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

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 𝑘 or 𝑚 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 𝑓(𝑢,𝑣)=1/(1+𝑚𝑢)(1+𝑘𝑣) 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:𝑢_𝑡−Δ𝑢=𝑢𝑎−𝑢−𝑏𝑣()𝑣1+𝑚𝑢)(1+𝑘𝑣,𝑥∈Ω,𝑡>0,_𝑡−Δ𝑣=𝑣𝑐−𝑣+𝑑𝑢()𝑢()1+𝑚𝑢)(1+𝑘𝑣,𝑥∈Ω,𝑡>0,𝑢=𝑣=0,𝑥∈𝜕Ω,𝑡>0,𝑥,0=𝑢₀(𝑥)≥0,𝑣(𝑥,0)=𝑣₀(𝑥)≥0,𝑥∈Ω. (1.1)

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):−Δ𝑢=𝑢𝑎−𝑢−𝑏𝑣(1+𝑚𝑢)(1+𝑘𝑣),𝑥∈Ω,−Δ𝑣=𝑣𝑐−𝑣+𝑑𝑢(1+𝑚𝑢)(1+𝑘𝑣),𝑥∈Ω,𝑢=𝑣=0,𝑥∈𝜕Ω, (1.2) where Ω is a bounded domain in 𝑅^𝑁(𝑁≥1) with smooth boundary 𝜕Ω;  𝑎,𝑏,𝑐 are positive constants; 𝑑 is constant; 𝑚 and 𝑘 are nonnegative constants.

If 𝑚=𝑘=0, 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 𝑛 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 𝑚>0 and 𝑘=0 was first studied by Blat and Brown [12]. In this case, the term 𝑓(𝑢,𝑣) 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 𝑚 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 𝑚.

However, the case when 𝑚>0 and 𝑘>0 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 𝑘 or 𝑚 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 𝑎 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 𝑘 or 𝑚 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 𝜆₁(𝑞)<𝜆₂(𝑞)≤𝜆₃(𝑞)≤⋯ be all eigenvalues of the following problem: −Δ𝜙+𝑞(𝑥)𝜙=𝜆𝜙,𝜙_𝜕Ω=0, (2.1) where 𝑞(𝑥)∈𝐶(Ω). It is easy to see that 𝜆₁(𝑞) is simple and 𝜆₁(𝑞) is strictly increasing in the sense that 𝑞₁≤𝑞₂ and 𝑞₁≢𝑞₂ imply 𝜆₁(𝑞₁)<𝜆₁(𝑞₂). When 𝑞(𝑥)≡0, we denote by 𝜆_𝑖𝜆_𝑖(0). Moreover, we denote by Φ₁ the eigenfunction corresponding to 𝜆₁ with normalization ‖Φ₁‖_∞=1 and positive in Ω.

Define 𝐶₀(Ω)={𝑢∈𝐶(Ω)∣𝑢=0on𝜕Ω}. It is well known that for any 𝑎>𝜆₁, the problem −Δ𝑢=𝑎𝑢−𝑢²,𝑢|_𝜕Ω=0 (2.2) has a unique positive solution which we denote by 𝜃_𝑎. It is well known that the mapping 𝑎→𝜃_𝑎 is strictly increasing, continuously differentiable from (𝜆₁,∞) to 𝐶²(Ω)∩𝐶₀(Ω) and that 𝜃_𝑎→0 uniformly on Ω as 𝑎→𝜆₁. Moreover, we have 0<𝜃_𝑎<𝑎 in Ω. It follows that (1.2) has two semitrivial solutions (𝜃_𝑎,0) and (0,𝜃_𝑐) if 𝑎,𝑐>𝜆₁.

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

Lemma 2.1. Any coexistence state (𝑢,𝑣) of (1.2) has an a priori boundary, that is, 𝑢≤𝑎,𝑣≤𝐵∶=𝑐+𝑎𝑑.1+𝑎𝑚 (2.3)

In the following, we set up the fixed point index theory for later use. Let 𝐸 be a Banach space. 𝑊⊂𝐸 is called a wedge if 𝑊 is a closed convex set and 𝛽𝑊⊂𝑊 for all 𝛽≥0. For 𝑦∈𝑊, we define 𝑊_𝑦={𝑥∈𝐸∶∃𝑟=𝑟(𝑥)>0,s.t.𝑦+𝑟𝑥∈𝑊}, 𝑆_𝑦={𝑥∈𝑊_𝑦∶−𝑥∈𝑊_𝑦}∶ we always assume that 𝐸=𝑊−𝑊. Let 𝑇∶𝑊_𝑦→𝑊_𝑦 be a compact linear operator on 𝐸. We say that 𝑇 has property 𝛼 on 𝑊_𝑦 if there exists 𝑡∈(0,1) and 𝜔∈𝑊_𝑦⧵𝑆_𝑦, such that 𝜔−𝑡𝑇𝜔∈𝑆_𝑦.

For any 𝛿>0 and 𝑦∈𝑊, we denote 𝐵⁺_𝛿(𝑦)=𝐵_𝛿(𝑦)∩𝑊. Assume that 𝐹∶𝐵⁺_𝛿(𝑦)→𝑊 is a compact operator and 𝑦 is an isolated fixed point of 𝐹. If 𝐹^′ is Fréchet differentiable at 𝑦, then the derivative 𝐹^′(𝑦) has the property that 𝐹^′(𝑦)∶𝑊_𝑦→𝑊_𝑦. We denote by index_𝑊(𝐹,𝑦) the fixed point index of 𝐹 at 𝑦 relative to 𝑊.

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

Lemma 2.2. Suppose that 𝐼−𝐿 is invertible on 𝑊_𝑦.(i) If 𝐿 has property 𝛼 on 𝑊_𝑦, then index_𝑊(𝐹,𝑦)=0.(ii) If 𝐿 does not have property 𝛼 on 𝑊_𝑦, then index_𝑊(𝐹,𝑦)=(−1)^𝜎, where 𝜎 is the sum of algebra multiplicities of the eigenvalues of 𝐿 which are greater than 1.

We introduce some notations as follows: 𝑋=𝐶¹₀(Ω)⨁𝐶¹₀(Ω), where 𝐶¹₀(Ω)={𝜔∈𝐶¹(Ω)∶𝜔∣_𝜕Ω=0}, 𝑊=𝐾⨁𝐾, where 𝐾={𝜑∈𝐶(Ω)∶𝜑(𝑥)≥0}, 𝐷∶={(𝑢,𝑣)∈𝑋∶𝑢≤𝑎+1,𝑣≤𝐵+1}, 𝐷^′∶=(int𝐷)⋂𝑊.

Define 𝐹_𝑡∶𝐷^′→𝑊 by 𝐹_𝑡(𝑢,𝑣)=(−Δ+𝑃)⁻¹𝑡𝑢𝑎−𝑢−𝑏𝑣(1+𝑚𝑢)(1+𝑘𝑣)+𝑃𝑢𝑡𝑣𝑐−𝑣+𝑑𝑢(1+𝑚𝑢)(1+𝑘𝑣)+𝑃𝑣, (2.4) where 𝑡∈[0,1] and 𝑃>max{𝑎+𝑏𝐵,𝑐+𝑑𝑎/(1+𝑚𝑎)}. It follows from Maximum Principle that (−Δ+𝑃)⁻¹ is a compact positive operator, 𝐹_𝑡 is complete continuous and Fréchet differentiable. Denote 𝐹₁=𝐹, observe that (1.2) has a positive solution in 𝑊 if and only if 𝐹_𝑡=𝐹₁ has a positive fixed point in 𝐷^′.

