# Bistable Wave Fronts in Integrodifference Equations

Authors: Shuxia Pan; Ping-An Zhang
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 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 form(1.1)vn+1(x)=Q[vn](x),n=0,1,2,…,
where x∈ℋ⊂ℝm, v∈ℝk, and Q is an ℝk-valued mapping, herein ℋ admits proper geometric structure such that it is a habitat (e.g., the integer lattice in ℝm; 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:(1.2)vn+1(x)=Q[vn](x)=∫R2m(x-y)g(vn(y))dy,n=0,1,2,…,
where x,y∈ℝ2,  vn∈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(vn) 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 (1.3)g(u)=λup1+(λ-1)up,λ>1,p>0,u≥0.Ifp=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:(1.4)vn+1(x)=∫Rk(x-y)g(vn(y))dy,x∈R,
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 ℝn. 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 thatk takes the form of the Gaussian in (1.4), namely; we will investigate the bistable wave fronts of the following integrodifference equation:(1.5)un+1(x)=Q[un](x)=14πd∫Re-(x-y)2/4dg(un(y))dy,n=0,1,2,…,
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 Section2, 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 inℝ. We also denote C(ℝ,ℝ) as follows: (2.1)C(R,R)={u∣u:R⟶Risuniformlycontinuousandbounded},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 (2.2)C[a,b]={u:u∈C(R,R),a≤u(x)≤b,∀x∈R}.For convenience, we list the assumptions ong, which will be imposed in what follows:(g1)
foru∈[0,1], g(u)=u exactly has three distinct zeros 0,1,a∈(0,1),(g2)
ifu∈[0,1], then g(u)∈C2; if u∈(0,1), then g′(u)>0,(g3)
there existρ>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].Letg(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)=maxx∈[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.
Atraveling wave solution of (1.5) is a special solution with the form un(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 Definition2.2, a traveling wave solution of (1.5) must satisfy(2.3)ϕ(t+c)=14πd∫Re-(t-y)2/4dg(ϕ(y))dy,t∈R,
and we are interested in the following asymptotic boundary conditions:(2.4)lim⁡t→-∞ϕ(t)=0,lim⁡t→∞ϕ(t)=1.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:(3.1)un+1(x)=Q[un](x),n=0,1,2,3…,u0(x)=u(x),
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,…,  un(x)∈C[0,1].  Moreover, un(x) is uniformly continuous  for all n  ≥    1 and x∈ℝ.Definition 3.2.
Assume thatvn(x)∈C[0,1] for all n=0,1,…,N.Then, vn(x) is called  an upper solution (a lower solution) of (3.1) if
(3.2)vn+1(x)≥(≤)Q[vn](x),n=0,1,…,N-1,v0(x)≥(≤)u(x).By the upper and lower solutions, we can establish the following comparison principle.Theorem 3.3.
Assume thatu¯n(x) and u̲n(x),n=0,1,…,N, are upper and lower solutions of (3.1), respectively. Then, the following items hold:  (i)
u¯n(x)≥u̲n(x),n=0,1,…,N,(ii)
u¯n(x)≥un(x)≥u̲n(x),n=0,1,…,N,(iii)
u¯n+1(x)-u̲n+1(x)≥14πd∫ℝe-(x-y)2/4d[g(u¯n(y))-g(u̲n(y))]dy,n=0,1,…,N-1.Assume thatϕ(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 thatϕ(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-∈ℝ(3.3)u¯n(x)=min⁡{ϕ(x+cn+ξ+-σδe-βn)+δe-βn,1},u̲n(x)=max⁡{ϕ(x+cn+ξ-+σδe-βn)-δe-βn,0}are the upper and lower solutions of (3.1) if u¯0(x)≥u(x)≥u̲0(x),x∈ℝ.Proof.
By Definition3.2, it suffices to prove that
(3.4)u¯n+1(x)≥Q[u¯n](x),u̲n+1(x)≤Q[u̲n](x),n=0,1,….
