Positive Solutions for a General Gause-Type Predator-Prey Model with Monotonic Functional Response

and Applied Analysis 3 Let λ1 h denote the principle eigenvalue of the following eigenvalue problem: −d2Δu h x u λu inΩ, u 0 on ∂Ω, 2.2 and denote λ1 0 , λ1 0 by λ1, λ ∗ 1 for simplicity. It is easy to know that λ1 h , λ ∗ 1 h 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 h ∈ C Ω 0 < α < 1 and M is a sufficiently large number such thatM > h x for all x ∈ Ω. Define a positive and compact operator L −d1Δ M −1 M − h x . Denote the spectral radius of L by r L . i λ1 h > 0 if and only if r L < 1; ii λ1 h < 0 if and only if r L > 1; iii λ1 h 0 if and only if r L 1. It is easy to see that the corresponding conclusions in Theorem 2.1 are also correct if the positive and compact operator L −d1Δ M −1 M − h x is replaced by L −d2Δ M −1 M − h x . From Theorem 2.1, we see that it is crucial to know the sign of the eigenvalue λ1,k h to determine the spectral radius of L. The following theorem give some sufficient conditions to determine the sign of the eigenvalue λ1,k h . Theorem 2.2 see 7, 9, 10, 23, 24 . Let h x ∈ L∞ Ω and φ ≥ 0, φ/≡ 0 in Ω with φ 0 on ∂Ω. Then one has i if 0/≡ −Δφ h x φ ≤ 0, then λ1 h x < 0; ii if 0/≡ −Δφ h x φ ≥ 0, then λ1 h x > 0; iii if −Δφ h x φ ≡ 0, then λ1 h x 0. Consider the following equation: −d1Δφ φg ( φ ) in Ω, φ 0 on ∂Ω, 2.3 where Ω is a bounded domain in R N ≥ 1 is an integer with a smooth boundary ∂Ω. Theorem 2.3 see 7, 23, 24 . Assume that the function g φ : Ω → R satisfies the following hypotheses: i g φ ∈ C1 Ω and gφ φ < 0 for all φ ≥ 0; ii g φ ≤ 0 for φ ≥ C, where C is a positive constant. Then, 2.3 has a unique positive solution if λ1 −g 0 < 0. 4 Abstract and Applied Analysis LetΘ g φ be the unique positive solution of 2.3 when the unique positive solution exists. Denote Θ g 0 by Θ for simplicity. Remark 2.4. It is easy to see that if the function g φ satisfies the hypothesis H1 , then it must satisfies the conditions i and ii in Theorem 2.3. We also point out that the condition λ1 −g 0 < 0 holds if and only if g 0 > λ1. Therefore, if the function g φ satisfies the hypothesis H1 and g 0 > λ1, 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 E be a real Banach space and let W ⊂ E be the natural positive cone of E. W ⊂ E is a closed convex set. W is called a total wedge if τW ⊂ W and W − W E. For y ∈ W, define Wy {x ∈ E : y γ ∈ W for some γ > 0} and Sy {x ∈ Wy : −x ∈ Wy}. Then, Wy is a wedge containing W, y, −y, while Sy is a closed subset of E containing y. Let T be a compact linear operator on E which satisfies T Wy ⊂ Wy. We say that T has property α on Wy if there is a t ∈ 0, 1 and an ω ∈ Wy \Sy such that I − tT ω ∈ Sy. LetA : W → W be a compact operator with a fixed point y ∈ W and A, a Fréchet differentiable at y. Let L A′ y be the Fréchet derivative ofA at y. Then, L maps Wy into itself. We denote by degW I −A,D the degree of I −A in D relative to W, indexW A, y the fixed point index ofA at y relative to W. Then, the following theorem can be obtained. Theorem 2.5 see 5, 11, 13 . Assume that I − L is invertible on Wy. i If L have property α on Wy, then indexW A, y 0; ii If L does not have property α on Wy, then indexW A, y −1 σ , where σ is the sum of algebraic multiplicities of the eigenvalues of L 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. LetU be an ordered Banach space whose positive cone P is normal and has nonempty interior, and consider the nonlinear abstract equation: F λ, u L λ u R λ, u , 2.4 where HL L λ : IU − N λ ∈ L U , λ ∈ R, is a compact and continuous operator pencil with a discrete set of singular values, denoted by G. HR R ∈ C R ×U;U is compact on bounded sets and lim u→ 0 R c, u ‖u‖C Ω 0 2.5 uniformly on compact intervals of R. HP The solutions of 2.4 satisfy the strong maximum principle in the sense that c, u ∈ R × P \ {0} , F c, u 0 ⇒ u ∈ IntP, 2.6 where IntP stands for the interior of the cone P . Abstract and Applied Analysis 5 Define the parity mapping C : G → {−1, 0, 1} byand Applied Analysis 5 Define the parity mapping C : G → {−1, 0, 1} by C σ : 1 2 lim ε →0 Ind 0,N σ ε − Ind 0,N σ − ε , σ ∈ G. 2.7 Then, thanks to 25, Theorem 6.2.1 , 2.4 possesses a component emanating from λ, 0 at λ0 if C λ0 ∈ {−1, 1}. Such a component will be subsequently denoted by Cλ0 . Then, the following abstract result hold. Theorem 2.6. Suppose that λ0 ∈ G satisfies C λ0 / 0, N L λ0 span [ φ0 ] , φ0 ∈ P \ {0}, 2.8 and N λ0 is strongly positive in the sense that N λ0 P \ {0} ⊂ IntP. 2.9 Then, there exists a subcomponent C λ0 of Cλ0 in R × IntP such that λ0, 0 ∈ C P λ0 . Moreover, if λ0 is the unique singular value for which 1 is an eigenvalue of N λ to a positive eigenvector, then CPλ0 must be unbounded in R ×U. Remark 2.7. When we are working in a product-ordered Banach space, the conditions 2.6 and 2.9 can be modified as c, u, v ∈ R × P \ {0} × P \ {0} , F c, u, v 0 ⇒ u, v ∈ IntP × IntP, N c P \ {0} × P \ {0} ⊂ IntP × IntP. 2.10 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 g 0 > λ1 and −λ1 < c < −λ1 −mp Θ . Proof. Assume u, v is a positive solution of 1.1 . Then, it is obvious that g 0 > λ1 and u < Θ by maximum principle. Because u, v satisfies −d2Δv −cv mp u v in Ω, v 0 on ∂Ω, 3.1 6 Abstract and Applied Analysis we have 0 λ1 ( c −mp u ) > λ1 ( c −mp Θ ) c λ1 (−mp Θ ), 0 λ1 ( c −mp u ) < λ1 c c λ1. 3.2 So, −λ1 < c < −λ1 −mp Θ . 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 c > −λ1 and u, v is a positive solution of 1.1 . Then, one has u ≤ g 0 , v ≤ g 0 ( cd1m d2 mg 0 ) ∥


