A Lotka-Volterra Competition Model with Cross-Diffusion

Abstract

A Lotka-Volterra competition model with cross-diffusions under homogeneous Dirichlet boundary condition is considered, where cross-diffusions are included in such a way that the two species run away from each other because of the competition between them. Using the method of upper and lower solutions, sufficient conditions for the existence of positive solutions are provided when the cross-diffusions are sufficiently small. Furthermore, the investigation of nonexistence of positive solutions is also presented.

1. Introduction

In this paper, we deal with the following Lotka-Volterra competition model with cross-diffusions: ()()()−Δ𝑢+𝛼𝑣=𝑢𝑎−𝑢−𝑐𝑣,𝑥∈Ω,−Δ𝛽𝑢+𝑣=𝑣𝑏−𝑣−𝑑𝑢,𝑥∈Ω,𝑢=𝑣=0,𝑥∈𝜕Ω, (1) where Ω is a bounded domain in 𝐑^𝑁 with smooth boundary 𝜕Ω and all parameters 𝑎,𝑏,𝑐,𝑑,𝛼,𝛽 are positive constants. 𝑢 and 𝑣 stand for the densities of the two competitors; 𝑎 and 𝑏 are the intrinsic growth rates of 𝑢 and 𝑣, respectively; 𝑐 and 𝑑 are the competitive parameters between the two species; Here 𝛼 and 𝛽 are referred to as cross-diffusions. Cross-diffusions express the two species run away from each other because of the competition between them. In this paper, the boundary condition is under homogeneous Dirichlet boundary condition which in biologically means that the boundary is not suitable for both species and they will all die on the boundary, and this is an ideal case.

In order to describe the meaning of cross-diffusions in this model (1) from the biological point, we give the general model with intrinsic diffusion and cross-diffusion: 𝜕𝑢𝑘𝜕𝑡=div₁₁(𝑢,𝑣)∇𝑢+𝑘₁₂(𝑢,𝑣)∇𝑣+𝑓(𝑢,𝑣),𝜕𝑣𝑘𝜕𝑡=div₂₁(𝑢,𝑣)∇𝑢+𝑘₂₂(𝑢,𝑣)∇𝑣+𝑔(𝑢,𝑣), (2) where 𝑢 and 𝑣 stand for the densities of the two species, intrinsic diffusion parameters 𝑘₁₁(𝑢,𝑣), 𝑘₂₂(𝑢,𝑣)>0, cross-diffusion parameters 𝑘₁₂(𝑢,𝑣),𝑘₂₁(𝑢,𝑣), 𝐽_𝑢=−𝑘₁₁(𝑢,𝑣)∇𝑢+𝑘₁₂(),𝐽𝑢,𝑣∇𝑣_𝑣=−𝑘₂₁(𝑢,𝑣)∇𝑢+𝑘₂₂(𝑢,𝑣)∇𝑣 (3) can be seen as the out-flux vector of 𝑢 and 𝑣 at 𝑥. The cross-diffusion parameters 𝑘₁₂(𝑢,𝑣),𝑘₂₁(𝑢,𝑣)≥0 imply that the two competitors 𝑢 and 𝑣 diffuse in the direction of lower contrary of their competitor to avoid each other. 𝑓(𝑢,𝑣),𝑔(𝑢,𝑣) are response function and in this paper the classical Logistic Type is considered and 𝛼,𝛽≥0. More biological meaning of the system can be seen in [1–3].

