Minimal Wave Speed in an Integrodifference System of Predator-Prey Type

)is 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.


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 K ∈ N is a positive constant, i ∈ 1, 2, . . . , K { }, n ∈ N, 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 K � 1, we see [5] u n+1 � u n e r 1− u n ( ) , has a positive invariant interval [0, L] with L � 1, r ∈ (0, 1], e r− 1 r , r > 1. However, (2) is not monotone in the interval [0, L] when r > 1, 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][2][3].
e 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: u n+1 � u n e r 1 1− u n − av n ( ) , u 0 > 0, n � 0, 1, 2, . . . , v n+1 � v n e r 2 1− v n +bu n ( ) , v 0 > 0, n � 0, 1, 2, . . . , ⎧ ⎨ ⎩ (4) in which all the parameters are positive. Regarding (4) as a model in population dynamics, then u n , v n stand for the densities of the prey and predator, respectively, r 1 > 0, r 2 > 0 depend on their intrinsic growth ratios, a reflects the capture rate, while b 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 predatorprey system for some parameters. More precisely, we see that (4) has a positive invariant region [0, Even if for n ∈ N, we cannot obtain u n , v n ≥ u n , v n ≥ u n , v n , n � 1, 2, . . . , when r 1 ∈ (0, 1], do not hold, which is similar to the possible nonmonotonicity in (2). e evolutionary process of many species can be modeled by the integrodifference equations involving birthdiffusion 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 x ∈ R, k 1 , k 2 , 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 k i is of Gaussian type, Li and Li [22] established the existence of traveling wave solutions connecting (0, 0) 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? irdly, what will happen if (8) does not hold? e 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: and there exists λ > 0 such that R k i (y)e λy dy < ∞ , i � 1, 2.
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 2 Discrete Dynamics in Nature and Society 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.

Preliminaries and Definitions
In this paper, we use the standard partial ordering in R 2 . at is, if then Moreover, C(R, R) denotes the set of uniformly continuous and bounded functions. When b > a > 0, then and C + is defined by We now recall the comparison principle and asymptotic spreading in Hsu and Zhao [19] and consider the initial value problem in which w(x) ∈ C + , k satisfies (A), and b: (14) admits a unique solution w n (x) ∈ C [0,B] , n ∈ N.
Remark 1. Due to the monotone condition, ω n (x) 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 by in which u, v ∈ C are the wave profile functions, while c > 0 is the wave speed. Let t � x + cn; then, u, v, and c must satisfy Similar to that in Li and Li [22], the purpose of this paper is to model that two species invade the same potential habitat, so ϕ, ψ satisfy for some δ > 0, of which the biological backgrounds are evident due to c > 0 and t � x + cn. us, 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 Discrete Dynamics in Nature and Society 3 systems without monotone conditions. By Lin [23], eorem 3.5, we present the following existence conclusion of (21). If for all t ∈ R and any u, v ∈ C + with ) are a pair of generalized upper and lower solutions of (21) without further monotone assumption. at is, the existence of (21) may be confirmed by the existence of generalized upper and lower solutions.

Existence of Nonconstant Traveling Wave Solutions
In this section, we shall show the existence of (21) by Lemma 2. Firstly, we define c * � max c * 1 , c * 2 by en, these constants satisfy the following lemma.
en, they satisfy the following facts for i � 1, 2: By these positive constants, we show the following conclusion.
are positive.
Proof. We now show (21) has a pair of generalized upper and lower solutions. Let ϵ > 0 and η > 1 such that Define continuous functions for t ∈ R, where p > L 1 + L 2 and q > 1 are positive constants clarified later. Evidently, if p and q are large, then we have Moreover, it is clear that For any fixed u(t), v(t) ∈ C with we first verify that If v(t + c) � L 2 , then it is clear that Otherwise, t < − c, and we have u(y) ≤ L 1 e ϵy , y ∈ R, and there exists some constant L > 0 such that us, it is sufficient to verify that there exists q > 1 large enough (such that − t > 0 is large enough) such that which is true provided that when qe ηλ c 21 (t+c) < L 2 . ese inequalities are evident for q > 1 large enough. After this, we may fix q > 1 that is independent on p.
If u(t + c) � L 1 e λ c 11 (t+c) − pe ηλ c 11 (t+c) > 0, then t + c < 0 by q > 1, and there exists K > 0 such that which is clear if p > L 1 is large. When When v(t + c) � L 2 e λ c 21 (t+c) − pe ηλ c 21 (t+c) > 0, then t + c < 0, and there exists A > 0 such that if p > 1 is large. e proof is complete.
Proof. We now prove the result by constructing generalized upper and lower solutions, which is similar to that in [24,25]. By the assumption, there exists S > 0 such that We first prove (1). Let L > 0 such that (− t + L)e λ c 1 t , t < 0, is strictly increasing and Define continuous functions for some p > 0, q > L 2 , η > 1. Evidently, we have if p and q are large. We now confirm that they are generalized upper and lower solutions of (21) if p and q are large, and the desirable inequalities are clear if they are constants. 6 Discrete Dynamics in Nature and Society For any fixed u(t), v(t) ∈ C with we first verify that By the definition of L > 0, we see that u(t + c) < L 1 implies − t 1 > S + 2c, and R u(y)e r 1 (1− u(y)− av(y)) k 1 (t − y)dy In the similar way, we may verify other inequalities.
In case (2), we define continuous functions by similar parameters in case (1). For case (3), we define by similar parameters in cases (1) and (2). By selecting large L, p, q, we may obtain the desirable inequalities. e proof is complete.

