Bistable Wave Fronts in Integrodifference Equations

Abstract

This paper is concerned with the bistable wave fronts of integrodifference equations. The existence, uniqueness, and asymptotic stability of bistable wave fronts for such an equation are proved by the squeezing technique based on comparison principle.

1. Introduction

In 1982, Weinberger [1] proposed a famous discrete-time recursion to model a class of biological processes, to which a number of evolutionary systems can be reduced. A typical recursion takes the following form𝑣_𝑛+1(𝑥)=𝑄𝑣_𝑛(𝑥),𝑛=0,1,2,…, (1.1) where 𝑥∈ℋ⊂ℝ^𝑚, 𝑣∈ℝ^𝑘, and 𝑄 is an ℝ^𝑘-valued mapping, herein ℋ admits proper geometric structure such that it is a habitat (e.g., the integer lattice in ℝ^𝑚; see [1]). Such a recursion can be defined for both discrete and continuous temporal-spatial variables, for example, the reaction-diffusion equations, lattice dynamical systems, and integrodifference equations [1–3]. And these models can be dealt with by the same theory scheme established for the abstract discrete-time recursions; see, for example, Liang and Zhao [2], Weinberger et al. [3]. In particular, Weinberger [1] derived a concrete population model as follows:𝑣_𝑛+1(𝐱)=𝑄𝑣_𝑛(𝐱)=_ℝ2𝑚(𝐱−𝐲)𝑔𝑣_𝑛(𝐲)𝑑𝐲,𝑛=0,1,2,…, (1.2) where 𝐱,𝐲∈ℝ²,  𝑣_𝑛∈ℝ denotes the gene fraction in the newly born individuals of the 𝑛-th generation, 𝑚 is the probability function describing the migration of the individuals, and 𝑔(𝑣_𝑛) means the gene fraction before the migration.

In the past decades, the traveling wave solutions and the spreading speeds of discrete-time recursions have been widely studied if they are monostable (analogous to reaction-diffusion equations), and we refer to [1–13]. As mentioned in Weinberger [1, Theorem 6.6 and Remark 11.4], maybe the traveling wave solutions of bistable recursions (analogous to reaction-diffusion equations) are totally different from those of the monostable equations, and we may also compare the spatial propagation modes in Kot et al. [14] and Wang et al. [15] to understand the differences between the monostable and the bistable. When (1.2) is concerned, an important 𝑔(𝑢) is the so-called Hill function 𝑔(𝑢)=𝜆𝑢^𝑝1+(𝜆−1)𝑢^𝑝,𝜆>1,𝑝>0,𝑢≥0. (1.3)

If 𝑝=1 and 𝜆>1, then it is the famous Beverton-Holt stock recruitment curve, and it is monostable (see Kot [16]); if 𝑝=2 and 𝜆>2, then it is bistable (see Wang et al. [15]). More precisely, 𝑔(𝑢)=𝑢 has three equilibria 0,1/(𝜆−1),1, and 0,1 are locally stable, while 1/(𝜆−1) is unstable in the sense of the corresponding difference equation. In population dynamics, the case 𝑝=2 with 𝜆>2 describes the famous Allen effect (see [15]). For the reader’s convenience in understanding the difference, we may see the following Figures 1 and 2 for 𝑝=1,2 with 𝜆=3, respectively.

Figure 1 

𝑔(𝑢)=3𝑢/(1+2𝑢).

Figure 2 

𝑔(𝑢)=3𝑢2/(1+2𝑢2).

For other special examples of the abstract discrete-time recursions, there are also some significant differences between the monostable system and the bistable one from the viewpoint of the traveling wave solutions such as reaction-diffusion equations [17–21] and lattice differential equations [22–25].

If the spatial variable is one-dimensional, Lui [26, 27] considered the bistable wave fronts of the following integrodifference equation:𝑣_𝑛+1(𝑥)=_ℝ𝑘𝑥−𝑦𝑔𝑣_𝑛𝑦𝑑𝑦,𝑥∈ℝ, (1.4) where 𝑘∶ℝ→ℝ⁺ is a probability function. In particular, the author proved that for a large class of initial data, the solution of (1.4) will be trapped between two suitable translates of the wave, and the author conjectured that the bistable wave front of (1.4) is stable indeed, namely, these translates would convergence to the same limit. For the case that the wave speed is 0, some results on the stability of the traveling wave solutions were given by Lui [28]. Creegan and Lui [29] further considered the existence and uniqueness of bistable wave fronts of an integrodifference equation in ℝ^𝑛. In this paper, we will prove the conjecture in Lui [26, 27] for the asymptotic stability of the bistable wave fronts in a position.

For the sake of simplicity, we first assume that 𝑘 takes the form of the Gaussian in (1.4), namely; we will investigate the bistable wave fronts of the following integrodifference equation:𝑢_𝑛+1(𝑥)=𝑄𝑢_𝑛𝑥)=1(√4𝜋𝑑^ℝ𝑒^{−(𝑥−𝑦)2/4𝑑}𝑔𝑢_𝑛𝑦𝑑𝑦,𝑛=0,1,2,…, (1.5) where 𝑥,𝑦∈ℝ, 𝑔∶ℝ→ℝ, and 𝑑>0 is a constant. We should note that there are some significant differences in investigating the traveling wave solutions of the bistable and monostable evolution equations; on the one hand, the process of constructing upper and lower solutions was completed by different techniques. On the other hand, it seems that the characteristic equation of the monostable equation plays a more important role than that of the bistable one, and we refer to [20–25]. So, we do not hope that the techniques used for monostable integrodifference equation in [9] can be applied to a bistable equation, and we will apply the squeezing technique based on comparison principle and upper and lower solutions, which was earlier used in Chen [22] and Fife and McLeod [30] and was further developed by Ma and Wu [31], Ma and Zou [24], Smith and Zhao [18], and Wang et al. [20]. In particular, our discussion is independent of the characteristic equation, so we also relax some requirements on 𝑔 which were used in Lui [26, 27] (see Remark 2.1). Moreover, Yagisita [32] established an abstract scheme to establish the existence of bistable wave fronts of monotone semiflows, and from Yagisita [32], Corollary 5 can be applied to our model if we can verify [32, Hypotheses 2-3]. In this paper, we could construct proper functions satisfying [32], Hypotheses 2-3 for (1.5), and so, their assumptions are reasonable and achievable, at least for (1.5).

Comparing our results with those in [1, 2, 9], a bistable integrodifference equation is also significantly different from a monostable one in the sense of the traveling wave solutions, such as the wave speed of bistable wave fronts of a bistable equation is unique while a monostable equation has infinite many wave speeds such that the system has a monostable wave front with each speed, and the wave profile of bistable wave front of bistable equation is unique in the sense of phase shift, while there are many different wave profiles of monostable wave fronts with different wave speeds of monostable equation.

The remainder of this paper is organized as follows. In Section 2, we will do some preliminaries for later sections. The corresponding initial value problem will be investigated in Section 3. In Section 4, the stability and uniqueness of the bistable wave fronts will be established. In Section 5, we show the existence of traveling wave solutions for (1.5). This paper ends up with the discussion on the bistable wave fronts of integrodifference equations.

2. Preliminaries

Throughout the paper, we will use the standard order and the interval notations in ℝ. We also denote 𝐶(ℝ,ℝ) as follows: 𝐶(ℝ,ℝ)=𝑢∣𝑢∶ℝ⟶ℝisuniformlycontinuousandbounded, (2.1)

which is a Banach space equipped with the super norm |⋅|. Assume that 𝑎<𝑏 with 𝑎,𝑏∈ℝ, then 𝐶_[𝑎,𝑏] will be interpreted as 𝐶_[𝑎,𝑏]={𝑢∶𝑢∈𝐶(ℝ,ℝ),𝑎≤𝑢(𝑥)≤𝑏,∀𝑥∈ℝ}. (2.2)

For convenience, we list the assumptions on 𝑔, which will be imposed in what follows:(g1)for 𝑢∈[0,1], 𝑔(𝑢)=𝑢 exactly has three distinct zeros 0,1,𝑎∈(0,1),(g2)if 𝑢∈[0,1], then 𝑔(𝑢)∈𝐶²; if 𝑢∈(0,1), then 𝑔^′(𝑢)>0,(g3)there exist 𝜌>0,𝜚>0 such that 𝑔^′(𝑢)<1−𝜚 for 𝑢∈[0,𝜌]∪[1−𝜌,1] and 𝑔^′(𝑢)>1+𝜚 for 𝑢∈[𝑎−𝜌,𝑎+𝜌].

