Traveling Wavefronts of Competing Pioneer and Climax Model with Nonlocal Diffusion

Abstract

We study a competing pioneer-climax species model with nonlocal diffusion. By constructing a pair of upper-lower solutions and using the iterative technique, we establish the existence of traveling wavefronts connecting the pioneer-existence equilibrium and the coexistence equilibrium. We also discuss the asymptotic behavior of the wave tail for the traveling wavefronts as 𝑠=𝑥+𝑐𝑡→−∞.

1. Introduction

As we know, the interactions among species are important in determining the process of evolution for the ecosystem, and the modeling accompanied with the mathematical analysis of the models can help people to understand and control the propagation of species. In general, the per capital growth rate (i.e., fitness) for a species in the model is assumed to be a function of a weighted total density of all interacting species. A well-known example is the standard Lotka-Volterra model; its fitness of a species is a linear function. It is natural to consider other kinds of fitness functions other than the linear one, because of the various species and interaction rules. In this paper, we will analyze a reaction-diffusion model describing pioneer and climax species. This model describes interaction among species with peculiar fitness functions.

A species is called a pioneer species if it thrives best at lower density but its fitness decreases monotonically with total population density for overcrowded. Thus, the fitness function of a pioneer species is assumed to be a decreasing function. Pine and yellow poplar are the species of this type. A species is called a climax species if its fitness increases up to a maximum value and then decreases of its total density. Hence, a climax population is assumed to have a nonmonotone, “one-humped” smooth fitness function. Oak and maple are the climax species.

A typical reaction-diffusion model for a pioneer-climax species is given by the following system: 𝜕𝑢𝜕𝑡=𝑑₁𝜕²𝑢𝜕𝑥²+𝑢𝑓𝑐₁₁𝑢+𝑐₁₂𝑣,𝜕𝑣𝜕𝑡=𝑑₂𝜕²𝑣𝜕𝑥²+𝑣𝑔𝑐₂₁𝑢+𝑐₂₂𝑣, (1) where 𝑢 and 𝑣 represent densities of the pioneer and climax species, respectively. 𝑓 and 𝑔 denote the pioneer fitness function and climax fitness function, respectively, 𝑐_𝑖𝑗>0(𝑖,𝑗=1,2). By making changes of variables 𝑢=𝑐₂₁𝑢, 𝑣̃=𝑐₁₂𝑣, system (1) changes into the form (the tildes of 𝑢 and 𝑣̃ are dropped out) 𝜕𝑢𝜕𝑡=𝑑₁𝜕²𝑢𝜕𝑥²+𝑢𝑓𝑐₁₁𝑢+𝑣,𝜕𝑣𝜕𝑡=𝑑₂𝜕²𝑣𝜕𝑥²+𝑣𝑔𝑢+𝑐₂₂𝑣, (2) where we still use 𝑐₁₁ and 𝑐₂₂ as the new coefficients without confusing.

From the previous introduction, we assume that the pioneer fitness function 𝑓 satisfies 𝑓^′(𝑧)<0,𝑓𝑧₀=0 (3) for some 𝑧₀>0, and the climax fitness function 𝑔 satisfies 𝑔𝑤₁=𝑔𝑤₂=0,0<𝑤₁<𝑤₂,𝑤^∗−𝑤𝑔^′(𝑤)>0for𝑤≠𝑤^∗∈𝑤₁,𝑤₂. (4) Ricker [1] used the fitness function 𝑓(𝑢)=𝑒^{𝑟(1−𝑢)}−𝑎, Hassell and Comins [2] used the fitness function 𝑓(𝑢)=(𝑟/(1+𝑏𝑢)^𝑝)−𝑎, and Cushing [3] used the fitness function 𝑔(𝑢)=𝑢𝑒^{𝑟(1−𝑢)}−𝑎. It is obvious that these 𝑓 and 𝑔 have the curves in Figure 1.

(a)

(b)

Figure 1 

Typical fitness functions for pioneer species and climax species.

There are some existing results about the stability and traveling wave solutions for (2) ([4–6]). About traveling wave solutions, Brown et al. [4] studied the traveling wave of (1) connecting two boundary equilibria by singular perturbation technique, and Yuan and Zou [6] obtained the existence of traveling wave solutions connecting a monoculture state and a coexistence state by upper-lower solution method combined with the Schauder fixed point theorem. Also see van Vuuren [7] for the existence of traveling plane waves in a general class of competition-diffusion systems and Murray [8] for more biological description of traveling wave solutions.

For system without spatial diffusion, the model will be 𝑑𝑢𝑐𝑑𝑡=𝑢𝑓₁₁𝑢+𝑣,𝑑𝑣𝑑𝑡=𝑣𝑔𝑢+𝑐₂₂𝑣. (5) Selgrade and Roberds [9], Sumner [10] analyzed the Hopf bifurcation of (5), and Selgrade and Namkoong [11], Sumner [12] considered the stable periodic behavior of (5). Because of the existence of rich equilibria and the various ranges of parameters, the dynamics of ordinary differential system (5) are complex, and a detailed review of all equilibrium types can be found in Buchanan [13, 14].

Although the Laplacian operator Δ∶=𝜕²/𝜕𝑥² is always used to model the diffusion of the species, it suggests that the population at the location 𝑥 can only be influenced by the variation of the population near 𝑥. As we know that the individuals can move freely, then the movement of individuals is bounded to affect the other individuals. So, the Laplacian operator may have some shortage to describe the diffusion. One way to deal with this problem is to replace the Laplacian operator with a convolution diffusion term ∫^∞_−∞𝐽(𝑥−𝑦)[𝑢(𝑡,𝑦)−𝑢(𝑡,𝑥)]𝑑𝑦. This implies that the probability distribution function for the population at location 𝑦 moving to the location 𝑥 is 𝐽(𝑥−𝑦). At time 𝑡, the total individuals that move from the whole space into the location 𝑥 will be ∫^∞_−∞𝐽(𝑥−𝑦)[𝑢(𝑡,𝑦)−𝑢(𝑡,𝑥)]𝑑𝑦. Therefore, one may call it as a nonlocal diffusion, and, correspondingly, call 𝜕²𝑢(𝑡,𝑥)/𝜕𝑥² as a local diffusion. During the recent years, the models with the nonlocal diffusion have been attracted much more attentions (see [15–18]).

In this paper, instead of (2), we will concentrate on the following pioneer-climax species with nonlocal diffusion: 𝜕𝑢(𝑡,𝑥)𝜕𝑡=𝑑₁^+∞_−∞𝐽𝑢(()𝑓𝑐𝑦−𝑥𝑡,𝑦−𝑢𝑡,𝑥)𝑑𝑦+𝑢𝑡,𝑥₁₁,()𝑢+𝑣𝜕𝑣𝑡,𝑥𝜕𝑡=𝑑₂^+∞_−∞𝐽𝑣(()𝑔𝑦−𝑥𝑡,𝑦−𝑣𝑡,𝑥)𝑑𝑦+𝑣𝑡,𝑥𝑢+𝑐₂₂𝑣, (6) where 𝑑₁, 𝑑₂ are positive constants accounting for the diffusivity, 𝐽(𝑧) is a kernel function which is continuous on ℝ satisfying _ℝ𝐽(𝑥)(𝑥)𝐽(𝑥)()𝑑𝑥=1,𝐽≥0,=𝐽−𝑥for𝑥∈ℝ,_ℝ𝐽(𝑥)𝑒^𝜈𝑥𝑑𝑥<∞foranyfixed𝜈∈[0,∞). (A1)

We are interested in traveling wavefronts accounting for a mildinvasion of the two species (traveling wavefronts connecting a boundary equilibrium and the coexistence equilibrium). For system (2), the sufficient condition for (2) to have a mildinvasion is 𝑑₂/𝑑₁≥1/2 (see [6]), but our condition in this paper for system (6) is 𝑑₂/𝑑₁≥1, which reveals a fact that the nonlocal diffusion of either the pioneer species or the climax species did affect the climax invasion and wave propagation. Please see Section 5 for the discussion.

The remaining of this paper is organized as follows. In Section 2, there are some preliminaries about the equilibria and the system is transformed into a cooperative one. In Section 3, we prove the existence of traveling wavefronts by using an iteration scheme combined with a pair of admissible upper and lower solutions, which can be constructed obviously, and thus a criterion of the existence for traveling wavefronts is obtained. We also give a discussion on asymptotic behavior for the traveling wavefront tail as 𝑠=𝑥+𝑐𝑡→−∞ in Section 4. At last, we give some concluding discussions in Section 5.