The method of upper and lower solutions is a useful tool to study the existence of solutions of elliptic systems. However, there are many difficulties in investigating the existences of positive solutions of strongly coupled elliptic systems. Recently, by changing general strongly coupled elliptic systems into weakly coupled ones, the author in paper [4] gives the method to judge the solutions existence of elliptic systems by using the Schauder theorem. Furthermore, the method can be used to solve the existence of solutions of strongly coupled elliptic systems. In [5] Ko and Ryu investigate Lotka-Volterra prey-predator model with cross-diffusion: −Δ𝑢=𝑢𝑎₁−𝑢−𝑏₁₂𝑣𝑎,𝑥∈Ω,−𝐷Δ𝑢−Δ𝑣=𝑣₂+𝑏₂₁𝑢−𝑣,𝑥∈Ω,𝑢=𝑣=0,𝑥∈𝜕Ω. (4) Here 𝐷 may be positive or negative. Using the developing method of upper and lower solutions in [4], the author gave a sufficient conditions for the existence of positive solutions to (4). Inspired by the paper [5], we investigate the existence and nonexistence of positive solutions to (1).

The main goal of this paper is to provide sufficient conditions for the existence of positive solutions to (1) when the cross-diffusions 𝛼 and 𝛽 are small. More precisely, we have the following theorem. Let 𝜆₁>0 be the principal eigenvalue of −Δ under homogeneous Dirichlet boundary condition. It is well known that the principal eigenfunction 𝜙 corresponding to 𝜆₁ does not change sign in Ω and ||𝜙||_∞=1.

Theorem 1. If min{𝑎−𝑐𝑏,𝑏−𝑑𝑎}>𝜆₁, then there exist positive constants 𝛼=𝛼(𝑎,𝑏,𝑐,𝑑,Ω),𝛽=𝛽(𝑎,𝑏,𝑐,𝑑,Ω), when 𝛼<𝛼,𝛽<𝛽, (1) has at least one positive solution.

For 𝛼=𝛽=0, (1) is the Lotka-Volterra competition model under homogeneous Dirichlet boundary condition. In [6, 7], the authors use different methods to prove the existence of positive solutions, a sufficient condition for the existence is min{𝑎−𝑐𝑏,𝑏−𝑑𝑎}>𝜆₁. The conclusion implies that weakly cross-diffusion does not affect the existence of positive solution.

This paper is organized as follows. In Section 2, the existence theorem of solutions for a general class of strongly coupled elliptic systems is presented using the method of upper and lower solutions. In Section 3, sufficient conditions for the existence and nonexistence of positive solutions of (1) are investigated. Moreover, we give the corresponding results simply if the competitive system only has one cross-diffusion.

2. The Existence Theorem of Solutions for a Class of Strongly Coupled Elliptic Systems

In this section, we presented the existence theorem of solutions for a general class of strongly coupled elliptic systems: −Δ𝐴(𝑢,𝑣)=𝑓₁()()𝑢,𝑣,𝑥∈Ω,−Δ𝐵𝑢,𝑣=𝑓₂(𝑢,𝑣),𝑥∈Ω,𝑢=𝑣=0,𝑥∈𝜕Ω. (5) Here let 𝐴,𝐵,𝑓₁,𝑓₂ satisfy the following hypotheses conditions.(H1)𝑈,𝑉 are domain in 𝐑², (0,0)∈𝑈. (𝐴,𝐵) is a 𝐶² function about (𝑢,𝑣) from 𝑈 to 𝑉, 𝐴(0,0)=𝐵(0,0)=0, and have a continuous inverse (𝐴^∗,𝐵^∗)∈𝐶²(𝑉,𝑈). Then for all (𝑢,𝑣)∈𝑈, let 𝑤=𝐴(𝑢,𝑣),𝑧=𝐵(𝑢,𝑣). (6) There exists only one (𝑤,𝑧)∈𝑉, satisfying 𝑢=𝐴^∗(𝑤,𝑧),𝑣=𝐵^∗(𝑤,𝑧). (7)(H2) The function 𝐴^∗ is increasing in 𝑤 and decreasing in 𝑧; 𝐵^∗ is decreasing in 𝑤 and increasing in 𝑧.(H3) The functions 𝑓₁(𝑢,𝑣),𝑓₂(𝑢,𝑣) are Lipschitz continuous in 𝑈, and there exist positive constants 𝑀₁,𝑀₂ such that for all (𝑢,𝑣)∈𝑈, the function 𝑓₁(𝑢,𝑣)+𝑀₁𝐴(𝑢,𝑣) is increasing in 𝑢; the function 𝑓₂(𝑢,𝑣)+𝑀₂𝐵(𝑢,𝑣) is increasing in 𝑣.

