PERIODIC TRAVELING WAVE SOLUTIONS OF PERIODIC INTEGRODIFFERENCE SYSTEMS

. This paper is concerned with the periodic traveling wave solutions of integrodiﬀerence systems with periodic parameters. Without the assump- tions on monotonicity, the existence of periodic traveling wave solutions is deduced to the existence of generalized upper and lower solutions by ﬁxed point theorem and an operator with multi steps. The asymptotic behavior of periodic traveling wave solutions is investigated by the stability of periodic solutions in the corresponding initial value problem or the corresponding dif-ference systems. To illustrate our conclusions, we study the periodic traveling wave solutions of two models including a scalar equation and a competitive type system, which do not generate monotone semiﬂows. The existence or nonexistence of periodic traveling wave solutions with all positive wave speeds is presented, which implies the minimal wave speeds of these models.


1.
Introduction. In population dynamics, there have been established a number of discrete time spatial contact models, for example, we refer to several earlier models and their analysis on changes of gene frequency by Lui [24,25,26,28], Slatkin [34]. Among these models, one well studied type is the integrodifference equations that may describe populations with discrete, nonoverlapping generations and well-defined growth and dispersal stages [10,27]. A typical integrodifference equation has the following form where x ∈ R, n ∈ Z, U(x, n) denotes the population density of the n-th generation at location x, f : R → R is often called the birth function while k(x) is a probability function on the spatial migration of individuals. In the past decades, much attention has been paid to the traveling wave solutions of (1). Here, a traveling wave solution of (1) is a special entire solution where φ is the wave profile that propagates through the one-dimensional spatial domain R at the constant wave speed c. By the above definition, we see that the corresponding wave system of (1) defines an operator, so the existence of traveling wave solutions can be studied by fixed point theorem, see Lin [17] and references cited therein for some results. Moreover, when an integrodifference equation/system satisfies proper monotone condition, the existence of traveling wave solutions may be studied by the theory of monotone dynamical systems, see Fang and Zhao [7], Liang and Zhao [16], Weinberger [39,40], Weinberger et al. [41]. In particular, the results established for abstract monotone integrodifference equations/systems have been applied to several other parabolic type systems in these works. Note that the wave profile is a function of single variable, then the limit behavior can be studied by fluctuation method, contracting rectangles and so on when the traveling wave solution is not monotone, see Hsu and Zhao [9], Li et al. [12], Lin [17], Wang and Castillo-Chavez [36]. The above model (1) was established in the homogeneous habitat and environment, in which the inhomogeneous habitat and seasonal change were not taken into account. In some evolutionary systems, it has been proved that the inhomogeneous habitat and seasonal change may lead to some new phenomena, see a survey paper by Xin [44] from the viewpoint of spatial propagation, and Cantrell and Cosner [3] from the viewpoint of persistence and extinction of population models. We now focus on the integrodifference systems. When the spatial inhomogeneous is involved, there are some conclusions on monotone/cooperative integrodifference systems in periodic habitat, see Ding et al. [4], Fang et al. [8], Weinberger [40], Wu and Zhao [43] and references cited therein. When the time periodic is concerned, Liang et al. [15] studied the asymptotic spreading and traveling wave solutions of an abstract monotone integrodifference equation, and the results can be applied to other models generating monotone semiflows.
