Abstract

We study a general Gause-type predator-prey model with monotonic functional response under Dirichlet boundary condition. Necessary and sufficient conditions for the existence and nonexistence of positive solutions for this system are obtained by means of the fixed point index theory. In addition, the local and global bifurcations from a semitrivial state are also investigated on the basis of bifurcation theory. The results indicate diffusion, and functional response does help to create stationary pattern.

1. Introduction

In this paper, we are interested in the following semilinear elliptic system with monotonic functional response under Dirichlet boundary condition: where is a bounded domain in is an integer) with a smooth boundary . The two functions and represent the densities of the prey and predator, respectively. The positive constants and are the diffusion coefficients of the corresponding species, is the death rate of the predator, and , which is assumed to be space dependent, represents the conversion rate of the prey to predators. The function denotes the growth rate of the prey species in the absence of predator. Throughout this paper, we impose the following hypotheses on the function . for all with a positive constant ; there exists a unique positive constant such that .Obviously, the classical Logistic growth rate satisfies . The function denotes the functional response of predators to prey. According to different biology backgrounds, the functional response may have several forms and many important results on the dynamics of predator-prey systems with different functional response have been obtained (see [1–20] and references therein). In many predator-prey interactions, the functional responses satisfies the following hypotheses. for all with a positive constant .It is easy to see that Holling-type , Holling-type , Holling-type , and Ivelev functional response satisfy hypothesis .

In this work, we aim to understand the influence of diffusion and functional response on pattern formation, that is, the positive solutions of (1.1). Throughout this paper, a solution of (1.1) is called a positive solution if for all and for all , where stand for the outward unit norm to at . As a consequence, the results indicate the stationary pattern arises when the diffusion coefficient enter into certain regions. In other words, we show that diffusion does help to create stationary pattern and diffusion and functional response can become determining factors in the formation pattern. Furthermore, we also investigate the properties of the nonconstant positive solution by using local bifurcation theory introduced by Crandall and Rabinowitz in [21] and global bifurcation theory introduced by López-Gómez and Molina-Meyer in [22]. We remark that problem (1.1) with Neumann boundary conditions was discussed in [5] recently. We point out that our results about the existence and nonexistence of positive solutions are different from [5] (see Corollary 3.8 and Remark 3.9).

The rest of this paper is organized as follows. In Section 2, some necessary preliminaries are introduced. In Section 3, we will give a priori upper bounds for positive solutions and investigate the existence and nonexistence of positive solutions of (1.1). In Section 4, the local bifurcations about parameter are investigated. Finally, the results about global bifurcations are obtained in Section 5.

2. Some Preliminaries

In order to give the main results and complete the corresponding proofs, we need to introduce some necessary notations and theorems as the following.

For each , let denote the principle eigenvalue of the following eigenvalue problem: Let denote the principle eigenvalue of the following eigenvalue problem: and denote by for simplicity. It is easy to know that is strictly increasing (see [23, 24]).

In order to calculate the indexes at the trivial and semitrivial states by means of the fixed point index theory, we also need to introduce the following theorem.

Theorem 2.1 (see [9, 13]). Assume and is a sufficiently large number such that for all . Define a positive and compact operator . Denote the spectral radius of by .(i) if and only if ;(ii) if and only if ;(iii) if and only if .

It is easy to see that the corresponding conclusions in Theorem 2.1 are also correct if the positive and compact operator is replaced by .

From Theorem 2.1, we see that it is crucial to know the sign of the eigenvalue to determine the spectral radius of . The following theorem give some sufficient conditions to determine the sign of the eigenvalue .

Theorem 2.2 (see [7, 9, 10, 23, 24]). Let and , in with on . Then one has(i)if , then ;(ii)if , then ;(iii)if , then .

Consider the following equation: where is a bounded domain in is an integer) with a smooth boundary .

Theorem 2.3 (see [7, 23, 24]). Assume that the function satisfies the following hypotheses:(i) and for all ;(ii) for , where is a positive constant.Then, (2.3) has a unique positive solution if .