If 𝑎>𝜆₁ and 𝑐>𝜆₁, then (0,0), (𝜃_𝑎,0), and (0,𝜃_𝑐) are the only nonnegative fixed points of 𝐹. Then index_𝑊(𝐹,(0,0)), index_𝑊(𝐹,(𝜃_𝑎,0)), and index_𝑊(𝐹,(0,𝜃_𝑐)) are well defined. We calculate the Fréchet operator of 𝐹 as follows: (−Δ+𝑃)⁻¹𝑎−2𝑢−𝑏𝑣(1+𝑚𝑢)²(1+𝑘𝑣)+𝑃−𝑏𝑢(1+𝑚𝑢)(1+𝑘𝑣)²𝑑𝑣(1+𝑚𝑢)²(1+𝑘𝑣)𝑐−2𝑣+𝑑𝑢(1+𝑚𝑢)(1+𝑘𝑣)²+𝑃. (2.5)

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 𝑎>𝜆₁, one has(i)deg_𝑊(𝐼−𝐹,𝐷^′)=1, where deg_𝑊(𝐼−𝐹,𝐷^′) is the degree of 𝑇−𝐹 in 𝐷^′ relative to 𝑊(ii) if 𝑐≠𝜆₁, then index_𝑊(𝐹,(0,0))=0;(iii) if 𝑐>𝜆₁(−𝑑𝜃_𝑎/(1+𝑚𝜃_𝑎)), then index_𝑊(𝐹,(𝜃_𝑎,0))=0;(iv) if 𝑐<𝜆₁(−𝑑𝜃_𝑎/(1+𝑚𝜃_𝑎)), then index_𝑊(𝐹,(𝜃_𝑎,0))=1.

Lemma 2.4. Suppose that 𝑐>𝜆₁, one has(i) if 𝑎>𝜆₁(𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐)), then index_𝑊(𝐹,(0,𝜃_𝑐))=0;(ii) if 𝑎<𝜆₁(𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐)), then index_𝑊(𝐹,(0,𝜃_𝑐))=1.

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

Theorem 2.5. (𝑖) If 𝑎≤𝜆₁, then (1.2) has no positive solution; if 𝑎≤𝜆₁ and 𝑐≤𝜆₁, then (1.2) has no nonnegative nonzero solution.
(𝑖𝑖) If 𝑐≤𝜆₁ and (1.2) has a positive solution, then 𝑎>𝜆₁,𝑐+𝑑𝑎/(1+𝑚𝑎)>𝜆₁.
(𝑖𝑖𝑖) If 𝑐>𝜆₁ and (1.2) has a positive solution, then 𝑎>𝜆₁(𝑏𝜃_𝑐/(1+𝑚𝜃_𝑎)(1+𝑘𝜃_𝑐)).

Proof. (i) First assume that (𝑢,𝑣) is a positive solution of (1.2), then (𝑢,𝑣) satisfies −Δ𝑢=𝑢𝑎−𝑢−𝑏𝑣(1+𝑚𝑢)(1+𝑘𝑣),𝑥∈Ω,𝑢=0,𝑥∈𝜕Ω, (2.6) and so 𝑎=𝜆₁(𝑢+𝑏𝑣/(1+𝑚𝑢)(1+𝑘𝑣)) by the eigenvalue problem. Due to the comparison principle for eigenvalues, we have 𝑎>𝜆₁, a contradiction. Next, assume that (𝑢,𝑣) is a nonnegative nonzero solution of (1.2). If 𝑢≢0 and 𝑣≡0, then 𝑎>𝜆₁ by the previous proof. We can also similarly derive 𝑐>𝜆₁ when 𝑢≡0 and 𝑣≢0, which is a contradiction again.
(ii) Assume that (𝑢,𝑣) is a positive solution of (1.2). Then 𝑎>𝜆₁ by (i), and so the positive semitrivial solution 𝜃_𝑎 exists. Since −Δ𝑢=𝑢𝑎−𝑢−𝑏𝑣()()1+𝑚𝑢)(1+𝑘𝑣≤𝑢𝑎−𝑢,𝑥∈Ω,𝑢=0,𝑥∈𝜕Ω, (2.7)𝑢 is a lower solution of (1.2). By the uniqueness of 𝜃_𝑎, 𝑢≤𝜃_𝑎. Furthermore, since 𝑣 satisfies the equation −Δ𝑣=𝑣𝑐−𝑣+𝑑𝑢(1+𝑚𝑢)(1+𝑘𝑣),𝑥∈Ω,𝑣=0,𝑥∈𝜕Ω, (2.8) one has 0=𝜆₁(−𝑐+𝑣−𝑑𝑢/(1+𝑚𝑢)(1+𝑘𝑣))>𝜆₁(−𝑐−𝑑𝑎/(1+𝑚𝑎)), which implies the result.
(iii) Let (𝑢,𝑣) be a positive solution of (1.2); then 𝜃_𝑎 exists with 𝑢≤𝜃_𝑎 as in (ii). Similarly, the given assumption 𝑐>𝜆₁ implies the existence of positive solution 𝜃_𝑐 of (1.2) with 𝜃_𝑐≤𝑣. Similar to the proof of (i), we have 𝑎=𝜆₁(𝑢+𝑏𝑣/(1+𝑚𝑢)(1+𝑘𝑣))>𝜆₁(𝑏𝜃_𝑐/(1+𝑚𝜃_𝑎)(1+𝑘𝜃_𝑐)). This follows since the function 𝑏𝑣/(1+𝑚𝑢)(1+𝑘𝑣) has a minimum at 𝑢=𝜃_𝑎 and 𝑣=𝜃_𝑐 (for 𝑢≤𝜃_𝑎 and 𝑣≥𝜃_𝑐).

Theorem 2.6. (𝑖) If 𝑐>𝜆₁ and 𝑎>𝜆₁(𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐)). Then (1.2) has at least a positive solution.
(𝑖𝑖) Suppose that 𝑐<𝜆₁. Then (1.2) has positive solution if and only if 𝑎>𝜆₁ and 𝑐>𝜆₁(−𝑑𝜃_𝑎/(1+𝑚𝜃_𝑎)).

Proof. (i) By Lemmas 2.3 and 2.4, we have deg_𝑊(𝐼−𝐹,𝐷)−index_𝑊𝑓,(0,0)−index_𝑊𝑓,𝜃_𝑎,0−index_𝑊𝑓,0,𝜃_𝑐=1. (2.9) So (1.2) has at least one positive solution.
(ii) We first prove the sufficiency. Since 𝑐<𝜆₁, (1.2) has no solution taking the form (0,𝑣) with 𝑣>0. If 𝑎>𝜆₁ and 𝑐>𝜆₁(−𝑑𝜃_𝑎/(1+𝑚𝜃_𝑎)), note that 𝑐<𝜆₁; from Lemma 2.4, we have deg_𝑊(𝐼−𝐹,𝐷)−index_𝑊𝑓,(0,0)−index_𝑊𝑓,𝜃_𝑎,0=1. (2.10) Hence (1.2) has at least one positive solution.
Conversely, suppose that (𝑢,𝑣) is a positive solution of (1.2). Then 𝑎>𝜆₁, and 𝑢<𝜃_𝑎. Since (𝑢,𝑣) satisfies −Δ𝑣=𝑣(𝑐−𝑣+𝑑𝑢/(1+𝑚𝑢)(1+𝑘𝑣)),𝑥∈Ω,𝑣=0,𝑥∈𝜕Ω. (2.11) It follows that 0=𝜆₁(−𝑐+𝑣−𝑑𝑢/(1+𝑚𝑢)(1+𝑘𝑣))>𝜆₁(−𝑐−𝑑𝑎/(1+𝑚𝑎)).