In this paper, we study the periodic traveling wave solutions of the following integrodifference system with time periodic parameters u i (x, n + 1) = R F i (n, u 1 (y, n), · · · , u m (y, n))k i (x − y)dy, where m ∈ N, i ∈ {1, · · · , m} := I, x, u i ∈ R, n ∈ Z, F i : R m+1 → R is continuous and N −periodic in n with some N ∈ N, k i (y) is the so called kernel function. Hereafter, a periodic traveling wave solution (for simplicity, a traveling wave solution) of (2) is a special entire solution defined for all n ∈ Z and taking the following form where Φ = (φ 1 , φ 2 , · · · , φ m ) is the wave profile and c is the wave speed. By the above definition, we may obtain the corresponding wave system of (2), which satisfies the following integral system φ i (ξ + c, n + 1) = R k i (y)F i (n, φ 1 (ξ − y, n), · · · , φ m (ξ − y, n))dy (3) for all i ∈ I, ξ ∈ R, n ∈ Z. Similar to the study on (1), to formulate a desired evolutionary process, the traveling wave solution must satisfy proper asymptotic behavior when ξ → ±∞. Intuitively, the existence of (3) can not be investigated by an operator similar to that in (1) since it also depends on n ∈ Z. Motivated by the periodic property, we try to construct an N −steps mapping to obtain the existence of traveling wave solutions. More precisely, we first introduce the definition of generalized upper and lower solutions of (3) although (3) does not satisfy monotone assumptions. Then we construct a potential wave profile set by the generalized upper and lower solutions, and discuss the existence of traveling wave solutions by Schauder's fixed point theorem and an N −steps operator. The process implies the existence of (3) can be obtained by the existence of generalized upper and lower solutions. Since we do not require the cooperative property on (3), our abstract conclusion can be applied to many models.
When the limit behavior of nonconstant solutions of (3) as ξ → ±∞ is concerned, there are also some difficulties since the traveling wave solutions may be nonmonotone. Because Φ(ξ, n) depends on two variables, it is difficult to analysis the limit behavior by the methods mentioned above. In literature, there are many results on the stability of periodic difference systems, can we use the known results to study the limit behavior? Since a traveling wave solution is a special solution, we try to estimate some properties of the corresponding initial value problem to present the limit behavior of traveling wave solutions as ξ → ∞. Moreover, the initial value problem may be studied by the dynamics of the corresponding difference systems, so we achieve our purpose. In particular, we do not consider the limit behavior when ξ → −∞ since the behavior may be obtained by the property of upper and lower solutions in many models.
To illustrate our theory, we study and u 1 (x, n + 1) = R (1+r1(n))u1(x−y,n) 1+r1(n)(u1(x−y,n)+a1(n)u2(x−y,n)) k 1 (y)dy, u 2 (x, n + 1) = R (1+r2(n))u2(x−y,n) 1+r2(n)(u2(x−y,n)+a2(n)u1(x−y,n)) k 2 (y)dy, in which all the parameters are periodic and will be illustrated in later sections, and we refer to Hsu and Zhao [9], Li et al. [13], Lin [17], Zhang and Pan [45] for the models with constant coefficients. The first equation may be nonmonotone, of which the monotonicity depends on the property of b. There are the so-called interspecific and intraspecific competitions in the second system, which can not be studied by the theory of monotone semiflows when the trivial steady state and coexistence state are concerned (we may refer to Fang and Zhao [6], Fang et al. [8], Weinberger et al. [40] for the dynamics between the resident and the invader, which can be changed into a cooperative system that admits ordered steady states). We shall present the existence or nonexistence of traveling wave solutions for all positive wave speed, and so obtain the minimal wave speeds. Our results could complete/improve some known conclusions in [17] when the coefficients are constant or N = 1.
The rest of this paper is organized as follows. In Section 2, we shall present some preliminaries including some notations, assumptions and known results. Section 3 is devoted to an abstract result on the existence of traveling wave solutions. In Section 4, we study the asymptotic behavior of traveling wave solutions when ξ → ∞. Two examples will be presented in Sections 5-6.
2. Preliminaries. In this paper, we shall utilize the standard partial ordering in R m . Define C by C = {U (ξ) : U (ξ) : R → R m is uniformly continuous in ξ and bounded} and in which M is a vector with positive components. Let · be the supremum norm in R m and µ > 0 be a constant. Define Then B µ , |·| µ is a Banach space. By the decay property of the norm, we have the following conclusion.
Proposition 1. Assume that D ⊂ C is bounded in the sense of · . If D is equicontinuous in the sense of · , then it is precompact in the sense of | · | µ .
On the nonlinearity F and kernel functions, we shall make the following assumptions that will be true in Sections 3-4 of this paper.