2. Preliminaries

It is evident that (0,0) is a trivial equilibrium of (6). The system (6) has at least four equilibria and at most six equilibria. The existence of nonnegative steady states depends on the locations of the three nullclines: 𝑐₁₁𝑢+𝑣=𝑧₀,𝑢+𝑐₂₂𝑣=𝑤₁,𝑢+𝑐₂₂𝑣=𝑤₂. (7) The long-term behavior of solutions to (6) can be qualitatively different caused by the different number, distribution, and types of equilibria. The dynamics of the system (6) are of course very rich and complex. However, in this paper, we will only consider the following case: 𝑧₀>𝑤₂𝑐₂₂,𝑤₁<𝑧₀𝑐₁₁<𝑤₂. (8) The condition 𝑐₁₁𝑐₂₂>1 follows as a sequence. Under the previous assumption, (6) has four nontrivial equilibria: (𝑧₀/𝑐₁₁,0), (0,𝑤₁/𝑐₂₂), (0,𝑤₂/𝑐₂₂), and (𝑢^∗,𝑣^∗) except for (0,0), where 𝑢^∗=𝑐₂₂𝑧₀−𝑤₂𝑐₁₁𝑐₂₂−1,𝑣^∗=𝑐₁₁𝑤₂−𝑧₀𝑐₁₁𝑐₂₂.−1 (9) It is obvious that 𝑢^∗<𝑧₀/𝑐₁₁. We further assume that 𝑤^∗≤𝑢^∗ (10) for the technical reason. See Figure 2 for this situation.

Figure 2 

One coexistence state for the two species with 𝑤∗≤𝑢∗.

As mentioned in the introduction, we are interested in the coexistence of the two species. That means that we will seek traveling wavefronts connecting equilibrium (𝑧₀/𝑐₁₁,0) and equilibrium (𝑢^∗,𝑣^∗).

By making changes of variables 𝑢=𝑧₀/𝑐₁₁−𝑢, 𝑣̃=𝑣 and dropping the tildes, system (6) becomes 𝜕𝑢(𝑡,𝑥)𝜕𝑡=𝑑₁^+∞_−∞𝐽𝑦−𝑥𝑢𝑡,𝑦𝑑𝑦−𝑑₁𝑢()+𝑢()−𝑧𝑡,𝑥𝑡,𝑥⁰𝑐₁₁𝑓𝑧₀−𝑐₁₁,()𝑢+𝑣𝜕𝑣𝑡,𝑥𝜕𝑡=𝑑₂^+∞_−∞𝐽𝑦−𝑥𝑣𝑡,𝑦𝑑𝑦−𝑑₂𝑣()()𝑔𝑧𝑡,𝑥+𝑣𝑡,𝑥⁰𝑐₁₁−𝑢+𝑐₂₂𝑣, (11) and the equilibria (𝑧₀/𝑐₁₁,0),  (𝑢^∗,𝑣^∗) are changed into (0,0), (𝑢⁺,𝑣⁺), respectively, where 𝑢⁺=𝑧₀/𝑐₁₁−𝑢^∗, 𝑣⁺=𝑣^∗.

3. Existence of Traveling Wavefronts

A traveling wavefront of (6) connecting equilibria (𝑧₀/𝑐₁₁,0) and (𝑢^∗,𝑣^∗) can be changed into a traveling wavefront of (11) connecting (0,0) and (𝑢⁺,𝑣⁺). Therefore, we consider the system (11) hereby.

A traveling wave solution of (11) is a solution with the form 𝑢(𝑡,𝑥)=𝜙(𝑥+𝑐𝑡)=𝜙(𝑠) and 𝑣(𝑡,𝑥)=𝜑(𝑥+𝑐𝑡)=𝜑(𝑠), where 𝑠=𝑥+𝑐𝑡 and 𝑐>0 is a wave speed. A traveling wavefront is a traveling wave solution (𝜙(𝑠),𝜑(𝑠)) which has finite limits (𝜙(±∞),𝜑(±∞)). Denoting the traveling wave coordinate 𝑥+𝑐𝑡 still by 𝑡, we derive the wave profile system from (11): 𝑐𝜙^′(𝑡)=𝑑₁^+∞_−∞𝐽𝑦−𝑡𝜙𝑦𝑑𝑦−𝑑₁𝜙(𝑡)+𝜙(𝑡)−𝑧₀𝑐₁₁𝑓𝑧₀−𝑐₁₁𝜙(𝑡)+𝜑(𝑡),𝑐𝜑^′(𝑡)=𝑑₂^+∞_−∞𝐽𝑦−𝑡𝜑𝑦𝑑𝑦−𝑑₂𝜑(𝑡)(𝑡)𝑔𝑧+𝜑⁰𝑐₁₁−𝜙(𝑡)+𝑐₂₂𝜑(𝑡). (12) Associated with (12), we consider its solutions subject to the following boundary value conditions: lim_{𝑡→−∞}𝜙(𝑡),𝜑(𝑡)=(0,0),lim_𝑡→+∞𝜙(𝑡),𝜑(𝑡)=𝑢⁺,𝑣⁺. (13)

For 𝑎=(𝑎₁,𝑎₂), 𝑏=(𝑏₁,𝑏₂)∈ℝ², 𝑎≤𝑏 implies 𝑎_𝑖≤𝑏_𝑖(𝑖=1,2); 𝑎<𝑏 implies 𝑎≤𝑏 but 𝑎≠𝑏; 𝑎≪𝑏 implies 𝑎_𝑖<𝑏_𝑖(𝑖=1,2). Furthermore, the norm ‖⋅‖ in ℝ² is the Euclidean norm. Define 𝔻=𝜙,𝜑∈𝐶ℝ,ℝ²∣0≤𝜙(𝑡)≤𝑢⁺,0≤𝜑(𝑡)≤𝑣⁺,𝑡∈ℝ. (14) For some constants 𝛽₁, 𝛽₂, letting 𝑏₁=𝛽₁−𝑑₁, 𝑏₂=𝛽₂−𝑑₂, we define an operator 𝐻=(𝐻₁,𝐻₂)∶𝔻→𝐶(ℝ,ℝ²) by 𝐻₁𝜙,𝜑(𝑡)=𝑑₁^+∞_−∞𝐽𝜙𝑦+𝜙(𝑡)−𝑧𝑦−𝑡𝑑𝑦₀𝑐₁₁𝑓𝑧₀−𝑐₁₁𝜙(𝑡)+𝜑(𝑡)+𝑏₁𝜙(𝑡),𝐻₂𝜙,𝜑(𝑡)=𝑑₂^+∞_−∞𝐽𝜑𝑦(𝑡)𝑔𝑧𝑦−𝑡𝑑𝑦+𝜑⁰𝑐₁₁−𝜙(𝑡)+𝑐₂₂𝜑(𝑡)+𝑏₂𝜑(𝑡). (15) Then, (12) can be written as an equivalent form: 𝑐𝜙^′(𝑡)=−𝛽₁𝜙(𝑡)+𝐻₁𝜙,𝜑(𝑡),𝑐𝜑^′(𝑡)=−𝛽₂𝜑(𝑡)+𝐻₂𝜙,𝜑(𝑡). (16) Denote 𝑄=(𝑄₁,𝑄₂) by 𝑄₁𝑡)=1𝜙,𝜑(𝑐𝑒^{−(𝛽1/𝑐)𝑡𝑡}_−∞𝑒^{(𝛽1/𝑐)𝑠}𝐻₁𝑠)𝑄𝜙,𝜑(𝑑𝑠,₂(𝑡)=1𝜙,𝜑𝑐𝑒^{−(𝛽2/𝑐)𝑡𝑡}_−∞𝑒^{(𝛽2/𝑐)𝑠}𝐻₂𝜙,𝜑(𝑠)𝑑𝑠. (17) It is obvious that a traveling wave solution of the problem (12) and (13) is a fixed point of 𝑄 and vice verse.

The following lemma states the monotone property of 𝐻.

Lemma 1. Assume that (A1) holds, for sufficiently large 𝛽₁,𝛽₂>0 and (𝜙_𝑖,𝜑_𝑖)∈𝔻, 𝑖=1,2 with 𝜙₁(𝑠)≤𝜙₂(𝑠) and 𝜑₁(𝑠)≤𝜑₂(𝑠),  𝑠∈ℝ; one has (i)𝐻₁(𝜙₂,𝜑₁)(𝑡)≥𝐻₁(𝜙₁,𝜑₁)(𝑡),  𝐻₂(𝜙₁,𝜑₂)(𝑡)≥𝐻₂(𝜙₁,𝜑₁)(𝑡),  (ii)𝐻₁(𝜙₁,𝜑₂)(𝑡)≥𝐻₁(𝜙₁,𝜑₁)(𝑡),  𝐻₂(𝜙₂,𝜑₁)(𝑡)≥𝐻₂(𝜙₁,𝜑₁)(𝑡)for all 𝑡∈ℝ.