Theorem 2.7. If one of the following conditions holds, then (1.2) has no positive solutions:(i)𝑏≥(1+𝑚𝑎)(1+𝑘𝐵) and 𝑎≤𝑐,(ii)𝑏<(1+𝑚𝑎)(1+𝑘𝐵) and 𝑐−𝑎≥(1−𝑏𝑎/(1+𝑚𝑎)(1+𝑘𝐵))𝐵.

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 {(0,𝜃_𝑐,𝑎)} by taking 𝑎 as a bifurcation parameter and fixing 𝑐>𝜆₁. Furthermore, we show that the existence of global bifurcation of (1.2) with respect to parameter 𝑎 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 𝑎 be the principal eigenvalue of the following problem: −Δ𝜙+𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜙=𝑎𝜙,𝜙_𝜕Ω=0, (3.1) and Φ is the corresponding eigenfunction with ‖Φ‖_∞=1.

Let 𝜔=𝑢,𝜒=𝑣−𝜃_𝑐; then 0≤𝜔≤𝜃_𝑎,𝜒≥0, and 𝜔,𝜒 satisfies −Δ𝜔=𝑎−𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜔+𝐹₁𝜔,𝜒,𝑥∈Ω,−Δ𝜒=𝑐−2𝜃_𝑐𝜒+𝑑𝜃_𝑐1+𝑘𝜃_𝑐𝜔+𝐹₂𝜔,𝜒,𝑥∈Ω,𝜔=𝜒=0,𝑥∈𝜕Ω, (3.2) where 𝐹₁𝜔,𝜒=𝑏𝜔𝜃_𝑐1+𝑘𝜃_𝑐−𝑏𝜔𝜒+𝜃_𝑐(1+𝑚𝜔)1+𝑘𝜒+𝜃_𝑐−𝜔²,𝐹₂𝜔,𝜒=𝑑𝜔𝜒+𝜃_𝑐(1+𝑚𝜔)1+𝑘𝜒+𝜃_𝑐−𝑑𝜔𝜃_𝑐1+𝑘𝜃_𝑐−𝜒². (3.3)

Clearly, 𝐹=(𝐹₁,𝐹₂) is continuous, 𝐹(0,0)=0, and the Fréchet derivative 𝐷_(𝜔,𝜒)𝐹∣_(0,0)=0. Let 𝐾 be the inverse of −Δ with Dirichlet boundary condition. Then, we have 𝜔=𝑎𝐾𝜔−𝑏𝐾𝜔𝜃_𝑐1+𝑘𝜃_𝑐+𝐾𝐹₁𝜔,𝜒,𝑥∈Ω,𝜒=𝑐𝐾𝜒−2𝐾𝜒𝜃_𝑐+𝑑𝐾𝜔𝜃_𝑐1+𝑘𝜃_𝑐+𝐾𝐹₂𝜔,𝜒,𝑥∈Ω,𝜔=𝜒=0,𝑥∈𝜕Ω. (3.4)

Define the operator 𝑇∶𝑅⁺×𝑋→𝑋 as follows: 𝑇𝑎;𝜔,𝜒=𝑎𝐾𝜔−𝑏𝐾𝜔𝜃_𝑐1+𝑘𝜃_𝑐+𝐾𝐹₁𝜔,𝜒𝑐𝐾𝜒−2𝐾𝜒𝜃_𝑐+𝑑𝐾𝜔𝜃_𝑐1+𝑘𝜃_𝑐+𝐾𝐹₂𝜔,𝜒, (3.5) then 𝑇(𝑎;𝜔,𝜒) is a compact operator on 𝑋. Let 𝐺(𝑎;𝜔,𝜒)=(𝜔,𝜒)^𝑇−𝑇(𝑎;𝜔,𝜒); then 𝐺 is continuous, and 𝐺(𝑎;0,0)=0. 𝐺(𝑎;𝜔,𝜒)=0 with 0≤𝜔≤𝜃_𝑎,𝜒≥0 if and only if (𝜔,𝜒+𝜃_𝑐,𝑎) is a nonnegative solution of (1.2).

Lemma 3.1. Assume that 𝑐>𝜆₁. Then (𝑎;𝑢,𝑣)=(𝑎;0,𝜃_𝑐) is a bifurcation point of (3.2), and there exist positive solutions of (3.2) in the neighborhood of (𝑎;0,𝜃_𝑐), where 𝑎=𝜆₁(𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐)).

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 𝛿>0 and 𝐶¹ continuous curve (𝑎(𝑠);𝜙(𝑠),𝜑(𝑠))∶(−𝛿,𝛿)→𝑅×𝑍 such that 𝑎(0)=𝑎,𝜙(0)=0,𝜑(0)=0, and (𝑎(𝑠);𝜔(𝑠),𝜒(𝑠))=(𝑎(𝑠);𝑠(Φ+𝜙(𝑠)),𝑠(Ψ+𝜑(𝑠))) satisfies 𝐺(𝑎(𝑠);𝜔(𝑠),𝜒(𝑠))=0, where 𝑋=𝑍⊕span{(Φ,Ψ)}. Hence (𝑎(𝑠);𝑈(𝑠),𝑉(𝑠))(|𝑠|<𝛿) is a bifurcation solution of (3.2), where 𝑈(𝑠)=𝑠(Φ+𝜙(𝑠)),𝑉(𝑠)=𝜃_𝑐+𝑠(Ψ+𝜑(𝑠)), Ψ=(−Δ−𝑐+2𝜃_𝑐)⁻¹(𝑑𝜃_𝑐/(1+𝑘𝜃_𝑐)Φ).
If we take 0<𝑠<𝛿, then the nontrivial nonnegative solutions of (1.2) close to (𝑎;0,𝜃_𝑐) are either on the branch {(𝑎;0,𝜃_𝑐)∶𝑎∈𝑅⁺} or the branch {(𝑎(𝑠);𝑈(𝑠),𝑉(𝑠))∶0<𝑠<𝛿}.
Let 𝑇∶𝑋×𝑅→𝑋 be a compact continuously differentiable operator such that 𝑇(0,𝑎)=0. Suppose that we can write 𝑇 as 𝑇(𝑢,𝑎)=𝐾(𝑎)𝑢+𝑅(𝑢,𝑎), where 𝐾(𝑎) is a linear compact operator and the Fréchet derivative 𝑅_𝑢(0,0)=0. If 𝑥₀ is an isolated fixed point of 𝑇, then we can define the index if 𝑇 at 𝑥₀ as index(𝑇,𝑥₀)=deg(𝐼−𝑇,𝑈_𝛿(𝑥₀),𝑥₀), where 𝑈_𝛿(𝑥₀) is a ball with center at 𝑥₀ such that 𝑥₀ is the only fixed point of 𝑇 in 𝑈_𝛿(𝑥₀). If 𝐼−𝑇^′(𝑥₀) is invertible, then 𝑥₀ is an isolated fixed point of 𝑇 and index(𝑇,𝑥₀)=deg(𝐼−𝑇,𝑈_𝛿(𝑥₀),𝑥₀)=deg(𝐼−𝑇^′(𝑥₀),𝑈_𝛿(𝑥₀),0). If 𝑥₀=0, then it is well known that the Leray-Schauder degree deg(𝐼−𝐾(𝑎),𝑈_𝛿(𝑥₀),0)=(−1)^𝜎, where 𝜎 is equal to the algebraic multiplicities of the eigenvalue of 𝐾 which is greater than one.
Next, we will extend the local bifurcation solution {(𝑎(𝑠);𝑈(𝑠),𝑉(𝑠))∶0<𝑠<𝛿} given by Lemma 3.1 to the global bifurcation.
Let 𝑃₁={𝑢∈𝐶¹₀(Ω)∶𝑢(𝑥)>0,𝑥∈Ω,𝜕𝑢/𝜕𝑛<0,𝑥∈𝜕Ω},𝑃={(𝑢,𝑣,𝑎)∈𝑋×𝑅⁺∶𝑢,𝑣∈𝑃₁}.