Let 𝑔(𝑢)=𝜆𝑢²/(1+(𝜆−1)𝑢²) with 𝜆>2 (see Wang et al. [15]) or 𝑔(𝑢)=(𝑠𝑢²+𝑢)/(1+𝑠𝑢²+𝜎(1−𝑢)²) with 0<𝜎≤𝑠 (see Weinberger [1]), then (g1)–(g3) hold.

Remark 2.1. In Lui [26, 27], 𝑔^′(𝑎)=max_𝑥∈[0,1]𝑔^′(𝑥) was required. Moreover, Lui [27] assumed that 𝑔^′(0)>0, since the characteristic equation was involved. It is clear that 𝑔(𝑢)=𝜆𝑢²/(1+(𝜆−1)𝑢²) does not satisfy these assumptions if 𝜆≠4. But Lui [26, 27] have less requirements on the kernel function.

Definition 2.2. A traveling wave solution of (1.5) is a special solution with the form 𝑢_𝑛(𝑥)=𝜙(𝑥+𝑐𝑛), in which −𝑐∈ℝ is the wave speed that the wave profile 𝜙∈𝐶(ℝ,ℝ) spreads in ℝ. In particular, if 𝜙(𝑡) is monotone in 𝑡∈ℝ, then it is a traveling wave front.

By Definition 2.2, a traveling wave solution of (1.5) must satisfy𝜙()=1𝑡+𝑐√4𝜋𝑑^ℝ𝑒^{−(𝑡−𝑦)2/4𝑑}𝑔𝜙𝑦𝑑𝑦,𝑡∈ℝ, (2.3) and we are interested in the following asymptotic boundary conditions:lim_{𝑡→−∞}𝜙(𝑡)=0,lim_𝑡→∞𝜙(𝑡)=1. (2.4)

Remark 2.3. Analogues to a parabolic system [19], we also call a traveling wave front satisfying (2.3)-(2.4) as a bistable wave front if (g1) and (g3) hold. Moreover, the smoothness effect of the Gaussian kernel (see [33]) implies that 𝜙(𝑡)∈𝐶¹(ℝ,ℝ) if 𝜙(𝑡)∈𝐶(ℝ,ℝ).

3. Initial Value Problem

In this part, we will show some results of the following Cauchy type problem:𝑢_𝑛+1(𝑥)=𝑄𝑢_𝑛𝑥)𝑢(,𝑛=0,1,2,3…,₀(𝑥)=𝑢(𝑥), (3.1) in which 𝑢(𝑥)∈𝐶_[0,1] and 𝑄 is given by (1.5). We also refer to Liang and Zhao [2], Weinberger [1]. In particular, the following result is obvious.

Theorem 3.1. For all =0,1,…, 𝑢_𝑛(𝑥)∈𝐶_[0,1]. Moreover, 𝑢_𝑛(𝑥) is uniformly continuous  for all n  ≥   1 and 𝑥∈ℝ.

Definition 3.2. Assume that 𝑣_𝑛(𝑥)∈𝐶_[0,1] for all 𝑛=0,1,…,𝑁. Then, 𝑣_𝑛(𝑥) is called  an upper solution (a lower solution) of (3.1) if 𝑣_𝑛+1(𝑥)≥(≤)𝑄𝑣_𝑛(𝑥),𝑛=0,1,…,𝑁−1,𝑣₀(𝑥)≥(≤)𝑢(𝑥). (3.2)

By the upper and lower solutions, we can establish the following comparison principle.

Theorem 3.3. Assume that 𝑢_𝑛(𝑥) and 𝑢_𝑛(𝑥),𝑛=0,1,…,𝑁, are upper and lower solutions of (3.1), respectively. Then, the following items hold: (i)𝑢_𝑛(𝑥)≥𝑢_𝑛(𝑥),𝑛=0,1,…,𝑁, (ii)𝑢_𝑛(𝑥)≥𝑢_𝑛(𝑥)≥𝑢_𝑛(𝑥),𝑛=0,1,…,𝑁, (iii)𝑢_𝑛+1(𝑥)−𝑢_𝑛+11(𝑥)≥√∫4𝜋𝑑^ℝ𝑒^{−(𝑥−𝑦)2/4𝑑}[𝑔(𝑢_𝑛(𝑦))−𝑔(𝑢_𝑛(𝑦))]𝑑𝑦,𝑛=0,1,…,𝑁−1.

Assume that 𝜙(𝑡) is a traveling wave front of (1.5) and satisfies (2.3)-(2.4). Then, the following result is clear by the item (iii) and the condition (g2).

Lemma 3.4. 𝜙(𝑡) is strictly monotone such that 𝜙^′(𝑡)>0 and 0<𝜙(𝑡)<1 for all 𝑡∈ℝ.

Lemma 3.5. Assume that 𝜙(𝑥+𝑐𝑛) is a bistable wave front of (1.5). Then, there exist positive numbers 𝛽,𝜎 and 𝛿₀<𝑚𝑖𝑛{𝑎,1−𝑎} such that for any 𝛿∈(0,𝛿₀] and 𝜉⁺,𝜉⁻∈ℝ𝑢_𝑛(𝑥)=min𝜙𝑥+𝑐𝑛+𝜉⁺−𝜎𝛿𝑒^−𝛽𝑛+𝛿𝑒^−𝛽𝑛,𝑢,1_𝑛(𝑥)=max𝜙𝑥+𝑐𝑛+𝜉⁻+𝜎𝛿𝑒^−𝛽𝑛−𝛿𝑒^−𝛽𝑛,0 (3.3)are the upper and lower solutions of (3.1) if u₀(𝑥)≥𝑢(𝑥)≥𝑢₀(𝑥),𝑥∈ℝ.