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
(3.5)ϕ(x+c(n+1)+ξ+-σδe-β(n+1))+δe-β(n+1)-ϕ(x+c(n+1)+ξ+-σδe-βn)≥14πd∫Re-(x-y)2/4d[g(u¯n(y))-g(ϕ(y+cn+ξ+-σδe-βn))]dy.
Define ι=maxu∈[0,1]g′(u). Let M>0 be large enough, then (g1)–(g3) imply that there exists ρ′∈(0,1) such that
(3.6)0≤g(u¯n(y))-g(ϕ(y+cn+ξ+-σδe-βn))≤1-ρ′-βn,(1-ρ′)+ι4πd∫-∞-Me-y2/4ddy<1,for all |y+cn+ξ+-σδe-βn|>M and δ∈(0,δ0]. Thus, it is clear that
(3.7)14πd∫Re-(x-y)2/4d[g(u¯n(y))-g(ϕ(y+cn+ξ+-σδe-βn))]dy≤[(1-ρ′)+ι4πd∫-∞-Me-y2/4ddy]δe-βn,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
(3.8)e-β>(1-ρ′)+ι4πd∫-∞-Me-y2/4ddy.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=mins∈[-2M,2M]ϕ′(s)>0, which is well defined by Lemma 3.4. Then, (3.5) holds if
(3.9)υσδe-βn[1-e-β]+δe-β(n+1)≥ιδe-βn.
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 thatu̲n(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 fixedδ0<min{a,1-a}, σ and β are uniform for all δ∈(0,δ0], which is very important in the following discussion.Letχ(x)∈C[0,1] be a fixed function such that(3.10)χ(x)=0ifx≤0,χ(x)=1ifx≥4,0<χ′(x)<1,|χ′′(x)|≤1ifx∈(0,4).Lemma 3.7.
For anyδ∈(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
(3.11)un+(x)=min⁡{(1+δ)-[1-(a-2δ)e-ϵn]χ(-ϵ(x-ξ++Cn)),1},un-(x)=max⁡{-δ+[1-(1-a-2δ)e-ϵn]χ(ϵ(x-ξ-+Cn)),0}.Then, un+(x) and un-(x) are the upper and lower solutions of (3.1) if u0+(x)≥u(x)≥u0-(x).Proof.
By Definition3.2, it suffices to prove that
(3.12)un+1+(x)≥Q[un+](x),un+1-(x)≤Q[un-](x),n=0,1,….
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=un+1+(x)≥Q[un+](x).
Ifχ(-ϵ(x-ξ++Cn))=1, then un+(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)∫M∞e-y2/4ddy<1. Similar to the discussion of (3.5), un+1+(x)≥Q[un+](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 thatun+1-(x)≤Q[un-](x),x∈ℝ,n≥0. The proof is complete.


## 4. Asymptotic Stability of Traveling Wave Fronts

Lemma3.5 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 existn*≥0,ξ∈ℝ,δ∈(0,δ0/2] and h>0 such that
(4.1)max⁡{ϕ(x+cn*+ξ)-δ,0}≤un*(x)≤min⁡{ϕ(x+cn*+ξ+h)+δ,1},x∈R.
Then, for every n≥n*+1, there exist ξ̂(n),δ̂(n),ĥ(n) satisfying
(4.2)ξ̂(n)∈[ξ-σ1δ,ξ+h+σ1δ],δ̂(n)≤e-β(n-n*-1)[δ+ε*min⁡{h,1}],ĥ(n)≤[h-σ1ε*min⁡{h,1}]+2σ1δ,such that (4.1) remains true if we replace (n*,ξ,δ,h) by (n,ξ̂(n),δ̂(n),ĥ(n)), herein β>0,σ=σ1 such that Lemma 3.5 holds.Proof.