According to the hypothesis (H1), (5) can be rewritten as the following equal PDE equations: −Δ𝑤+𝑀₁𝑤=𝑓₁(𝑢,𝑣)+𝑀₁𝐴(𝑢,𝑣),𝑥∈Ω,−Δ𝑧+𝑀₂𝑧=𝑓₂(𝑢,𝑣)+𝑀₂𝐵(𝑢,𝑣),𝑥∈Ω,𝑢=𝐴^∗(𝑤,𝑧),𝑣=𝐵^∗(𝑤,𝑧),𝑥∈Ω,𝑤=𝑧=0,𝑥∈𝜕Ω. (8)

Remark 2. According to the hypothesis (H1), (5) can also be equal to the following weakly coupled elliptic equations: −Δ𝑤=𝑓₁𝐴^∗(𝑤,𝑧),𝐵^∗(𝑤,𝑧)∶=𝑔₁(𝑤,𝑧),𝑥∈Ω,−Δ𝑧=𝑓₂𝐴^∗(𝑤,𝑧),𝐵^∗(𝑤,𝑧)∶=𝑔₂(𝑤,𝑧),𝑥∈Ω,𝑤=𝑧=0,𝑥∈𝜕Ω. (9)

In its pure form, (9) is simpler than (8). However, due to the complicity of mixed functions 𝑔₁(𝑤(𝑥),𝑧(𝑥)) and 𝑔₂(𝑤(𝑥),𝑧(𝑥)), it is difficult to find the solutions of (9) directly. Therefore, we discuss (8).

Assume functions 𝑢,𝑣,𝑢,𝑣∈𝐶(Ω),𝑤,𝑧,𝑤,𝑧∈𝐶^𝛼(Ω)⋂𝐶²(Ω), the values of functions (𝑢,𝑣) and (𝑢,𝑣) are in 𝑉 and the values of functions (𝑤,𝑧) and (𝑤,𝑧) are in 𝑈. To describe easily, let 𝑈=𝑢∈𝐶Ω∶𝑢(𝑥)≤𝑢(𝑥)≤𝑢(𝑥),𝑉=𝑢∈𝐶Ω∶𝑣(𝑥)≤𝑣(𝑥)≤𝑣(𝑥). (10)

According the definition of upper and lower solutions in [4] and conditions (H1)–(H3), we give the definition of upper and lower solutions of (5).

Definition 3. A pair of functions ((𝑢,𝑣,𝑤,𝑧),(𝑢,𝑣,𝑤,𝑧)) are called upper and lower solutions of (9) provided that they satisfy the relation (𝑢,𝑣,𝑤,𝑧)≥(𝑢,𝑣,𝑤,𝑧), and for all (𝑢,𝑣)∈𝑈×𝑉, satisfy the following inequalities: −Δ𝑤+𝑀₁𝑤≥𝑓₁(𝑢,𝑣)+𝑀₁𝐴(𝑢,𝑣),𝑥∈Ω,−Δ𝑧+𝑀₂𝑧≥𝑓₂(𝑢,𝑣)+𝑀₂𝐵(𝑢,𝑣),𝑥∈Ω,−Δ𝑤+𝑀₁𝑤≤𝑓₁𝑢,𝑣+𝑀₁𝐴𝑢,𝑣,𝑥∈Ω,−Δ𝑧+𝑀₂𝑧≤𝑓₂𝑢,𝑣+𝑀₂𝐵𝑢,𝑣,𝑥∈Ω,𝑢≥𝐴^∗𝑤,𝑧,𝑣≥𝐵^∗𝑤,𝑧𝑢,𝑥∈Ω,≤𝐴^∗𝑤,𝑧,𝑣≤𝐵^∗𝑤,𝑧,𝑥∈Ω,𝑤≥0≥𝑤,𝑧≥0≥𝑧,𝑥∈𝜕Ω. (11)

We can have the following conclusion from [4, Theorem 2.1].