Proof. By Definition 3.2, it suffices to prove that 𝑢_𝑛+1(𝑥)≥𝑄𝑢_𝑛(𝑥),𝑢_𝑛+1(𝑥)≤𝑄𝑢_𝑛(𝑥),𝑛=0,1,…. (3.4) If 𝑢_𝑛+1(𝑥)=1 holds, then the result is clear by the assumptions (g1) and (g2).  If 𝑢_𝑛+1(𝑥)<1, by Definition 2.2, we only need to prove that 𝜙𝑥+𝑐(𝑛+1)+𝜉⁺−𝜎𝛿𝑒^{−𝛽(𝑛+1)}+𝛿𝑒^{−𝛽(𝑛+1)}−𝜙𝑥+𝑐(𝑛+1)+𝜉⁺−𝜎𝛿𝑒^−𝛽𝑛≥1√4𝜋𝑑^ℝ𝑒^{−(𝑥−𝑦)2/4𝑑}𝑔𝑢_𝑛𝑦−𝑔𝜙𝑦+𝑐𝑛+𝜉⁺−𝜎𝛿𝑒^−𝛽𝑛𝑑𝑦. (3.5) Define 𝜄=max_𝑢∈[0,1]𝑔^′(𝑢). Let 𝑀>0 be large enough, then (g1)–(g3) imply that there exists 𝜌^′∈(0,1) such that 0≤𝑔𝑢_𝑛𝑦−𝑔𝜙𝑦+𝑐𝑛+𝜉⁺−𝜎𝛿𝑒^−𝛽𝑛≤1−𝜌^{′−𝛽𝑛},1−𝜌^′+𝜄√4𝜋𝑑^{−𝑀−∞}𝑒^{−𝑦2/4𝑑}𝑑𝑦<1, (3.6)for all |𝑦+c𝑛+𝜉⁺−𝜎𝛿𝑒^−𝛽𝑛|>𝑀 and 𝛿∈(0,𝛿₀]. Thus, it is clear that 1√4𝜋𝑑^ℝ𝑒^{−(𝑥−𝑦)2/4𝑑}𝑔𝑢_𝑛𝑦−𝑔𝜙𝑦+𝑐𝑛+𝜉⁺−𝜎𝛿𝑒^−𝛽𝑛≤𝑑𝑦1−𝜌^′+𝜄√4𝜋𝑑^{−𝑀−∞}𝑒^{−𝑦2/4𝑑}𝑑𝑦𝛿𝑒^−𝛽𝑛, (3.7)for |𝑥+𝑐(𝑛+1)+𝜉⁺−𝜎𝛿𝑒^{−𝛽(𝑛+1)}|>2𝑀 or |𝑥+𝑐(𝑛+1)+𝜉⁺−𝜎𝛿𝑒^−𝛽𝑛|≥2𝑀 with 𝑀>0 large enough. Let 𝛽>0 be small enough such that 𝑒^−𝛽>1−𝜌^′+𝜄√4𝜋𝑑^{−𝑀−∞}𝑒^{−𝑦2/4𝑑}𝑑𝑦. (3.8)Then, the result is clear if |𝑥+𝑐(𝑛+1)+𝜉⁺−𝜎𝛿𝑒^{−𝛽(𝑛+1)}|>2𝑀 or |𝑥+𝑐(𝑛+1)+𝜉⁺−𝜎𝛿𝑒^−𝛽𝑛|≥2M hold.
If |𝑥+𝑐(𝑛+1)+𝜉⁺−𝜎𝛿𝑒^{−𝛽(𝑛+1)}|≤2𝑀 and |𝑥+𝑐(𝑛+1)+𝜉⁺−𝜎𝛿𝑒^−𝛽𝑛|≤2𝑀, define 𝜐=min_{𝑠∈[−2𝑀,2𝑀]}𝜙^′(𝑠)>0, which is well defined by Lemma 3.4. Then, (3.5) holds if 𝜐𝜎𝛿𝑒^−𝛽𝑛1−𝑒^−𝛽+𝛿𝑒^{−𝛽(𝑛+1)}≥𝜄𝛿𝑒^−𝛽𝑛. (3.9) Let 𝜎>0 be large enough, then the above inequality is clear. Thus, 𝑢_𝑛(𝑥) is an upper solution of (1.5) if 𝛽>0,(1/𝜎)>0 are small enough.
Similarly, we can prove that 𝑢_𝑛(𝑥) is a lower solution of (1.5) if 𝛽>0 is small enough while 𝜎>0 is large enough. The proof is complete.

Remark 3.6. We should note that for any fixed 𝛿₀<min{𝑎,1−𝑎}, 𝜎 and 𝛽 are uniform for all 𝛿∈(0,𝛿₀], which is very important in the following discussion.

Let 𝜒(𝑥)∈𝐶_[0,1] be a fixed function such that𝜒(𝑥)=0if𝑥≤0,𝜒(𝑥)=1if𝑥≥4,0<𝜒^′(𝑥)<1,𝜒^′′(𝑥)≤1if𝑥∈(0,4). (3.10)

Lemma 3.7. For any 𝛿∈(0,𝛿₀] with 2𝛿₀<min{𝑎,1−𝑎,𝜌} (𝜌 is defined by (g3)), there exist two positive numbers 𝜖=𝜖(𝛿) and 𝐶=𝐶(𝛿) such that for every 𝜉⁺,𝜉⁻∈ℝ, define the continuous functions 𝑢⁺_𝑛(𝑥)=min(1+𝛿)−1−(𝑎−2𝛿)𝑒^−𝜖𝑛𝜒−𝜖𝑥−𝜉⁺,𝑢+𝐶𝑛,1_−𝑛(𝑥)=max−𝛿+1−(1−𝑎−2𝛿)𝑒^−𝜖𝑛𝜒(𝜖(𝑥−𝜉⁻+𝐶𝑛)),0. (3.11)Then, 𝑢⁺_𝑛(𝑥) and 𝑢⁻_𝑛(𝑥) are the upper and lower solutions of (3.1) if 𝑢⁺₀(𝑥)≥𝑢(𝑥)≥𝑢⁻₀(𝑥).

Proof. By Definition 3.2, it suffices to prove that 𝑢⁺_𝑛+1(𝑥)≥𝑄𝑢⁺_𝑛(𝑥),𝑢⁻_𝑛+1(𝑥)≤𝑄𝑢⁻_𝑛(𝑥),𝑛=0,1,…. (3.12) If 𝜒(−𝜖(𝑥−𝜉⁺+𝐶𝑛))∈[0,1), then −𝜖(𝑥−𝜉⁺+𝐶𝑛)∈(−∞,4). Let 𝐶>0 be large enough such that 𝐶𝜖≥4, then it is clear that 𝜒(−𝜖(𝑥−𝜉⁺+𝐶(𝑛+1)))=0 such that 1=𝑢⁺_𝑛+1(𝑥)≥𝑄[𝑢⁺_𝑛](𝑥).
If 𝜒(−𝜖(𝑥−𝜉⁺+𝐶𝑛))=1, then 𝑢⁺_𝑛(𝑥)=𝛿+(𝑎−2𝛿)𝑒^−𝜖𝑛≤𝑎−𝛿. For any given 𝛿>0, there exists 𝜖^′>0 such that 𝑔(𝑢)≤(1−𝜖^′)𝑢,𝑢∈[0,𝑎−(𝛿/2)]. Moreover, there exists 𝑀>0 such that (1−𝜖^′)+(𝜄/√4𝜋𝑑)∫^∞_𝑀𝑒^{−𝑦2/4𝑑}𝑑𝑦<1. Similar to the discussion of (3.5), 𝑢⁺_𝑛+1(𝑥)≥𝑄[𝑢⁺_𝑛](𝑥) can be verified if 𝜖=𝜖(𝛿)>0 is small enough and 𝐶=𝐶(𝛿)>0 is large enough.
In a similar way, we can prove that 𝑢⁻_𝑛+1(𝑥)≤𝑄[𝑢⁻_𝑛](𝑥),𝑥∈ℝ,𝑛≥0. The proof is complete.

4. Asymptotic Stability of Traveling Wave Fronts

Lemma 3.5 implies a rough result on the long-time behavior between a traveling wave front and the solution of the initial value problem (3.1) if the initial value satisfies proper assumptions, which indicates that the solution can be controlled by two different phase shifts of the same traveling wave front. In this section, we will prove these phase shifts will convergence to the same limit, and this will affirm the stability of a bistable wave front. We first show the following result, which means that the phase shifts in Lemma 3.5 will be smaller if 𝑛 is larger.

Lemma 4.1. Assume that there exist 𝑛^∗≥0,𝜉∈ℝ,𝛿∈(0,𝛿₀/2] and ℎ>0 such that max𝜙𝑥+𝑐𝑛^∗+𝜉−𝛿,0≤𝑢_𝑛∗(𝑥)≤min𝜙𝑥+𝑐𝑛^∗+𝜉+ℎ+𝛿,1,𝑥∈ℝ. (4.1) Then, for every 𝑛≥𝑛^∗+1, there exist 𝜉̂(𝑛),𝛿(𝑛),ℎ(𝑛) satisfying 𝜉̂(𝑛)∈𝜉−𝜎₁𝛿,𝜉+ℎ+𝜎₁𝛿,𝛿(𝑛)≤𝑒^{−𝛽(𝑛−𝑛∗−1)}𝛿+𝜀^∗{,ℎ(𝑛)≤minℎ,1}ℎ−𝜎₁𝜀^∗min{ℎ,1}+2𝜎₁𝛿, (4.2)such that (4.1) remains true if we replace (𝑛^∗,𝜉,𝛿,ℎ) by (𝑛,𝜉̂(𝑛),𝛿(𝑛),ℎ(𝑛)), herein 𝛽>0,𝜎=𝜎₁ such that Lemma 3.5 holds.