Without loss of generality, we assume thatξ=n*=0, and denote u0(x)=u(x). By Theorem 3.3, we may obtain
(4.3)max⁡{ϕ(x+cn-σ1δ(1-e-βn))-δe-βn,0}≤un(x)≤min⁡{ϕ(x+cn+σ1δ(1-e-βn)+h)+δe-βn,1},x∈R,n≥0.
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 limsupx→-∞u(x)≤δ0/2, limsupx→∞u(x)≥1-(δ0/2) and (2.4) hold). Without loss of generality, we further assume that I=[0,1]. Define constants
(4.4)h¯=min⁡{h,1},ε1=12min⁡ξ∈[0,2]ϕ′(ξ)>0.
Then,
(4.5)∫01[ϕ(y+h¯)-ϕ(y)]dy=∫01[ϕ(y+h¯)-u(y)+u(y)-ϕ(y)]dy≥2ε1h¯,which implies that at least one of the following is true:
(4.6)(i)∫01[u(y)-ϕ(y)]dy≥ε1h¯;(ii)∫01[ϕ(y+h¯)-u(y)]dy≥ε1h¯.
Let M>0 be large enough such that
(4.7)ϕ′(t)<12σ1,|t|>M.If the case (i) holds, then |x|≤M+1+|c| with M>0 large enough indicates that
(4.8)u1(x)-max⁡{ϕ(x+ξ1)-δe-β,0}≥η4πde(M+2+|c|)2/-4d∫01[u(y)-max⁡{ϕ(y)-δ,0}]dy>η4πde(M+2+|c|)2/-4d∫01[u(y)-ϕ(y)]dy≥ηε1h¯4πde(M+2+|c|)2/-4d,
where ξ1=c-σ1δ(1-e-β) and η=minu∈[ϑ,1-ϑ]g′(u)>0. Set
(4.9)ε*=min⁡{δ02,12σ1,min⁡|x|≤M+2+|c|ηε12σ1ϕ′(x)4πde(M+2+|c|)2/-4d}.Then, there exists θ∈[x+ξ1,x+ξ1+2σ1ε*h¯] such that
(4.10)ϕ(x+ξ1+2σ1ε*h¯)-ϕ(x+ξ1)=2ϕ′(θ)σ1ε*h¯≤ηε1h¯4πde(M+2+|c|)2/-4d,for all |x|≤M+1+|c|. Thus, (4.8) indicates that
(4.11)u1(x)≥max⁡{ϕ(x+ξ1+2σ1ε*h¯)-δe-β,0},|x|≤M+1+|c|.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
(4.12)ϕ(x+ξ1)≥ϕ(x+ξ1+2σ1ε*h¯)-ε*h¯.Combining this with (3.10), it is clear that
(4.13)u1(x)≥max⁡{ϕ(x+ξ1+2σ1ε*h¯)-[δe-β+ε*h¯],0},x∈R.
Since δe-β+ε*h¯≤δ0 holds, then Theorem 3.3 and Lemma 3.4 indicate that
(4.14)un+1(x)≥max⁡{ϕ(x+cn+ξ1+2σ1ε*h¯-σ1(δe-β+ε*h¯)(1-e-βn))-(δe-β+ε*h¯)e-βn,0}≥max⁡{ϕ(x+c+cn+σ1ε*h¯-σ1δ)-(δ+ε*h¯)e-βn,0}.Therefore, the left side of (4.1) with (n,ξ̂(n),δ̂(n),ĥ(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),δ̂(n),ĥ(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 thatϕ(x+cn) is a monotone solution of (2.3)-(2.4) and u(x)∈C[0,1] satisfies
(4.15)lim⁡inf⁡x→∞u(x)>a,limsup⁡x→-∞u(x)<a.Let un(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
(4.16)sup⁡x∈R|un(x)-ϕ(x+cn+ξ)|≤Ke-κn,n≥0.Proof.
By what we have done (Lemmas3.5 and 4.1), the key of proof is the precise estimate of phase shifts mentioned above.