Introduction
In this paper, we are interested in the following semilinear elliptic system with monotonic functional response under Dirichlet boundary condition: where Ω is a bounded domain in R N N ≥ 1 is an integer with a smooth boundary ∂Ω.The two functions u and v represent the densities of the prey and predator, respectively.The positive constants d 1 and d 2 are the diffusion coefficients of the corresponding species, c is the death rate of the predator, and m x , which is assumed to be space dependent, represents the conversion rate of the prey to predators.The function g u denotes the growth rate of the prey species in the absence of predator.Throughout this paper, we impose the following hypotheses on the function g u .
H1 g ∈ C 1 0, ∞ , g 0 > 0, − g < g u u < 0 for all u ≥ 0 with a positive constant g; there exists a unique positive constant K such that g K 0.
Obviously, the classical Logistic growth rate g u r 1 − u/K satisfies H1 .The function p u denotes the functional response of predators to prey.According to different biology backgrounds, the functional response p u 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.
It is easy to see that Holling-type I, Holling-type II, Holling-type III, and Ivelev functional response satisfy hypothesis H2 .
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 u, v of 1.1 is called a positive solution if u x > 0, v x > 0 for all x ∈ Ω and ∂ w /∂ ν x , ∂ v /∂ ν x < 0 for all x ∈ ∂Ω, where ∂ ν x stand for the outward unit norm to Ω at x.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 L ópez-G ómez 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 c are investigated.Finally, the results about global bifurcations are obtained in Section 5.