Proof. Without loss of generality, we assume that 𝜉=𝑛^∗=0, and denote 𝑢₀(𝑥)=𝑢(𝑥). By Theorem 3.3, we may obtain max𝜙𝑥+𝑐𝑛−𝜎₁𝛿1−𝑒^−𝛽𝑛−𝛿𝑒^−𝛽𝑛,0≤𝑢_𝑛(𝑥)≤min𝜙𝑥+𝑐𝑛+𝜎₁𝛿1−𝑒^−𝛽𝑛+ℎ+𝛿𝑒^−𝛽𝑛,1,𝑥∈ℝ,𝑛≥0. (4.3) Moreover, it is clear that there exist 𝜗∈(0,1/2) and an interval 𝐼⊂ℝ such that 𝑢(𝑦),𝜙(𝑦)∈[𝜗,1−𝜗] for all 𝑦∈𝐼 (since limsup_{𝑥→−∞}𝑢(𝑥)≤𝛿₀/2, limsup_𝑥→∞𝑢(𝑥)≥1−(𝛿₀/2) and (2.4) hold). Without loss of generality, we further assume that 𝐼=[0,1]. Define constants ℎ=min{ℎ,1},𝜀₁=12min_[]𝜉∈0,2𝜙^′(𝜉)>0. (4.4) Then, ¹₀𝜙𝑦+ℎ−𝜙𝑦𝑑𝑦=¹₀𝜙𝑦+ℎ−𝑢𝑦+𝑢𝑦−𝜙𝑦𝑑𝑦≥2𝜀₁ℎ, (4.5)which implies that at least one of the following is true: (i)¹₀𝑢𝑦−𝜙𝑦𝑑𝑦≥𝜀₁ℎ;(ii)¹₀𝜙𝑦+ℎ−𝑢𝑦𝑑𝑦≥𝜀₁ℎ. (4.6)   Let 𝑀>0 be large enough such that 𝜙^′(𝑡)<12𝜎₁,|𝑡|>𝑀. (4.7)If the case (i) holds, then |𝑥|≤𝑀+1+|𝑐| with 𝑀>0 large enough indicates that 𝑢₁(𝑥)−max𝜙𝑥+𝜉₁−𝛿𝑒^−𝛽≥𝜂,0√𝑒4𝜋d^{(𝑀+2+|𝑐|)2/−4𝑑10}𝑢𝑦𝜙𝑦>𝜂−max−𝛿,0𝑑𝑦√𝑒4𝜋𝑑^{(𝑀+2+|𝑐|)2/−4𝑑10}𝑢𝑦−𝜙𝑦𝑑𝑦≥𝜂𝜀₁ℎ√𝑒4𝜋𝑑^{(𝑀+2+|𝑐|)2/−4𝑑}, (4.8) where 𝜉₁=𝑐−𝜎₁𝛿(1−𝑒^−𝛽) and 𝜂=min_{𝑢∈[𝜗,1−𝜗]}𝑔^′(𝑢)>0. Set 𝜀^∗𝛿=min⁰2,12𝜎₁,min_{|𝑥|≤𝑀+2+|𝑐|}𝜂𝜀₁2𝜎₁𝜙^′(𝑥)√𝑒4𝜋𝑑^{(𝑀+2+|𝑐|)2/−4𝑑}. (4.9)Then, there exists 𝜃∈[𝑥+𝜉₁,𝑥+𝜉₁+2𝜎₁𝜀^∗ℎ] such that 𝜙𝑥+𝜉₁+2𝜎₁𝜀^∗ℎ−𝜙𝑥+𝜉₁=2𝜙^′(𝜃)𝜎₁𝜀^∗ℎ≤𝜂𝜀₁ℎ√𝑒4𝜋𝑑^{(𝑀+2+|𝑐|)2/−4𝑑}, (4.10)for all |𝑥|≤𝑀+1+|𝑐|. Thus, (4.8) indicates that 𝑢₁(𝑥)≥max𝜙𝑥+𝜉₁+2𝜎₁𝜀^∗ℎ−𝛿𝑒^−𝛽,0,|𝑥|≤𝑀+1+|𝑐|. (4.11)If the case (i) is false while the case (ii) is true, then we can also establish a similar result and the discussion is omitted.
If |𝑥|≥𝑀+1+|𝑐|, then the definition of 𝑀 implies that 𝜙𝑥+𝜉₁≥𝜙𝑥+𝜉₁+2𝜎₁𝜀^∗ℎ−𝜀^∗ℎ. (4.12)Combining this with (3.10), it is clear that 𝑢₁(𝑥)≥max𝜙𝑥+𝜉₁+2𝜎₁𝜀^∗ℎ−𝛿𝑒^−𝛽+𝜀^∗ℎ,0,𝑥∈ℝ. (4.13) Since 𝛿𝑒^−𝛽+𝜀^∗ℎ≤𝛿₀ holds, then Theorem 3.3 and Lemma 3.4 indicate that 𝑢_𝑛+1(𝑥)≥max𝜙𝑥+𝑐𝑛+𝜉₁+2𝜎₁𝜀^∗ℎ−𝜎₁𝛿𝑒^−𝛽+𝜀^∗ℎ1−𝑒^−𝛽𝑛−𝛿𝑒^−𝛽+𝜀^∗ℎ𝑒^−𝛽𝑛𝜙,0≥max𝑥+𝑐+𝑐𝑛+𝜎₁𝜀^∗ℎ−𝜎₁𝛿−𝛿+𝜀^∗ℎ𝑒^−𝛽𝑛,0. (4.14)Therefore, the left side of (4.1) with (𝑛,𝜉̂(𝑛),𝛿(𝑛),ℎ(𝑛)) is true.
In a similar way, we can verify the right side of (4.1) if we replace (𝑛^∗,𝜉,𝛿,ℎ) by (𝑛,𝜉̂(𝑛),𝛿(𝑛),ℎ(𝑛)). The proof is complete.

The following theorem is the main result of this section, which further implies the asymptotic stability of a bistable wave front of (1.5).

Theorem 4.2. Assume that 𝜙(𝑥+𝑐𝑛) is a monotone solution of (2.3)-(2.4) and 𝑢(𝑥)∈𝐶_[0,1] satisfies liminf_𝑥→∞𝑢(𝑥)>𝑎,limsup_{𝑥→−∞}𝑢(𝑥)<𝑎. (4.15)Let 𝑢_𝑛(𝑥) be defined by (3.1). Then, there exist 𝜉=𝜉(𝑢(𝑥))∈ℝ,𝐾=𝐾(𝑢(𝑥))>0 and 𝜅>0 (𝜅 is independent of the initial value) such that sup_𝑥∈ℝ𝑢_𝑛(𝑥)−𝜙(𝑥+𝑐𝑛+𝜉)≤𝐾𝑒^−𝜅𝑛,𝑛≥0. (4.16)