For any givenδ>0, Lemmas 3.7 and 4.1 imply that
(4.17)max⁡{ϕ(x+cN-H2)-δ,0}≤uN(x)≤min⁡{ϕ(x+cN+H2)+δ,1},x∈R,
for N>0 and H>0 large enough. Define constants
(4.18)δ*:=min⁡{δ02,ε*4},κ*:=σ1ε*-2σ1δ*≥σ1ε*2>0.We further choose n*≥2 such that
(4.19)e-β(n*-1)[1+ε*δ*]≤1-κ*.In (4.17), let δ=δ*, and denote the corresponding constants H and N by h0 and N0.
Claim 1.
There exists a natural numberN1>N0 such that (4.1) holds for n=N1,δ=δ*,h<1 and some ξ∈ℝ.
In fact, lettingn=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
(4.20)e-β(n*-1)[δ*+ε*]≤δ*,h0-σ1ε*+2σ1δ*≤h0-κ*.We now repeat the above process N times (see (4.4), so we can repeat the process for any h≥1) such that
(4.21)h0-(N-1)κ*≥1,h0-Nκ*<1.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 everyk≥0, (4.1) holds for some ξk∈ℝ and
(4.22)n=Nk:=N1+kn*,δ=δk:=(1-κ*)kδ*,h=hk:=(1-κ*)k.
In fact, the result withk=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 (Nk+1,ξ̂,δ̂,ĥ), herein
(4.23)ξ̂∈[ξk-σ1δk,ξk+σ1δk],δ̂≤e-β(n*-1)(δk+ε*hk)=(1-κ*)kδ*e-β(n*-1)(1+ε*δ*)≤(1-κ*)k+1δ*,ĥ≤hk-σ1ε*hk+2σ1δk=(1-κ*)k(1-σ1ε*+2σ1δ*)=(1-κ*)k+1,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 (Nk,ξk,δk,hk) for all k=0,1,…. The comparison principle further indicates that (4.1) holds for all n≥Nk,δ=δn,h=hn+2σ1δk,ξ=ξk-σ1δk with k=0,1,….
Defineδ(n)=δk,ξ(n)=ξk-σ1δk,andh(n)=hk+2σ1δk for all Nk≤n<Nk+1 and k=0,1,…. Then,
(4.24)ϕ(x+cn+ξ(n))-δ(n)≤un(x)≤ϕ(x+cn+ξ(n)+h(n))+δ(n),for all x∈ℝ,n≥N1. The definitions of δ(n) and h(n) also imply that
(4.25)δ(n)=δn=(1-κ*)nδ*≤δ*e(((n-N1)/n*)-1)ln⁡(1-κ*),n≥N1,h(n)=hn+2σ1δk≤(1+2σ1δ*)e(((n-N1)/n*)-1)ln⁡(1-κ*),n≥N1,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
(4.26)|ξ(n)-ξ(n*)|≤h(n)+2σ1δ(n*),which indicates that ξ(n) is bounded and limn→∞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 Lemma3.5 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 thatϕ(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 Lemma3.5, 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 Theorem5.1 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 problem(5.1)wn+1(x)=Q[wn](x),w0(x)=χ(x),n=0,1,…,
herein χ(x) is same to that in Section 3.Lemma 5.2.
For any givenα∈(0,1) and n=0,1,…, there exists z(α,n) such that wn(z(α,n))=α.Proof.
The proof is essentially based on the implicit function theorem. For the case ofn=0, the result is clear by the strict monotonicity of χ(x),x∈(0,4). We claim that wn(x),n≥1 is strictly monotone in x∈ℝ. For any x>y, it is clear that
(5.2)w1(x)-w1(y)=14πd∫Re-z2/4d[g(χ(x+z))-g(χ(y+z))]dz>0,by the definition of χ and the assumption (g2). Then, w1(x) is strictly monotone in x∈ℝ. Assume that wn(x) is strictly monotone for some n≥1, then we can prove that wn+1(x) is also strictly monotone in x∈ℝ, and the proof is similar to the discussion of w1. By mathematical induction, wn(x),n≥1 is strictly monotone in x∈ℝ. Moreover, it is also clear that limx→-∞wn(x)=0,limx→∞wn(x)=1 for any fixed n≥0. Combining the implicit function theorem with the strict monotonicity of wn(x), then the lemma is obvious. The proof is complete.Lemma 5.3.