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 h ∈ C α Ω 0 < α < 1 , let λ 1 h denote the principle eigenvalue of the following eigenvalue problem:

2.1
Abstract and Applied Analysis 3 Let λ * 1 h denote the principle eigenvalue of the following eigenvalue problem: and denote λ 1 0 , λ * 1 0 by λ 1 , λ * 1 for simplicity.It is easy to know that λ 1 h , λ * 1 h 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 h ∈ C α Ω 0 < α < 1 and M is a sufficiently large number such that M > h x for all x ∈ Ω. Define a positive and compact operator It is easy to see that the corresponding conclusions in Theorem 2.1 are also correct if the positive and compact operator L From Theorem 2.1, we see that it is crucial to know the sign of the eigenvalue λ 1,k h to determine the spectral radius of L. The following theorem give some sufficient conditions to determine the sign of the eigenvalue λ 1,k h .Theorem 2.2 see 7, 9, 10, 23, 24 .Let h x ∈ L ∞ Ω and ϕ ≥ 0, ϕ / ≡ 0 in Ω with ϕ 0 on ∂Ω.Then one has Consider the following equation: where Ω is a bounded domain in R N N ≥ 1 is an integer with a smooth boundary ∂Ω.
Let Θ g ϕ be the unique positive solution of 2.3 when the unique positive solution exists.Denote Θ g 0 by Θ for simplicity.
Remark 2.4.It is easy to see that if the function g ϕ satisfies the hypothesis H1 , then it must satisfies the conditions i and ii in Theorem 2.3.We also point out that the condition λ 1 −g 0 < 0 holds if and only if g 0 > λ 1 .Therefore, if the function g ϕ satisfies the hypothesis H1 and g 0 > λ 1 , 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 E be a real Banach space and let W ⊂ E be the natural positive cone of E. W ⊂ E is a closed convex set.W is called a total wedge if τW ⊂ W and W − W E. For y ∈ W, define W y {x ∈ E : y γ ∈ W for some γ > 0} and S y {x ∈ W y : −x ∈ W y }.Then, W y is a wedge containing W, y, −y, while S y is a closed subset of E containing y.Let T be a compact linear operator on E which satisfies T W y ⊂ W y .We say that T has property α on W y if there is a t ∈ 0, 1 and an ω ∈ W y \ S y such that I − tT ω ∈ S y .Let A : W → W be a compact operator with a fixed point y ∈ W and A, a Fréchet differentiable at y. Let L A y be the Fréchet derivative of A at y.Then, L maps W y into itself.We denote by deg W I − A, D the degree of I − A in D relative to W, index W A, y the fixed point index of A at y relative to W.Then, the following theorem can be obtained.
, where σ is the sum of algebraic multiplicities of the eigenvalues of L which are greater than 1.
Finally, we introduce a result about global bifurcation, which was introduced by L ópez-G ómez and Molina-Meyer in 22 and we state here for convenience.
Let U be an ordered Banach space whose positive cone P 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 G.
uniformly on compact intervals of R.
HP The solutions of 2.4 satisfy the strong maximum principle in the sense that where Int P stands for the interior of the cone P .
Theorem 2.6.Suppose that and N λ 0 is strongly positive in the sense that Then, there exists a subcomponent 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.

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.Proof.Assume u, v is a positive solution of 1.1 .Then, it is obvious that g 0 > λ 1 and u < Θ by maximum principle.Because u, v satisfies 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 c > −λ *
1 and u, v is a positive solution of 1.1 .Then, one has Proof.It is obvious that u x ≤ g 0 by the maximum principle.From 1.1 , we can find that and hence