Proof. In order to prove (i), let 𝑓₁(𝑥,𝑦)=(𝑥−𝑧₀/𝑐₁₁)𝑓(𝑧₀−𝑐₁₁𝑥+𝑦)+𝑏₁𝑥, 𝑔₁(𝑥,𝑦)=𝑦𝑔(𝑧₀/𝑐₁₁−𝑥+𝑐₂₂𝑦)+𝑏₂𝑦. Then 𝜕𝑓₁𝑧𝜕𝑥=𝑓₀−𝑐₁₁+𝑧𝑥+𝑦₀−𝑐₁₁𝑥𝑓^′𝑧₀−𝑐₁₁𝑥+𝑦+𝛽₁−𝑑₁,𝜕𝑔₁𝑧𝜕𝑦=𝑔⁰𝑐₁₁−𝑥+𝑐₂₂𝑦+𝑐₂₂𝑦𝑔^′𝑧₀𝑐₁₁−𝑥+𝑐₂₂𝑦+𝛽₂−𝑑₂. (18) For 0≤𝑥≤𝑢⁺, 0≤𝑦≤𝑣⁺ and sufficiently large 𝛽₁,𝛽₂>0, it follows that 𝜕𝑓₁/𝜕𝑥≥0, 𝜕𝑔₁/𝜕𝑦≥0. Thus, if 𝜙₁(𝑠)≤𝜙₂(𝑠) and 𝜑₁(𝑠)≤𝜑₂(𝑠) for 𝑠∈ℝ, we have 𝐻₁𝜙₂,𝜑₁(𝑡)−𝐻₁𝜙₁,𝜑₁(𝑡)=𝑑₁^+∞_−∞𝐽𝑦−𝑡𝜙₂𝑦−𝜙₁𝑦𝑑𝑦+𝑓₁𝜙₂,𝜑₁−𝑓₁𝜙₁,𝜑₁𝐻≥0,₂𝜙₁,𝜑₂(𝑡)−𝐻₂𝜙₁,𝜑₁(𝑡)=𝑑₂^+∞_−∞𝐽𝑦−𝑡𝜑₂𝑦−𝜑₁𝑦𝑑𝑦+𝑔₁𝜙₁,𝜑₂−𝑔₁𝜙₁,𝜑₁≥0. (19)
For (ii), we know that 0≤𝜙₁(𝑡)≤𝑢⁺=𝑧₀/𝑐₁₁−𝑢^∗≤𝑧₀/𝑐₁₁ and 𝑓^′(𝑦)<0 for 𝑦∈ℝ. It follows that 𝐻₁𝜙₁,𝜑₂(𝑡)−𝐻₁𝜙₁,𝜑₁(𝑡)=𝜙₁(𝑡)−𝑧₀𝑐₁₁𝑓𝑧₀−𝑐₁₁𝜙₁(𝑡)+𝜑₂(𝑡)−𝑓𝑧₀−𝑐₁₁𝜙₁(𝑡)+𝜑₁(𝑡)≥0. (20) From 0≤𝜑₁(𝑡)≤𝑣⁺=𝑣^∗, 0≤𝜙₁(𝑡)≤𝜙₂(𝑡)≤𝑢⁺=𝑧₀/𝑐₁₁−𝑢^∗, we have 𝑤^∗≤𝑢^∗=𝑧₀𝑐₁₁−𝑧₀𝑐₁₁−𝑢^∗≤𝑧₀𝑐₁₁−𝜙₂(𝑡)+𝑐₂₂𝜑₁(𝑡)≤𝑧₀𝑐₁₁−𝜙₁(𝑡)+𝑐₂₂𝜑₁(𝑡). (21) Note that 𝑔^′(𝑤)<0 for 𝑤>𝑤^∗. This leads to 𝐻₂𝜙₂,𝜑₁(𝑡)−𝐻₂𝜙₁,𝜑₁(𝑡)=𝜑₁(𝑡𝑔𝑧)⁰𝑐₁₁−𝜙₂(𝑡)+𝑐₂₂𝜑₁(𝑡)𝑧−𝑔⁰𝑐₁₁−𝜙₁(𝑡)+𝑐₂₂𝜑₁(𝑡)≥0. (22) The proof is complete.

The conclusion of the following lemma is direct.

Lemma 2. Assume that 𝛽₁>0, 𝛽₂>0 are sufficiently large. For (𝜙,𝜑)∈𝔻 with 𝜙(𝑡),𝜑(𝑡) nondecreasing on 𝑡∈ℝ, 𝐻₁(𝜙,𝜑)(𝑡), 𝐻₂(𝜙,𝜑)(𝑡) are also nondecreasing on 𝑡∈ℝ.

We can easily see that 𝑄=(𝑄₁,𝑄₂) also enjoys the same properties as those for 𝐻=(𝐻₁,𝐻₂) settled in Lemmas 1 and 2.

Let 𝜇∈(0,min{𝛽₁/𝑐,𝛽₂/𝑐}) and 𝐵_𝜇ℝ,ℝ²=𝜙∈𝐶ℝ,ℝ²∣sup_𝑡∈ℝ𝜙(𝑡)𝑒^{−𝜇|𝑡|}<∞. (23) It is clear that (𝐵_𝜇(ℝ,ℝ²),|⋅|_𝜇) is a Banach space equipped with the norm |⋅|_𝜇 defined by |𝜙|_𝜇=sup_𝑡∈ℝ|𝜙(𝑡)|𝑒^{−𝜇|𝑡|}.

Definition 3. A pair of continuous functions Φ(𝑡)=(𝜙(𝑡),𝜑(𝑡)), Φ(𝑡)=(𝜙(𝑡),𝜑(𝑡)) is called an upper solution and a lower solution of (12), respectively, if there exists a set Γ={𝑇_𝑖∈ℝ,𝑖=1,…,𝑘} such that Φ and Φ are differentiable in ℝ⧵Γ and the essential bounded functions Φ^′,Φ^′ satisfy 𝑑₁^+∞_−∞𝐽𝑦−𝑡𝜙𝑦𝑑𝑦−𝑑₁𝜙(𝑡)−𝑐𝜙^′(𝑡)+𝜙(𝑡)−𝑧₀𝑐₁₁𝑓𝑧₀−𝑐₁₁𝜙(𝑡)+𝜑(𝑡)𝑑≤0,_2+∞−∞𝐽𝑦−𝑡𝜑𝑦𝑑𝑦−𝑑₂𝜑(𝑡)−𝑐𝜑^′(𝑡)+𝜑(𝑡)𝑔𝑧₀𝑐₁₁−𝜙(𝑡)+𝑐₂₂𝜑(𝑡)𝑑≤0,_1+∞−∞𝐽𝑦−𝑡𝜙𝑦𝑑𝑦−𝑑₁𝜙(𝑡)−𝑐𝜙^′(𝑡)+𝜙(𝑡)−𝑧₀𝑐₁₁𝑓𝑧₀−𝑐₁₁𝜙(𝑡)+𝜑(𝑡)𝑑≥0,_2+∞−∞𝐽𝑦−𝑡𝜑𝑦𝑑𝑦−𝑑₂𝜑(𝑡)−𝑐𝜑^′(𝑡)+𝜑(𝑡)𝑔𝑧₀𝑐₁₁−𝜙(𝑡)+𝑐₂₂𝜑(𝑡)≥0 (24) for 𝑡∈ℝ⧵Γ.

In what follows, we assume that (12) has an upper solution (𝜙(𝑡),𝜑(𝑡)) and a lower solution (𝜙(𝑡),𝜑(𝑡)), such that(P1)(0,0)≪(𝜙(𝑡),𝜑(𝑡))≤(𝜙(𝑡),𝜑(𝑡))≤(𝑢⁺,𝑣⁺) for 𝑡∈ℝ;(P2)lim_{𝑡→−∞}(𝜙(𝑡),𝜑(𝑡))=(0,0), lim_𝑡→+∞(𝜙(𝑡),𝜑(𝑡))=(𝑢⁺,𝑣⁺);(P3)𝜙(𝑡) and 𝜑(𝑡) are nondecreasing.

Define the following profile set Λ=Λ(Φ,Φ) by ΛΦ,Φ=𝜙,𝜑∈𝔻∣(i)𝜙(𝑡)≤𝜙(𝑡)≤𝜙(𝑡),𝜑(𝑡)≤𝜑(𝑡)≤𝜑(𝑡),(ii)𝜙(𝑡),𝜑(𝑡)arenondecreasingfor𝑡∈ℝ. (25) It is obvious that Λ(Φ,Φ) is nonempty.