Consider the following integrodifference equation [15,39] w(x, n + 1) = R f (n, w(y, n))k(y)dy, x ∈ R, n = 0, 1, · · · , w( where f, k, ω satisfy the following assumptions w := f n exists and is positive, and f (n, w) = w, w ∈ (0, b] has at most one root, there also exist L > 0, α > 1 such that We shall denote N N 1 f n := f in what follows. Because of (f2), we have the following comparison principle.
Consider the corresponding difference equation which is well defined for all n ∈ N if w 0 ∈ (0, b]. Moreover, for such an equation (5), if w n is an N −periodic solution, we say it is globally stable in the sense of By the boundedness and monotonicity, we have the following conclusion.
Lemma 2.2. Assume that f > 1. Then (5) has a globally asymptotic stable positive N −periodic solution w n .
ln f R e λy k(y)dy λ .
Then the following items are true.
In Liang et al. [15], the authors studied periodic monotone systems. By [15, Theorem 2.1], we have the following result.  3. Existence of traveling wave solutions. To investigate the existence of (3) or an equivalent wave system we shall use the generalized upper and lower solutions, which is partially motivated by the invariant set.
are a pair of generalized upper and lower solutions of (6) if and for all i ∈ I, ξ ∈ R and any uniformly continuous functions φ i (ξ, n) satisfying Theorem 3.2. Assume that (6) has a pair of generalized upper and lower solutions Φ(ξ, n), Φ(ξ, n). Then (6) has a solution Φ(ξ, n) such that Proof. Define Γ as follows for any i ∈ I, Φ(ξ) ∈ Γ, ξ ∈ R. By the boundedness and (K), we see that F i (Φ)(ξ, n), i ∈ I, n ∈ J, are equicontinuous in ξ ∈ R. In fact, if Φ ∈ Γ, then for any ξ , ξ ∈ R, i ∈ I. By (K), we see that F i (Γ)(ξ, 1), i ∈ I are equicontinuous in the sense of super norm. Moreover, the definition of upper and lower solutions implies that Repeating the above process N −times, we see that for any Φ(ξ) ∈ Γ, ξ ∈ R, and F i (Γ)(ξ, N ) is equicontinuous in the sense of super norm for all ξ ∈ R, i ∈ I. By Proposition 1, we see that F : Γ → Γ is compact in the sense of | · | µ . We now investigate the continuity of F. Let µ > 0 be a constant such that which is admissible by (K). For any i ∈ I and and so which further indicates that So the mapping F i (Φ)(ξ, 1) is continuous in the sense of |·| µ for every i ∈ I. Repeating the above process N −times, then F is completely continuous in the sense of |·| µ .
Using Schauder's fixed point theorem, F admits a fixed point Φ * (ξ) ∈ Γ, and we denote it by Then we obtain a periodic traveling wave solution, which completes the proof.
Remark 1. When the periodic reaction-diffusion systems with proper monotone conditions are concerned, there are some results on the existence of periodic traveling wave solutions by constructing upper and lower solutions and applying monotone iteration or fixed point theorem, we may refer to Bo et al. [1] and Zhao and Ruan [46,47] for Lotka-Volterra type competitive systems, Wang et al. [38] for an SIR model. 4. Asymptotic behavior of traveling wave solutions. In this section, we study the asymptotic behavior of traveling wave solutions. Consider the initial value problem for i ∈ I, x ∈ R, n + 1 ∈ N, where the initial value is uniformly continuous. We assume that the corresponding difference system admits a positive N −periodic solution For the initial value problem (10), we make the following assumption.
(C): Suppose that u i (x, n), i ∈ I, x ∈ R, n ∈ N are defined by (10). If there exists δ > 0 such that then lim n→∞ sup x∈R,i∈I Remark 2. In fact, the convergence condition (C) may be verified by the properties of the corresponding difference systems. For example, we shall show the property in Section 6 for a competitive system under weaker initial condition than (12).
then lim ξ→∞ sup i∈I,n∈J Proof. For any given > 0, we shall prove that there exists ξ 0 such that sup ξ>ξ0,i∈I,n∈J From the definition of traveling wave solutions, φ i (x + cn, n) satisfies of which the corresponding equality is monotone and takes the following form Select λ > 0, C > 0, D > 0, E > 0 such that By direct calculation, we see that We end this section by making the following remarks.
