Bistable Wave Fronts in Integrodifference Equations

and Applied Analysis 3


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 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: where x, y ∈ R 2 , v n ∈ R denotes the gene fraction in the newly born individuals of the nth 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 discretetime recursions have been widely studied if they are monostable analogous to reactiondiffusion 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 p 1 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.
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:

Abstract and Applied Analysis
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: 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 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.

Preliminaries
Throughout the paper, we will use the standard order and the interval notations in R. We also denote C R, R as follows: 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 ,

Initial Value Problem
In this part, we will show some results of the following Cauchy type problem: 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.
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 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: 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 δ 0 < min{a, 1 − a} such that for any δ ∈ 0, δ 0 and ξ , ξ are the upper and lower solutions of 3.
Proof.By Definition 3.2, it suffices to prove that 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 4πd R e − x−y 2 /4d g u n y − g φ y cn ξ − σδe −βn dy.

3.8
Then, the result is which is well defined by Lemma 3.4.Then, 3.5 holds if 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 δ 0 < min{a, 1 − a}, σ and β are uniform for all δ ∈ 0, δ 0 , which is very important in the following discussion.
Let χ x ∈ C 0,1 be a fixed function such that

3.11
Then, u n x and u − n x are the upper and lower solutions of 3.
Proof.By Definition 3.2, it suffices to prove that ∞ M e −y 2 /4d dy < 1.Similar to the discussion of 3.5 , u n 1 x ≥ Q u n x can be verified if δ > 0 is small enough and C C δ > 0 is large enough.In a similar way, we can prove that u

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.
Proof.Without loss of generality, we assume that ξ n * 0, and denote u 0 x u x .By Theorem 3.3, we may obtain Moreover, it is clear that there exist ϑ ∈ 0, 1/2 and an interval which implies that at least one of the following is true: Let M > 0 be large enough such that If the case i holds, then |x| ≤ M 1 |c| with M > 0 large enough indicates that where ξ 1 c − σ 1 δ 1 − e −β and η min u∈ ϑ,1−ϑ g u > 0. Set Then, there exists θ ∈ x ξ 1 , x ξ 1 2σ 1 ε * h such that for all |x| ≤ M 1 |c|.Thus, 4.8 indicates that 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 Combining this with 3.10 , it is clear that Since δe −β ε * h ≤ δ 0 holds, then Theorem 3.3 and Lemma 3.4 indicate that

4.14
Therefore, the left side of 4.1 with n, ξ 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, ξ 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 .
for N > 0 and H > 0 large enough.Define constants We further choose n * ≥ 2 such that In 4.17 , let δ δ * , and denote the corresponding constants H and N by h 0 and N 0 .
Claim 1.There exists a natural number N 1 > N 0 such that 4.1 holds for n N 1 , δ δ * , h < 1 and some ξ ∈ R.
Claim 2. For every k ≥ 0, 4.1 holds for some ξ k ∈ R and 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 , ξ, δ, h , herein for all x ∈ R, n ≥ N 1 .The definitions of δ n and h n also imply that 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.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.

Existence of Traveling Wave Fronts
We first list the main result of this section as follows.
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 problem herein χ x is same to that in Section 3.
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 by the definition of χ and the assumption g2 .Then, w 1 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 w 1 .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 v 1 n and w 1 n are defined by the following linear equations:

5.3
Then, there exist ii There exists δ 1 > 0 small enough such that the solutions to iii There exists h 1 > 0 such that the solutions to Abstract and Applied Analysis satisfy 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 , Then, for each n ≥ 1, there exist ξ − n , ξ n such that
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 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

5.13
We now prove that v 2 n , v 2 n are a pair of upper and lower solutions if x ∈ R, 0 ≤ n ≤ n 1 .The initial value conditions and v 2 n 1 ≥ Q v 2 n are clear, so we need to prove that

5.14
Namely, we will verify that

5.15
By the Taylor's formula, there exists s s g u ≥ 0 for u ∈ 0, 1 see g2 such that .

5.16
Thus, the definition of v 1 n x implies that we only need to prove dy, 5.17 which is true if .

5.18
Note that δ 1 > 0 is small enough and K > 0 is large enough, then the result is clear.This completes the proof of v 2 n 1 x 1 ≥ a 2δ 1 .
In a similar way, we can prove that w 2 n 1 x 1 ≤ a − 2δ 1 .
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 δ 1 and C δ 1 such that In particular, 5.19 is true if n n 1 .We now prove that By translation and symmetry, we can assume that for any given n ≥ 0,

5.21
Set h max{ξ n 1 , h 1 }, then u n x h ≥ v 3 0 x for all x ∈ R. By the comparison principle, u n n 1 x h ≥ v 3 n 1 x .Thus, u n n 1 x 1 h ≥ a δ 1 , which implies that ξ n n 1 ≤ x 1 h .Set h − max{ξ n 1 − ξ − n 1 , h 1 }.Similar to the above discussion, we can verify that ξ − n n 1 ≥ −h − ξ n 1 x 1 , which further implies that

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 δ 1 and a large positive constant h 2 ≥ 1 such that Moreover, for any δ ∈ 0, δ 1 /2 , there exists m 1 δ > 0 such that Proof.The former is clear by Lemma 5.3.We now prove the latter of the lemma.By Lemma 3.7, there exist δ , C C δ such that

5.25
Thus, n > −1 | ln δ| : Δ implies that in which x denotes the maximal integer less than x 1.Note that 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 Proof.By the smoothness of the Gaussian, the existence of . By the definition of h 2 , it is clear that for any ξ ∈ −h 2 , h 2 − 1 and n ≥ 0, which further implies that then the result is clear.The proof is complete.
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
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 ε > 0, there exists J > 0 such that for any m > J see Lemma 5.8 ,

5.34
We further have

5.36
Let m 0 > 0 be large enough such that

5.37
Then, Lemma 5.7 and comparison principle imply that
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 4πd R e − t−y 2 /4d g φ y dy, 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.

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 su 2 u / 1 su 2 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.

Theorem 4 . 2 .Let u n x be defined by 3 . 1 . 16 Proof.
Assume that φ x cn is a monotone solution of 2.3 -2.4 and u x ∈ C 0,1 satisfies 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∈R u n x − φ x cn ξ ≤ Ke −κn , n ≥ 0.4.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 max

Theorem 4 . 3 .
Assume that φ x cn is a bistable wave front of 1.5 .If φ 1 x c 1 n is another traveling wave solution of 1.5 satisfying 2.3 -2.4 , then c c 1 holds and there exists a constant h ∈ R such that φ ξ φ 1 ξ h , ξ ∈ R.
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.
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.bistable wave front if g1 and g3 hold.Moreover, the smoothness effect of the Gaussian kernel see 33 implies that φ 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