Let be the unique positive solution of (2.3) when the unique positive solution exists. Denote by for simplicity.

Remark 2.4. It is easy to see that if the function satisfies the hypothesis , then it must satisfies the conditions (i) and (ii) in Theorem 2.3. We also point out that the condition holds if and only if . Therefore, if the function satisfies the hypothesis and , then (2.3) has a unique positive solution.
Now, we introduce the fixed point index theory which plays an important role in finding the sufficient conditions for the existence of positive solutions of model (1.1).
Let be a real Banach space and let be the natural positive cone of . is a closed convex set. is called a total wedge if and . For , define and . Then, is a wedge containing , , , while is a closed subset of containing . Let be a compact linear operator on which satisfies . We say that has property on if there is a and an such that . Let be a compact operator with a fixed point and , a Fréchet differentiable at . Let be the Fréchet derivative of at . Then, maps into itself. We denote by the degree of in relative to , the fixed point index of at relative to . Then, the following theorem can be obtained.

Theorem 2.5 (see [5, 11, 13]). Assume that is invertible on . (i)If have property on , then ;(ii)If does not have property on , then , where is the sum of algebraic multiplicities of the eigenvalues of which are greater than 1.

Finally, we introduce a result about global bifurcation, which was introduced by López-Gómez and Molina-Meyer in [22] and we state here for convenience.

Let be an ordered Banach space whose positive cone is normal and has nonempty interior, and consider the nonlinear abstract equation: where , is a compact and continuous operator pencil with a discrete set of singular values, denoted by . is compact on bounded sets and uniformly on compact intervals of .The solutions of (2.4) satisfy the strong maximum principle in the sense that where stands for the interior of the cone .

Define the parity mapping by Then, thanks to [25, Theorem??6.2.1], 2.4 possesses a component emanating from at if . Such a component will be subsequently denoted by . Then, the following abstract result hold.

Theorem 2.6. Suppose that satisfies , and is strongly positive in the sense that Then, there exists a subcomponent of in such that .
Moreover, if is the unique singular value for which 1 is an eigenvalue of to a positive eigenvector, then must be unbounded in .

Remark 2.7. When we are working in a product-ordered Banach space, the conditions (2.6) and (2.9) can be modified as For the technical details, one can refer to [25, Theorem??7.2.2] and [26, Proposition??2.2]. To avoid a repetition, we omitted it herein.

3. Existence and Nonexistence of Stationary Pattern

At first, we introduce the following lemma which gives the necessary condition for (1.1) to have positive solutions.

Lemma 3.1. If problem (1.1) has a positive solution, then and .

Proof. Assume is a positive solution of (1.1). Then, it is obvious that and by maximum principle. Because satisfies we have So, .

In the rest of this section, we shall prove that the necessary conditions in Lemma 3.1 are also sufficient conditions by means of the fixed point index theory. So, we need to obtain a priori bound for the positive solutions of (1.1).

Theorem 3.2. Assume and is a positive solution of (1.1). Then, one has

Proof. It is obvious that by the maximum principle. From (1.1), we can find that and hence Therefore,

Now, we introduce the following notations:

where . Take sufficiently large with such that and are, respectively, monotone increasing with respect to and for all .

Define a positive and compact operator by

Remark 3.3. () By the maximum principle, it is easy to see that if in in system (1.1). On the other hand, if , then we have in and on . From the assumption , we see that is the only semitrivial solution of (1.1) if . Moreover, (1.1) does not have any other constant solution except the trivial solution .
() Observe that (1.1) is equivalent to . Then, it is sufficient to prove that has a nonconstant positive fixed point in to show that (1.1) has a positive solution.
() From the Remarks (i) and (ii), we can see that it is necessary to calculate the fixed point index of at and . By Kronecker's existence theorem [23], we also need to calculate the topological degree of in to prove that the necessary conditions in Lemma 3.1 are also sufficient.
At first, we shall calculate the topological degree of the operator in and the fixed point index of the operator at , that is, and . It is easy to see that has no fixed point on . Then, the is well defined.
For , we define a positive and compact operator by Observe that and , ; we can obtain the following lemma and we omit the proofs because the calculations are standard.