Theorem 3.3. Suppose that 𝑐>𝜆₁; then the global bifurcation 𝐶 of bifurcating branch of positive solutions of (1.2) becomes unbounded by 𝑎 going to infinity in 𝑃.

Proof. Let 𝑇^′(𝑎)⋅𝜔,𝜒=𝐷_(𝜔,𝜒)𝑇()⋅=𝑎;0,0𝜔,𝜒𝑎𝐾𝜔−𝑏𝐾𝜔𝜃_𝑐1+𝑘𝜃_𝑐,𝑐𝐾𝜒−2𝐾𝜒𝜃_𝑐+𝑑𝐾𝜔𝜃_𝑐1+𝑘𝜃_𝑐. (3.6) Suppose that 𝜇≥1 is an eigenvalue of 𝑇^′(𝑎). Then we have −𝜇Δ𝜔=𝑎−𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜔,𝑥∈Ω,−𝜇Δ𝜒=𝑐−2𝜃_𝑐𝜒+𝑑𝜃_𝑐1+𝑘𝜃_𝑐𝜔,𝑥∈Ω,𝜔=𝜒=0,𝑥∈𝜕Ω. (3.7) Clearly 𝜔≢0, otherwise, 𝜔≡0, since all eigenvalue of the operator (−𝜇Δ−𝑐+2𝜃_𝑐) is greater than 0, so 𝜒≡0, a contradiction. Therefore, for some 𝑖 such that 𝑎=𝑎_𝑖(𝜇) is the eigenvalue of the following problem: −𝜇Δ𝜔+𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜔=𝑎𝜔,𝜔|_𝜕Ω=0. (3.8) It is well known that 𝑎_𝑖(𝜇) is increasing with respect to 𝜇 on [1,+∞) and can be ordered as 0<𝑎₁𝜇<𝑎₂𝜇≤𝑎₃𝜇≤⋯→∞,𝑎₁(1)=𝑎. (3.9) On the other hand, if 𝜇≥1, then all eigenvalues of (−𝜇Δ−𝑐+2𝜃_𝑐) are greater than 0; furthermore, 𝜒=(−𝜇Δ−𝑐+2𝜃_𝑐)⁻¹(𝑑𝜃_𝑐/(1+𝑘𝜃_𝑐)𝜔). Thus, 𝜇≥1 is the eigenvalue of 𝑇^′(𝑎) if and only if there exists some 𝑖, such that 𝑎=𝑎_𝑖(𝜇).
Suppose that 𝑎<𝑎. Then for any 𝜇≥1,𝑖≥1,𝑎<𝑎₁(1)≤𝑎_𝑖(𝜇). Hence, 𝑇^′(𝑎) has no eigenvalue greater than 1, and index(𝑇(𝑎;⋅),0)=1 as 𝑎<𝑎.
Suppose that 𝑎<𝑎<𝑎₂(1). Then for any 𝜇≥1,𝑖≥2,𝑎<𝑎_𝑖(𝜇). Since 𝑎₁(1)=𝑎, lim_𝜇→∞𝑎₁(𝜇)=+∞, and 𝑎₁(𝜇) is increasing with respect to 𝜇. Hence, there exists a unique 𝜇₁>1, such that 𝑎=𝑎₁(𝜇₁). So 𝑁(𝜇₁𝐼−𝑇^′(𝑎))=span{(𝜔,𝜒)},dim𝑁(𝜇₁𝐼−𝑇^′(𝑎))=1, where 𝜔>0 is the principal eigenvalue of the following problem: 𝜇₁Δ𝜔+𝑎−𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜔=0,𝜔|_𝜕Ω=0, (3.10) where 𝜒=(−𝜇₁Δ−𝑐+2𝜃_𝑐)⁻¹(𝑑𝜃_𝑐/(1+𝑘𝜃_𝑐)𝜔).
In the following, we will prove that 𝑅(𝜇₁𝐼−𝑇^′(𝑎))∩𝑁(𝜇₁𝐼−𝑇^′(𝑎))=0. In fact, if the assertion is false, we may assume that (𝜔,𝜒)∈𝑅(𝜇₁𝐼−𝑇^′(𝑎)). Then there exists (𝜔,𝜒)∈𝑋, such that (𝜇₁𝐼−𝑇^′(𝑎))(𝜔,𝜒)=(𝜔,𝜒), that is, 𝜇₁Δ𝜔+𝑎−𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜔=Δ𝜔,𝜔|_𝜕Ω=0. (3.11) Multiplying the equation by 𝜔, integrating over Ω, and using Green's formula, we obtain _Ω𝜔Δ𝜔=_Ω𝜇₁Δ𝜔+𝑎𝜔−𝑏𝜔𝜃_𝑐1+𝑘𝜃_𝑐𝜔=_Ω𝜇₁Δ𝜔+𝑎𝑏𝜔−𝜔𝜃_𝑐1+𝑘𝜃_𝑐𝜔=0, (3.12) which leads to ∫_Ω(1/𝜇₁)(𝑎−𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐))𝜔²=0, a contradiction. This proves the assertion, so it is verified that the multiplicity of 𝜇₁ is one and index(𝑇(𝑎;⋅),0)=−1 for 𝑎<𝑎<𝑎₂(1). According to global bifurcation theory [12], there exists a continuum 𝐶₀ of zeros of 𝐺(𝑎;𝜔,𝜒)=0 in 𝑅⁺×𝑋 bifurcating from (𝑎;0,0), and all zeros of 𝐺(𝑎;𝜔,𝜒) close to (𝑎;0,0) lie on the curve whose existence was proved by Lemma 3.1. Let 𝐶₁ be the maximal continuum defined by 𝐶₁=𝐶₀−{(𝑎(𝑠);𝑠(Φ+𝜙(𝑠)),𝑠(Ψ+𝜑(𝑠)))∶−𝛿<𝑠<0}. Then, 𝐶₁ consists of the curve {(𝑎(𝑠);𝑠(Φ+𝜙(𝑠)),𝑠(Ψ+𝜑(𝑠)))∶−𝛿<𝑠<0} in the neighborhood of the bifurcation point (𝑎;0,0). Let 𝐶={(𝑎;𝑢,𝑣)∶𝑈=𝜔,𝑉=𝜃_𝑐+𝜒,(𝜔,𝜒)∈𝐶₁}. Then 𝐶 is the solution branch of (1.2) which bifurcates from (𝑎;0,𝜃_𝑐) and remains positive in a small neighborhood of (𝑎;0,𝜃_𝑐) and 𝐶⊂𝑃. Thus the continuum 𝐶−{(𝑎;0,𝜃_𝑐)} must satisfy one of the following three alternatives.(i)𝐶 contains in its closure points (𝑎;0,𝜃_𝑐) and (𝑎;0,𝜃_𝑐), where 𝐼−𝑇^′(𝑎) is not invertible, and 𝑎≠𝑎.(ii)𝐶 is joining up from (𝑎;0,𝜃_𝑐) to ∞ in 𝑅×𝑋.(iii)𝐶 is containing points of the form (𝑎;𝑢,𝜃_𝑐+𝑣) and (𝑎;−𝑢,𝜃_𝑐−𝑣), where (𝑢,𝑣)≠(0,0).
