Tao SU (蘇濤) Guobao ZHANG (張國寶)?

Colloge of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

E-mail : 970913788@qq.com; zhanggb2011@nwnu.edu.cn

Here, uj(t) and vj(t) are the population densities of the prey and predator species at niche j and time t, respectively. A, a, d and B are positive constants. The parameter d > 0 is a rescaled diffusion coefficient of the predator species while the diffusion coefficient for the prey is rescaled to be 1. The functional response of predator to prey is givenwhich is ratio-dependent Holling type II. The parameter A is the capturing rate, and a is the halfcapturing saturation constant. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of the predator species follows a logistic dynamic with a varying carrying capacity proportional to the density of prey. B denotes the intrinsic growth rate of the predator.

It is easy to see that system (1.1) has two nonnegative spatially constant equilibria (1,0)and (k,k), where k = 1 ?A/(a+1) when A < a+1. Clearly, (1,0) is unstable and (k,k)is stable. This implies that we can observe the invading coexistence phenomenon between the resident (the prey) and the invader (the predator). To describe such an invading coexistence phenomenon, the traveling wave solution plays an important role. Here, a traveling wave solution (traveling wave, for short) of (1.1) is a special translation invariant solution of the form

where c > 0 is the wave speed at which the wave profile (φ1,φ2) ∈ C(R, R2) propagates in spatial media R. Thus, (φ1,φ2) with c>0 must satisfy

and the asymptotic boundary conditions

Note that the existence of such traveling waves with c>0 means the successful invasion of the predator. Therefore, we call a solution of (1.2) satisfying (1.3) an invasion traveling wave of (1.1). The main purpose of this paper is to prove the existence and nonexistence of invasion traveling waves of (1.1).

Traveling waves of lattice dynamical systems have been widely studied for the past several years; see [1, 3, 4, 6, 20–22, 24, 27, 28, 30]for scalar lattice differential equations, and [5, 9–12, 14, 16, 19, 33]for lattice differential systems. It is well known that if the nonlinearity of lattice dynamical systems enjoy the monotone property,then the theory of monotone dynamical systems provides a powerful tool to prove the existence of traveling wave solutions[17]. However,predator-prey systems do not generate monotone semi-flows,so these theories cannot be applied.To the best of our knowledge, there are very few research results on the traveling waves of predator-prey systems on a lattice. Huang, Lu and Ruan [14]investigated a system of delayed lattice differential equations with partial monotonicity. By using Schauder’s fixed point theorem and a new cross-iteration scheme, they established the existence of traveling wave solutions. In Remark 4.9 of [14], the authors pointed out that the existence of traveling wave solutions of some discrete predator-prey models with time delays can be obtained by finding a suitable pair of upper and lower solutions. More recently, Chen et al. [5]considered a lattice predator-prey dynamical system without delay:

where the functional response of predator to prey is given by rkuj, which is of Lotka-Volterra type. By applying Schauder’s fixed point theory with the help of suitable upper and lower solutions, the existence of traveling waves of system (1.4) with speed c ≥ c?has been proven.Moreover, the existence of traveling waves for the continuous diffusion case of (1.4) has also been studied ([5]). For the existence of traveling waves of single equations and other predatorprey systems with local and nonlocal diffusion, we can refer the reader to [7, 8, 13, 15, 18, 25,26, 29, 31, 32, 34].

Motivated by [5], in this article, we still use Schauder’s fixed point theory to prove the existence of invasion traveling waves of system (1.1). We should point out that although this method is very standard, the construction of suitable upper and lower solutions of (1.2) is not trivial. Because the functional response of predator to prey in (1.1) is more complicated than that of (1.4), we construct a pair of upper and lower solutions that are different from those in[5]. In order to establish the existence of invasion traveling waves with critical speed, we take a limiting argument. The key step in adopting this argument is to derive the tail behavior of the solution. Finally, when the wave speed is smaller than the threshold, the nonexistence of invasion traveling waves is established by the theory of asymptotic spreading developed by Liu and Weng [20].

In order to obtain the existence of invasion traveling waves, we impose the following assumption on a and A:

Now we give the main results of this paper.

Theorem 1.1Suppose that(H)holds. Then,for each c ≥ c?,system(1.1)has a positive traveling wave(uj(t),vj(t))=(φ1(j+ct),φ2(j+ct)) connecting the equilibria(1,0)and(k,k).

Theorem 1.2Suppose that (H) holds. Then, for 0 < c < c?, system (1.1) has no nonnegative traveling wave connecting (1,0) and (k,k).

Remark 1.3Theorems 1.1 and 1.2 show that c?is the minimal wave speed.

The article is organized as follows: in Section 2, we give a general result on the existence of traveling wave solutions. In Section 3, we study the existence and nonexistence of invasion traveling waves of (1.1).

2 A General Result

In this section, we present a general existence result for solutions of system (1.2). For convenience, we introduce some notations. Define

which is a Banach space with the maximum norm |·|. We also define

It is clear that (1.2) can be rewritten as follows:

Define an operator G =(G1,G2):

Clearly, a fixed point of G = (G1,G2) is a solution of (2.2). We prove the existence of the fixed point by Schauder’s fixed point theorem.

We first give the definition of a pair of coupled upper and lower solutions of (2.2).

Definition 2.1are called a pair of upper and lower solutions of (2.2) ifexist and the inequalities

hold for ξ ∈ RT with some finite set T={Ti∈ R:i=1,2,···,n}.

