EXISTENCE AND NON-MONOTONICITY OF TRAVELING WAVE SOLUTIONS FOR GENERAL DIFFUSIVE PREDATOR-PREY MODELS

. This paper is concerned with the existence and non-monotonicity of traveling wave solutions for general diﬀusive predator-prey models. By us- ing Schauder’s ﬁxed point theorem and the existence of contracting rectangles, we obtain the existence result. Then we investigate the asymptotic behavior of positive monotone traveling wave solutions by using the modiﬁed Ikehara’s Theorem. With the help of their asymptotic behavior, we provide a suﬃ- cient condition which guarantee that all positive traveling wave solutions of the system are non-monotone. Furthermore, to illustrate our main results, the existence and non-monotonicity of traveling wave solutions of Lotka-Volterra predator-prey model and modiﬁed Leslie-Gower predator-prey models with dif- ferent kinds of functional responses are also discussed.


1.
Introduction. The purpose of this work is to investigate the existence and nonmonotonicity of traveling wave solutions for general diffusive predator-prey models governed by the following equations where d 1 , d 2 > 0 are diffusion coefficients, and f n : R 2 → R are reaction functions for n = 1, 2. We can regard u n = u n (x, t) as the population density of species u n at time t and position x, then system (1) can be used to describe the interaction of species u 1 and u 2 . In this article we assume the interaction between species u 1 and u 2 belongs to the predator-prey relationship, i.e. one species (predator) captures and feeds on the other species (prey). Such a interaction is governed by the functional responses f 1 and f 2 .
In past decades, traveling wave solutions for system (1) with different functional responses have been extensively studied by many researchers. For examples, Dunbar [9,10,11] proved the existence of traveling wave solutions for system (1) with the specific functional responses: • f 1 (u 1 , u 2 ) = u 1 (A − Bu 2 ), f 2 (u 1 , u 2 ) = u 2 (Eu 1 − C); • f 1 (u 1 , u 2 ) = Au 1 (1 − u 1 /K) − Bu 1 u 2 , f 2 (u 1 , u 2 ) = u 2 (Eu 1 − C); (2) by using the shooting method, Wazewski Theorem, Liapunov function and LaSalle's invariance principle. Different to Dunbar's method, Gardner [14] used a modification of Conley index (called the connection index) to get the existence of traveling wave solutions of (1) for more abstract functional responses. However, many concrete models may not fit well with the frame established in Gardner's abstract approaches. Later, the authors of the works [1,12,15,18,19,20,23,26] improved the technique of Dunbar's approach to investigate the traveling wave solutions of system (1) with different types of functional responses. For examples, Huang [20] used a geometric approach to derive the existence of traveling waves for some classes of non-monotone reaction-diffusion systems. Ai et al [1] also used the squeeze method and a Lyapunov function method to study the traveling waves for a generalized Holling-Tanner predator-prey model. Additionally, with the help of the upperlower solutions and Schauder's fixed point theorem, Ma [30] studied the existence of traveling wavefronts for reaction-diffusion systems with quasimonotonicity reactions. In 2010, Li et al [25] also proved the existence of such solutions by applying the cross iteration method. Improving the method of Li et al [25], Lin and Ruan [28] investigated the existence of traveling wave solutions for the following delayed reaction-diffusion systems where x ∈ R, t > 0, d i > 0 for i = 1, · · · , n, v t (x) := v(x, t + s) for s ∈ [−τ, 0] and τ > 0 is the time delay. Applying the Schauder's fixed point theorem, the traveling wave problem of (3) can be reduced to the existence of a pair of upper and lower solutions. Using the technique of contracting rectangles (see Definitions 2.1 and 2.4), the asymptotic behavior of traveling wave solutions for delayed diffusive systems is then obtained. Furthermore, they applied their results to Lotka-Volterra reaction-diffusion system with distributed delay and discussed the existence of nonmonotone traveling wave solutions of the system. Recently, Pan [31] also used contracting rectangles to study the convergence and traveling wave solutions for a predator-prey system with distributed delays. For the other related references, we refer to [6,16,17,27,29,34,35] and the references cited therein. Although there are several literature related to the existence of traveling wave solutions, it is worthy mentioning that there are few papers considering the non-monotonicity of traveling wave solutions for predator-prey models.