LetH(x) be the Heaviside function. Then, the following results hold.
(i) Assume thatvn1 and wn1 are defined by the following linear equations:
(5.3)vn+11(x)=g′(a)4πd∫Re-(x-y)2/4dvn1(y)dy,n=0,1,…,v01(x)=H(x),wn+11(x)=g′(a)4πd∫Re-(x-y)2/4dwn1(y)dy,n=0,1,…,w01(x)=-1+2H(x).Then, there exist x1∈ℝ,n1≥1 such that
(5.4)vn11(x1)≥3,wn11(x1)≤-3.(ii) There exists δ1>0 small enough such that the solutions to
(5.5)vn+12(x)=Q[vn2](x),n=0,1,…,v02(x)=a+δ1H(x),wn+12(x)=Q[wn2](x),n=0,1,…,w02(x)=a+δ1[H(x)-H(-x)]
satisfy
(5.6)vn12(x1)≥a+2δ1,wn12(x1)≤a-2δ1.(iii) There exists h1>0 such that the solutions to
(5.7)vn+13(x)=Q[vn3](x),v03(x)=a+δ1H(x)-aH(-h1-x),wn+13(x)=Q[wn3](x),w03(x)=a+δ1[H(x)-H(-x)]+(1-a-δ1)H(x-h1)
satisfy
(5.8)vn13(x1)≥a+δ1,wn13(x1)≤a-δ1.(iv) Let un(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),
(5.9)u(ξ-(0))≤a-δ1,u(ξ+(0))≥a+δ1.
Then, for each n≥1, there exist ξ-(n),ξ+(n) such that
(5.10)un(ξ-(n))=a-δ1,un(ξ+(n))=a+δ1,ξ+(n)-ξ-(n)≤max⁡{ξ+(0)-ξ-(0)+8ϵ-1(δ1)+2C(δ1)n1,2h1},
where ϵ-1(δ1),C(δ1) are as in Lemma 3.7.Proof.
For each fixedn>0, it is clear that vni,wni,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 fixedn>0, we know that vn(x)∈C(ℝ,ℝ) and
(5.11)lim⁡x→-∞vn1(x)=0,lim⁡x→∞vn1(x)=[g′(a)]n.Let n1>0 such that [g′(a)]n1≥9 and [g′(a)]n1-1<9. Then, there exists x1∈ℝ such that vn11(x1)=3. Since wn1=-[g′(a)]n+2vn1, then wn11(x1)≤-3.Proof of (ii).
LetK>0 be a large constant and δ1>0  small enough such that Kδ1<1,δ1[g′(a)]2n1+2<1 and
(5.12)0≤a+δ1vn1(x)-Kδ12[g′(a)]2n,a+δ1vn1(x)≤1,∀x∈R,0≤n≤n1,
which is admissible, since vn1(x) is bounded for allx∈ℝ,0≤n≤n1. Define
(5.13)v̲n2(x)=a+δ1vn1(x)-Kδ12[g′(a)]2n,v¯n2(x)=1,x∈R.