Proposition 4. Assume that (8) has coupled upper and lower solutions ((𝑢,𝑣,𝑤,𝑧),(𝑢,𝑣,𝑤,𝑧)), then there exists at least one solution (𝑢,𝑣,𝑤,𝑧), satisfying the relation 𝑢,𝑣,𝑤,𝑧≤(𝑢,𝑣,𝑤,𝑧)≤(𝑢,𝑣,𝑤,𝑧). (12) Furthermore, (𝑢,𝑣) is the solution of (5).

Next, if 𝑢,𝑣,𝑢,𝑣 satisfy 𝑢=𝐴^∗𝑤,𝑧,𝑣=𝐵^∗𝑤,𝑧,𝑢=𝐴^∗𝑤,𝑧,𝑣=𝐵^∗𝑤,𝑧, (13) then 𝑤=𝐴𝑢,𝑣,𝑧=𝐵𝑢,𝑣,𝑤=𝐴𝑢,𝑣,𝑧=𝐵𝑢,𝑣, (14) (11) can be rewritten as −Δ𝐴𝑢,𝑣+𝑀₁𝐴𝑢,𝑣≥𝑓₁(𝑢,𝑣)+𝑀₁𝐴()𝑢𝑢,𝑣,𝑥∈Ω,−Δ𝐵,𝑣+𝑀₂𝐵𝑢,𝑣≥𝑓₂(𝑢,𝑣)+𝑀₂𝐵(𝑢,𝑣)𝑢,𝑥∈Ω,−Δ𝐴,𝑣+𝑀₁𝐴𝑢,𝑣≤𝑓₁𝑢,𝑣+𝑀₁𝐴𝑢,𝑣,𝑥∈Ω,−Δ𝐵𝑢,𝑣+𝑀₂𝐵𝑢,𝑣≤𝑓₂𝑢,𝑣+𝑀₂𝐵𝑢,𝑣𝐴,𝑥∈Ω,𝑢,𝑣≥0≥𝐴𝑢,𝑣,𝐵𝑢,𝑣≥0≥𝐵𝑢,𝑣,𝑥∈𝜕Ω. (15) Synthetically, we have the following result.

Theorem 5. If there is a pair of functions ((𝑢,𝑣),(𝑢,𝑣)), satisfying 𝑢,𝑣,𝐴𝑢,𝑣,𝐵𝑢,𝑣≥𝑢,𝑣,𝐴𝑢,𝑣,𝐵𝑢,𝑣, (16) and for all (𝑢,𝑣)∈𝑈×𝑉, (15) is satisfied, then (5) has at least one solution (𝑢,𝑣), satisfying the relation (𝑢,𝑣)≤(𝑢,𝑣)≤(𝑢,𝑣).

To make sure the upper and lower solutions reasonable, we give the following two lemmas; more details can be found in [8, 9].

Lemma 6. If the functions 𝑢,𝑣∈𝐶¹(Ω) satisfy 𝑢|_𝜕Ω=𝑣|_𝜕Ω=0,𝑢|_Ω>0,  (𝜕𝑢/𝜕𝜈)|_𝜕Ω<0, 𝜈 is the outer unit normal vector of 𝜕Ω, then there exists positive constant 𝜀, such that 𝑢(𝑥)>𝜀𝑣(𝑥), for all 𝑥∈Ω.

For the equation: −Δ𝑢=𝑢(𝑎−𝑢),𝑥∈Ω,𝑢=0,𝑥∈𝜕Ω. (17)

Lemma 7. If 𝑎>𝜆₁, then (17) has a unique positive solution 𝜃_𝑎 satisfying 𝜃_𝑎≤𝑎. In addition, 𝜃_𝑎 is increasing with respect to 𝑎.

3. A Lotka-Volterra Competition Model with Two Cross-Diffusions

In this section, the existence of positive solutions of (1) corresponding to 𝛼≥0,𝛽≥0, is investigated by applying Theorem 5 to prove Theorem 1.