Since system (1) is not quasi-monotone, it becomes harder to study the traveling wave solutions and some new phenomena might occur, e.g., the existence of nonmonotone traveling wave solutions. Motivated by above mentioned literature, we are concerned with the existence and non-monotonicity of traveling wave solutions for system (1) with more general functional responses. In this article, we assume the functional responses satisfy the following assumptions.
In the final section, we will illustrate some models which have functional responses satisfying the above assumptions, e.g. the Lotka-Volterra predator-prey model and diffusive predator-prey models with modified Leslie-Gower term and different kinds of functional response. Here we point out the significance of assumptions (H1)-(H3) to the proof of the main result. First, by (H1) and (H2) we know that g 1 (0) > 0 and g 2 (0) > 0 which implies The conditions (5) will be used in proving the non-existence of traveling wave solutions (see Subsection 4.1). In addition, by (H1) and (H3) and the condition g(α) = 0, we have Therefore, by Intermediate Value Theorem, there exists a number k 1 ∈ (0, α) such that g 2 (k 1 ) = g 1 (k 1 ). Let us define k 2 := g 2 (k 1 ) = g 1 (k 1 ), then it is clear that Hence (k 1 , k 2 ) is a positive equilibrium of system (1). Moreover, by (6), it is clear that (k 1 , k 2 ) is the unique positive equilibrium (see Figure 1). By assumption (H2), it's easy to verify that the equilibria (0, 0) is unstable, and (k 1 , k 2 ) is stable for system of (8) with d 1 = d 2 = 0. Due to the importance of coexistence in population dynamics, we will look for the existence of monostable traveling wave solutions for system (1) which connect these two equilibria. Figure 1: graphs of functions g 1 (·) and g 2 (·).

CHENG-HSIUNG HSU AND JIAN-JHONG LIN
A solution (u 1 (x, t), u 2 (x, t)) of system (1) is called a traveling wave solution if there exist c > 0 and smooth functions φ 1 (·), φ 2 (·) such that Note that φ 1 (·), φ 2 (·) are called the wave profiles, c is the wave speed and ξ := x+ct is the moving coordinate. Substituting the ansätzes of (7) into system (1), we can obtain the profile equations In this paper, we look for the solutions of system (8) satisfying the conditions By using the Schauder's fixed point theory via the existence of contracting rectangles (cf. [33,Theorem 5.2.5], Lin and Ruan [28] and Lemma 2.6), we can obtain the existence of traveling wave solutions of system (1) provided the wave speed is larger than the minimal speed c * . In addition, to study the monotonicity of traveling wave solutions, we first apply a modified version of Ikehara's Theorem (see Lemma 3.1) to investigate the asymptotic behavior of positive monotone traveling wave solutions.
With the help of their asymptotic behavior, we can derive the non-monotonicity of traveling wave solutions of system. Our main results are stated as follows.
Moreover, we have the following existence result of traveling wave solution when the wave speed is equal to the minimal speed c = c * .
Noting that although we follow the arguments similar to [28], there are some differences between our work and theirs. First, Lin and Ruan [28] proved the existence of strict contracting rectangle for a specific model. In this paper we can construct the strict contracting rectangle for more general models. Hence, our proof becomes more difficult and significantly different from that in [28]. Additionally, we provide some sufficient conditions which guarantee that any positive solution of system (8) satisfying (9) is non-monotone. To the best our knowledge, the proof arguments for this part are new. Moreover, our result can be applied to more ecology models, e.g. the Lotka-Volterra predator-prey model, the modified Leslie-Gower predator-prey system with different functional responses, and so on.