For 𝑡∈ℝ and 𝑛≥2, define (𝜙₁,𝜑₁)(𝑡)=(𝜙,𝜑)(𝑡), (𝜙₁,𝜑₁)(𝑡)=(𝜙,𝜑)(𝑡) and 𝜙_𝑛(𝑡)=𝑄₁𝜙_𝑛−1,𝜑_𝑛−1(𝑡),𝜑_𝑛(𝑡)=𝑄₂𝜙_𝑛−1,𝜑_𝑛−1𝑡),𝜙_𝑛((𝑡)=𝑄₁𝜙_𝑛−1,𝜑_𝑛−1(𝑡),𝜑_𝑛(𝑡)=𝑄₂𝜙_𝑛−1,𝜑_𝑛−1(𝑡). (26)

Lemma 4. For 𝑛≥2, the functions 𝜙_𝑛(𝑡), 𝜑_𝑛(𝑡) and 𝜙_𝑛(𝑡), 𝜑_𝑛(𝑡) defined by (26) satisfy(i)(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡))∈Λ; (ii)(𝜙_𝑛−1(𝑡),𝜑_𝑛−1(𝑡))≤(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡))≤(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡))≤(𝜙_𝑛−1(𝑡),𝜑_𝑛−1(𝑡)) for 𝑡∈ℝ;(iii)(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)) and (𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)) is a pair of upper and lower solutions of (12);(iv)(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)) and (𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)) are continuously differentiable on 𝑡∈ℝ.

Proof. We only give the argument for 𝑛=2, and the situation for 𝑛≥3 can be obtained by mathematical induction. From Definition 3, we obtain 𝐻₁𝜙₁,𝜑₁(𝑡)≤𝑐𝜙^′₁(𝑡)+𝛽₁𝜙₁(𝑡)for𝑡∈ℝ⧵Γ. (27) Let 𝑇₀=−∞, 𝑇_𝑘+1=+∞. For any 𝑡∈ℝ, there exists some 𝑖 such that 𝑡∈(𝑇_𝑖−1,𝑇_𝑖](𝑖=1,2,⋅,𝑘) or 𝑡∈(𝑇_𝑘,∞). We then derive 𝜙₂(𝑡)=𝑄₁𝜙₁,𝜑₁𝑡)=1(𝑐^𝑡_−∞𝑒^{−(𝛽1/𝑐)(𝑡−𝑠)}𝐻₁𝜙₁,𝜑₁𝑠)≤1(𝑑𝑠𝑐^𝑖−1_𝑗=1^𝑇𝑗_𝑇𝑗−1+^𝑡_𝑇𝑖−1𝑒^{−(𝛽1/𝑐)(𝑡−𝑠)}×𝑐𝜙^′₁(𝑠)+𝛽₁𝜙₁(𝑠)𝑑𝑠=𝜙₁(𝑡), (28) where 𝑖=1,2,…,𝑘+1.
By similar arguments, we can get 𝜙₁(𝑡),𝜑₁(𝑡)≤𝜙₂(𝑡),𝜑₂(𝑡)≤𝜙₂(𝑡),𝜑₂(𝑡)≤𝜙₁(𝑡),𝜑₁(𝑡)for𝑡∈ℝ. (29) The previous arguments implies that the conclusion (ii) holds.
From the monotone property of 𝑄, we can easily obtain that 𝜙₂(𝑡), 𝜑₂(𝑡) are nondecreasing for 𝑡∈ℝ, and therefore the conclusion (i) holds.
For 𝜙₂(𝑡)=𝑄₁(𝜙₁,𝜑₁)(𝑡), by Lemma 1, we have −𝛽₁𝜙₂(𝑡)−𝑐𝜙^′₂(𝑡)+𝐻₁𝜙₂,𝜑₂(𝑡)≤−𝛽₁𝜙₂(𝑡)−𝑐𝜙^′₂(𝑡)+𝐻₁𝜙₁,𝜑₁(𝑡)=0for𝑡∈ℝ⧵Γ. (30) In a similar way, we can prove that −𝛽₂𝜑₂(𝑡)−𝑐𝜑^′₂(𝑡)+𝐻₂𝜙₂,𝜑₂(𝑡)≤0,−𝛽₁𝜙₂(𝑡)−𝑐𝜙^′₂(𝑡)+𝐻₁𝜙₂,𝜑₂(𝑡)≥0,−𝛽₂𝜑₂(𝑡)−𝑐𝜑^′₂(𝑡)+𝐻₂𝜙₂,𝜑₂(𝑡)≥0 (31) for 𝑡∈ℝ⧵Γ. This indicates that (𝜙₂(𝑡),𝜑₂(𝑡)) and (𝜙₂(𝑡),𝜑₂(𝑡)) are a pair of upper and lower solutions of (12).
The conclusion (iv) is obvious. The proof is complete.

Lemma 5. lim_𝑛→∞(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡))=(𝜙^∗(𝑡),𝜑^∗(𝑡))∈Λ, and the convergence is uniform with respect to the decay norm |⋅|_𝜇.

Proof. We have from Lemma 4 that the following limit exists: lim_𝑛→∞𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)=𝜙^∗(𝑡),𝜑^∗(𝑡). (32) It is easy to know that Λ is a closed and convex set. By the nondecreasing property of (𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)), we have (𝜙^∗(𝑡),𝜑^∗(𝑡))∈Λ. In the following, we prove that the convergence is uniform with respect to the decay norm.
Since 𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)−𝜙^∗(𝑡),𝜑^∗(𝑡)≤2𝑢⁺+𝑣⁺for𝑛≥1, (33) for any 𝜀>0, there exists a 𝑇=𝑇(𝜀)>0, such that for all 𝑛≥1, sup_|𝑇|𝑡>𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)−𝜙^∗(𝑡),𝜑^∗(𝑡)𝑒^{−𝜇|𝑡|}<𝜀. (34) Now, we consider the sequences {(𝜙_𝑛,𝜑_𝑛)}^∞_𝑛=1 for 𝑡∈[−𝑇,𝑇]. Note 𝜙_𝑛(𝑡) is nondecreasing on 𝑡, and thus 0≤𝜙^′_𝑛(𝑡)=𝑄^′₁𝜙_𝑛−1,𝜑_𝑛−1(𝑡)=1𝑐𝐻₁𝜙_𝑛−1,𝜑_𝑛−1𝑡)−𝛽(¹𝑐²𝑒^{−(𝛽1/𝑐)𝑡}×^𝑡_−∞𝑒^{(𝛽1/𝑐)𝑠}𝐻₁𝜙_𝑛−1,𝜑_𝑛−1(𝑠)≤1𝑑𝑠𝑐𝐻₁𝜙_𝑛−1,𝜑_𝑛−1(𝑡)≤1𝑐𝐻₁𝜙₁,𝜑₁(𝑡). (35) By (𝜙₁(𝑡),𝜑₁(𝑡))∈𝐷, there exists a positive constant 𝑀₁, such that |𝜙^′_𝑛(𝑡)|≤𝑀₁ for 𝑡∈[−𝑇,𝑇]. Similarly, we can prove that there exists a positive number 𝑀₂, such that |𝜑^′_𝑛(𝑡)|≤𝑀₂ for 𝑡∈[−𝑇,𝑇].
From the previous estimates, we know that {(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡))}^∞_𝑛=1 is equicontinuous on [−𝑇,𝑇] with respect to the supremum norm. On the other hand, we have from Lemma 4 (ii) that {(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡))}^∞_𝑛=1 is uniformly bounded. By Arzéla-Ascoli theorem, there exist subsequences of {(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡))}^∞_𝑛=1 which are uniformly convergent in 𝑡∈[−𝑇,𝑇]. Without loss of generality, we still express this subsequence as {(𝜙_𝑛(𝑡),𝜑_𝑛(𝑡))}^∞_𝑛=1. Thus, there exists a positive integer 𝑁^∗>2, such that sup_{[]𝑡∈−𝑇,𝑇}𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)−𝜙^∗(𝑡),𝜑^∗(𝑡)<𝜀for𝑛>𝑁^∗. (36) Furthermore, we have sup_{[]𝑡∈−𝑇,𝑇}𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)−𝜙^∗(𝑡),𝜑^∗(𝑡)𝑒^{−𝜇|𝑡|}<𝜀for𝑛>𝑁^∗. (37)
Summarizing the previous arguments, for 𝑛>𝑁^∗ we have sup_𝑡∈ℝ𝜙_𝑛,𝜑_𝑛−𝜙^∗,𝜑^∗_𝜇=sup_𝑡∈ℝ𝜙_𝑛(𝑡),𝜑_𝑛(𝑡)−𝜙^∗(𝑡),𝜑^∗(𝑡)𝑒^{−𝜇|𝑡|}<𝜀. (38) The proof is complete.

Theorem 6. Assume that (A1) holds; if (12) has a pair of upper and lower solutions that satisfy (P1)–(P3), then the system (11) has a traveling wavefront satisfying (13).