Proof. We seek some positive constants 𝑅,𝐾,𝛿,𝑅,𝐾>𝜆₁ sufficiently large and 𝛿 sufficiently small, Lemma 6 may guarantee the existence of 𝜃_𝑅 and 𝜃_𝐾. It can be easily known from Hopf boundary lemma: 𝜕𝜙(𝑥)𝜕𝜈<0,𝜕𝜃_𝑅(𝑥)𝜕𝜈<0,𝜕𝜃_𝐾(𝑥)𝜕𝜈<0,∀𝑥∈𝜕Ω. (18) Observe that min{𝑎−𝑐𝑏,𝑏−𝑑𝑎}>𝜆₁, using Lemma 7, we can have 𝑅,𝐾,𝛿,  𝑎<𝑅<(𝑏−𝜆₁)/𝑑,𝑏<𝐾<(𝑎−𝜆₁)/𝑐, satisfying the following three conditions:(i)𝛿𝜙(𝑥)<𝜃_𝑅(𝑥),𝛿𝜙(𝑥)<𝜃_𝐾(𝑥), for all 𝑥∈Ω;(ii)(𝜕(𝜃_𝑅−𝛿𝜙)/𝜕𝜈)(𝑥)<0,(𝜕(𝜃_𝐾−𝛿𝜙)/𝜕𝜈)(𝑥)<0;(iii)𝛿<min{𝑎−𝜆₁−𝑐𝐾,𝑏−𝜆₁−𝑑𝑅}.
Let 𝑀₁=2𝑅+𝑐𝐾,𝑀₂=2𝐾+𝑑𝑅. Using Lemma 7 again, there exist 𝛼=𝛼(𝑎,𝑏,𝑐,𝑑,Ω)<1,𝛽=𝛽(𝑎,𝑏,𝑐,𝑑,Ω)<1, for all (𝜌,𝜏)∈[0,𝛼)×[0,𝛽), for all 𝑥∈Ω, satisfying(iv)𝜃_𝑅−𝛿𝜙>𝜌(𝜃_𝐾−𝛿𝜙),𝜃_𝐾−𝛿𝜙>𝜏(𝜃_𝑅−𝛿𝜙);(v)(𝑅−𝑎)𝜃_𝑅>𝜌[𝑀₁𝜃_𝐾−(𝑀₁+𝜆₁)𝛿𝜙],(𝐾−𝑏)𝜃_𝐾>𝜏[𝑀₂𝜃_𝑅−(𝑀₂+𝜆₁)𝛿𝜙];(vi)(𝑎−𝜆₁−𝛿−𝑐𝐾)𝛿𝜙>𝜌[(𝐾+𝑀₁−𝜃_𝐾)𝜃_𝐾−𝑀₁𝛿𝜙]; (vii)(𝑏−𝜆₁−𝛿−𝑑𝑅)𝛿𝜙>𝜏[(𝑅+𝑀₂−𝜃_𝑅)𝜃_𝑅−𝑀₂𝛿𝜙].
We will verify 𝛼,𝛽 satisfying Theorem 5. Suppose that (𝛼,𝛽)∈[0,𝛼)×[0,𝛽). Then we construct a pair of upper and lower solutions of the form (𝑢,𝑣)=𝜃_𝑅,𝜃_𝐾,𝑢,𝑣=𝛿𝜙,𝛿𝜙, (19) where 𝛿 satisfies conditions (i)–(iii). Let 𝐴(𝑢,𝑣)=𝑢+𝛼𝑣,𝐵(𝑢,𝑣)=𝛽𝑢+𝑣. (20) Then 𝐴^∗(𝑤,𝑧)=𝑤−𝛼𝑧1−𝛼𝛽,𝐵^∗(𝑤,𝑧)=𝑧−𝛽𝑤.1−𝛼𝛽 (21) By simply computing, (H1) and (H2) are satisfied, where 𝑈=[0,𝑅]×[0,𝐾],𝑉=[0,𝑅+𝛼𝐾]×[0,𝐾+𝛽𝑅].
