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 that M > 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, LmapsWy into itself. We denote by degW I −A,D the degree of I −A in D relative toW, indexW A, y the fixed point index ofA at y relative toW. 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 ) ∥