-Gómez and R. Pardo, “Existence and uniqueness of coexistence states for the predator-prey model with diffusion: the scalar case,” Differential and Integral Equations, vol. 6, no. 5, pp. 1025–1031, 1993.

E. N. Dancer, “On uniqueness and stability for solutions of singularly perturbed predator-prey type equations with diffusion,” Journal of Differential Equations, vol. 102, no. 1, pp. 1–32, 1993.

N. Lakos, “Existence of steady-state solutions for a one-predator–two-prey system,” SIAM Journal on Mathematical Analysis, vol. 21, no. 3, pp. 647–659, 1990.

J. Blat and K. J. Brown, “Global bifurcation of positive solutions in some systems of elliptic equations,” SIAM Journal on Mathematical Analysis, vol. 17, no. 6, pp. 1339–1353, 1986.

Y. H. Du and Y. Lou, “S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model,” Journal of Differential Equations, vol. 144, no. 2, pp. 390–440, 1998.

Y. H. Du and Y. Lou, “Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation,” Proceedings of the Royal Society of Edinburgh A, vol. 131, no. 2, pp. 321–349, 2001.

J. H. Wu, “Global bifurcation of coexistence state for the competition model in the chemostat,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal A, vol. 39, no. 7, pp. 817–835, 2000.

H. Nie and J. Wu, “Positive solutions of a competition model for two resources in the unstirred chemostat,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 231–242, 2009.

H. Nie and J. Wu, “Asymptotic behavior of an unstirred chemostat model with internal inhibitor,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 889–908, 2007.

Y. H. Du and Y. Lou, “Some uniqueness and exact multiplicity results for a predator-prey model,” Transactions of the American Mathematical Society, vol. 349, no. 6, pp. 2443–2475, 1997.

M. X. Wang and Q. Wu, “Positive solutions of a prey-predator model with predator saturation and competition,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 708–718, 2008.

E. N. Dancer, “On the indices of fixed points of mappings in cones and applications,” Journal of Mathematical Analysis and Applications, vol. 91, no. 1, pp. 131–151, 1983.

M. G. Crandall and P. H. Rabinowitz, “Bifurcation from simple eigenvalues,” Journal of Functional Analysis, vol. 8, no. 2, pp. 321–340, 1971.

M. G. Crandall and P. H. Rabinowitz, “Bifurcation, perturbation of simple eigenvalues and linearized stability,” Archive for Rational Mechanics and Analysis, vol. 52, pp. 161–180, 1973.

Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, China, 1990.

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, NY, USA, 1966.

K. Wonlyul and R. Kimun, “Coexistence states of a predator-prey system with non-monotonic functional response,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 8, no. 3, pp. 769–786, 2007.