Proof. By the Lebesgue's dominated convergence theorem and the iteration scheme (26), we have 𝜙^∗(𝑡),𝜑^∗(𝑡)=𝑄₁𝜙^∗(𝑡),𝜑^∗(𝑡),𝑄₂𝜙^∗(𝑡),𝜑^∗(𝑡). (39) Therefore, (𝜙^∗(𝑡),𝜑^∗(𝑡)) is a fixed point of 𝑄, which also satisfies (12). Furthermore, (P2) indicates that (𝜙^∗(𝑡),𝜑^∗(𝑡)) satisfy lim_{𝑡→−∞}(𝜙^∗(𝑡),𝜑^∗(𝑡))=(0,0). On the other hand, we have from the monotonicity of (𝜙^∗(𝑡),𝜑^∗(𝑡)) that lim_{𝑡→−∞}(𝜙^∗(𝑡),𝜑^∗(𝑡))=(𝑢,𝑣) exists. Furthermore, since (0,0)≪(𝜙(𝑡),𝜑(𝑡))≤(𝜙^∗(𝑡),𝜑^∗(𝑡)), we know that (𝑢,𝑣)≫(0,0). By using L’Hôspital’s rule, we obtain 𝑢=lim_𝑡→∞𝜙^∗(𝑡)=lim_𝑡→∞𝑄₁𝜙^∗(𝑡),𝜑^∗(𝑡)=lim_𝑡→∞(1/𝑐)∫^𝑡_−∞𝑒^{(𝛽1/𝑐)𝑠}𝐻₁𝜙^∗,𝜑^∗(𝑠)𝑑𝑠𝑒^{(𝛽1/𝑐)𝑡}=1𝛽₁𝐻₁(𝑢,𝑣). (40) Similarly, one can obtain 𝑣=(1/𝛽₂)𝐻₂(𝑢,𝑣). That is, (𝑢,𝑣) is an equilibrium of (12). Note that the assumption (10) implies that there is only one positive equilibrium (𝑢⁺,𝑣⁺) of (12) satisfying (0,0)<(𝑢,𝑣)≤(𝑢⁺,𝑣⁺). Therefore, (𝑢,𝑣)=(𝑢⁺,𝑣⁺) and (𝜙^∗(𝑡),𝜑^∗(𝑡)) is a traveling wavefront satisfying (13). The proof is complete.

In order to construct a pair of admissible upper-lower solutions for (12), we linearize (12) at (0,0) and obtain 𝑑₁^+∞_−∞𝐽𝑦−𝑡𝜙𝑦𝑑𝑦−𝑑₁𝜙(𝑡)−𝑐𝜙^′(𝑡)𝑑=0,_2+∞−∞𝐽𝑦−𝑡𝜑𝑦𝑑𝑦−𝑑₂𝜑(𝑡)−𝑐𝜑^′(𝑡)(𝑡)𝑔𝑧+𝜑⁰𝑐₁₁=0. (41) Thus, we consider the following characteristic equation: 𝐹(𝜆,𝑐)∶=𝑑₂^+∞_−∞𝐽𝑦𝑒^𝜆𝑦𝑑𝑦−𝑑₂𝑧−𝑐𝜆+𝑔⁰𝑐₁₁=0. (42) Note that 𝐹()()𝜆,0>0forany𝜆>0by𝐴1and𝑒^𝑦+𝑒^−𝑦,𝐹()𝐹()𝑧≥2𝜆,+∞=−∞forany𝜆>0,0,𝑐=𝑔⁰𝑐₁₁()>0,𝜕𝐹𝜆,𝑐𝜕𝜆=𝑑₂^+∞_−∞𝑦𝐽𝑦𝑒^𝜆𝑦𝜕𝑑𝑦−𝑐,₂𝐹(𝜆,𝑐)𝜕𝜆²=𝑑₂^+∞_−∞𝑦²𝐽𝑦𝑒^𝜆𝑦()𝑑𝑦>0forany𝜆∈ℝ,𝜕𝐹𝜆,𝑐𝜕𝑐=−𝜆<0forany𝜆>0. (43) The convex property of 𝐹 leads to 𝐹(+∞,𝑐)=+∞ for any 𝑐>0. In view of the previous observation, we have the following lemma directly.

Lemma 7. The following conclusions are true. (i)There exists a (𝜆^∗,𝑐^∗), 𝜆^∗>0, 𝑐^∗>0 such that 𝐹𝜆^∗,𝑐^∗()=0,𝜕𝐹𝜆,𝑐𝜕𝜆_(𝜆^∗_,𝑐^∗₎=0. (44)(ii)For 0<𝑐<𝑐^∗, 𝐹(𝜆,𝑐)>0 for 𝜆∈ℝ.(iii)For 𝑐>𝑐^∗, the equation 𝐹(𝜆,𝑐)=0 has two zeros 0<𝜆₁<𝜆₂ such that 𝐹(𝜆,𝑐)<0𝑓𝑜𝑟𝜆₁<𝜆<𝜆₂. (45)

Now we are ready to construct the upper solution of (12).

Lemma 8. Define 𝜙(𝑡)=min𝑒^𝜆1𝑡,𝑢⁺,𝜑(𝑡)=min𝑐₁₁𝑒^𝜆1𝑡,𝑣⁺. (46) Then for 𝑑₂≥𝑑₁, (𝜙(𝑡),𝜑(𝑡)) is an upper solution of (12).

Proof. Let 𝑡₁,𝑡₂ be such that 𝑒^𝜆1𝑡1=𝑢⁺,𝑐₁₁𝑒^𝜆1𝑡2=𝑣⁺. (47) Notice that 𝑣⁺=𝑐₁₁𝑢⁺, we have 𝑡₀∶=𝑡₁=𝑡₂=(1/𝜆₁)ln(𝑣⁺/𝑐₁₁).
If 𝑡<𝑡₀, 𝜙(𝑡)=𝑒^𝜆1𝑡, 𝜑(𝑡)=𝑐₁₁𝑒^𝜆1𝑡, we have from the fact 𝜙(𝑡)≤𝑒^𝜆1𝑡 for 𝑡∈ℝ and 𝑤₁<𝑤^∗≤𝑢^∗<𝑧₀/𝑐₁₁<𝑤₂ that 𝑑₁^+∞_−∞𝐽𝑦−𝑡𝜙𝑦𝑑𝑦−𝑑₁𝜙(𝑡)−𝑐𝜙^′(𝑡)+𝜙(𝑡)−𝑧₀𝑐₁₁𝑓𝑧₀−𝑐₁₁𝜙(𝑡)+𝜑(𝑡)≤𝑑₁^+∞_−∞𝐽𝑦−𝑡𝑒^𝜆1𝑦𝑑𝑦−𝑑₁𝑒^𝜆1𝑡−𝑐𝜆₁𝑒^𝜆1𝑡+𝑒^𝜆1𝑡−𝑧₀𝑐₁₁𝑓𝑧₀=𝑑₁^+∞_−∞𝐽𝑦𝑒^𝜆1𝑦𝑑𝑦−𝑑₁−𝑐𝜆₁𝑒^𝜆1𝑡=𝑑₁𝑑₂𝑑₂^+∞_−∞𝐽𝑦𝑒^𝜆1𝑦𝑑𝑦−𝑑₂−𝑐𝜆₁+𝑐𝜆₁−𝑐𝜆₁𝑒^𝜆1𝑡=−𝑑₁𝑑₂𝑔𝑧₀𝑐₁₁+𝑑₁𝑑₂𝑐𝜆₁−𝑐𝜆₁𝑒^𝜆1𝑡𝑑≤−¹𝑑₂𝑔𝑧₀𝑐₁₁𝑒^𝜆1𝑡<0. (48) Similarly, we have from the fact 𝜑(𝑡)≤𝑐₁₁𝑒^𝜆1𝑡 for 𝑡∈ℝ, 𝑐₁₁𝑐₂₂>1 and 𝑔^′(𝑤)<0 for 𝑤>𝑤^∗ that 𝑑₂^+∞_−∞𝐽𝑦−𝑡𝜑𝑦𝑑𝑦−𝑑₂𝜑(𝑡)−𝑐𝜑^′(𝑡)+𝜑(𝑡)𝑔𝑧₀𝑐₁₁−𝜙(𝑡)+𝑐₂₂𝜑(𝑡)≤𝑑₂^+∞_−∞𝐽𝑦−𝑡𝑐₁₁𝑒^𝜆1𝑦𝑑𝑦−𝑑₂𝑐₁₁𝑒^𝜆1𝑡−𝑐𝑐₁₁𝜆₁𝑒^𝜆1𝑡+𝑐₁₁𝑒^𝜆1𝑡𝑔𝑧₀𝑐₁₁−𝑒^𝜆1𝑡+𝑐₁₁𝑐₂₂𝑒^𝜆1𝑡<𝑑₂^+∞_−∞𝐽𝑦𝑒^𝜆1𝑦𝑑𝑦−𝑑₂−𝑐𝜆₁𝑧+𝑔⁰𝑐₁₁𝑐₁₁𝑒^𝜆1𝑡=0. (49)
If 𝑡>𝑡₀, 𝜙(𝑡)=𝑢⁺, 𝜑(𝑡)=𝑣⁺, we have 𝑑₁^+∞_−∞𝐽𝑦−𝑡𝜙𝑦𝑑𝑦−𝑑₁𝜙(𝑡)−𝑐𝜙^′(𝑡)+𝜙(𝑡)−𝑧₀𝑐₁₁𝑓𝑧₀−𝑐₁₁𝜙(𝑡)+𝜑(𝑡)≤0;𝑑₂^+∞_−∞𝐽𝑦−𝑡𝜑𝑦𝑑𝑦−𝑑₂𝜑(𝑡)−𝑐𝜑^′(𝑡)+𝜑(𝑡)𝑔𝑧₀𝑐₁₁−𝜙(𝑡)+𝑐₂₂𝜑(𝑡)≤𝑣⁺𝑔𝑧₀𝑐₁₁−𝑢⁺+𝑐₂₂𝑣⁺=𝑣⁺𝑔𝑤₂=0. (50)(51) The proof is complete.