The rest of this paper is organized as follows. In section 2, we construct a pair of upper and lower solutions of system (8) and prove the existence of the strict contracting rectangle. In section 3, we apply the Ikehara's theorem to study the asymptotic behavior of positive monotone solutions for system (8) satisfying (9). Then we prove the results of Theorem 1 in Section 4. In the final section, we apply our theorem to some predator-prey models.

Preliminaries.
To prove Theorem 1.1 (1) and (2), we need to construct a pair of upper and lower solutions of system (8) and establish the strictly contracting rectangle. First we give the definition of the upper and lower solutions of system (8).

Construction of upper and lower solutions.
To construct a pair of upper and lower solutions, we first consider the characteristic functions of system (8) given by ∆ n (λ) = d n λ 2 − cλ + f n (0, 0), for any 1 ≤ n ≤ 2. (16) It is clear that the function ∆ n (λ) arises from the linearization of system (8) about the equilibrium (0, 0). One can see later that real roots of the characteristic functions play an important role in the construction of upper and lower solution. The following lemma follows from (16) obviously. (1) For any 0 < c < 2 d n f n (0, 0), the function ∆ n (λ) has no real root.

CHENG-HSIUNG HSU AND JIAN-JHONG LIN
Based on Lemma 2.2, we can denote the threshold wave speed Next, let c > c * and η be chosen such that Then, let q > 0 be fixed, we set the function n (ξ) and constantst n , t * n by n (ξ) := e λnξ − qe ηλnξ ,t n := It is easy to check that n (ξ) has a global maximum at ξ =t n with n (t * n ) = 0,t n < t * n and lim Furthermore, let t 4 be the number satisfying e λ2t4 + qe ηλ2t4 = g 2 (α).
By Lemma 2.3, we can prove the existence of solutions for system (8) (in Section 4) by using crossing iteration method and Schauder's fixed point theorem. However, from the formulas (29) and (30), we only know that Φ(−∞) = Φ(−∞) = (0, 0), and which implies the solutions of (8) satisfying the left condition of (9). To guarantee the solutions satisfying the right condition of (9), we need to establish the strictly contracting rectangles in the following subsection.

2.2.
Existence of strictly contracting rectangles. First, we give the definition of the strictly contracting rectangle (cf. [33, Section 5.2]).
is called a strict contracting rectangle if the following conditions hold: a n (s) > 0 and b n (s) < 0, for any s ∈ (0, 1), n = 1, 2, f 2 (x, b 2 (s)) < 0, for any s ∈ (0, 1) and Let's remark that the notation a ≤ b means the standard partial ordering in R 2 and the notation [a, b] stands for the rectangle {x ∈ R 2 | a ≤ x ≤ b}.
In the sequel, to prove the existence of the strictly contracting rectangle, we define the regions (see .
Then we have the following result.
(2) Let L 2 be the line segment connecting (2α, 0) and (k 1 , k 2 ), i.e., the graph of the set {(b 1 (s), a 2 (s)) | 0 ≤ s ≤ 1}. Note that the slope of the tangent L 2T to the graph of function g 1 at (k 1 , k 2 ) is g 1 (k 1 ). Since g 1 is decreasing and g 2 is increasing, by (H3) and the fact k 1 < α, it is easy to see that Then the slope of line L 2T is less than the slope of line L 2 . Since g 1 is concave, L 2 lies above the graph of function g 1 on [k 1 , 2α] except the point (k 1 , k 2 ). Thus, it is easy to check that (b 1 (s), a 2 (s)) ∈ Ω 2 for any s ∈ (0, 1). See the right graph of Figure 3.
(3) Let L 3 be the line segment connecting (2α, 2g 2 (α)) and (k 1 , k 2 ), i.e., the graph of the set Since g 2 is convex, L 3 lies above the graph of function g 2 on [k 1 , 2α] except the point (k 1 , k 2 ). Then one can easily see that desired result is true. See the left graph of Figure 4.