Proof. By what we have done (Lemmas 3.5 and 4.1), the key of proof is the precise estimate of phase shifts mentioned above.
For any given 𝛿>0, Lemmas 3.7 and 4.1 imply that 𝜙𝐻max𝑥+𝑐𝑁−2−𝛿,0≤𝑢_𝑁(𝑥)𝜙𝐻≤min𝑥+𝑐𝑁+2+𝛿,1,𝑥∈ℝ, (4.17) for 𝑁>0 and 𝐻>0 large enough. Define constants 𝛿^∗𝛿∶=min⁰2,𝜀^∗4,𝜅^∗∶=𝜎₁𝜀^∗−2𝜎₁𝛿^∗≥𝜎₁𝜀^∗2>0. (4.18)We further choose 𝑛^∗≥2 such that 𝑒^{−𝛽(𝑛∗−1)}𝜀1+^∗𝛿^∗≤1−𝜅^∗. (4.19)In (4.17), let 𝛿=𝛿^∗, and denote the corresponding constants 𝐻 and 𝑁 by ℎ₀ and 𝑁₀.
Claim 1. There exists a natural number 𝑁₁>𝑁₀ such that (4.1) holds for 𝑛=𝑁₁,𝛿=𝛿^∗,ℎ<1 and some 𝜉∈ℝ.
In fact, letting 𝑛=𝑁₀,𝜉=−ℎ₀/2,ℎ=ℎ₀, and 𝛿=𝛿^∗ in (4.1), then Lemma 4.1 implies that (4.1) holds with 𝑛=𝑁₀+𝑛^∗, some 𝜉∈[(−ℎ₀/2)−𝜎₁𝛿^∗,ℎ₀/2+𝜎₁𝛿^∗],𝛿=𝛿^∗, and ℎ=ℎ₀−𝜅^∗, since the definitions of 𝑛^∗ and 𝜅^∗ imply that 𝑒^{−𝛽(𝑛∗−1)}𝛿^∗+𝜀^∗≤𝛿^∗,ℎ₀−𝜎₁𝜀^∗+2𝜎₁𝛿^∗≤ℎ₀−𝜅^∗. (4.20)We now repeat the above process 𝑁 times (see (4.4), so we can repeat the process for any ℎ≥1) such that ℎ₀−(𝑁−1)𝜅^∗≥1,ℎ₀−𝑁𝜅^∗<1. (4.21)Then, (4.1) holds for 𝑛=𝑁₀+𝑁𝑛^∗,𝛿=𝛿^∗,ℎ=ℎ₀−𝑁𝜅^∗<1, and some 𝜉=𝜉₀∈ℝ. Therefore, the claim is true.Claim 2. For every 𝑘≥0, (4.1) holds for some 𝜉_𝑘∈ℝ and 𝑛=𝑁_𝑘∶=𝑁₁+𝑘𝑛^∗,𝛿=𝛿^𝑘∶=1−𝜅^∗𝑘𝛿^∗,ℎ=ℎ^𝑘∶=1−𝜅^∗𝑘. (4.22)
In fact, the result with 𝑘=0 is clear by what we have done. Assume that the claim holds for some 𝑘≥0, and we now prove the claim for 𝑘+1. By applying Lemma 4.1, (4.1) holds with (𝑛^∗,𝜉,𝛿,ℎ) replaced by (𝑁_𝑘+1,𝜉̂,𝛿,ℎ), herein 𝜉̂∈𝜉_𝑘−𝜎₁𝛿^𝑘,𝜉_𝑘+𝜎₁𝛿^𝑘,𝛿≤𝑒^{−𝛽(𝑛∗−1)}𝛿^𝑘+𝜀^∗ℎ^𝑘=1−𝜅^∗𝑘𝛿^∗𝑒^{−𝛽(𝑛∗−1)}𝜀1+^∗𝛿^∗≤1−𝜅^∗𝑘+1𝛿^∗,ℎ≤ℎ^𝑘−𝜎₁𝜀^∗ℎ^𝑘+2𝜎₁𝛿^𝑘=1−𝜅^∗𝑘1−𝜎₁𝜀^∗+2𝜎₁𝛿^∗=1−𝜅^∗𝑘+1, (4.23)by the definitions of 𝛿^∗,𝜅^∗ and 𝑛^∗. Thus, (4.1) holds for 𝑘+1. The proof of Claim 2 is completed by the mathematical induction.
By what we have done, (4.1) holds if (𝑛^∗,𝜉,𝛿,ℎ) is replaced by (𝑁_𝑘,𝜉^𝑘,𝛿^𝑘,ℎ^𝑘) for all 𝑘=0,1,…. The comparison principle further indicates that (4.1) holds for all 𝑛≥𝑁_𝑘,𝛿=𝛿_𝑛,ℎ=ℎ^𝑛+2𝜎₁𝛿^𝑘,𝜉=𝜉^𝑘−𝜎₁𝛿^𝑘 with 𝑘=0,1,….
Define 𝛿(𝑛)=𝛿^𝑘,𝜉(𝑛)=𝜉^𝑘−𝜎₁𝛿^𝑘,andℎ(𝑛)=ℎ^𝑘+2𝜎₁𝛿^𝑘 for all 𝑁_𝑘≤𝑛<𝑁_𝑘+1 and 𝑘=0,1,…. Then, 𝜙(𝑥+𝑐𝑛+𝜉(𝑛))−𝛿(𝑛)≤𝑢_𝑛(𝑥)≤𝜙(𝑥+𝑐𝑛+𝜉(𝑛)+ℎ(𝑛))+𝛿(𝑛), (4.24)for all 𝑥∈ℝ,𝑛≥𝑁₁. The definitions of 𝛿(𝑛) and ℎ(𝑛) also imply that 𝛿(𝑛)=𝛿^𝑛=1−𝜅^∗𝑛𝛿^∗≤𝛿^∗𝑒^{(((𝑛−𝑁1)/𝑛∗)−1)ln(1−𝜅∗)},𝑛≥𝑁₁,ℎ(𝑛)=ℎ^𝑛+2𝜎₁𝛿^𝑘≤1+2𝜎₁𝛿^∗𝑒^{(((𝑛−𝑁1)/𝑛∗)−1)ln(1−𝜅∗)},𝑛≥𝑁₁, (4.25)in which 𝑘 is the largest integer no bigger than (𝑛−𝑁₁)/𝑛^∗.
Furthermore, 𝑛≥𝑛^∗≥𝑁₁ and 𝜉(𝑛)∈[𝜉(𝑛^∗)−𝜎₁𝛿(𝑛^∗),𝜉(𝑛^∗)+ℎ(𝑛^∗)+𝜎₁𝛿(𝑛^∗)] imply that 𝜉(𝑛)−𝜉𝑛^∗≤ℎ(𝑛)+2𝜎₁𝛿𝑛^∗, (4.26)which indicates that 𝜉(𝑛) is bounded and lim_𝑛→∞𝜉(𝑛) exists (since ℎ(𝑛)→0,𝛿(𝑛)→0 if 𝑛→∞). Let 𝜅=−(ln(1−𝜅^∗))/𝑛^∗ be fixed and 𝐾>0 be large enough, then the result is clear. This completes the proof.

By Lemma 3.5 and Theorem 4.2, the following result is evident, which indicates that the wave speed of the bistable wave fronts of (1.5) is unique and the wave profile is also unique in the sense of phase shift.

Theorem 4.3. Assume that 𝜙(𝑥+𝑐𝑛) is a bistable wave front of (1.5). If 𝜙₁(𝑥+𝑐₁𝑛) is another traveling wave solution of (1.5) satisfying (2.3)-(2.4), then 𝑐=𝑐₁ holds and there exists a constant ℎ∈ℝ such that 𝜙(𝜉)=𝜙₁(𝜉+ℎ),𝜉∈ℝ.

Due to our estimates in Lemma 3.5, the proof is similar to that of Smith and Zhao [18, Theorem 3.4], and we omit it here.

5. Existence of Traveling Wave Fronts

We first list the main result of this section as follows.

Theorem 5.1. Assume that (𝑔1)–(𝑔3) hold. Then (1.5) admits a traveling wave front. Namely, there exist 𝑐∈ℝ and 𝜙∈𝐶_[0,1] such that (2.3)-(2.4) hold.

We will split the proof of Theorem 5.1 into several lemmas, of which the motivation is the stability of traveling wave front. Namely, a bistable wave front can be approached by the solution of (1.5) if its initial value satisfies proper assumptions. For this purpose, we first consider the initial value problem𝑤_𝑛+1(𝑥)=𝑄𝑤_𝑛(𝑥),𝑤₀(𝑥)=𝜒(𝑥),𝑛=0,1,…, (5.1) herein 𝜒(𝑥) is same to that in Section 3.

Lemma 5.2. For any given 𝛼∈(0,1) and 𝑛=0,1,…, there exists 𝑧(𝛼,𝑛) such that 𝑤_𝑛(𝑧(𝛼,𝑛))=𝛼.

Proof. The proof is essentially based on the implicit function theorem. For the case of 𝑛=0, the result is clear by the strict monotonicity of 𝜒(𝑥),𝑥∈(0,4). We claim that 𝑤_𝑛(𝑥),𝑛≥1 is strictly monotone in 𝑥∈ℝ. For any 𝑥>𝑦, it is clear that 𝑤₁(𝑥)−𝑤₁𝑦=1√4𝜋𝑑^ℝ𝑒^{−𝑧2/4𝑑}𝑔𝜒(𝑥+𝑧)−𝑔𝜒𝑦+𝑧𝑑𝑧>0, (5.2)by the definition of 𝜒 and the assumption (g2). Then, 𝑤₁(𝑥) is strictly monotone in 𝑥∈ℝ. Assume that 𝑤_𝑛(𝑥) is strictly monotone for some 𝑛≥1, then we can prove that 𝑤_𝑛+1(𝑥) is also strictly monotone in 𝑥∈ℝ, and the proof is similar to the discussion of 𝑤₁. By mathematical induction, 𝑤_𝑛(𝑥),𝑛≥1 is strictly monotone in 𝑥∈ℝ. Moreover, it is also clear that lim_{𝑥→−∞}𝑤_𝑛(𝑥)=0,lim_𝑥→∞𝑤_𝑛(𝑥)=1 for any fixed 𝑛≥0. Combining the implicit function theorem with the strict monotonicity of 𝑤_𝑛(𝑥), then the lemma is obvious. The proof is complete.