Let 𝜂∈(1,min{2,𝜆₂/𝜆₁}) satisfying 𝜆₁<𝜂𝜆₁<𝜆₂, we can obtain from (45) that 𝑑₂^+∞_−∞𝐽𝑦𝑒^{𝜂𝜆1𝑦}𝑑𝑦−𝑑₂−𝑐𝜂𝜆₁𝑧+𝑔⁰𝑐₁₁<0. (52)

Lemma 9. Define 𝜙_∗(𝑡)=0,𝜑_∗(𝑡)=max𝑐₁₁𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡},0, (53) where 𝑞>1 is a constant to be chosen later. Then (𝜙_∗(𝑡),𝜑_∗(𝑡)) is a lower solution of (12). Furthermore, 𝜙(𝑡),𝜑(𝑡)∶=𝑄₁𝜙_∗(𝑡),𝜑_∗(𝑡),𝑄₂𝜙_∗(𝑡),𝜑_∗(𝑡) (54) is a lower solution of (12) satisfying 𝜙(𝑡),𝜑(𝑡)≫(0,0)𝑓𝑜𝑟𝑡∈ℝ. (55)

Proof. Let 𝑡₃ be such that 𝑐₁₁(𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡})=0, it follows that 𝑡₃=1𝜂−1𝜆₁1ln𝑞<0. (56)
If 𝑡<𝑡₃<0, 𝜙_∗(𝑡)=0, 𝜑_∗(𝑡)=𝑐₁₁(𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}), and 𝜑_∗(𝑡)≥𝑐₁₁(𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}) for 𝑡∈ℝ, we have 𝑑₁^+∞_−∞𝐽𝑦−𝑡𝜙_∗𝑦𝑑𝑦−𝑑₁𝜙_∗(𝑡)−𝑐𝜙^′_∗(𝑡)+𝜙_∗(𝑡)−𝑧₀𝑐₁₁𝑓𝑧₀−𝑐₁₁𝜙_∗(𝑡)+𝜑_∗(𝑡)𝑧=−⁰𝑐₁₁𝑓𝑧₀+𝜑_∗(𝑡)𝑧≥−⁰𝑓𝑧₀𝑐₁₁=0. (57)
Let 𝑚∶=min_{𝑤∈[𝑧0/𝑐11,𝑧0/𝑐11+𝑐22𝑣+]}𝑔^′(𝑤)<0. By the fact that 𝑐₁₁𝑐₂₂𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}≥0for𝑡<𝑡₃, (58) we have 𝑔𝑧₀𝑐₁₁+𝑐₁₁𝑐₂₂𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}=𝑔𝑧₀𝑐₁₁+𝑐₁₁𝑐₂₂𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}𝑧−𝑔⁰𝑐₁₁𝑧+𝑔⁰𝑐₁₁≥𝑚𝑐₁₁𝑐₂₂𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}𝑧+𝑔⁰𝑐₁₁. (59) Note that (58) leads to 𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}>0 for 𝑡<𝑡₃; it follows that 𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡2}≤𝑒^2𝜆1𝑡, (60) and thus 𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}𝑔𝑧₀𝑐₁₁+𝑐₁₁𝑐₂₂𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}𝑧≥𝑔⁰𝑐₁₁𝑒^𝜆1𝑡+𝑚𝑐₁₁𝑐₂₂𝑒^2𝜆1𝑡𝑧−𝑔⁰𝑐₁₁𝑞𝑒^{𝜂𝜆1𝑡}. (61) Therefore, 𝑑₂^+∞_−∞𝐽𝑦−𝑡𝜑_∗𝑦𝑑𝑦−𝑑₂𝜑_∗(𝑡)−𝑐𝜑^′_∗(𝑡)+𝜑_∗(𝑡)𝑧×𝑔⁰𝑐₁₁−𝜙_∗(𝑡)+𝑐₂₂𝜑_∗(𝑡)≥𝑐₁₁𝑑₂^+∞_−∞𝐽𝑦−𝑡𝑒^𝜆1𝑦−𝑞𝑒^{𝜂𝜆1𝑦}𝑑𝑦−𝑑₂𝑐₁₁×𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}−𝑐𝑐₁₁𝜆₁𝑒^𝜆1𝑡−𝑞𝜂𝜆₁𝑒^{𝜂𝜆1𝑡}+𝑐₁₁𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}𝑧×𝑔⁰𝑐₁₁+𝑐₁₁𝑐₂₂𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}≥𝑐₁₁𝑒^𝜆1𝑡𝑑₂^+∞_−∞𝐽𝑦𝑒^𝜆1𝑦𝑑𝑦−𝑑₂−𝑐𝜆₁𝑧+𝑔⁰𝑐₁₁−𝑐₁₁𝑒^{𝜂𝜆1𝑡}𝑞𝑑₂^+∞_−∞𝐽𝑦𝑒^{𝜂𝜆1𝑦}𝑑𝑦−𝑑₂−𝑐𝜂𝜆₁𝑧+𝑔⁰𝑐₁₁−𝑚𝑐₁₁𝑐₂₂𝑒^{(2−𝜂)𝜆1𝑡}≥−𝑐₁₁𝑒^{𝜂𝜆1𝑡}𝑞𝑑₂^+∞_−∞𝐽𝑦𝑒^{𝜂𝜆1𝑦}𝑑𝑦−𝑑₂−𝑐𝜂𝜆₁𝑧+𝑔⁰𝑐₁₁−𝑚𝑐₁₁𝑐₂₂. (62) Let 𝑞>1 sufficiently large; we can have 𝑞𝑑₂^+∞_−∞𝐽𝑦𝑒^{𝜂𝜆1𝑦}𝑑𝑦−𝑑₂−𝑐𝜂𝜆₁𝑧+𝑔⁰𝑐₁₁−𝑚𝑐₁₁𝑐₂₂<0, (63) hence, 𝑑₂^+∞_−∞𝐽𝑦−𝑡𝜑_∗𝑦𝑑𝑦−𝑑₂𝜑_∗(𝑡)−𝑐𝜙^′_∗(𝑡)+𝜑_∗(𝑡)𝑔𝑧₀𝑐₁₁−𝜙_∗(𝑡)+𝑐₂₂𝜑_∗(𝑡)≥0. (64)
If 𝑡>𝑡₃, 𝜙_∗(𝑡)=0, 𝜑_∗(𝑡)=0, and 𝜑_∗(𝑡)≥0 for 𝑡∈ℝ, we have 𝑑₁^+∞_−∞𝐽𝑦−𝑡𝜙_∗𝑦𝑑𝑦−𝑑₁𝜙_∗(𝑡)−𝑐𝜙^′_∗(𝑡)+𝜙_∗(𝑡)−𝑧₀𝑐₁₁𝑓𝑧₀−𝑐₁₁𝜙_∗(𝑡)+𝜑_∗(𝑡)=0. (65)
From the previous arguments, we obtain that (𝜙_∗(𝑡),𝜑_∗(𝑡)) is a lower solution of (12). By Lemma 4, we can get that (𝜙(𝑡),𝜑(𝑡)) is also a lower solution. Furthermore, for 𝑡<𝑡₃, 𝜑_∗(𝑡)>0, by direct calculation, we have 𝐻₁(𝜙_∗,𝜑_∗)(𝑡)>0 for 𝑡<𝑡₃. Therefore, 𝜙(𝑡)=𝑄₁(𝜙_∗,𝜑_∗)(𝑡)>0 for 𝑡∈ℝ. Similarly, we have 𝜑(𝑡)>0 for 𝑡∈ℝ. The proof is complete.