We now prove that v¯n2,v̲n2 are a pair of upper and lower solutions if x∈ℝ,0≤n≤n1. The initial value conditions and v¯n+12≥Q[v¯n2] are clear, so we need to prove that
(5.14)v̲n+12(x)≤Q[v̲n2](x),0≤n≤n1,x∈R.Namely, we will verify that
(5.15)a+δ1vn+11(x)-Kδ12[g′(a)]2n+2≤14πd∫Re-(x-y)2/4dg(a+δ1vn1(y)-Kδ12[g′(a)]2n)dy.By the Taylor's formula, there exists s=s(g(u))≥0 for u∈[0,1] (see (g2)) such that
(5.16)g(a+δ1vn1(y)-Kδ12[g′(a)]2n)≥g(a)+g′(a){δ1vn1(y)-Kδ12[g′(a)]2n}-s{δ1vn1(y)-Kδ12[g′(a)]2n}2.Thus, the definition of vn1(x) implies that we only need to prove
(5.17)-(g′(a)-1)Kδ12[g′(a)]2n+1≤-s4πd∫Re-(x-y)2/4d{δ1vn1(y)-Kδ12[g′(a)]2n}2dy,which is true if
(5.18)(g′(a)-1)Kδ12[g′(a)]2n+1≥2s[δ1[g′(a)]n]2,(g′(a)-1)Kδ12[g′(a)]2n+1≥2s[Kδ12[g′(a)]2n]2.Note that δ1>0 is small enough and K>0 is large enough, then the result is clear. This completes the proof of vn12(x1)≥a+2δ1.
In a similar way, we can prove thatwn12(x1)≤a-2δ1.Proof of (iii).
Note thatlimN→∞1/4πd∫-NNe-y2/4ddy=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 ofu(x), it is clear that un(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
(5.19)ξ+(n)-ξ-(n)≤ξ+(0)-ξ-(0)+8ϵ-1(δ1)+2C(δ1)n.
In particular, (5.19) is true if n=n1. We now prove that
(5.20)ξ+(n+n1)-ξ-(n+n1)≤max⁡{ξ+(n1)-ξ-(n1),2h1}foranyn≥0.By translation and symmetry, we can assume that for any given n≥0,
(5.21)ξ-(n)<0<ξ+(n),ξ+(n)≥-ξ-(n).
Seth+=max{ξ+(n1),h1}, then un(x+h+)≥v03(x) for all x∈ℝ. By the comparison principle, un+n1(x+h+)≥vn13(x). Thus, un+n1(x1+h+)≥a+δ1, which implies that ξ+(n+n1)≤x1+h+.
Seth-=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
(5.22)ξ+(n+n1)-ξ-(n+n1)≤h++h--ξ-(n1)≤max⁡{ξ+(n1)-ξ-(n1),2h1}≤max⁡{ξ+(0)-ξ-(0)+8ϵ-1(δ1)+2C(δ1)n1,2h1}.
The proof is complete.Lemma 5.4.
Assume thatz(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
(5.23)z(a+δ1,n)-z(a-δ1,n)≤h2,∀n≥0.Moreover, for any δ∈(0,δ1/2], there exists m1(δ)>0 such that
(5.24)z(1-δ,n)-z(δ,n)≤m1(δ),n≥0.Proof.
The former is clear by Lemma5.3. We now prove the latter of the lemma. By Lemma 3.7, there exist ϵ=ϵ(δ),C=C(δ) such that
(5.25)z(1-2δ,n)≤z(a+δ,n1)+4ϵ-1+C(n-n1)foranyn-n1>ϵ-1|ln⁡δ|,z(2δ,n)≥z(a-δ,n1)-4ϵ-1-C(n-n1)foranyn-n1≥ϵ-1|ln⁡δ|.Thus, n>ϵ-1|lnδ|=:Δ implies that
(5.26)z(1-2δ,n)-z(2δ,n)≤z(a+δ,n-⌊Δ⌋+1)-z(a-δ,n-⌊Δ⌋+1)+8ϵ-1+2C(⌊Δ⌋+1),in which ⌊x⌋ denotes the maximal integer less than x+1. Note that
(5.27)z(a+δ1,n-⌊Δ⌋+1)-z(a-δ1,n-⌊Δ⌋+1)≤h2,∀n-⌊Δ⌋+1≥0.Then, the result is clear, and the proof is complete.Remark 5.5.