Lemma 5.3. Let 𝐻(𝑥) be the Heaviside function. Then, the following results hold.
(i) Assume that 𝑣¹_𝑛 and 𝑤¹_𝑛 are defined by the following linear equations: 𝑣¹_𝑛+1(𝑥)=𝑔^′(𝑎)√4𝜋𝑑^ℝ𝑒^{−(𝑥−y)2/4𝑑}𝑣¹_𝑛𝑦𝑑𝑦,𝑛=0,1,…,𝑣¹₀(𝑥)(𝑥),𝑤=𝐻_1𝑛+1(𝑥)=𝑔^′(𝑎)√4𝜋𝑑^ℝ𝑒^{−(𝑥−𝑦)2/4𝑑}𝑤¹_𝑛𝑦𝑑𝑦,𝑛=0,1,…,𝑤¹₀(𝑥)=−1+2𝐻(𝑥). (5.3)Then, there exist 𝑥₁∈ℝ,𝑛₁≥1 such that 𝑣¹_𝑛1𝑥₁≥3,𝑤¹_𝑛1𝑥₁≤−3. (5.4)(ii) There exists 𝛿₁>0 small enough such that the solutions to 𝑣²_𝑛+1(𝑥)=𝑄𝑣²_𝑛(𝑥),𝑛=0,1,…,𝑣²₀(𝑥)=𝑎+𝛿₁𝐻(𝑥),𝑤²_𝑛+1(𝑥)=𝑄𝑤²_𝑛(𝑥),𝑛=0,1,…,𝑤²₀(𝑥)=𝑎+𝛿₁[𝐻(𝑥)−𝐻(−𝑥)] (5.5) satisfy 𝑣²_𝑛1𝑥₁≥𝑎+2𝛿₁,𝑤²_𝑛1𝑥₁≤𝑎−2𝛿₁. (5.6)(iii) There exists ℎ₁>0 such that the solutions to 𝑣³_𝑛+1(𝑥)=𝑄𝑣³_𝑛(𝑥),𝑣³₀(𝑥)=𝑎+𝛿₁𝐻(𝑥)−𝑎𝐻−ℎ₁,𝑤−𝑥_3𝑛+1(𝑥)=𝑄𝑤³_𝑛(𝑥),𝑤³₀(𝑥)=𝑎+𝛿₁[𝐻(𝑥)−𝐻(−𝑥)]+1−𝑎−𝛿₁𝐻𝑥−ℎ₁ (5.7) satisfy 𝑣³_𝑛1𝑥₁≥𝑎+𝛿₁,𝑤³_𝑛1𝑥₁≤𝑎−𝛿₁. (5.8)(iv) Let 𝑢_𝑛(𝑥) be the solution of (1.5) with uniformly continuous and nondecreasing initial data 𝑢(𝑥) satisfying 0≤𝑢(𝑥)≤1 and for some finite 𝜉₋(0) and 𝜉₊(0), 𝑢𝜉₋(0)≤𝑎−𝛿₁,𝑢𝜉₊(0)≥𝑎+𝛿₁. (5.9) Then, for each 𝑛≥1, there exist 𝜉₋(𝑛),𝜉₊(𝑛) such that 𝑢_𝑛𝜉₋(𝑛)=𝑎−𝛿₁,𝑢_𝑛𝜉₊(𝑛)=𝑎+𝛿₁,𝜉₊(𝑛)−𝜉₋(𝑛)≤max𝜉₊(0)−𝜉₋(0)+8𝜖⁻¹𝛿₁+2𝐶𝛿₁𝑛₁,2ℎ₁, (5.10) where 𝜖⁻¹(𝛿₁),𝐶(𝛿₁) are as in Lemma 3.7.