(4) Let L 4 be the line segment connecting (0, 0) and (k 1 , k 2 ), i.e., the graph of the set {(a 1 (s), a 2 (s)) | 0 ≤ s ≤ 1}. Note also that the slope of the tangent L 4T to the graph of function g 2 at (k 1 , k 2 ) is g 2 (k 1 ). It follows from (H3) that This implies that the slope of line L 4T is less than the slope of the line L 4 . Since g 2 is convex, L 4 lies below the graph of function g 2 on [0, k 1 ] except the point (k 1 , k 2 ). Therefore, it is easy to see that desired result is true. See the right graph of Figure  4. The proof is complete. Figure 3: The regions of Ω 1 , Ω 2 , line segments L 1 , L 2 and tangent line L 2T .
x 1 3. Asymptotic behavior of traveling wave solutions. In this section, we will investigate the asymptotic behavior of positive monotonic traveling wave solutions of system (1), provided that such solutions exist. With the help of their asymptotic behavior, we can prove the statement of part (3) of Theorem 1.1 (see Section 5.) To this end, we first introduce the following modified Ikehara's Theorem. Assume that system (8) admits a positive monotone solution (φ 1 (ξ), φ 2 (ξ)) satisfying (9). Then we have the following properties.
Therefore, we investigate the properties of roots for ∆ 3 (λ) in the following lemma.
(2) One of the following statements holds.
Since (ψ 1 (ξ), ψ 2 (ξ)) satisfies (48), one can verify that Then it follows from the stable manifold theorem that λ < 0 ψ n (ξ) = O(e λξ ) and ψ n (ξ) = O(e λξ ) for ξ 1 = ∅, for any 1 ≤ n ≤ 2. This implies that Λ n and Λ n are well defined because of λ < 0 ψ n (ξ) = O(e λξ ) and ψ n (ξ) = O(e λξ ) for ξ 1 Now we multiply both sides of the first and second equations in system (48) by e −λξ and integrate them over (−∞, s). By integration by parts, one can see (67) where p n (λ) is defined by (51) and Let λ ∈ C with max{Λ 1 , Λ 2 } < Reλ < 0 be fixed and s tend to positive infinity. Then one can see where In the following, we want to use (66), (67) and (68) to claim Λ 1 = Λ 2 = Λ 1 = Λ 2 . To this end, assume the desired result is not true. Here we just focus on the condition Λ 1 < Λ 2 because the other conditions can be proved by similar arguments. Additionally, due to Λ 2 ≤ Λ 2 , we divide this proof into two parts.
Based on the previous claim, let's define Now we show that Λ > −∞. From (68), it is easy to check that for any Λ < Reλ < 0 with ∆ 3 (λ) = 0. Add the first equation in (73) to second equation in (73) and then we get for any Λ < Reλ < 0. If the case Λ = −∞ holds, one can choose λ < 0 with |λ| being large such that the sum of the integrand in each term of equation (74) is positive. This leads a contradiction. Thus, we have that Λ > −∞. Additionally, it follows from (72) and the definition of Λ n that (64) is true. Finally, we claim that Λ is a negative root of ∆ 3 (λ). By (73) again, it is clear that for any Λ < Reλ < 0 and ∆ 3 (λ) = 0. From the definition of Λ and the property of Laplace transforms [23, page 58], one can see the left side of equation (75) is singular at Λ. But, if the condition ∆ 3 (Λ) = 0 holds, it is easy to check that the right side of equation (75) is non-singular at Λ, which gives a contradiction. The proof is complete.
It's obvious that any fixed points of operator G are solutions of system (8). Hence our goal is to prove the existence of fixed points of G. (1) and (2) of Theorem 1.1.