Note 𝑓₁(𝑢,𝑣)=𝑢(𝑎−𝑢−𝑐𝑣),𝑓₂(𝑢,𝑣)=𝑣(𝑏−𝑣−𝑑𝑢). (22) And for all (𝑢,𝑣)∈𝑈, we have 𝑓₁(𝑢,𝑣)+𝑀₁𝐴(𝑢,𝑣)_𝑢=𝑎−2𝑢−𝑐𝑣+𝑀₁≥−2𝑅−𝑐𝐾+𝑀₁𝑓=0,₂(𝑢,𝑣)+𝑀₂𝐵(𝑢,𝑣)_𝑣=𝑏−2𝑣−𝑑𝑢+𝑀₂≥−2𝐾−𝑑𝑅+𝑀₂=0. (23) So (H3) is satisfied; observer that 𝑢|_𝜕Ω=𝑣|_𝜕Ω=𝑢|_𝜕Ω=𝑣|_𝜕Ω=0, (𝑢,𝑣)≥(𝑢,𝑣) and (iv) and (15) and the boundary conditions of (16) can be checked. Therefore, if we want to obtain the existence of solutions through [4, Theorem 2.1], we should only verify for all (𝑢,𝑣)∈𝑈×𝑉, −Δ𝐴𝑢,𝑣+𝑀₁𝐴𝑢,𝑣≥𝑓₁(𝑢,𝑣)+𝑀₁𝐴()𝑢𝑢,𝑣,𝑥∈Ω,−Δ𝐵,𝑣+𝑀₂𝐵𝑢,𝑣≥𝑓₂(𝑢,𝑣)+𝑀₂𝐵(𝑢,𝑣)𝑢,𝑥∈Ω,−Δ𝐴,𝑣+𝑀₁𝐴𝑢,𝑣≤𝑓₁𝑢,𝑣+𝑀₁𝐴𝑢,𝑣,𝑥∈Ω,−Δ𝐵𝑢,𝑣+𝑀₂𝐵𝑢,𝑣≤𝑓₂𝑢,𝑣+𝑀₂𝐵𝑢,𝑣,𝑥∈Ω. (24) Because 𝑓₁ is decreasing in 𝑣, 𝑓₂ is decreasing in 𝑢, and 𝐴(𝑢,𝑣) is increasing in 𝑣, 𝐵(𝑢,𝑣) is increasing in 𝑢, only to verify the following inequations: −Δ𝐴𝑢,𝑣+𝑀₁𝐴𝑢,𝑣≥𝑓₁𝑢,𝑣+𝑀₁𝐴(𝑢,𝑣)𝑢,𝑥∈Ω,−Δ𝐵,𝑣+𝑀₂𝐵𝑢,𝑣≥𝑓₂𝑢,𝑣+𝑀₂𝐵(𝑢,𝑣)𝑢,𝑥∈Ω,−Δ𝐴,𝑣+𝑀₁𝐴𝑢,𝑣≤𝑓₁𝑢,𝑣+𝑀₁𝐴𝑢,𝑣,𝑥∈Ω,−Δ𝐵𝑢,𝑣+𝑀₂𝐵𝑢,𝑣≤𝑓₂𝑢,𝑣+𝑀₂𝐵𝑢,𝑣,𝑥∈Ω. (25) It is easy to check (25) by (v), (vi), and (vii). So from [4, Theorem 2.1], (1) has a solution (𝑢,𝑣), in addition (𝑢,𝑣)≥(𝑢,𝑣)≥(𝑢,𝑣)>(0,0).