Lemma 3.4. Assume that and . Then, one has(i),(ii).

Now, we need to calculate the fixed point index of the operator at , that is, .

Lemma 3.5. Assume that and , Then, one has (i)if , then ; (ii)if , then .

Proof. (i) Observe . Let . Then, Assume for some . Then, Taking account of , if , then we can see from the second equation of (3.12) that . This contradicts . So, . Then, we can get from the first equation of (3.12) that If , then . On the other hand, , which is a contradiction. Therefore, and is invertible on .
We claim that has property on . In fact, set Since , we can see that is an eigenvalue of with a corresponding eigenfunction by Theorem 2.1. Because , we know that . Then, we have This establishes our claim. Hence, .
(i)From Remark 3.3, we know that the unique nonnegative solutions of (1.1) are and if . Thus, we have From Lemma 3.4, we know that and . Therefore, we have .

Now, we can prove that and are also the sufficient conditions for model (1.1) to have a positive solution.

Lemma 3.6. If and , then model (1.1) has at least one positive solution.

Proof. If and , by Lemmas 3.4 and 3.5, then we have Hence, model (1.1) has at least one positive solution by Kronecker's existence theorem [8].

From Lemmas 3.1 and 3.6, we can get the following theorem.

Theorem 3.7. Problem (1.1) has at least one positive solution if and only if and .
Let denote the principle eigenvalue of the following eigenvalue problem: Then, it is easy to see that the condition is equivalent to and the condition is equivalent to . Therefore, one can get the following corollary from Theorem 3.7.

Corollary 3.8. Problem (1.1) has no positive solution if one of the following conditions hold:(i); (ii);(iii).

Remark 3.9. From Corollary 3.8, we can see that if the prey diffuses so rapidly that , then no positive solution exists. On the other hand, if the predator diffuses so rapidly that or diffuses so slowly that , then we can also observe the same phenomena. These results are different from the corresponding results in paper [5]. In paper [5], if the predator diffuse so rapidly that , where is a constant, then the corresponding model has at least one positive solution (see [5], Theorem??3.8). How to explain these differences? The key point, we think, lies in the boundary conditions. Different from the reflecting boundary conditions, that is, Neumann boundary condition in [5], the prey and the predator in our model both face lethal boundary conditions, that is, Dirichlet conditions in our model. Therefore, the more rapidly the prey or the predator diffuses, the more possibly they encounter the lethal boundary and then the more possibly they cannot coexist.

4. Local Bifurcation

In this subsection, we will employ the local bifurcation theory [21] to investigate the positive solution branches of (1.1) which bifurcate from the semitrivial solution if . We choose as the bifurcation parameter and denote by the semitrivial solution set with the parameter . The next proposition gives the local bifurcation branch of positive solution of (1.1).

Theorem 4.1. Assume that . A branch of positive solutions of (1.1) bifurcates from if and only if . More precisely, there exists a positive number such that when , the local bifurcation positive solutions from have the following form: where with between and and is the positive eigenfunction corresponding to of the following eigenvalue problem with : Furthermore, the bifurcation is subcritical, that is, .