Proof. For each fixed 𝑛>0, it is clear that 𝑣^𝑖_𝑛,𝑤^𝑖_𝑛,𝑖=1,2,3 are uniformly continuous, and the comparison principle remains true even if the initial value is a discontinuous function. More precisely, if 0≤𝑢(𝑦)≤𝑣(𝑦)≤1,𝑦∈ℝ such that 𝑄[𝑢](𝑥),𝑄[𝑣](𝑥),𝑥∈ℝ are well defined, then 0≤𝑄[𝑢](𝑥)≤𝑄[𝑣](𝑥)≤1,𝑥∈ℝ. We now prove the items (i)–(iv) one by one.
Proof of (i). For each fixed 𝑛>0, we know that 𝑣_𝑛(𝑥)∈𝐶(ℝ,ℝ) and lim_{𝑥→−∞}𝑣¹_𝑛(𝑥)=0,lim_𝑥→∞𝑣¹_𝑛(𝑥)=𝑔^′(𝑎)^𝑛. (5.11)Let 𝑛₁>0 such that [𝑔^′(𝑎)]^𝑛1≥9 and [𝑔^′(𝑎)]^𝑛1−1<9. Then, there exists 𝑥₁∈ℝ such that 𝑣¹_𝑛1(𝑥₁)=3. Since 𝑤¹_𝑛=−[𝑔^′(𝑎)]^𝑛+2𝑣¹_𝑛, then 𝑤¹_𝑛1(𝑥₁)≤−3.Proof of (ii). Let 𝐾>0 be a large constant and 𝛿₁>0 small enough such that 𝐾𝛿₁<1,𝛿₁[𝑔^′(𝑎)]^2𝑛1+2<1 and 0≤𝑎+𝛿₁𝑣¹_𝑛(𝑥)−𝐾𝛿²₁𝑔^′(𝑎)^2𝑛,𝑎+𝛿₁𝑣¹_𝑛(𝑥)≤1,∀𝑥∈ℝ,0≤𝑛≤𝑛₁, (5.12) which is admissible, since 𝑣¹_𝑛(𝑥) is bounded for all𝑥∈ℝ,0≤𝑛≤𝑛₁. Define 𝑣²_𝑛(𝑥)=𝑎+𝛿₁𝑣¹_𝑛(𝑥)−𝐾𝛿²₁𝑔^′(𝑎)^2𝑛,𝑣²_𝑛(𝑥)=1,𝑥∈ℝ. (5.13) We now prove that 𝑣²_𝑛,𝑣²_𝑛 are a pair of upper and lower solutions if 𝑥∈ℝ,0≤𝑛≤𝑛₁. The initial value conditions and 𝑣²_𝑛+1≥𝑄[𝑣²_𝑛] are clear, so we need to prove that 𝑣²_𝑛+1(𝑥)≤𝑄𝑣²_𝑛(𝑥),0≤𝑛≤𝑛₁,𝑥∈ℝ. (5.14)Namely, we will verify that 𝑎+𝛿₁𝑣¹_𝑛+1(𝑥)−𝐾𝛿²₁𝑔^′(𝑎)^2𝑛+2≤1√4𝜋𝑑^ℝ𝑒^{−(𝑥−𝑦)2/4𝑑}𝑔𝑎+𝛿₁𝑣¹_𝑛𝑦−𝐾𝛿²₁𝑔^′(𝑎)^2𝑛𝑑𝑦. (5.15)By the Taylor's formula, there exists 𝑠=𝑠(𝑔(𝑢))≥0 for 𝑢∈[0,1] (see (g2)) such that 𝑔𝑎+𝛿₁𝑣¹_𝑛𝑦−𝐾𝛿²₁𝑔^′(𝑎)^2𝑛≥𝑔(𝑎)+𝑔^′(𝑎)𝛿₁𝑣¹_𝑛𝑦−𝐾𝛿²₁𝑔^′(𝑎)^2𝑛−𝑠𝛿₁𝑣¹_𝑛𝑦−𝐾𝛿²₁𝑔^′(𝑎)^2𝑛2. (5.16)Thus, the definition of 𝑣¹_𝑛(𝑥) implies that we only need to prove −𝑔^′(𝑎)−1𝐾𝛿²₁𝑔^′(𝑎)^2𝑛+1≤−𝑠√4𝜋𝑑^ℝ𝑒^{−(𝑥−𝑦)2/4𝑑}𝛿₁𝑣¹_𝑛𝑦−𝐾𝛿²₁𝑔^′(𝑎)^2𝑛2𝑑𝑦, (5.17)which is true if 𝑔^′(𝑎)−1𝐾𝛿²₁𝑔^′(𝑎)^2𝑛+1≥2𝑠𝛿₁𝑔^′(𝑎)^𝑛2,𝑔^′(𝑎)−1𝐾𝛿²₁𝑔^′(𝑎)^2𝑛+1≥2𝑠𝐾𝛿²₁𝑔^′(𝑎)^2𝑛2. (5.18)Note that 𝛿₁>0 is small enough and 𝐾>0 is large enough, then the result is clear. This completes the proof of 𝑣²_𝑛1(𝑥₁)≥𝑎+2𝛿₁.
In a similar way, we can prove that 𝑤²_𝑛1(𝑥₁)≤𝑎−2𝛿₁.Proof of (iii). Note that lim_𝑁→∞1/√4𝜋𝑑∫^𝑁_−𝑁𝑒^{−𝑦2/4𝑑}𝑑𝑦=1 and 𝑎−2𝛿₁<𝑎−𝛿₁,𝑎+2𝛿₁>𝑎+𝛿₁, then the result is clear by letting ℎ₁>0 large enough.Proof of (iv). By the definition of 𝑢(𝑥), it is clear that 𝑢_𝑛(𝑥) are strictly monotone in 𝑥∈ℝ for every 𝑛≥1. Thus, the existence and uniqueness of 𝜉₋(𝑛),𝜉₊(𝑛) are obvious. By Lemma 3.7, there exist 𝜖(𝛿₁) and 𝐶(𝛿₁) such that 𝜉₊(𝑛)−𝜉₋(𝑛)≤𝜉₊(0)−𝜉₋(0)+8𝜖⁻¹𝛿₁+2𝐶𝛿₁𝑛. (5.19) In particular, (5.19) is true if 𝑛=𝑛₁. We now prove that 𝜉₊𝑛+𝑛₁−𝜉₋𝑛+𝑛₁≤max𝜉₊𝑛₁−𝜉₋𝑛₁,2ℎ₁forany𝑛≥0. (5.20)By translation and symmetry, we can assume that for any given 𝑛≥0, 𝜉₋(𝑛)<0<𝜉₊(𝑛),𝜉₊(𝑛)≥−𝜉₋(𝑛). (5.21)
Set ℎ₊=max{𝜉₊(𝑛₁),ℎ₁}, then 𝑢_𝑛(𝑥+ℎ₊)≥𝑣³₀(𝑥) for all 𝑥∈ℝ. By the comparison principle, 𝑢_𝑛+𝑛1(𝑥+ℎ₊)≥𝑣³_𝑛1(𝑥). Thus, 𝑢_𝑛+𝑛1(𝑥₁+ℎ₊)≥𝑎+𝛿₁, which implies that 𝜉₊(𝑛+𝑛₁)≤𝑥₁+ℎ₊.
Set ℎ₋=max{𝜉₊(𝑛₁)−𝜉₋(𝑛₁),ℎ₁}. Similar to the above discussion, we can verify that 𝜉₋(𝑛+𝑛₁)≥−ℎ₋+𝜉₊(𝑛₁)+𝑥₁, which further implies that 𝜉₊𝑛+𝑛₁−𝜉₋𝑛+𝑛₁≤ℎ₊+ℎ₋−𝜉₋𝑛₁≤max𝜉₊𝑛₁−𝜉₋𝑛₁,2ℎ₁≤max𝜉₊(0)−𝜉₋(0)+8𝜖⁻¹𝛿₁+2𝐶𝛿₁𝑛₁,2ℎ₁. (5.22) The proof is complete.

Lemma 5.4. Assume that 𝑧(𝛼,𝑛) is defined by Lemma 5.2. Then, there exist a small positive constant 𝛿₁ and a large positive constant ℎ₂≥1 such that 𝑧𝑎+𝛿₁,𝑛−𝑧𝑎−𝛿₁,𝑛≤ℎ₂,∀𝑛≥0. (5.23)Moreover, for any 𝛿∈(0,𝛿₁/2], there exists 𝑚₁(𝛿)>0 such that 𝑧(1−𝛿,𝑛)−𝑧(𝛿,𝑛)≤𝑚₁(𝛿),𝑛≥0. (5.24)

Proof. The former is clear by Lemma 5.3. We now prove the latter of the lemma. By Lemma 3.7, there exist 𝜖=𝜖(𝛿),𝐶=𝐶(𝛿) such that 𝑧(1−2𝛿,𝑛)≤𝑧𝑎+𝛿,𝑛₁+4𝜖⁻¹+𝐶𝑛−𝑛₁forany𝑛−𝑛₁>𝜖⁻¹||,𝑧()ln𝛿2𝛿,𝑛≥𝑧𝑎−𝛿,𝑛₁−4𝜖⁻¹−𝐶𝑛−𝑛₁forany𝑛−𝑛₁≥𝜖⁻¹|ln𝛿|. (5.25)Thus, 𝑛>𝜖⁻¹|ln𝛿|=∶Δ implies that 𝑧(1−2𝛿,𝑛)−𝑧(2𝛿,𝑛)≤𝑧(𝑎+𝛿,𝑛−⌊Δ⌋+1)−𝑧(𝑎−𝛿,𝑛−⌊Δ⌋+1)+8𝜖⁻¹+2𝐶(⌊Δ⌋+1), (5.26)in which ⌊𝑥⌋ denotes the maximal integer less than 𝑥+1. Note that 𝑧𝑎+𝛿₁,𝑛−⌊Δ⌋+1−𝑧𝑎−𝛿₁,𝑛−⌊Δ⌋+1≤ℎ₂,∀𝑛−⌊Δ⌋+1≥0. (5.27)Then, the result is clear, and the proof is complete.

Remark 5.5. Lemma 5.4 gives the estimate of variations of 𝑧(𝑎,𝑛), which will ensure that the solution of (5.1) converges to a nontrivial traveling wave solution in the following discussion.

Lemma 5.6. For every 𝑀>0, there exists a constant 𝜂(𝑀)>0 such that 𝑑𝑣_𝑛+1(𝑡)𝑑𝑡_{𝑡=𝑥+𝑧(𝑎,𝑛)}>𝜂(𝑀)>0,forany𝑛≥1,𝑥∈[−𝑀,𝑀]. (5.28)

Proof. By the smoothness of the Gaussian, the existence of (𝑑𝑣_𝑛+1(𝑡))/𝑑𝑡,𝑡∈ℝ is clear. Let 𝜂(𝑥)=(1/√4𝜋𝑑)𝑒^{−𝑥2/4𝑑}. By the definition of ℎ₂, it is clear that 𝑣_𝑛(𝜉+1+𝑧(𝑎,𝑛))−𝑣_𝑛((≥𝛿𝜉+𝑧𝑎,𝑛))¹ℎ₂, (5.29) for any 𝜉∈[−ℎ₂,ℎ₂−1] and 𝑛≥0, which further implies that min_−𝑥∈𝑀^,𝑀𝑑𝑣_𝑛+1(𝑡)𝑑𝑡_{𝑡=𝑥+𝑧(𝑎,𝑛)}≥min_−𝑥∈𝑀^−ℎ₂^,𝑀^+ℎ₂𝜂𝑀+ℎ₂𝛿₁ℎ₂, (5.30)for any 𝑛≥0. Since |𝑧(𝑎,𝑛+1)−𝑧(𝑎,𝑛)|≤ℎ₂+8𝜖⁻¹(𝛿₁)+2𝐶(𝛿₁), taking 𝑀=𝑀+ℎ₂+8𝜖⁻¹(𝛿₁)+2𝐶(𝛿₁) and 𝜂(𝑀)=𝜂(𝑀+ℎ₂), then the result is clear. The proof is complete.