Next, we prove that 𝐶−{(𝑎;0,𝜃_𝑐)}⊂𝑃. Assume that 𝐶−{(𝑎;0,𝜃_𝑐)}⊆𝑃̸. Then there exists (𝑎;𝑢,𝑣)∈(𝐶−{(𝑎;0,𝜃_𝑐)})∩𝜕𝑃 and sequence {(𝑎_𝑛;𝑢_𝑛,𝑣_𝑛)}⊂𝐶∩𝑃,𝑢_𝑛>0,𝑣_𝑛>0 such that (𝑎_𝑛;𝑢_𝑛,𝑣_𝑛)→(𝑎;𝑢,𝑣) when 𝑛→∞. It is easy to get that 𝑢∈𝜕𝑃₁ or 𝑣∈𝜕𝑃₁. Suppose 𝑢∈𝜕𝑃₁, then 𝑢≥0,𝑥∈Ω. Hence, we find either 𝑥₀∈Ω such that 𝑢(𝑥₀)=0 or 𝑥₀∈𝜕Ω such that 𝜕𝑢/𝜕𝑛∣_𝑥0=0. Since 𝑢 satisfies 𝑏𝑣−Δ𝑢=𝑎−𝑢−(𝑢𝑣)𝑢|1+𝑚)(1+𝑘𝑢,_𝜕Ω=0. (3.13) It follows from the maximum principle that 𝑢≡0. Similarly, we can show that 𝑣≡0 for 𝑣∈𝜕𝑃₁.
Thus we only need the following three cases: (i)(𝑢,𝑣)≡𝜃𝑎,0,(ii)(𝑢,𝑣)≡0,𝜃_𝑐,(iii)(𝑢,𝑣)≡(0,0). (3.14)
Suppose that (𝑢,𝑣)≡(𝜃𝑎,0). Then (𝑎_𝑛;𝑢_𝑛,𝑣_𝑛)→(𝑎;𝜃𝑎,0) when 𝑛→∞. Let 𝑉_𝑛=𝑣_𝑛/||𝑣_𝑛||_∞; then 𝑉_𝑛 satisfies −Δ𝑉_𝑛=𝑐−𝑣_𝑛+𝑑𝑢_𝑛1+𝑚𝑢_𝑛1+𝑘𝑣_𝑛𝑉_𝑛,𝑉_𝑛𝜕Ω=0. (3.15) Thanks to 𝐿^𝑃 estimates and Sobolev embedding theorem, there exists a convergent subsequence of 𝑉_𝑛, which we still denote by 𝑉_𝑛, such that 𝑉_𝑛→𝑉 in 𝐶¹₀(Ω) as 𝑛→∞, and 𝑉≥0,≢0,𝑥∈Ω because of ‖𝑉‖=1. So taking the limit in (3.15) as 𝑛→∞, we get−Δ𝑉=𝑐+𝑑𝜃𝑎𝑎|1+𝑚𝜃𝑉,𝑉_𝜕Ω=0. (3.16) It follows from the maximum principle that 𝑉>0,𝑥∈Ω, which implies 𝑐=𝜆₁(−𝑑𝜃𝑎/(1+𝑚𝜃𝑎)). This contradicts 𝑐>𝜆₁.
Suppose that (𝑢,𝑣)≡(0,𝜃_𝑐). Then (𝑎_𝑛;𝑢_𝑛,𝑣_𝑛)→(𝑎;0,𝜃_𝑐) as 𝑛→∞. Let 𝑈_𝑛=𝑢_𝑛/||𝑢_𝑛||_∞; then 𝑈_𝑛 satisfies −Δ𝑈_𝑛=𝑎_𝑛−𝑢_𝑛−𝑏𝑣_𝑛1+𝑚𝑢_𝑛1+𝑘𝑣_𝑛𝑈_𝑛,𝑈_𝑛𝜕Ω=0. (3.17) Similarly, By 𝐿^𝑃 estimates and and Sobolev embedding theorem, there exists a convergent subsequence of 𝑈_𝑛, which we still denote by 𝑈_𝑛, such that 𝑈_𝑛→𝑈 in 𝐶¹₀(Ω) as 𝑛→∞, and 𝑈≥0,≢0,𝑥∈Ω because of ‖𝑉‖=1. So taking limit in (3.17) as 𝑛→∞, we obtain−Δ𝑈=𝑎−𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝑈,𝑈|_𝜕Ω=0. (3.18) It follows from the maximum principle that 𝑈>0,𝑥∈Ω. Hence 𝑎=𝜆₁(𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐)). A contradiction with 𝑎≠𝑎.
Suppose that (𝑢,𝑣)≡(0,0). Similar to the previously mentioned, we can get contradiction.
Thus 𝐶−{(𝑎;0,𝜃_𝑐)}⊂𝑃. By Lemma 2.1, we have 0≤𝑈≤𝑎,𝜃_𝑐≤𝑉≤𝑐+𝑑𝑎/(1+𝛼𝑎). Thanks to 𝐿^𝑃 estimates and and Sobolev embedding theorem, then there exists a constant 𝑀>0 such that ||𝑈||_𝐶¹,||𝑉||_𝐶¹≤𝑀. Hence the global bifurcation 𝐶 of positive solutions of (1.2) bifurcating at (𝑎;0,𝜃_𝑐) contains points with 𝑎 is arbitrarily large in 𝑃.

In the following, we will study the stability of the bifurcation solution. Let 𝑋₁=[𝐶^2,𝛼(Ω)×𝐶^2,𝛼(Ω)∩𝑋],𝑌=[𝐶^𝛼(Ω)×𝐶^𝛼(Ω)], where 0<𝛼<1.𝑖∶𝑋₁→𝑌 is the inclusion mapping. Since 𝐿₁ is the linearized operator at (𝑎;0,𝜃_𝑐) for (1.2). By the proof of Lemma 3.1, we have 𝑁(𝐿₁)= span{(Φ,Ψ)}, Codim𝑅(𝐿₁)=1, and 𝑅(𝐿₀)={(𝑢,𝑣)∈𝑋∶∫_Ω𝑢Φ𝑑𝑥=0}. Since 𝑖(Φ,Ψ)∈𝑅(𝐿₁), so it follows from [21] that 0 is an 𝑖-simple eigenvalue of 𝐿₁.

Lemma 3.4. 0 is the eigenvalue of 𝐿₁ with the largest real part, and all the other eigenvalue of 𝐿₁ lie in the left half complex plane.