3.6
Abstract and Applied Analysis 7 Now, we introduce the following notations: C Ω .Take q sufficiently large with q > max{g 0 p u 0 R, −c p 0 } such that ug u −p u v qu and −cv p u v qv are, respectively, monotone increasing with respect to u and v for all u, v ∈ 0, K × 0, R .
Define a positive and compact operator R : Remark 3.3.i By the maximum principle, it is easy to see that v ≡ 0 if u ≡ 0 in Ω in system 1.1 .On the other hand, if v ≡ 0, then we have −d 1 Δu ug u in Ω and u 0 on ∂Ω.From the assumption H1 , we see that Θ, 0 is the only semitrivial solution of 1.1 if g 0 > λ 1 .Moreover, 1.1 does not have any other constant solution except the trivial solution 0, 0 .
ii Observe that 1.1 is equivalent to u, v R u, v .Then, it is sufficient to prove that R has a nonconstant positive fixed point in D to show that 1.1 has a positive solution.
iii From the Remarks i and ii , we can see that it is necessary to calculate the fixed point index of R at 0, 0 and Θ, 0 .By Kronecker's existence theorem 23 , we also need to calculate the topological degree of R in D to prove that the necessary conditions in Lemma 3.1 are also sufficient.
At first, we shall calculate the topological degree of the operator R in D and the fixed point index of the operator at 0, 0 , that is, deg W I − R, D and index W R, 0, 0 .It is easy to see that R has no fixed point on ∂D.Then, the deg W I − R, D is well defined.

3.18
Then, it is easy to see that the condition g 0 > λ 1 is equivalent to d 1 < g 0 /λ 0 and the condition mp Θ /λ 0 .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:
Remark 3.9.From Corollary 3.8, we can see that if the prey diffuses so rapidly that d 1 > g 0 /λ 0 , then no positive solution exists.On the other hand, if the predator diffuses so rapidly that d 2 > λ * 1 mp Θ /λ 0 or diffuses so slowly that d 2 < λ * 1 −mp Θ mp Θ /λ 0 , 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 d 2 > D d 1 , where D d 1 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.

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 Θ, 0 if g 0 > λ 1 .We choose c as the bifurcation parameter and denote by Γ u { c, Θ, 0 : c ∈ R} the semitrivial solution set with the parameter c.The next proposition gives the local bifurcation branch of positive solution of 1.1 .
* with ξ between Θ and u and φ * is the positive eigenfunction corresponding to c −λ * 1 −mp Θ of the following eigenvalue problem with Ω φ 2 1:
Proof.Let us introduce the change of variable w Θ − u, which shifts the semitrivial solution Θ, 0 to 0, 0 .Introduce an operator Φ : R × where ξ is between Θ and u.We will seek for the degenerate point of the linearized operator Φ w,v c, 0, 0 .By a simple calculation, we have φ ψ 0 on ∂Ω.

4.7
Then, multiplying the second equation of 4.7 by ψ * and integrating the resulting expression, we obtain that Ω ψ * 2 dx 0, which obviously yields a contradiction.Consequently, we can apply the local bifurcation theorem to Φ at c, 0, 0 .Furthermore, by virtue of the Krein-Rutman theorem, we know that the possibility of other bifurcation points except c c * is excluded.
In order to investigate the bifurcation direction from −λ * 1 −mp Θ , Θ, 0 , substituting c s , u s , v s into the second equation of 1.1 and differentiating it with respect to s, setting s 0, we have Multiplying 4.8 by φ and applying divergence theorem, we obtain By 4.2 , the terms including v ss 0 in 4.10 can be dropped out.Then, we can get According to hypothesis H2 , we have p u Θ > 0 and c 1 < 0.Then, we know that the bifurcation direction from −λ * 1 −mp Θ , Θ, 0 is subcritical.

Global Bifurcation
In this subsection, basing on the results in Theorem 4.1, we can obtain the following results about global bifurcation from Θ, 0 by using the global bifurcation theory introduced by L ópez-G ómez, Molina-Meyer in 22 .Proof.Let w Θ − u.Then, 1.1 is equivalent to the following problem: where ξ is between Θ and u.Introduce an operator F : R × E → E as the following: for every c ∈ R and w, v ∈ E. Obviously, F c, 0, 0 0 for all c ∈ R and by elliptic regularity F c, w, v 0 ⇔ w, v is a classic solution of 5.2 .Subsequently, for every w, v ∈ E, we consider

5.4
It is easy to see that R c, 0, 0 0 and D w,v R c, 0, 0 0, 0 .Then, we have

5.5
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 L c I − N c , we can see that L c 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 HL
w v 0 on ∂Ω.