Lemma 5.7. Let 𝑀₃=𝑚₁(𝛿₀)>0,𝜎₂>0 be large enough and 𝛽>0 small enough. Then, for every 𝛿∈(0,𝛿₀] and 𝜉∈ℝ, the continuous functions 𝑊⁺_𝑛(𝑥)∶=min𝑣_𝑛+1𝑥+𝜉+𝜎₂𝛿1−𝑒^−𝛽𝑛+𝛿𝑒^−𝛽𝑛,𝑊,1_−𝑛(𝑥)∶=max𝑣_𝑛+1𝑥+𝜉−𝜎₂𝛿1−𝑒^−𝛽𝑛−𝛿𝑒^−𝛽𝑛,0 (5.31)are the upper and lower solutions of (1.5) if 𝑊⁺₀(𝑥)≥𝑣₀(𝑥)≥𝑊⁻₀(𝑥).

By the estimate on the derivative of 𝑣_𝑛 formulated by Lemma 5.6, namely, the strict monotonicity of 𝑣_𝑛, the proof is similar to that of Lemma 3.5, so we omit it here. In particular, these parameters have a property similar to Remark 3.6.

Lemma 5.8. There exists a nondecreasing function 𝜙(𝜉) such that 𝑣_𝑛(𝜉+𝑧(𝑎,𝑛))⟶𝜙(𝜉),𝑛⟶∞,∀𝜉∈ℝ, (5.32)and lim_{𝜉→−∞}𝜙(𝜉)=0,lim_𝜉→∞𝜙(𝜉)=1.

Proof. The existence of 𝜙(𝜉) is clear by the boundedness, uniform continuity, and monotonicity of the sequences, and 𝜙(0)=𝑎 is also obvious. Note that 𝜙(𝜉) is monotone, then lim_𝜉→±∞𝜙(𝜉)=𝜙_± exist. Lemmas 5.4 and 5.6 further imply that 𝜙₋<𝑎 and 𝜙₊>𝑎. Then, 𝜙₋=0,𝜙₊=1 by (g1). The proof is complete.

Lemma 5.9. 𝜙 is the profile of a traveling wave front.

Proof. Let 𝜙_𝑛(𝑥) be the solution of (1.5) with the initial value 𝜙(𝑥), and we will prove that 𝑣_𝑚+𝑛(𝑥+𝑧(𝑎,𝑚))→𝜙_𝑛(𝑥),𝑚→∞. (5.33)In fact, from the monotonicity of 𝑣_𝑛(𝑥) and 𝜙(𝜉), for any 𝜀̂>0, there exists 𝐽>0 such that for any 𝑚>𝐽 (see Lemma 5.8), 𝑣_𝑚(𝑥−𝜀̂+𝑧(𝑎,𝑚))−𝜀̂≤𝜙(𝑥)≤𝑣_𝑚(𝑥+𝜀̂+𝑧(𝑎,𝑚))+𝜀.̂ (5.34)We further have 𝑣_𝑚+𝑛𝑥−𝜀̂+𝑧(𝑎,𝑛)−𝜎₂𝜀̂1−𝑒^−𝛽𝑛−𝜀𝑒̂^−𝛽𝑛≤𝜙_𝑛(𝑥)≤𝑣_𝑚+𝑛𝑥+𝜀̂+𝑧(𝑎,𝑛)+𝜎₂𝜀̂1−𝑒^−𝛽𝑛+𝜀𝑒̂^−𝛽𝑛, (5.35)for all 𝑛≥0 and 𝑚>𝐽. Sending 𝑚→∞ first and then 𝜀̂→0 (see Remark 3.6), we immediately get lim_𝑚→∞𝑣_𝑚+𝑛(𝑥+𝑧(𝑎,𝑚))=𝜙_𝑛(𝑥),𝑥∈ℝ. (5.36) Let 𝑚₀>0 be large enough such that 𝑣₁𝑥−𝑚₀−𝛿₀≤𝑣₀(𝑥)≤𝑣₁𝑥+𝑚₀+𝛿₀. (5.37)Then, Lemma 5.7 and comparison principle imply that 𝜙₁𝑥−𝑚₀−𝜎₂𝛿₀≤𝜙(𝑥)≤𝜙₁𝑥+𝑚₀+𝜎₂𝛿₀. (5.38)Define constants 𝜉_∗=sup𝜉∶𝜙₁(𝑥+𝜉)≤𝜙(𝑥),𝜉^∗=inf𝜉∶𝜙₁(𝑥+𝜉)≥𝜙(𝑥). (5.39)It is clear that 𝜉^∗≥𝜉_∗ holds and 𝜉_∗,𝜉^∗ are well defined. In particular, if 𝜉_∗=𝜉^∗, then we complete the proof.
Due to a prior estimate formulated by Lemma 5.6 which is similar to that of Lemma 3.4 established for the bistable wave front, the proof of 𝜉_∗=𝜉^∗ is similar to that of Theorem 4.2 in estimating the convergence of phase shift, and we omit it here. We also refer to Chen [22, Lemma 4.2] for some details. This implies that 𝜙(𝑡) satisfies 𝜙𝑡+𝜉^∗=1√4𝜋𝑑^ℝ𝑒^{−(𝑡−𝑦)2/4𝑑}𝑔𝜙𝑦𝑑𝑦,𝑡∈ℝ. (5.40) The proof is complete.

Lemma 5.10. lim_𝑛→∞[𝑧(𝑎,𝑛)−𝑧(𝑎,𝑛+1)] exists. Let 𝑐=lim_𝑛→∞[𝑧(𝑎,𝑛)−𝑧(𝑎,𝑛+1)], then 𝜙(𝑥+𝑐𝑛) is a bistable wave front of (1.5).

Note that the convergence in Lemma 5.8 is also locally uniform by Ascoli-Arzela lemma, then the proof of Lemma 5.10 is clear by Lemma 5.9 and the Lebesgue’s dominated convergence theorem, so we omit it here.

6. Discussion

In this paper, we consider the bistable wave fronts of (1.5), the existence, uniqueness and asymptotic stability of traveling wave fronts as well as the uniqueness of wave speed are established. Our results answer the conjecture of the stability of the bistable wave fronts in Lui [26], and we give more weaker assumptions on 𝑔 than that in [26–28] for the Gaussian kernel function. Very likely, our methods and results can be extended to more general kernel functions.

In particular, for such a bistable wave front, it is proved that both the wave speed and the wave profile are unique. By Theorem 3.3 and Lemma 3.5, it is clear that the wave speed can characterize the invasion speed (or extinction speed, and this depends on the sign of the wave speed [15]) when a single species with Allen effect is involved in population dynamics. Especially, such a speed formulates the evolution of a single species that admits the Allen effect, and the initial population density on a half space is large (e.g., let the initial value of (3.1) be the Heaviside function), which is different from the spreading speed of monostable case (see [1–3, 5]) that describes the population invasion process from finite domain to the whole space. Moreover, the stability of the bistable wave fronts also implies that the wave profile can further formulate the invasion process. Thus, our results provide further theory illustration for part numerical results (e.g., Wang et al. [15]). In Weinberger [1], the author derived a population genetics model (1.2) with 𝑔(𝑢)=(𝑠𝑢²+𝑢)/(1+𝑠𝑢²+𝜎(1−𝑢)². If 0<𝜎≤𝑠 holds, then it also satisfies (g1)–(g3). Thus, we also give the theory illustration for such a population genetics model with gaussian kernel.

Note that Lemmas 5.8–5.10 indicate the estimates of the wave profile and the wave speed, then it is possible to study the wave speed and the wave profile numerically. Since there are several results on such a numerical simulation (see Wang et al. [15]), we omit it here.

Acknowledgment

This work is supported by the Development Program for Outstanding Young Teachers in Lanzhou University of Technology (1010ZCX019).