Minimal Wave Speed
In this part, we shall prove that c * is the minimal wave speed if which will be true throughout this section.

Theorem 3.
If c > (<)c * , then (21) and (22) have a strictly positive solution (60) If k 1 , k 2 admit the compact supports, then c ≥ c * if and only if (21) and (22) have a strictly positive solution In the following, we prove eorem 3 by three lemmas. then (62) e lemma is evident by (A) and the definition of traveling wave solutions, and we omit the proof. Clearly, this lemma implies a positive solution of (21) which is strictly positive. Furthermore, Lemma 4 implies the existence of nonconstant traveling wave solutions u, v of (21) if c ≥ c * , which satisfy (63) erefore, we need to show the asymptotic behavior of traveling wave solutions when t ⟶ ∞ and the nonexistence of traveling wave solutions when c < c * . On the limit behavior, we have the following conclusion.
is nondecreasing and continuous such that and there exists u * ∈ (0, we see that Similarly, we have for some δ > 0. e proof is complete. □ Lemma 6. If c < c * , then (21) does not have a positive solution satisfying (22).
is nondecreasing and continuous such that and there exists v * ∈ (0, From Lemma 1, we have lim inf erefore, we obtain lim inf by t � x + cn. A contradiction occurs. Similarly, we may prove that c < c * 1 is impossible. e proof is complete. □ Remark 3. If (8) and (59) hold, then we can obtain lim t⟶∞ u(t), lim t⟶∞ v(t) by the simple fluctuation technique. In fact, let (80) en, they are positive constants by Lemma 5, and the monotonicity and Fatou lemma further imply that Discrete Dynamics in Nature and Society and so u � u, v � v.

Discussion
Minimal wave speed of traveling wave solutions of evolutionary systems is very important in modeling the spatial expansion of individuals under consideration [26,27]. However, there are many open problems on the minimal wave speed of traveling wave solutions of different noncooperative systems. In this paper, we obtain the threshold for a predator-prey system without further monotone assumptions, which completes/improves the known results. We shall further utilize the techniques in this paper to more nonmonotone models including predator-prey systems and epidemic systems. For example, Lutscher ([15], Chapter 14) and Murray ([4], Sections 3.9 and 3.10) introduced some discrete time models; we will try to investigate the spatiotemporal modes of these models by developing the method in this paper.
In the literature, the minimal wave speed of traveling wave solutions may be the spreading speed of some systems [14,[16][17][18][19][20][21]. Here, the spreading speed is an important threshold of the corresponding initial value problems with some special initial conditions [28]. For some monotone semiflows, it has been proved the equivalence of these thresholds, see Liang and Zhao [18]. However, for nonmonotone systems, very likely the above conclusion fails, and we may refer to two papers [29,30] that studied a reaction-diffusion system of the predator-prey type. When the interspecific action between the predator and the prey is concerned, one typical problem is that the predator invades the habitat of the prey [31][32][33][34]. Due to the deficiency of the comparison principle, to further present the invasion threshold in such a nonmonotone system (9) is a challenging problem.

Data Availability
e data used to support the findings of this study are included within the article.