Research Article | Open Access
Bistable Wave Fronts in Integrodifference Equations
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.
In 1982, Weinberger  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 , , and is an -valued mapping, herein admits proper geometric structure such that it is a habitat (e.g., the integer lattice in ; see ). 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 , Weinberger et al. . In particular, Weinberger  derived a concrete population model as follows: where , denotes the gene fraction in the newly born individuals of the -th generation, is the probability function describing the migration of the individuals, and 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.  and Wang et al.  to understand the differences between the monostable and the bistable. When (1.2) is concerned, an important is the so-called Hill function
If and , then it is the famous Beverton-Holt stock recruitment curve, and it is monostable (see Kot ); if and , then it is bistable (see Wang et al. ). More precisely, has three equilibria , and are locally stable, while is unstable in the sense of the corresponding difference equation. In population dynamics, the case with describes the famous Allen effect (see ). For the reader’s convenience in understanding the difference, we may see the following Figures 1 and 2 for with , 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: where 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 . Creegan and Lui  further considered the existence and uniqueness of bistable wave fronts of an integrodifference equation in . 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 takes the form of the Gaussian in (1.4), namely; we will investigate the bistable wave fronts of the following integrodifference equation: where , , and 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  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  and Fife and McLeod  and was further developed by Ma and Wu , Ma and Zou , Smith and Zhao , and Wang et al. . In particular, our discussion is independent of the characteristic equation, so we also relax some requirements on which were used in Lui [26, 27] (see Remark 2.1). Moreover, Yagisita  established an abstract scheme to establish the existence of bistable wave fronts of monotone semiflows, and from Yagisita , 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 , 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.
Throughout the paper, we will use the standard order and the interval notations in . We also denote as follows:
which is a Banach space equipped with the super norm . Assume that with , then will be interpreted as
For convenience, we list the assumptions on , which will be imposed in what follows:(g1)for , exactly has three distinct zeros ,(g2)if , then ; if , then ,(g3)there exist such that for and for .
Remark 2.1. In Lui [26, 27], was required. Moreover, Lui  assumed that , since the characteristic equation was involved. It is clear that does not satisfy these assumptions if . 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 , in which is the wave speed that the wave profile spreads in . In particular, if is monotone in , then it is a traveling wave front.
Remark 2.3. Analogues to a parabolic system , 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 ) implies that if .
3. Initial Value Problem
In this part, we will show some results of the following Cauchy type problem: in which and is given by (1.5). We also refer to Liang and Zhao , Weinberger . In particular, the following result is obvious.
Theorem 3.1. For all , . Moreover, is uniformly continuous for all n ≥ 1 and .
Definition 3.2. Assume that for all . Then, 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 and , are upper and lower solutions of (3.1), respectively. Then, the following items hold: (i), (ii), (iii).
Lemma 3.4. is strictly monotone such that and for all .
Proof. By Definition 3.2, it suffices to prove that
If holds, then the result is clear by the assumptions (g1) and (g2). If , by Definition 2.2, we only need to prove that
Define . Let be large enough, then (g1)–(g3) imply that there exists such that
for all and . Thus, it is clear that
for or with large enough. Let be small enough such that
Then, the result is clear if or hold.
If and , define , which is well defined by Lemma 3.4. Then, (3.5) holds if Let be large enough, then the above inequality is clear. Thus, is an upper solution of (1.5) if are small enough.
Similarly, we can prove that is a lower solution of (1.5) if is small enough while is large enough. The proof is complete.
Remark 3.6. We should note that for any fixed , and are uniform for all , which is very important in the following discussion.
Let be a fixed function such that
Lemma 3.7. For any with ( is defined by (g3)), there exist two positive numbers and such that for every , define the continuous functions Then, and are the upper and lower solutions of (3.1) if .
Proof. By Definition 3.2, it suffices to prove that
If , then . Let be large enough such that , then it is clear that such that .
If , then . For any given , there exists such that . Moreover, there exists such that . Similar to the discussion of (3.5), can be verified if is small enough and is large enough.
In a similar way, we can prove that . 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 is larger.
