Abstract

This article studies the minimal wave speed of traveling wave solutions in an integrodifference predator-prey system that does not have the comparison principle. By constructing generalized upper and lower solutions and utilizing the theory of asymptotic spreading, we show the minimal wave speed of traveling wave solutions modeling the invasion process of two species by presenting the existence and nonexistence of nonconstant traveling wave solutions with any wave speeds.

1. Introduction

Discrete time systems are widely used to model the evolution of species without overlapping generations, and some novel dynamics including chaos have been presented comparing with the corresponding continuous time models; see some models by Beddington et al. [1], Hofbauer et al. [2], and Weide et al. [3] and monographs by Murray [4]. One typical discrete time model takes the form as follows [2]:in which is a positive constant, and the other parameters are given. Although the form of (1) seems to be simple, it may be nonmonotone in its positive invariant region. For example, when , we see [5]has a positive invariant interval with

However, (2) is not monotone in the interval when , which may reflect the phenomenon of overcompensatory growth in population dynamics (see Murray [4], Section 2.3). In the literature, the deficiency of monotonicity may lead to complex dynamics including the existence of nontrivial periodic solutions (the periodic solution is larger than 1) and other complex dynamics even in the scalar equations [4, 6, 7]. For coupled system (1), we may observe the plentiful dynamics by [1–3].

The energy transfer is one of the fundamental phenomena in the real world, and many predator-prey systems are established to model the process; we may refer to Murray [4] and Ruan [8] for some classical models and theoretical results. From (1), we may obtain different predator-prey systems by selecting proper parameters. In particular, after rescaling, one predator-prey system takes the form as follows:in which all the parameters are positive. Regarding (4) as a model in population dynamics, then stand for the densities of the prey and predator, respectively, depend on their intrinsic growth ratios, reflects the capture rate, while describes the energy conversion ratio. Similar to (2), we see that (4) may be nonmonotone since it does not admit the comparison principle appealing to the predator-prey system for some parameters. More precisely, we see that (4) has a positive invariant region defined by

Even iffor , we cannot obtainwhendo not hold, which is similar to the possible nonmonotonicity in (2).

The evolutionary process of many species can be modeled by the integrodifference equations involving birth-diffusion process, in which birth and diffusion occur in different stages of individuals [9]. Some important examples of integrodifference equations can be found in Bourgeois et al. [10], Jacobsen et al. [11], Kot [12, 13], Lui [14], and a recent book by Lutcher [15]. Since Weinberger [16], the propagation dynamics of integrodifference equations has been widely studied, of which the abstract results can be applied to continuous time models [17]. In particular, Liang and Zhao [18] and Weinberger et al. [17] studied the propagation dynamics of monotone abstract integrodifference systems and applied the abstract results to several parabolic-type systems generating monotone semiflows. We also refer to Hsu and Zhao [19], Wang and Castillo-Chave [20], and Yi et al. [21] for some integrodifference systems that are local nonmonotone.

When the birth-diffusion process is concerned in (4), Li and Li [22] considered the following integrodifference system:in which , which are called kernel functions, reflect the random walk of individuals, and monotone condition (8) holds. By direct calculation, when (8) is true, then (4) or (9) admits the comparison principle appealing to predator-prey systems and may have a coexistence steady state. When is of Gaussian type, Li and Li [22] established the existence of traveling wave solutions connecting with the coexistence state if the wave speed is larger than the threshold, which is finished by constructing upper and lower solutions.

Although Li and Li [22] investigated the traveling wave solutions of (9), there are some open problems for this system. Firstly, only the large wave speed is investigated, what is the threshold on the existence of nontrivial traveling wave solutions? Secondly, the movement law in [22] is too special, is it possible to consider the equation with more kernel functions? Thirdly, what will happen if (8) does not hold? The purpose of this paper is to further investigate the nontrivial (nonconstant) traveling wave solutions of (9) motivated by the above three questions. For example, we only make the following assumption on kernel functions:(A), is Lebesgue-measurable and integrable such that and there exists such that .

Evidently, (A) contains the Gaussian kernel as a special case. We also remove condition (8) to study the threshold that determines the existence or nonexistence of nontrivial traveling wave solutions for all positive wave speeds.

In Section 2, we shall give the definition and recall some known results. In Section 3, the existence of nonconstant traveling wave solutions is established by constructing generalized upper and lower solutions. Furthermore, we show the asymptotic behavior of traveling wave solutions and nonexistence of traveling wave solutions, which indicate a minimal wave speed of nontrivial traveling wave solutions. Finally, a further discussion is presented in Section 5.

2. Preliminaries and Definitions

In this paper, we use the standard partial ordering in . That is, ifthen

Moreover, denotes the set of uniformly continuous and bounded functions. When , thenand is defined by

We now recall the comparison principle and asymptotic spreading in Hsu and Zhao [19] and consider the initial value problemin which , satisfies (A), and such that(B)There exists such that is continuous, and satisfying

Lemma 1. If , then (14) admits a unique solution .(1)Assume that is nondecreasing. If , such that then .(2)Define(a)If admits nonempty compact support, then(b)If admits nonempty support, then

Remark 1. Due to the monotone condition, is called an upper solution if we take in the first item of Lemma 1, and we obtain a lower solution by taking .
To study the traveling wave solutions, we first give the following definition. Here, a traveling wave solution of (9) is a special entire solution defined byin which are the wave profile functions, while is the wave speed. Let ; then, must satisfySimilar to that in Li and Li [22], the purpose of this paper is to model that two species invade the same potential habitat, so satisfyfor some , of which the biological backgrounds are evident due to and . Thus, we need to investigate the existence of an integral system satisfying asymptotic boundary conditions.
In [23], Lin investigated the existence and asymptotic behavior of traveling wave solutions of integrodifference systems without monotone conditions. By Lin [23], Theorem 3.5, we present the following existence conclusion of (21).

Lemma 2. Assume that such thatIffor all and any withthen (21) has a solution such that

Remark 2. In Lemma 2, are a pair of generalized upper and lower solutions of (21) without further monotone assumption. That is, the existence of (21) may be confirmed by the existence of generalized upper and lower solutions.

3. Existence of Nonconstant Traveling Wave Solutions

In this section, we shall show the existence of (21) by Lemma 2. Firstly, we define by

Then, these constants satisfy the following lemma.

Lemma 3. . LetThen, they satisfy the following facts for : (c1) If , then has two positive roots such that with any . (c2) If , then . (c3) If , then . Moreover, has a unique solution such thatBy these positive constants, we show the following conclusion.

Theorem 1. For any fixed , (21) has a positive solution such thatare positive.

Proof. We now show (21) has a pair of generalized upper and lower solutions. Let and such thatDefine continuous functionsfor , where are positive constants clarified later. Evidently, if are large, then we haveMoreover, it is clear thatFor any fixed withwe first verify thatwhich is clear if . Otherwise, we haveIf , then it is clear thatOtherwise, , and we have , and there exists some constant such thatThus, it is sufficient to verify that there exists large enough (such that is large enough) such thatwhich is true provided thatwhen . These inequalities are evident for large enough. After this, we may fix that is independent on .
When , it is clear thatIf , then by , and there exists such thatifwhich is clear if is large.
When it is clear thatWhen , then , and there exists such thatif is large. The proof is complete.