In addition, by Lemma 2.6, we know that there exists a strict contracting rectangle. Note that Φ * (ξ) and Φ * (ξ) are uniform bounded. Then, following the same argument as that in [28, Theorem 3.2], we can conclude that Φ * (∞) = (k 1 , k 2 ). The proof is complete.
Case 2. Assume that  in Lemma 3.3 holds. The proof for this part is similar to previous condition and hence the proof is skipped.
Case 3. Assume that (2)(3) in Lemma 3.3 holds. Under this condition, we know that ∆ 3 (λ) has no negative root. However, it follows from Lemma 3.4 that Λ is a negative root of ∆ 3 (λ). This leads a contradiction.
Next, motivated by the works of [16,24,31,32], we prove the results of Theorem 1.2 as follows.
5. Some applications. In this section, we will apply our main theorem to some diffusive predator-prey models.
5.1. The Lotka-Volterra Predator-Prey model. The Lotka-Volterra predatorprey model is described as the following equations where u n := u n (x, t) : R 2 → R and all parameters are positive. The traveling wave solutions for this model with delays has been considered in Lin et al [25]. Now let us set α = 1 a 11 , g 1 (x) = 1 a 12 − a 11 a 12 x and g 2 (x) = 1 a 22 + a 21 a 22 x.
Hence, as a consequence of Theorem 1.1, we have the following result. we also obtain the assertion (3) of Theorem 1.1.

5.2.
The modified Leslie-Gower Predator-Prey system with different functional responses. The diffusive predator-prey models with modified Leslie-Gower term and functional response F (u 1 ) can be described by the following equations where u n := u n (x, t) : R 2 → R, all parameters are positive and γ is called the Leslie-Gower term. The non-diffusive system (94) with Holling-type II functional response F (u 1 , u 2 ) = a 1 u 1 /(γ 1 + u 1 ) was first proposed by Aziz-Alaoui and Okiye [2]. From their work, we know that the parameter r 1 means the growth rate of prey u 1 ; b measures the strength of competition among individuals of species u 1 ; a 1 is the maximum value which per capita reduction rate of u 1 can attain; γ 1 (respectively, γ) measures the extent to which environment provides protection to prey u 1 (respectively, to predator u 2 ); r 2 means the growth rate of u 2 and a 2 has a similar meaning to a 1 . Furthermore, the authors considered the boundedness of solutions, existence of an attracting set and global stability of the coexisting interior equilibrium. Additionally, Zhou [38] investigated the existence, multiplicity and stability of positive steady-state solutions of system (94) According to the notations used in Section 1, we set Then g 1 (0) = γ 1 /a 1 , g 2 (0) = γ/a 2 , In addition, g 2 (x) > xg 2 (x) for all x ∈ [0, 2α], and g 2 (2α) < 2g 2 (α) < −αg 1 (0) when 2 + bγ a 2 b < 2(1 + bγ) Then one can easily verify that (H1)∼(H3) hold under the assumption Moreover, by direct computations, the conditions of (10) are equivalent to Since k 1 and k 2 are independent of r 1 and r 2 , we can find parameters satisfying (96). Hence, as a consequence of Theorem 1.1, we have the following result.  Similarly, we set α = 1/b, g 1 (x) = (1 − bx)(γ 1 + γ 2 x 2 ) a 1 x , g 2 (x) = x + γ a 2 , f 1 (x 1 , x 2 ) = r 1 (1 − bx 1 − a 1 x 1 x 2 γ 1 + γ 2 x 2 1 ) and f 2 (x 1 , x 2 ) = r 2 1 − a 2 x 2 γ + x 1 .
In fact, (97) and (98) hold for all b when 4a 2 1 γ 2 < a 2 γ 1 (a 2 γ 2 − 4a 1 ). (2) No matter the functional response is Holling-type II or III, system (94) always admit traveling wave solutions when the strength of competition among individuals of species u 1 (i.e. b) is large enough.
Since k 1 and k 2 are independent of r 1 and r 2 , we can find parameters satisfying the above inequality. Therefore, we have the following results.