Proof. We assume that 𝜇₀ is the eigenvalue of 𝐿₁ with the largest real part Re𝜇₀>0, and (𝜉,𝜂) is the corresponding eigenfunction; then 𝐿₁(𝜉,𝜂)=𝜇₀(𝜉,𝜂), equivalently, Δ𝜉+𝑎−𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜉=𝜇₀𝜉,𝑥∈Ω,Δ𝜂+𝑐−2𝜃_𝑐𝜂+𝑑𝜃_𝑐1+𝑘𝜃_𝑐𝜉=𝜇₀𝜂,𝑥∈Ω,𝜂=𝜉=0,𝑥∈𝜕Ω. (3.19)
Suppose that 𝜉≡0. Then 𝜇₀ is an eigenvalue of the operator (Δ+(𝑐−2𝜃_𝑐)𝐼), and then 𝜇₀∈𝑅 and 𝜇₀<0, a contradiction. Thus Φ≡0. It follows that 𝜇₀ is an eigenvalue of the operator (Δ+(𝑎−𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐))𝐼). Since 𝑎=𝜆₁(𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐)),0 is the principal eigenvalue of the operator (Δ+(𝑎−𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐))𝐼); furthermore, 𝜇₀≤0. This contradicts the assumption, so assumption does not hold. Which proves our conclusion.

We will use the linearized stability theory from [22]. Let 𝐿(𝑢(𝑠),𝑣(𝑠),𝑎(𝑠)), and 𝐿(𝑎;0,𝜃_𝑐) be the linearized operators of (1.2) at (𝑢(𝑠),𝑣(𝑠),𝑎(𝑠)), and (𝑎;0,𝜃_𝑐), 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 𝐶¹-function: 𝑎→(𝑀(𝑎),𝛾(𝑎)), and 𝑠→(𝑁(𝑠),𝜋(𝑠)), defined from the neighborhood of 𝑎 and 0 into 𝑋₁×𝑅, respectively, such that 𝛾(𝑎)=𝜋(0)=0,𝑀(𝑎)=𝑁(0)=(Φ,Ψ) and 𝐿𝑎;0,𝜃_𝑐𝑀(𝑎)(𝑎)𝑀(𝑎)|𝑎|𝐿(𝑢(𝑠)(𝑠)(𝑠𝑁(𝑠)(𝑠)𝑁(𝑠)|𝑠|=𝛾,𝑓𝑜𝑟𝑎−≪1,,𝑣,𝑎))=𝜋,𝑓𝑜𝑟≪1, (3.20) where 𝑀(𝑎)=(𝜙₁(𝑎),𝜙₂(𝑎)),𝑁(𝑠)=(𝜑₁(𝑎)𝜑₂(𝑎)). Moreover 𝛾^′(𝑎)≠0, whence 𝜋(𝑠)≠0,𝜋(𝑠) and −𝑠𝑎^′(𝑠)𝛾^′(𝑎) have the same sign for |𝑠|≪1. Where 𝛾^′(𝑎) is the derivative of 𝛾(𝑎) with respect to 𝑎 at 𝑎=𝑎, and 𝑎^′(𝑠) is the derivative of 𝑎(𝑠) with respect to 𝑠.

Lemma 3.6. The derivative of 𝛾(𝑠) with respect to 𝑎 at 𝑎 is positive.

Proof. It follows from 𝐿(𝑎;0,𝜃_𝑐)𝑀(𝑎)=𝛾(𝑎)𝑀(𝑎),for,|𝑎−𝑎|≪1, that is, Δ𝜙₁+𝑎−𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜙₁=𝛾(𝑎)𝜙₁,𝑥∈Ω,Δ𝜙₂+𝑐−2𝜃_𝑐𝜙₂+𝑑𝜃_𝑐1+𝑘𝜃_𝑐𝜙₁=𝛾(𝑎)𝜙₂𝜙,𝑥∈Ω,₁=𝜙₂=0,𝑥∈𝜕Ω. (3.21) Since |𝑎−𝑎|≪1, so |𝛾(𝑎)|≪1. Clearly, 𝜙₁≢0, otherwise, 𝜙₁≡0, then 𝜙₂≡0, a contradiction. Hence 𝛾(𝑎) is an eigenvalue of the operator (Δ+(𝑎−𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐))𝐼). We consider Φ>0, then 𝜙₁=𝜙₁(𝑎)>0, as |𝑎−𝑎|≪1. So 𝛾(𝑎) is the principal eigenvalue of the operator (Δ+(𝑎−𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐))𝐼), and 𝛾(𝑎) is increasing with respect to 𝑎 as |𝑎−𝑎|≪1. Moreover, 𝛾^′(𝑎)≠0. Hence 𝛾^′(𝑎)>0.

Lemma 3.7. The derivative of 𝑎(𝑠) with respect to 𝑠 at 𝑠=0 satisfies 𝑎^′(0)_ΩΦ²𝑑𝑥=_Ω1−𝑏𝑚𝜃_𝑐1+𝑘𝜃_𝑐Φ³𝑑𝑥+_Ω𝑏Ψ1+𝑘𝜃_𝑐²Φ²𝑑𝑥. (3.22)

Proof. By substituting (𝑢(𝑠),𝑣(𝑠),𝑎(𝑠)) into (1.2), differentiating with respect to 𝑠, and then setting 𝑠=0, we find that −Δ𝜙^′(0)=𝑎−𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜙^′(0)+𝑎^′(0)−Φ−𝑏Ψ−𝑚𝜃_𝑐Φ1+𝑘𝜃_𝑐1+𝑘𝜃_𝑐²Φ, (3.23) where 𝜙^′(0) is the derivative of 𝜙 with respect to 𝑠 at 𝑠=0.
Taking the inner product with Φ, using Green's formula, and noting the definition of Φ, we have 𝑎^′(0)_ΩΦ²𝑑𝑥=_Ω1−𝑏𝑚𝜃_𝑐1+𝑘𝜃_𝑐Φ³𝑑𝑥+_Ω𝑏Ψ1+𝑘𝜃_𝑐²Φ²𝑑𝑥. (3.24)

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

Theorem 3.8. Let 𝜎=∫_Ω(1−𝑏𝑚𝜃_𝑐/(1+𝑘𝜃_𝑐))Φ³𝑑𝑥+∫_Ω(𝑏Ψ/(1+𝑘𝜃_𝑐)²)Φ²𝑑𝑥. If 𝜎>0, then bifurcation solution (𝑢(𝑠),𝑣(𝑠)) is stable; If 𝜎<0, then bifurcation solution (𝑢(𝑠),𝑣(𝑠)) 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 𝑎>𝜆₁(𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐)) and 𝑎>𝜆₁(𝑏𝜃_𝑐/(1+𝑚𝜃_𝑎)(1+𝑘𝜃_𝑐)) when 𝑐>𝜆₁. Next, we will consider the multiplicity, stability, and uniqueness of positive solutions in the gap.

Theorem 3.9. Assume 𝑐>𝜆₁ and ∫_Ω(1−𝑏𝑚𝜃_𝑐/(1+𝑘𝜃_𝑐))Φ³𝑑𝑥<0. Then there exist 𝜀>0 such that bifurcation positive solution (𝑢(𝑠),𝑣(𝑠)) is nondegenerate and unstable for 𝑎∈(𝑎−𝜀,𝑎) and 𝑑≪1. Moreover, Problem (1.2) has at least two positive solutions.