In the end, before investigating the nonexistence of positive solutions of (1), we give its priori bound of positive solutions.

Theorem 8. Any positive solutions (𝑢,𝑣) of (1) have a priori bound; that is 𝑢(𝑥)≤𝑏𝑑(𝑥)≤𝑎,𝑣𝑐. (26)

Proof. Let 𝑤=𝑢+𝛼𝑣,𝑧=𝛽𝑢+𝑣; then 𝑢=𝑤−𝛼𝑧1−𝛼𝛽,𝑣=𝑧−𝛽𝑤.1−𝛼𝛽 (27) Equation (1) can be rewritten as −Δ𝑤=𝑤−𝛼𝑧1−𝛼𝛽𝑎−𝑤−𝛼𝑧1−𝛼𝛽−𝑐𝑧−𝛽𝑤1−𝛼𝛽,𝑥∈Ω,−Δ𝑧=𝑧−𝛽𝑤1−𝛼𝛽𝑏−𝑧−𝛽𝑤1−𝛼𝛽−𝑑𝑤−𝛼𝑧()=()1−𝛼𝛽,𝑥∈Ω,𝑤,𝑧0,0,𝑥∈𝜕Ω. (28) Since (𝑢,𝑣)>(0,0), it easily follows that 𝑤−𝛼𝑧>0,𝑧−𝛽𝑤>0. Assume that 𝑧(𝑥) attains its positive maximum at 𝑥₀∈Ω, then 𝑎1−𝛼𝛽−𝑤𝑥₀+𝛼𝑧𝑥₀−𝑐𝑧𝑥₀+𝑐𝛽𝑤𝑥₀𝑎𝑥>01−𝛼𝛽−𝑐𝑧₀+𝑐𝛽𝛼𝑧𝑥₀𝑧(𝑥)𝑥>0,≤𝑧₀≤𝑎𝑐 (29) so that 𝑣=𝑧−𝛽𝑢≤𝑧𝑥₀≤𝑎𝑐. (30) Similarly, we can obtain the desired result 𝑏𝑢≤𝑑. (31)

Theorem 9. If one of the following conditions:(i)𝑏≤𝑎𝑑,𝜆₁≥(𝑏+𝑐𝛽(𝑏/𝑑))/(1−𝛼𝛽);(ii)(1−(𝛼+𝛽)/2)𝜆₁≥max{𝑎,𝑏};is satisfied, then (1) with 𝛼<𝛼,𝛽<𝛽 has no positive solution.