Theorem 10. Assume that (A1) and 𝑑₂≥𝑑₁ hold. Then for any 𝑐≥𝑐^∗, the system (11) has a traveling wavefront with speed 𝑐, which connects (0,0) and (𝑢⁺,𝑣⁺).

Proof. The conclusion for 𝑐>𝑐^∗ can be obtained from the previous discussions. We only need to establish the existence of wave fronts when 𝑐=𝑐^∗.
Let 𝑐_𝑘⊂(𝑐^∗,𝑐^∗+1) with lim_𝑘→∞𝑐_𝑘=𝑐^∗. For 𝑐_𝑘>𝑐^∗, (12) with 𝑐=𝑐_𝑘 admits a nondecreasing solution (𝜙_𝑘(𝑡),𝜑_𝑘(𝑡)) such that lim_{𝑡→−∞}𝜙_𝑘(𝑡),𝜑_𝑘(𝑡)=(0,0),lim_𝑡→+∞𝜙_𝑘(𝑡),𝜑_𝑘(𝑡)=𝑢⁺,𝑣⁺. (66) Without loss of generality, we assume that (𝜙_𝑘(0),𝜑_𝑘(0))=(𝑢⁺/2,𝑣⁺/2). Obviously, |𝜙_𝑘(𝑡)|≤𝑢⁺, |𝜑_𝑘(𝑡)|≤𝑣⁺, and 𝜙_𝑘(𝑡),𝜑_𝑘(𝑡) satisfy 𝜙_𝑘(𝑡)=1𝑐_𝑘𝑒^{−(𝛽1/𝑐𝑘)𝑡𝑡}_−∞𝑒^{(𝛽1/𝑐𝑘)𝑠}𝐻₁𝜙_𝑘,𝜑_𝑘𝑠)𝜑(𝑑𝑠,_𝑘(𝑡)=1𝑐_𝑘𝑒^{−(𝛽2/𝑐𝑘)𝑡𝑡}_−∞𝑒^{(𝛽2/𝑐𝑘)𝑠}𝐻₂𝜙_𝑘,𝜑_𝑘(𝑠)𝑑𝑠. (67)
As the same argument in Lemma 5, we can obtain that {(𝜙_𝑘(𝑡),𝜑_𝑘(𝑡))} is uniformly bounded and equicontinuous on ℝ; using Arzéla-Ascoli theorem and the standard diagonal method, we can obtain a subsequence of {(𝜙_𝑘(𝑡),𝜑_𝑘(𝑡))}, still denoted by {(𝜙_𝑘(𝑡),𝜑_𝑘(𝑡))}, such that (𝜙_𝑘(𝑡),𝜑_𝑘(𝑡))→(𝜙^∗(𝑡),𝜑^∗(𝑡)) uniformly for 𝑡 in any bounded subset of ℝ, as 𝑘→∞. Clearly, (𝜙^∗(𝑡),𝜑^∗(𝑡)) is nondecreasing and (𝜙^∗(0),𝜑^∗(0))=(𝑢⁺/2,𝑣⁺/2).
By the dominated convergence theorem and (67), it follows that 𝜙^∗(𝑡)=1𝑐^∗𝑒^{−(𝛽1/𝑐∗)𝑡𝑡}_−∞𝑒^{(𝛽1/𝑐∗)𝑠}𝐻₁𝜙^∗,𝜑^∗𝑠)𝜑(𝑑𝑠,_∗(𝑡)=1𝑐^∗𝑒^{−(𝛽2/𝑐∗)𝑡𝑡}_−∞𝑒^{(𝛽2/𝑐∗)𝑠}𝐻₂𝜙^∗,𝜑^∗(𝑠)𝑑𝑠. (68) Since lim_𝑡→±∞𝜙^∗(𝑡) and lim_𝑡→±∞𝜑^∗(𝑡) exist, using L’Hôspital rule leads to lim_{𝑡→−∞}𝜙^∗(𝑡),𝜑^∗(𝑡)=(0,0),lim_𝑡→+∞𝜙^∗(𝑡),𝜑^∗(𝑡)=𝑢⁺,𝑣⁺. (69) Thus, (𝜙^∗(𝑡),𝜑^∗(𝑡)) is a traveling wavefront of the system (11) connecting (0,0) and (𝑢⁺,𝑣⁺).

Remark 11. We say that the 𝑐^∗ is the minimal wave speed in the sense that (11) has no traveling wavefront with 𝑐∈(0,𝑐^∗). We could briefly explain this in the following. In fact, the linearization of (12) at zero solution is (41), and the function 𝐹(𝜆,𝑐) is obtained by substituting 𝑒^𝜆𝑡 in the second equation of (41). For 0<𝑐<𝑐^∗, we know from (ii) of Lemma 7 that 𝐹(𝜆,𝑐)>0 for any 𝜆∈ℝ. We have from the second equation of (12) and the second equation of (41) that (12) cannot have a solution (𝜙(𝑡),𝜑(𝑡)) that satisfies lim_{𝑡→−∞}(𝜙(𝑡),𝜑(𝑡))=(0,0).

Theorem 12. Assume that (A1) and 𝑑₂≥𝑑₁ hold. Then for any 𝑐≥𝑐^∗, the system (6) has a traveling wavefront with speed c, which connects (𝑧₀/𝑐₁₁,0) and (𝑢^∗,𝑣^∗).

4. Asymptotic Behavior for Traveling Wavefronts

In this section, we discuss the asymptotic behavior for the traveling wavefronts obtained in the previous section as 𝑡→−∞.

Theorem 13. Let (𝜙(𝑡),𝜓(𝑡)) be a traveling wavefront of (11) decided by Theorem 10; then lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡,𝜑(𝑡)𝑒^−𝜆1𝑡=𝑑₂𝑧₀𝑓^′𝑧₀𝑐𝜆₁𝑑₁−𝑑₂−𝑑₁𝑔𝑧₀/𝑐₁₁+𝑑₂𝑧₀𝑓^′𝑧₀,𝑐₁₁,lim_{𝑡→−∞}𝜙^′(𝑡)𝑒^−𝜆1𝑡,𝜑^′(𝑡)𝑒^−𝜆1𝑡=𝜆₁𝑑₂𝑧₀𝑓^′𝑧₀𝑐𝜆₁𝑑₁−𝑑₂−𝑑₁𝑔𝑧₀/𝑐₁₁+𝑑₂𝑧₀𝑓^′𝑧₀,𝑐₁₁𝜆₁, (70) where 𝜆₁ is the smallest root of the characteristic equation (42).

Proof. Note that 𝑐₁₁𝑒^𝜆1𝑡−𝑞𝑒^{𝜂𝜆1𝑡}≤𝜑(𝑡)≤𝑐₁₁𝑒^𝜆1𝑡for𝑡∈ℝ. (71) Then we have lim_{𝑡→−∞}𝜑(𝑡)𝑒^−𝜆1𝑡−𝑐₁₁≤lim_{𝑡→−∞}𝑐₁₁𝑞𝑒^{(𝜂−1)𝜆1𝑡}=0, (72) which implies that lim_{𝑡→−∞}𝜑(𝑡)𝑒^−𝜆1𝑡=𝑐₁₁. (73)
Note that we have from 0≤𝜑(𝑡)𝑒^−𝜆1𝑡≤𝑐₁₁ and (A1) that 𝑒^{−𝜆1𝑡+∞}_−∞𝐽𝜑𝑦=𝑦−𝑡𝑑𝑦_+∞−∞𝐽𝑦𝑒^𝜆1𝑦𝜑𝑦+𝑡𝑒^{−𝜆1(𝑦+𝑡)}𝑑𝑦<∞, (74) which is uniformly on 𝑡. By the second equation of (12), we have from (73), 𝐹(𝜆₁,𝑐)=0 and convergence theorem that lim_{𝑡→−∞}𝜑^′(𝑡)𝑒^−𝜆1𝑡=1𝑐𝑐₁₁𝑔𝑧₀𝑐₁₁−𝑑₂+𝑑₂lim_{𝑡→−∞}𝑒^{−𝜆1𝑡+∞}_−∞𝐽𝜑𝑦=1𝑦−𝑡𝑑𝑦𝑐𝑐₁₁𝑔𝑧₀𝑐₁₁−𝑑₂+𝑑₂lim_{𝑡→−∞}𝑒^{−𝜆1𝑡+∞}_−∞𝐽𝑦𝜑=1𝑦+𝑡𝑑𝑦𝑐𝑐₁₁𝑔𝑧₀𝑐₁₁−𝑑₂+𝑑₂^+∞_−∞𝐽𝑦𝑒^𝜆1𝑦lim_{𝑡→−∞}𝜑𝑦+𝑡𝑒^{−𝜆1(𝑦+𝑡)}=1𝑑𝑦𝑐𝑐₁₁𝑔𝑧₀𝑐₁₁−𝑑₂+𝑐₁₁𝑑₂+𝑐𝜆₁𝑧−𝑔⁰𝑐₁₁=𝑐₁₁𝜆₁. (75)