Proof. Let us introduce the change of variable , which shifts the semitrivial solution to .
Introduce an operator as the following: where is between and . We will seek for the degenerate point of the linearized operator . By a simple calculation, we have When , it is easy to show that KerSpan, where in .
If Range?, then there exist such that By the Fredholm alternative theorem, it is easy to see that the first equation of (4.5) is solvable if and only if .
For such a solution, the first equation enables us to obtain . Therefore, we know that codim Range. In order to use the local bifurcation theorem [21] at the degenerate point, we need to verify that Range? Range. Here, it can be calculated that Suppose for contradiction that Range. By (4.4) and (4.6), there exist such that Then, multiplying the second equation of (4.7) by and integrating the resulting expression, we obtain that , which obviously yields a contradiction. Consequently, we can apply the local bifurcation theorem to at . Furthermore, by virtue of the Krein-Rutman theorem, we know that the possibility of other bifurcation points except is excluded.
In order to investigate the bifurcation direction from , substituting into the second equation of (1.1) and differentiating it with respect to , setting , we have Multiplying (4.8) by and applying divergence theorem, we obtain By (4.2), the terms including in (4.10) can be dropped out. Then, we can get According to hypothesis , we have and . Then, we know that the bifurcation direction from is subcritical.

Remark 4.2. According to the theory of Rabinowitz [27], we can see that there is a continuum of the set of non-trivial solutions of (1.1) with under the conditions of Theorem 4.1 and the continuum consists of two subcontinua: , filled in by coexistence states, and , filled in by component-wise negative solution pairs in a neighborhood of . However, this does not necessarily implies that the subcontinuum satisfies the global alternative of Rabinowitz [27] by the reasons already explained by Dancer [12] and López-Gómez and molina-meyer [22]. Instead, the existence of a global subcontinuum of the set of positive solutions with follows by slightly adapting [22, Theorem??1.1]. Therefore, in the following subsection, we shall study the global bifurcation from by using the global bifurcation theory of [22].

5. Global Bifurcation

In this subsection, basing on the results in Theorem 4.1, we can obtain the following results about global bifurcation from by using the global bifurcation theory introduced by López-Gómez, Molina-Meyer in [22].

Theorem 5.1. Assume that . Then, if one chooses as the main continuation parameter of (1.1), there exists an unbounded component of the set of positive solutions of (1.1) such that where stands for the projection operator into the c-component of the tern. Moreover, must bifurcate from infinity at .

Proof. Let . Then, (1.1) is equivalent to the following problem: where is between and . Introduce an operator as the following: for every and . Obviously, for all and by elliptic regularity is a classic solution of (5.2).
Subsequently, for every , we consider It is easy to see that and . Then, we have Define an operator By the Ascoli-Arzelá theorem and the classical Schauder estimates, we know that (5.6) is a compact linear operator. Owing to , we can see that is Fredholm of index zero.
In order to complete the proof of Theorem 5.1, we shall use [22, Theorem??1.1]. So, it is necessary to check the assumptions in Theorem??2.6.
Proof of . Since is Fredholm of index zero, we know that if and only if is an eigenvalue of , that is, if dim?. Note that dim? if and only if there exists such that If , then and hence (if , then we have , a contradiction). Then, we must have . So, dim? if and only if is an eigenvalue of in . Consequently, the set of singular values of is indeed discrete and hence the assumption is fulfilled.Proof of . From the definition of the operator , it is easy to see that the assumption follows directly by a simple calculation.Proof of . It is easy to see that can be regarded as an ordered Banach Space with respect to the order induced by the product cone . Using the the strong maximum principle, we can show that imply that for all and . The assumption is fulfilled.Now, we can prove Theorem 5.1 according to the general framework of [22]. Firstly, note that if and only if Ind changes as crosses ; we can see that from Theorem 4.1. Considering the operator defined by (5.6), it is not difficult to check that is the unique value of for which 1 is an eigenvalue of to a positive eigenfunction and where are the corresponding eigenfunctions defined in Theorem 4.1. At last, for and , we can see that Following from [22, Theorem??1.1], we know that there exists an unbounded component of the set of positive solutions of (1.1) such that and due to Theorem 3.7.To complete the proof of Theorem 5.1, we suppose that and let be a positive solution of (1.1). Then, by Theorem 3.2, we have for all and Note that Therefore, we know that must bifurcate from infinity at . The proof of Theorem 5.1 is completed.

Acknowledgment

This work was supported by the Fundamental Research Funds for the Central Universities (no. XDJK2009C152).