Proof. We first prove that bifurcation positive solution (𝑢(𝑠),𝑣(𝑠)) is nondegenerate and unstable. To this end, it suffices to show that there exists a sufficiently small 𝜀>0 such that for 𝑎∈(𝑎−𝜀,𝑎), any positive solution (𝑢(𝑠),𝑣(𝑠)) of (1.2) is nondegenerate and the linearized eigenvalue problem: 𝑎(𝑠)(𝑠)−(𝑠)−Δ𝜉−−𝑢𝑏𝑣(1+𝑚𝑢(𝑠))²((𝑠(𝑠)1+𝑘𝑣))𝜉+𝑏𝑢(1+𝑚𝑢(𝑠))(1+𝑘𝑣(𝑠))²(𝑠)+(𝑠)𝜂=𝜇𝜉,𝑥∈Ω,−Δ𝜂−𝑐−2𝑣𝑑𝑢(1+𝑚𝑢(𝑠))(1+𝑘𝑣(𝑠))²(𝑠)𝜂−𝑑𝑣(1+𝑚𝑢(𝑠))²(1+𝑘𝑣(𝑠))=𝜇𝜂,𝑥∈Ω,𝜉=𝜂=0,𝑥∈𝜕Ω, (3.25) has a unique eigenvalue 𝜇_∗ and Re(𝜇_∗)<0 with algebra multiplicity one.
Let {𝜀_𝑖>0} and {𝑑_𝑖>0} be sequences which approach 0 as 𝑖→∞. Due to 𝑎=𝑎+𝑎^′(0)𝑠+𝑂(𝑠²), we can set sequences {𝜀_𝑖>0} and {𝑎_𝑖} such that 𝑎_𝑖∈(𝑎−𝜀_𝑖,𝑎) and 𝑠_𝑖→0 as 𝑖→∞. It follows that (𝑢_𝑖,𝑣_𝑖) is a solution of (1.2). Then the corresponding linearized problem (3.25) can become the following form: 𝐿_𝑖𝜉_𝑖𝜂_𝑖=𝜇_𝑖𝜉_𝑖𝜂_𝑖,𝐿_𝑖=𝑀_𝑖¹¹𝑀_𝑖¹²𝑀_𝑖²¹𝑀_𝑖²², (3.26) where (𝜉_𝑖,𝜂_𝑖)≢(0,0) and 𝑀_𝑖¹¹=−Δ−𝑎_𝑖−2𝑢_𝑖−𝑏𝑣_𝑖1+𝑚𝑢_𝑖²1+𝑘𝑣_𝑖,𝑀_𝑖¹²=𝑏𝑢_𝑖1+𝑚𝑢_𝑖1+𝑘𝑣_𝑖²,𝑀_𝑖²¹=−𝑑𝑣_𝑖1+𝑚𝑢_𝑖²1+𝑘𝑣_𝑖,𝑀_𝑖²²=−Δ−𝑐−2𝑣_𝑖+𝑑𝑢_𝑖1+𝑚𝑢_𝑖1+𝑘𝑣_𝑖². (3.27) Observe that as 𝑖→∞,  𝐿_𝑖^𝜉𝑖_𝜂𝑖 converges to𝐿₀𝜉𝜂=−Δ𝜉−𝑎−𝑏𝜃_𝑐1+𝑘𝜃_𝑐𝜉00−Δ𝜂−𝑐−2𝜃_𝑐𝜂. (3.28) It is easy to get that 0 is a simple eigenvalue of the operator 𝐿₀ with corresponding eigenfunction (𝜉,𝜂)^𝑇=(Φ,0)^𝑇. Moreover, all the other eigenvalues of 𝐿₀ are positive and stand apart from 0. Therefore, using perturbation theory [24], we get that for large 𝑖,𝐿_𝑖 has a unique eigenvalue 𝜇_𝑖 which is close to zero. In addition, all the other eigenvalues of 𝐿_𝑖 have positive real parts and stand apart from 0. Note that 𝜇_𝑖 is simple real eigenvalue which converges to zero, and we can take the corresponding eigenfunction (𝜉_𝑖,𝜂_𝑖)^𝑇 such that (𝜉_𝑖,𝜂_𝑖)→(Φ,0) as 𝑖→∞. If we show that Re𝜇_𝑖<0 for large 𝑖, then the result follows. By multiplying Φ to the first equation of 𝐿_𝑖(𝜉_𝑖,𝜂_𝑖)^𝑇=𝜇_𝑖(𝜉_𝑖,𝜂_𝑖)^𝑇 and integrating over Ω, we obtain −_ΩΦΔ𝜉_𝑖−_Ω𝑎_𝑖−2𝑢_𝑖−𝑏𝑣_𝑖1+𝑚𝑢_𝑖²1+𝑘𝑣_𝑖Φ𝜉_𝑖+_Ω𝑏𝑢_𝑖Φ𝜂_𝑖1+𝑚𝑢_𝑖1+𝑘𝑣_𝑖²=_Ω𝜇_𝑖Φ𝜉_𝑖. (3.29) Multiplying the first equation of (1.2) with (𝑎,𝑢,𝑣)=(𝑎_𝑖,𝑢_𝑖,𝑣_𝑖) by 𝜉_𝑖 and integrating, we have −_Ω𝜉_𝑖Δ𝑢_𝑖−_Ω𝑎_𝑖−𝑢_𝑖−𝑏𝑣_𝑖1+𝑚𝑢_𝑖1+𝑘𝑣_𝑖𝑢_𝑖𝜉_𝑖. (3.30) Due to 𝑢_𝑖=𝑠_𝑖Φ+𝑂(𝑠²_𝑖), the previous equation becomes −_ΩΦΔ𝜉_𝑖−_Ω𝜉_𝑖Φ𝑎_𝑖−𝑢_𝑖−𝑏𝑣_𝑖1+𝑚𝑢_𝑖1+𝑘𝑣_𝑖+𝑂𝑠²_𝑖. (3.31) Using (3.29) and (3.31), we have _Ω1−𝑏𝑚𝑣_𝑖1+𝑚𝑢_𝑖²1+𝑘𝑣_𝑖𝑢_𝑖Φ𝜉_𝑖+_Ω𝑏𝑢_𝑖Φ𝜂_𝑖1+𝑚𝑢_𝑖1+𝑘𝑣_𝑖²=_Ω𝑢_𝑖Φ𝜉_𝑖. (3.32) Recall that (𝑢_𝑖,𝑣_𝑖)=(𝑠_𝑖Φ+𝑂(𝑠²_𝑖),𝜃_𝑐+𝑠_𝑖Ψ_𝑑𝑖+𝑂(𝑠²_𝑖)), here Ψ_𝑑𝑖 is Ψ defined in Remark 3.2, and so dividing the previous equation by 𝑠_𝑖 and taking the limit, we have lim_𝑖→∞𝜇_𝑖𝑠_𝑖=∫_Ω1−𝑏𝑚𝜃_𝑐/1+𝑘𝜃_𝑐Φ³∫_ΩΦ²,<0 (3.33) which implies that Re𝜇_𝑖<0 for large 𝑖. 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 (𝑢,𝑣), then this solution must be bifurcated from (0,𝜃_𝑐). Since there exists a positive solution near 𝑎 by the local bifurcation theory. So (𝑢,𝑣) is nondegenerate, and the corresponding linearized eigenvalue problem has a unique eigenvalue 𝜇 with algebra multiplicity one such that Re𝜇<0. Due to these facts, it is easy to show that 𝐼−𝐹^′(𝑢,𝑣) is invertible and does not have property 𝛼 on 𝑊₍𝑢_,𝑣₎; it follows that index(𝐹,(𝑢,𝑣))=(−1)¹=−1 by Lemma 2.2 (ii). Finally, Using Lemmas 2.3–2.4 and the additivity property of the index, we obtain 1=index_𝑊(𝐹,𝐷)=index_𝑊(𝐹,(0,0))+index_𝑊𝐹,𝜃_𝑎,0+index_𝑊𝐹,0,𝜃_𝑐+index_𝑊(𝐹,(𝑢,𝑣))=0+0+1−1=0, (3.34) which derives a contradiction. Hence the Proof is complete.