Proof. Multiplying 𝑢 and 𝑣 to the first and second equations in (1), and integrating these equations on Ω, we have _Ω|∇𝑢|²𝑑𝑥+𝛼_Ω∇𝑢∇𝑣𝑑𝑥=_Ω𝑢²()𝛼𝑎−𝑢−𝑐𝑣𝑑𝑥,_Ω|∇𝑣|²𝑑𝑥+_Ω∇𝑢∇𝑣𝑑𝑥=_Ω𝑢𝑣(𝑎−𝑢−𝑐𝑣)𝑑𝑥,𝛽_Ω|∇𝑢|²𝑑𝑥+_Ω∇𝑢∇𝑣𝑑𝑥=_Ω𝑢𝑣(𝑏−𝑣−𝑑𝑢)𝑑𝑥,_Ω|∇𝑣|²𝑑𝑥+𝛽_Ω∇𝑢∇𝑣𝑑𝑥=_Ω𝑣²(𝑏−𝑣−𝑑𝑢)𝑑𝑥. (32)
(i) Suppose, by contradiction that (1) has a positive solution (𝑢,𝑣), then the second and fourth equations in (32) yield _Ω𝑣³𝑑𝑥+𝛽_Ω𝑢𝑣(𝑎−𝑢)𝑑𝑥=−1−𝛼𝛽_Ω|∇𝑣|²𝑑𝑥+𝛽_Ω𝑐𝑢𝑣²𝑑𝑥+_Ω𝑣²(𝑏−𝑑𝑢)𝑑𝑥. (33) Since 𝑢≤𝑏/𝑑 by Theorem 8, the left-hand side of (33) must be positive. On the other hand, the Poincare inequality, ‖∇𝑣‖²_𝐿2≥𝜆₁‖𝑣‖²_𝐿2, for 𝑣∈𝑊¹₂(Ω) and the given assumption shows the following contradiction: −1−𝛼𝛽_Ω|∇𝑣|²𝑑𝑥+𝛽_Ω𝑐𝑢𝑣²𝑑𝑥+_Ω𝑣²()𝜆𝑏−𝑑𝑢𝑑𝑥≤−1−𝛼𝛽₁𝑏−𝑐𝛽𝑑−𝑏_Ω𝑣²𝑑𝑥≤0. (34)
(ii) A contraction argument is also used assuming that (1) has a positive solution (𝑢,𝑣). Adding the first equation to the fourth equation, and then subtracting 𝑎∫_Ω𝑢²𝑑𝑥+𝑏∫_Ω𝑣²𝑑𝑥 from the both sides, the following identity is obtained: _Ω|∇𝑢|²𝑑𝑥+𝛼+𝛽_Ω+∇𝑢∇𝑣𝑑𝑥_Ω|∇𝑣|²𝑑𝑥−𝑎_Ω𝑢²𝑑𝑥−𝑏_Ω𝑣²𝑑𝑥=−_Ω𝑢²(𝑢+𝑐𝑣)𝑑𝑥−_Ω𝑣²(𝑣+𝑑𝑢)𝑑𝑥. (35) Since 2∇𝑢∇𝑣=|∇(𝑢+𝑣)|²−|∇𝑢|²−|∇𝑣|² and (1−(𝛼+𝛽)/2)𝜆₁≥max{𝑎,𝑏}, the Poincare inequality shows that the left-hand side of (35) must be nonnegative, more precisely, _Ω|∇𝑢|²𝑑𝑥+𝛼+𝛽_Ω+∇𝑢∇𝑣𝑑𝑥_Ω|∇𝑣|²𝑑𝑥−𝑎_Ω𝑢²𝑑𝑥−𝑏_Ω𝑣²=𝑑𝑥1−𝛼+𝛽2_Ω|∇𝑢|²𝑑𝑥+𝛼+𝛽2_Ω|∇(𝑢+𝑣)|²+𝑑𝑥1−𝛼+𝛽2_Ω|∇𝑣|²𝑑𝑥−𝑎_Ω𝑢²𝑑𝑥−𝑏_Ω𝑣²≥𝑑𝑥1−𝛼+𝛽2𝜆₁−𝑎_Ω𝑢²𝑑𝑥+𝛼+𝛽2⋅_Ω|∇(𝑢+𝑣)|²𝑑𝑥+1−𝛼+𝛽2𝜆₁−𝑏_Ω𝑣²≥𝑑𝑥1−𝛼+𝛽2𝜆₁−𝑎_Ω𝑢²+𝑑𝑥1−𝛼+𝛽2𝜆₁−𝑏_Ω𝑣²𝑑𝑥≥0. (36) However, this results in a contradiction since the right-hand side of (35) is clearly strictly negative by the positivity of 𝑢 and 𝑣.

Remark 10. Before closing this section, more sufficient conditions of the nonexistence of positive solutions of (1) with 𝛼+𝛽>0, 𝛼𝛽=0 are investigated. Take 𝛼=0,𝛽>0 for example, then (1) may be reduced as ()()()=()−Δ𝑢=𝑢𝑎−𝑢−𝑐𝑣,𝑥∈Ω,−Δ𝛽𝑢+𝑣=𝑣𝑏−𝑣−𝑑𝑢,𝑥∈Ω,𝑢,𝑣0,0,𝑥∈𝜕Ω. (37) Using the same method, we can obtain that (37) has no positive solution, if one of the following conditions is satisfied:(i)𝜆₁≥𝑏+𝛽𝑐𝑎;(ii)𝜆₁≥𝑎;(iii)(1−𝛽/2)𝜆₁≥max{𝑎,𝑏};(iv)𝑐<1<𝑎/𝑏 and (1−𝑑)/𝛽≤𝜆₁/(𝑏+𝛽𝑎)≤1.