Remark 3. When the traveling wave solutions of reaction-diffusion systems with periodic parameters are concerned, Bo et al. [1], Lin [18] investigated the limit behavior of periodic traveling wave solutions without monotonicity.

Remark 4.
We have mentioned in the study of limit behavior, the deficiency of the monotonicity of systems often leads to the difficulty. In some nonmonotone systems, it is possible to obtain the existence of monotone traveling wave solutions, see Kwong and Ou [11], Wang et al. [37], Wu and Zou [42] for delayed reaction-diffusion systems, and Lin and Wang [22] for a diffusion equation with state-dependent delay. But there are a few results on the existence of nonmonotone traveling wave solutions. On the one hand, it has been proved that the existence of both monotone and nonmonotone traveling wave solutions in some evolutionary systems, see Lin and Ruan [20], Ni and Taniguchi [29], Tang  5. Application to a scalar equation. In this section, we consider the following integrodifference equation which satisfies (A1): b(n, ·) = b(n + N, ·) for all n ∈ R and some N ∈ N; (A2): there exists w * > 0 such that b(·, w) : [0, w * ] → [0, w * ] is continuous and b(·, 0) = 0, let r(n) = b (n, w)| w=0 , n ∈ J, then r(i) > 0, i ∈ J, N i=1 r(i) > 1; (A3): 0 < b(n, w) ≤ r(n)w, w ∈ (0, w * ], n ∈ J and there exist L > 0, δ ∈ (0, 1] such that 0 ≤ r(n)w − b(n, w) ≤ Lw 1+δ , w ∈ (0, w * ], n ∈ J, and b(n, u) = u, u ∈ (0, w * ] has at most one root; (A4): k satisfies (K) in Section 2.
In the above assumptions, we do not require the monotonicity on b. Due to the possible nonmonotonicity, the traveling wave solution can not be studied by Liang et al. [15]. One typical example is the periodic version of Hsu and Zhao [9,Example 4.2] w(x, n + 1) = R w(y, n)e rn−w(y,n) k(x − y)dy, (17) in which Σ N n=1 r n > 0. When r n is a constant, we see its propagation dynamics by Hsu and Zhao [9] and Lin and Su [21]. For (17), let w * > 0 such that of which the admissibility is clear since lim w→∞ [we rn−w ] = 0 for all n ∈ J. Then it satisfies (A1)-(A3).
If φ, c > 0 are wave profile and wave speed of (16), respectively, then Moreover, we also require the following asymptotic behavior which may formulate the successful biology invasion in population dynamics. The remainder of this section is to study the existence or nonexistence of positive solutions of (18)- (19). Define Lemma 5.1. φ(ξ, n), φ(ξ, n) are a pair of generalized upper and lower solutions of (18). Therefore, for any c > c , (18)- (19) admits a nonconstant solution φ(ξ, n) such that To prove the existence of (18), we directly verify the definition of generalized upper and lower solutions for any φ(y, n) ≥ φ(y, n) ≥ φ(y, n), y ∈ R, n ∈ Z.
Summarizing what we have done, we have the following conclusion.
Then c is the minimal wave speed of (16). Remark 6. In population dynamics, (19) could formulate a success biology invasion. For the difference equation w n+1 = b(n, w n ), there are some results on the global stability of unique positive periodic solution w n , see Zhou and Zou [49] and references therein for (17). In this paper, we do not focus on the precise condition on lim ξ→∞ φ(ξ, n) = w n , n ∈ J, which at least holds for r n ≤ 1 such that (17) is monotone by selecting w * = 1.
Consider Γ i (γ, c) = r i R k i (y)e γ(y−c) dy, i = 1, 2, then similar to that in Section 2, we have the following conclusion.   Therefore, |y|>ci k i (y)dy > 0 holds.
We define c * = max{c 1 , c 2 } and show the main results of this section. Theorem 6.2. Assume that (P) holds.