Proof. Without loss of generality, we assume that , and denote . By Theorem 3.3, we may obtain
Moreover, it is clear that there exist and an interval such that for all (since , ) and (2.4) hold). Without loss of generality, we further assume that . Define constants
which implies that at least one of the following is true:
Let be large enough such that
If the case (i) holds, then with large enough indicates that
where and . Set
Then, there exists such that
for all . 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 , then the definition of implies that Combining this with (3.10), it is clear that Since holds, then Theorem 3.3 and Lemma 3.4 indicate that Therefore, the left side of (4.1) with is true.
In a similar way, we can verify the right side of (4.1) if we replace by . 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).
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 , Lemmas 3.7 and 4.1 imply that for and large enough. Define constants We further choose such that In (4.17), let , and denote the corresponding constants and by and .
Claim 1. There exists a natural number such that (4.1) holds for and some .
In fact, letting , and in (4.1), then Lemma 4.1 implies that (4.1) holds with , some , and , since the definitions of and imply that We now repeat the above process times (see (4.4), so we can repeat the process for any ) such that Then, (4.1) holds for , and some . Therefore, the claim is true.Claim 2. For every , (4.1) holds for some and
In fact, the result with is clear by what we have done. Assume that the claim holds for some , and we now prove the claim for . By applying Lemma 4.1, (4.1) holds with replaced by , herein by the definitions of and . Thus, (4.1) holds for . The proof of Claim 2 is completed by the mathematical induction.
By what we have done, (4.1) holds if is replaced by for all . The comparison principle further indicates that (4.1) holds for all with .
Define for all and . Then, for all . The definitions of and also imply that in which is the largest integer no bigger than .
Furthermore, and imply that which indicates that is bounded and exists (since if ). Let be fixed and 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.
5. 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 is same to that in Section 3.
Lemma 5.2. For any given and , there exists such that .
Proof. The proof is essentially based on the implicit function theorem. For the case of , the result is clear by the strict monotonicity of . We claim that is strictly monotone in . For any , it is clear that by the definition of and the assumption (g2). Then, is strictly monotone in . Assume that is strictly monotone for some , then we can prove that is also strictly monotone in , and the proof is similar to the discussion of . By mathematical induction, is strictly monotone in . Moreover, it is also clear that for any fixed . Combining the implicit function theorem with the strict monotonicity of , then the lemma is obvious. The proof is complete.
Lemma 5.3. Let be the Heaviside function. Then, the following results hold.
(i) Assume that and are defined by the following linear equations: Then, there exist such that (ii) There exists small enough such that the solutions to satisfy (iii) There exists such that the solutions to satisfy (iv) Let be the solution of (1.5) with uniformly continuous and nondecreasing initial data satisfying and for some finite and , Then, for each , there exist such that where are as in Lemma 3.7.
Proof. For each fixed , it is clear that are uniformly continuous, and the comparison principle remains true even if the initial value is a discontinuous function. More precisely, if such that are well defined, then . We now prove the items (i)–(iv) one by one.
Proof of (i). For each fixed , we know that and Let such that and . Then, there exists such that . Since , then .Proof of (ii). Let be a large constant and small enough such that and which is admissible, since is bounded for all. Define We now prove that are a pair of upper and lower solutions if . The initial value conditions and are clear, so we need to prove that Namely, we will verify that By the Taylor's formula, there exists for (see (g2)) such that Thus, the definition of implies that we only need to prove which is true if Note that is small enough and is large enough, then the result is clear. This completes the proof of .
In a similar way, we can prove that .Proof of (iii). Note that and , then the result is clear by letting large enough.Proof of (iv). By the definition of , it is clear that are strictly monotone in for every . Thus, the existence and uniqueness of are obvious. By Lemma 3.7, there exist and such that In particular, (5.19) is true if . We now prove that By translation and symmetry, we can assume that for any given ,
Set , then for all . By the comparison principle, . Thus, , which implies that .
Set . Similar to the above discussion, we can verify that , which further implies that The proof is complete.
Lemma 5.4. Assume that is defined by Lemma 5.2. Then, there exist a small positive constant and a large positive constant such that Moreover, for any , there exists 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 such that Thus, implies that in which denotes the maximal integer less than . Note that Then, the result is clear, and the proof is complete.
Lemma 5.6. For every , there exists a constant such that
Proof. By the smoothness of the Gaussian, the existence of is clear. Let . By the definition of , it is clear that for any and , which further implies that for any . Since , taking