Remark 3.10. In Theorem 3.9, the multiplicity can be shown easily when 𝑚≥0. Note that 𝑎^′(0)<0 for a sufficiently small 𝑑, since ∫_Ω(1−𝑏𝑚𝜃_𝑐/(1+𝑘𝜃_𝑐))Φ³<0, and so 𝑎=𝑎(𝑠)∈(𝜆₁,𝑎). Since there is no positive solution of (1.2) if 𝑎≤𝜆₁(𝑏𝜃_𝑐/(1+𝑚𝜃_𝑎)(1+𝑘𝜃_𝑐)) by Theorem 2.5 (i) when 𝑚≥0 and 𝑐>𝜆₁. Therefore we easily see that there must be at least two positive solutions for 𝑎∈(𝑎^∗,𝑎) and some 𝑎^∗∈(𝜆₁(𝑏𝜃_𝑐/(1+𝑚𝜃_𝑎)(1+𝑘𝜃_𝑐)),𝜆₁(𝑏𝜃_𝑐/(1+𝑘𝜃_𝑐))).

4. Multiplicity, Uniqueness and Stability as 𝑘 or 𝑚 Is Large

In this section, taking 𝑘 or 𝑚 as a parameter, we investigate the multiplicity, stability, and uniqueness of positive solutions of (1.2) as 𝑘 or 𝑚 is large. In the following, we will always assume that 𝑐>𝜆₁, and let 𝑏,𝑐,𝑑 be fixed, unless otherwise specified.

Firstly, we consider the case that 𝑘 is large and 𝑎 is bounded away from 𝜆₁. Hence the upper solution 𝜃_𝑎 for 𝑢 and the lower solution 𝜃_𝑐 for 𝑣 do not depend on 𝑘 when 𝑎>𝜆₁ and 𝑐>𝜆₁.

Lemma 4.1. For any small 𝜀>0, there exists 𝐾(𝜀) such that for 𝑘≥𝐾(𝜀), (1.2) has at least one positive solution (𝑢,𝑣) which satisfies 𝜃_{(𝑎−𝜀)}≤𝑢≤𝜃_𝑎,𝜃_𝑐≤𝑣≤𝜃_(𝑐+𝜀). (4.1)

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

Lemma 4.2. (𝑖) If 𝑘→∞, then any positive solution of (1.2) approaches (𝜃_𝑎,𝜃_𝑐).
(𝑖𝑖) There exists a large 𝐾(𝜖) such that any positive solution of (1.2) is nondegenerate and linearly stable when 𝑘>𝐾(𝜖).

Proof. (i) Let 𝑘→∞, we show that the compact operator 𝐹(𝑢,𝑣) converges to 𝐹(𝑢,𝑣), where 𝐹(𝑢,𝑣)=(−Δ+𝑃)⁻¹𝑢()𝑣()𝑎−𝑢+𝑃𝑢𝑐−𝑣+𝑃𝑣. (4.2) It follows that any positive solution of (1.2) converges to the fixed point of 𝐹(𝑢,𝑣) in this case. It is easy to see that (𝜃_𝑎,𝜃_𝑐) is a unique fixed point of 𝐹(𝑢,𝑣), so the positive solutions of (1.2) could not converge to semitrivial solutions when 𝑘→∞. Thus the conclusion is complete.
(ii) We use a contradiction method. Assume that there exists 𝑘_𝑖→∞, 𝜇_𝑖 with Re𝜇_𝑖≤0 and (𝜉_𝑖,𝜂_𝑖)≢(0,0) with ‖𝜉_𝑖‖²_𝐿2+‖𝜂_𝑖‖²_𝐿2=1 such that −Δ𝜉_𝑖−𝑎−2𝑢_𝑖−𝑏𝑣_𝑖1+𝑚𝑢_𝑖²1+𝑘_𝑖𝑣_𝑖𝜉_𝑖+𝑏𝑢_𝑖1+𝑚𝑢_𝑖1+𝑘_𝑖𝑣_𝑖²𝜂_𝑖=𝜇_𝑖𝜉_𝑖,𝑥∈Ω,−Δ𝜂_𝑖−𝑐−2𝑣_𝑖+𝑑𝑢_𝑖1+𝑚𝑢_𝑖1+𝑘_𝑖𝑣_𝑖²𝜂_𝑖−𝑑𝑣_𝑖1+𝑚𝑢_𝑖²1+𝑘_𝑖𝑣_𝑖𝜉_𝑖=𝜇_𝑖𝜂_𝑖𝜉,𝑥∈Ω,_𝑖=𝜂_𝑖=0,𝑥∈𝜕Ω, (4.3) where (𝑢_𝑖,𝑣_𝑖) is a positive solution of (1.2) with 𝑘=𝑘_𝑖. By computing, we have 𝜇_𝑖=_Ω∇𝜉_𝑖²−_Ω𝑎−2𝑢_𝑖−𝑏𝑣_𝑖1+𝑚𝑢_𝑖²1+𝑘_𝑖𝑣_𝑖𝜉_𝑖²+_Ω𝑏𝑢_𝑖𝜂_𝑖𝜉_𝑖1+𝑚𝑢_𝑖1+𝑘_𝑖𝑣_𝑖²+_Ω∇𝜂_𝑖²−_Ω𝑑𝑣_𝑖𝜉_𝑖𝜂_𝑖1+𝑚𝑢_𝑖²1+𝑘_𝑖𝑣_𝑖−_Ω𝑐−2𝑣_𝑖+𝑑𝑢_𝑖1+𝑚𝑢_𝑖1+𝑘_𝑖𝑣_𝑖²𝜂_𝑖², (4.4) where 𝜉_𝑖,𝜂_𝑖 are the complex conjugates of 𝜉_𝑖,𝜂_𝑖. From Lemma 2.1, we know that 0≤𝑢_𝑛≤𝜃_𝑎 and 𝜃_𝑐≤𝑣_𝑖≤𝑐+𝑑𝑎/(1+𝑚𝑎). It follows that Im𝜇_𝑖 is bounded and Re𝜇_𝑖 is bounded from the following. Thus 𝜇_𝑖 is bounded as we assume Re𝜇_𝑖≤0. Hence we can suppose that 𝜇_𝑖→𝜇 and Re𝜇≤0. Thanks to 𝐿^𝑝 estimate, we obtain ‖𝜉_𝑖‖_𝑊2,2 and ‖𝜂_𝑖‖_𝑊2,2 are bounded. Hence we may assume that 𝜉_𝑖→𝜉 and 𝜂_𝑖→𝜂 in 𝐻¹₀ strongly, here