Proof. We prove the result for any fixed c > c * by constructing proper generalized upper and lower solutions. Define two N −periodic sequences {h i (n)} n∈Z by Select a constant η ∈ (1, 2) such that Define continuous functions where q > 1 is a constant. In particular, there exists q > 1 such that we obtain a pair of generalized upper and lower solutions of (24), which will be verified in the Appendix of this paper. By Theorem 3.2, we complete the proof. Lemma 6.6. Assume that k 1 (y), k 2 (y) admit compact supports. If c = c * , then (24) has a positive solution (φ 1 (ξ, n), φ 2 (ξ, n)) such that lim ξ→−∞ (φ 1 (ξ, n), φ 2 (ξ, n)) = (0, 0).
Proof. By the property of k 1 (y), k 2 (y), there exists D > 0 such that When c 1 = c 2 , we may assume that c 1 > c 2 and c * = c 1 without loss of generality, then we define two N −periodic sequences e 1 (0) = 1, e 1 (n + 1) = (1 + r 1 (n))e 1 (n) R k 1 (y)e γ1(y−c) dy, n ∈ Z, and h 2 (n) as that in the proof of Lemma 6.5. Let L 1 be a large constant such that where ξ 1,2 (n), ξ 1,1 (n) are the real roots of −L 1 ξe 1 (n)e γ1ξ = 1. Select P 1 > 0 be a constant and define continuous functions as follows 1 /L 2 1 , and φ 2 (ξ, n), φ 2 (ξ, n) are similar to that in the proof of Lemma 6.5. In the Appendix, we shall confirm that we obtain a pair of generalized upper and lower solutions of (24) by selecting P 1 > 0 large enough.
If c 1 = c 2 , by notations similar to the above, we may define In the Appendix, we shall confirm that we obtain a pair of generalized upper and lower solutions of (24) by selecting P 1 > 0, P 2 > 0 large enough. By Theorem 3.2, the proof is complete.
According to what we have done, we may finish the proof of Theorem 6.2 by verifying the following limit behavior as ξ → ∞ and applying Theorem 4.1.
Before ending this paper, we make the following remarks.
Remark 8. From Lemmas 6.5-6.6, when ξ → −∞, then the precise limit behavior of traveling wave solutions of (24) with minimal wave speed is different from that of large wave speed. In this paper, λ 1 and λ 2 may be finite. So, it is weaker than Lin [17,Section 5.2]. By similar discussion, we can improve the conditions in Lin [17,Section 5.3].
Remark 9. Similar to that in Lemma 6.6, if k(y) admits compact support, we may prove the conclusion of Lemma 5.2 by constructing upper and lower solutions, and such a traveling wave solution does not decay exponentially when ξ → −∞. Moreover, it is evident that our methods can be applied to l−species competitive systems.
Remark 10. Besides the minimal wave speed of traveling wave solutions, another important propagation threshold is the spreading speed, which has been widely studied for monotone semiflows or local monotone semiflows. For some nonmonotone systems [5,19,23,32,30], it has been proven that the minimal wave speed may describe the spreading phenomena of the corresponding initial value problems. Very likely the minimal wave speed of (21) may play such a role, and we shall further investigate the question.
Appendix. In this part, we shall complete the verification of Lemmas 6.5-6.6 by two lemmas, of which the verification is partly motivated by that in nonmonotone delayed equation, and we refer to a very recent paper [14].
Proof. We directly verify the desired inequalities for i = 1 or 2 and If φ i (ξ + c, n + 1) = 1, then the result is clear. Otherwise, we have which implies the conclusion on upper solution.
On the lower solution, it is clear if φ i (ξ + c, n + 1) = 0. Otherwise, let i ∈ (0, 1) such that γ i,1 (c) + i γ 3−i,1 (c) = ηγ i,1 (c). In particular, we have By directly direction, we have Note that the second line of the above inequality is a bounded constant, then the above is true by selecting large but finite q > 1 if which is also a constant. In fact, (28) is true since by (1) of Lemma 6.1. The proof is complete.
When c 1 < c 2 or c 1 > c 2 , the verification is similar to what we have done. We omit it and complete the proof.