Since 𝜙(𝑡)=𝑄₁(𝜙,𝜑)(𝑡), by (17), we know that 𝜙(𝑡)=1𝑐𝑒^{−(𝛽1/𝑐)𝑡𝑡}_−∞𝑒^{(𝛽1/𝑐)𝑠}𝐻₁𝜙,𝜑(𝑠)𝑑𝑠. (76) On the other hand, by (76), (15), (42), and (75), noting 𝑓(𝑧₀)=0 and using L’Hôspital’s rule, we get lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡=1𝑐lim_{𝑡→−∞}𝑒^{−(𝜆1+𝛽1/𝑐)𝑡𝑡}_−∞𝑒^{(𝛽1/𝑐)𝑠}𝐻₁(𝑠)=1𝜙,𝜑𝑑𝑠𝑐lim_{𝑡→−∞}𝑒^{(𝛽1/𝑐)𝑡}𝐻₁𝜙,𝜑(𝑡)𝜆₁+𝛽₁/𝑐𝑒^{(𝜆1+𝛽1/𝑐)𝑡}=1𝑐𝜆₁+𝛽₁𝑑₁lim_{𝑡→−∞}𝑒^{−𝜆1𝑡+∞}_−∞𝐽𝜙𝑦−𝑧𝑦−𝑡𝑑𝑦₀𝑐₁₁lim_{𝑡→−∞}𝑓𝑧₀−𝑐₁₁𝜙(𝑡)+𝜑(𝑡)𝑒^𝜆1𝑡+𝑏₁lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡=1𝑐𝜆₁+𝛽₁𝑑₁𝑑₂𝑑₂+𝑐𝜆₁𝑧−𝑔⁰𝑐₁₁lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡−𝑧₀𝑓^′𝑧₀𝑐₁₁𝜆₁lim_{𝑡→−∞}−𝑐₁₁𝜙^′(𝑡)+𝜑^′(𝑡)𝑒^𝜆1𝑡+𝑏₁lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡=1𝑐𝜆₁+𝛽₁𝑏₁+𝑑₁+𝑐𝜆₁𝑑₁𝑑₂−𝑑₁𝑑₂𝑔𝑧₀𝑐₁₁×lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡+𝑧₀𝑓^′𝑧₀𝜆₁×lim_{𝑡→−∞}𝜙^′(𝑡)𝑒^−𝜆1𝑡−𝑧₀𝑓^′𝑧₀. (77) By the first equation of (12), we have from (75) and 𝐹(𝜆₁,𝑐)=0 that 𝑐lim_{𝑡→−∞}𝜙^′(𝑡)𝑒^−𝜆1𝑡=𝑑₁lim_{𝑡→−∞}𝑒^{−𝜆1𝑡+∞}_−∞𝐽𝑦−𝑡𝜙𝑦𝑑𝑦−𝑑₁×lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡−𝑧₀𝑐₁₁lim_{𝑡→−∞}𝑓𝑧₀−𝑐₁₁𝜙(𝑡)+𝜑(𝑡)𝑒^𝜆1𝑡=𝑑₁𝑑₂𝑑₂+𝑐𝜆₁𝑧−𝑔⁰𝑐₁₁lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡−𝑑₁lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡+𝑧₀𝑓^′𝑧₀𝜆₁×lim_{𝑡→−∞}𝜙^′(𝑡)𝑒^−𝜆1𝑡−𝑧₀𝑓^′𝑧₀=𝑑₁𝑑₂𝑐𝜆₁𝑧−𝑔⁰𝑐₁₁lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡+𝑧₀𝑓^′𝑧₀𝜆₁lim_{𝑡→−∞}𝜙^′(𝑡)𝑒^−𝜆1𝑡−𝑧₀𝑓^′𝑧₀. (78) Combining (77) and (78), and noting that 𝛽₁=𝑏₁+𝑑₁, we obtain from a direct calculation that lim_{𝑡→−∞}𝜙(𝑡)𝑒^−𝜆1𝑡=𝑑₂𝑧₀𝑓^′𝑧₀𝑐𝜆₁𝑑₁−𝑑₂−𝑑₁𝑔𝑧₀/𝑐₁₁+𝑑₂𝑧₀𝑓^′𝑧₀,lim_{𝑡→−∞}𝜙^′(𝑡)𝑒^−𝜆1𝑡=𝜆₁𝑑₂𝑧₀𝑓^′𝑧₀𝑐𝜆₁𝑑₁−𝑑₂−𝑑₁𝑔𝑧₀/𝑐₁₁+𝑑₂𝑧₀𝑓^′𝑧₀. (79)

Now, letting (𝜁(𝑡),𝜍(𝑡)) be a traveling wavefront of (6) decided by Theorem 12, by the equivalence between (6) and (11), we know that (𝜁(𝑡),𝜍(𝑡))=(𝑧₀/𝑐₁₁−𝜙(𝑡),𝜑(𝑡)) and thus obtain the following theorem.

Theorem 14. Let (𝜁(𝑡),𝜍(𝑡)) be a traveling wavefront of (6) decided by Theorem 12; then lim_{𝑡→−∞}𝜁(𝑡)𝑒^−𝜆1𝑡,𝜍(𝑡)𝑒^−𝜆1𝑡=∞,𝑐₁₁,lim_{𝑡→−∞}𝜁^′(𝑡)𝑒^−𝜆1𝑡,𝜍^′(𝑡)𝑒^−𝜆1𝑡=𝜆₁𝑑₂𝑧₀𝑓^′𝑧₀𝑐𝜆₁𝑑₂−𝑑₁+𝑑₁𝑔𝑧₀/𝑐₁₁−𝑑₂𝑧₀𝑓^′𝑧₀,𝑐₁₁𝜆₁, (80) where 𝜆₁ is the smallest root of the characteristic equation (42).

5. Concluding Discussions

We have considered a reaction model with nonlocal diffusion for competing pioneer-climax species. Some recent works (see Brown et al. [4], Yuan and Zou [6]) showed that the model with local diffusion (expressed by Laplacian operator) supports traveling wavefronts connecting two boundary equilibria or one boundary equilibrium and the coexisting equilibrium under some restrictions on the parameters. In Yuan and Zou [6], the sufficient condition for (2) to have the traveling wavefront connecting (𝑧₀/𝑐₁₁,0) and (𝑢^∗,𝑣^∗) is 𝑑₂/𝑑₁≥1/2, but ours for (6) with nonlocal diffusion is 𝑑₂/𝑑₁≥1. For a fixed 𝑑₂, to shift 𝑑₂/𝑑₁≥1/2 to 𝑑₂/𝑑₁≥1, one needs a smaller 𝑑₁. But for a fixed 𝑑₁, to shift 𝑑₂/𝑑₁≥1/2 to 𝑑₂/𝑑₁≥1, one needs a larger 𝑑₂. These facts imply that the nonlocal diffusion of the pioneer species accelerates the mild wave propagation, while the nonlocal diffusion of the climax species defers the mild wave propagation. That is to say, the nonlocal diffusion did affect the wave propagation of these two competitive species.

The generalized boundary conditions, lim_{𝑡→−∞}𝑧₀𝑐₁₁(𝑡)(𝑡)=𝑧−𝜙,𝜑⁰𝑐₁₁,0,lim_𝑡→∞𝑧₀𝑐₁₁−𝜙(𝑡),𝜑(𝑡)=𝑢^∗,𝑣^∗, (81) lead to an explanation that the climax species starts its invasion after that the pioneer species has achieved its steady state, and also the competition between the two species was not intense; they can achieve a coexistence state finally. See [19] for the significance of biological invasion.

We believe that this is the first time that the dynamics of the pioneer-climax competition model with nonlocal diffusion are studied. This model has complicated equilibrium structure, and we only considered one possible case about the traveling wavefront connecting one boundary equilibrium and the coexisting equilibrium. Some other situations about the species invasions and propagation of waves would be of great interest for further research.

Acknowledgments

The authors are very grateful to the anonymous referees for careful reading and helpful suggestions which led to an improvement of their original paper. Research is supported by the Natural Science Foundation of China (11171120), the Doctoral Program of Higher Education of China (20094407110001), and the Natural Science Foundation of Guangdong Province (10151063101000003).