Lemma5.4 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 everyM>0, there exists a constant η̂(M)>0 such that
(5.28)dvn+1(t)dt|t=x+z(a,n)>η̂(M)>0,for anyn≥1,x∈[-M,M].Proof.
By the smoothness of the Gaussian, the existence of(dvn+1(t))/dt,t∈ℝ is clear. Let η(x)=(1/4πd)e-x2/4d. By the definition of h2, it is clear that
(5.29)vn(ξ+1+z(a,n))-vn(ξ+z(a,n))≥δ1h2,
for any ξ∈[-h2,h2-1] and n≥0, which further implies that
(5.30)min⁡x∈[-M̂,M̂]dvn+1(t)dt|t=x+z(a,n)≥min⁡x∈[-M̂-h2,M̂+h2]η(M̂+h2)δ1h2,for any n≥0. Since |z(a,n+1)-z(a,n)|≤h2+8ϵ-1(δ1)+2C(δ1), taking M̂=M+h2+8ϵ-1(δ1)+2C(δ1) and η̂(M)=η(M̂+h2), then the result is clear. The proof is complete.Lemma 5.7.
LetM3=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
(5.31)Wn+(x):=min⁡{vn+1(x+ξ+σ2δ(1-e-βn))+δe-βn,1},Wn-(x):=max⁡{vn+1(x+ξ-σ2δ(1-e-βn))-δe-βn,0}are the upper and lower solutions of (1.5) if W0+(x)≥v0(x)≥W0-(x).By the estimate on the derivative ofvn formulated by Lemma 5.6, namely, the strict monotonicity of vn, 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 functionϕ(ξ) such that
(5.32)vn(ξ+z(a,n))⟶ϕ(ξ),n⟶∞,∀ξ∈R,and limξ→-∞ϕ(ξ)=0,limξ→∞ϕ(ξ)=1.Proof.
The existence ofϕ(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 limξ→±∞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.
Letϕ̃n(x) be the solution of (1.5) with the initial value ϕ(x), and we will prove that
(5.33)vm+n(x+z(a,m))→ϕ̃n(x),m→∞.In fact, from the monotonicity of vn(x) and ϕ(ξ), for any ε̂>0, there exists J>0 such that for any m>J (see Lemma 5.8),
(5.34)vm(x-ε̂+z(a,m))-ε̂≤ϕ(x)≤vm(x+ε̂+z(a,m))+ε̂.We further have
(5.35)vm+n(x-ε̂+z(a,n)-σ2ε̂(1-e-βn))-ε̂e-βn≤ϕ̃n(x)≤vm+n(x+ε̂+z(a,n)+σ2ε̂(1-e-βn))+ε̂e-βn,for all n≥0 and m>J. Sending m→∞ first and then ε̂→0 (see Remark 3.6), we immediately get
(5.36)lim⁡m→∞vm+n(x+z(a,m))=ϕ̃n(x),x∈R.
Let m0>0 be large enough such that
(5.37)v1(x-m0)-δ0≤v0(x)≤v1(x+m0)+δ0.Then, Lemma 5.7 and comparison principle imply that
(5.38)ϕ̃1(x-m0-σ2δ0)≤ϕ(x)≤ϕ̃1(x+m0+σ2δ0).Define constants
(5.39)ξ*=sup⁡{ξ:ϕ̃1(x+ξ)≤ϕ(x)},ξ*=inf⁡{ξ:ϕ̃1(x+ξ)≥ϕ(x)}.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 Lemma5.6 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
(5.40)ϕ(t+ξ*)=14πd∫Re-(t-y)2/4dg(ϕ(y))dy,t∈R.
The proof is complete.Lemma 5.10.
limn→∞[z(a,n)-z(a,n+1)] exists. Let c=limn→∞[z(a,n)-z(a,n+1)], then ϕ(x+cn) is a bistable wave front of (1.5).Note that the convergence in Lemma5.8 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 Theorem3.3 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 Lemmas5.8–5.10 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.



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