Theorem 4 . 1 .
Assume that g 0 > λ 1 .A branch of positive solutions of 1.1 bifurcates from Γ u if and only if c −λ * 1 −mp Θ .More precisely, there exists a positive number δ such that when 0 < s < δ, the local bifurcation positive solutions c s , u s , v s from −λ * 1 −mp Θ , Θ, 0 have the following form:

Remark 4 . 2 .
According to the theory of Rabinowitz 27 , we can see that there is a continuum C c * of the set of non-trivial solutions of 1.1 with c * , 0, 0 ∈ C c * under the conditions of Theorem 4.1 and the continuum C c * consists of two subcontinua: C c * , filled in by coexistence states, and C − c * , filled in by component-wise negative solution pairs in a neighborhood of −λ * 1 −mp Θ , Θ, 0 .However, this does not necessarily implies that the subcontinuum C c * satisfies the global alternative of Rabinowitz 27 by the reasons already explained by Dancer 12 and L ópez-G ómez and molina-meyer 22 .Instead, the existence of a global subcontinuum C c * of the set of positive solutions with −λ * 1 −mp Θ , Θ, 0 ∈ C c * follows by slightly adapting 22, Theorem 1.1 .Therefore, in the following subsection, we shall study the global bifurcation from Θ, 0 by using the global bifurcation theory of 22 .

Theorem 5 . 1 .
Assume that g 0 > λ 1 .Then, if one chooses c as the main continuation parameter of 1.1 , there exists an unbounded componentC c * ⊂ R × E of the set of positive solutions of 1.1 such that c, u, v −λ * 1 −mp Θ , Θ, 0 ∈ C c * , P c C c * −λ * 1 , −λ * 1 −mp Θ , 5.1where P c stands for the projection operator into the c-component of the tern.Moreover, C c * must bifurcate from infinity at c −λ * 1 .

5 . 7 If v 0, then −d 1 10 Following from 22 ,
Δw g Θ Θg u ξ w in Ω, w 0 on ∂Ω 5.8and hence w 0 if w / 0, then we have 0 λ 1 −g Θ − g u ξ > λ 1 −g Θ 0, a contradiction .Then, we must have v / 0. So, dimN L c * ≥ 1 if and only if −c is an eigenvalue of −d 2 Δ − mp Θ in Ω.Consequently, the set of singular values of L c is indeed discrete and hence the assumption HL is fulfilled.Proof of HR .From the definition of the operator R, it is easy to see that the assumption follows directly by a simple calculation.Proof of HP .It is easy to see that E can be regarded as an ordered Banach Space with respect to the order induced by the product cone P. Using the the strong maximum principle, we can show that c, w, v ∈ R × P \ {0} × P \ {0} ∩ Γ −1 0 imply that w, v > 0 for all x ∈ Ω and ∂ w /∂ ν x , ∂ v /∂ ν x < 0. The assumption HP is fulfilled.Now, we can prove Theorem 5.1 according to the general framework of 22 .Firstly, note that C σ / 0 if and only if Ind 0, N λ changes as λ crosses σ; we can see that C c * / 0 from Theorem 4.1.Considering the operator N c defined by 5.6 , it is not difficult to check that c −λ * 1 −mp Θ is the unique value of c for which 1 is an eigenvalue of N c to a positive eigenfunction andN L −λ * 1 mp Θ N I − N −λ * 1 mp Θ span φ, ψ , 5.9where φ, ψ are the corresponding eigenfunctions defined in Theorem 4.1.At last, for g 0 > λ 1 and −λ * 1 < c < −λ * 1 −mp Θ , we can see that N c P \ {0} × P \ {0} ⊂ Int P Int P × Int P. 5.Theorem 1.1 , we know that there exists an unbounded component C c * ⊂ R × E of the set of positive solutions of 1.1 such that c, u, v −λ * 1 −mp Θ , Θ, 0 ∈ C c * and P c C c * −λ * 1 , −λ * 1 −mp Θ due to Theorem 3.7.To complete the proof of Theorem 5.1, we suppose that −λ * 1 < c < −λ * 1 −mp Θ and let u, v be a positive solution of 1.1 .Then, by Theorem 3.2, we have u x ≤ g 0 for all x ∈ Ω and v C Ω ≤ g 0 cd 1 know that C c * must bifurcate from infinity at c −λ * 1 .The proof of Theorem 5.1 is completed.