Now we are in position to state the general existence result of solution of (2.2) by using the upper-lower solutions.

Proposition 2.2Assume that (H) holds and system (2.2) has a pair of upper and lower solutionsas defined in Definition 2.1 withThen, for each c>0, (2.2) admits a positive solution (φ1,φ2) such that

ProofFor anydefine

It is easy to see that Γ is a nonempty bounded closed convex set with respect to the weighted norm |·|μ.

We claim that (i) G(Γ) ? Γ; (ii) G : Γ → Γ is completely continuous with respect to the weighted norm. Then, the proposition can be proved by using Schauder’s fixed point theorem.By the monotone condition on F1and F2, claim (i) can be proved by a similar argument as to that in [5, Lemma 3.4]. The proof of claim (ii) is similar to that of [14, Lemmas 3.5 and 3.7].We omit these here. The proof is complete.

3 Traveling Waves of (1.1)

In this section, we first give a pair of upper and lower solutions of (2.2), and then prove the existence and nonexistence of invasion traveling waves of (1.1).

3.1 Upper and lower solutions

Define

for any c>0 and λ ∈ R. It is easy to see that the following result holds:

Lemma 3.1There exist c?>0 and λ?>0 such that the following items hold:

(ii) For any given c > c?, ?(λ,c) = 0 has two distinct positive roots λ2(c) and λ3(c).Moreover,assume that 0< λ2(c)< λ3(c) holds. Then,

(iii) If 00, then ?(λ,c)=0 has no real root.

For convenience, in what follows, we denote λi(c) by λi, i=2,3, respectively. Define

where λ2is the minimal positive root of (3.1), and λ1is chosen such that

ProofNote that λ1, λ2, η >0 are well-defined because c>c?, and it is easy to see thatIn what follows,we need to verify the conditions of upper and lower solutions separately.

First, we prove that (2.4) holds. Becausewe have thatandHence, we obtain

Secondly, we show that (2.5) holds. It is clear that

where we used the fact that the functionis increasing in x, where m > 0, and the fact thatHence, by (3.4), we obtain

Thirdly, we prove that (2.6) holds. It is easy to see that

Finally, we show that (2.7) holds. Note that

3.2 Existence and nonexistence of traveling waves

where D, B, K are positive constants. Denote that BN= {j ∈ N||j| ≤ N,N ∈ N}, uj(t) =u(j,t), j ∈ Z, U(t)=U(·,t)={uj}j∈Z, suppU(·,t)=is the support of U(·,t).

The following asymptotic result of solutions of (3.7) has been proved in [20, Theorem 5.2]:

Lemma 3.3Assume that Z(t) = {zj(t)}j∈Zis any solution of (3.7), and satisfies that:(1) Z0= {zj0∈ [0,M]}j∈Zis isotropic, where M > K is a some constant; (2) there exists N ∈ N such that suppZ0(·)? BNand zj0>0 for |j|≤ N. Then, for any c2>c?>c1>0, we have

To prove the existence and nonexistence of invasion traveling waves of (1.1), we need the following comparison principle, which can be proved by using an argument used in [23,Lemma 4.1](for the reader’s convenience, we give the complete proof here):

Lemma 3.4Assume that zj(t) is the bounded solution of (3.7), and(t)is bounded for j ∈ Z, t ≥ 0, differentiable in t>0, and satisfies

ProofBecause zj(t) andare bounded, there exist Mi∈ R, i = 1,··· ,4, such that M1≤zj(t) ≤M3and?j ∈ Z, t > 0. Then, wj(0) ≤ 0 for j ∈ Z, and wj(t) is continuous and bounded. LetSuppose that the assertion of the lemma is not true. Let κ>0 be such thatThen, there exists t0>0 such that ω(t0)>0 and

Now we are ready to prove Theorem 1.1. We divide the proof of Theorem 1.1 into two theorems: Theorems 3.5 and 3.7.

Theorem 3.5Suppose that (H) holds. Then, for each c > c?, system (1.1) admits a positive traveling wave (uj(t),vj(t)) = (φ1(j +ct),φ2(j +ct)) ∈connecting the equilibria(1,0) and (k,k).

ProofFrom Lemma 3.2, we can see that there exists a pair of upper and lower solutionsfor the system (2.2). By Proposition 2.2, we obtain that there exists a solution (φ1(ξ),φ2(ξ)) of (2.2) satisfyingMoreover,it is easy to see that

Then, by Lemmas 3.3 and 3.4, we obtain that when |j|≤ c1t with c1

Clearly, when |j|≤ c1t, j+ct> ?c1t+ct=(c ? c1)t → +∞ as t → +∞, because c>c?>c1.Hence, we have

and ξ1>M1+N withsuch that

Because(φ1(ξ),φ2(ξ))is a fixed point of G defined by(2.3)and based on the mixed monotonicity of F1and F2, we have the following inequality:

By the arbitrariness of ε, we obtain

Similarly, we can obtain

By (3.12) and (3.15), and consideringone has

Similarly, we can obtain from (3.13) and (3.14) that

Adding together (3.16) and (3.17) yields

By the assumption (H), we obtain

Substituting (3.19) into (3.12) and (3.13), one has

Therefore,when c>c?,the system(1.1)admits a traveling wave solution connecting(1,0)and(k,k).

Next, we shall use a limiting argument to prove the existence of solutions of (1.2)and(1.3)with speed c=c?.

Lemma 3.6Let (φ1n,φ2n) ∈be the solutions of (1.2) and (1.3) with speed cn>c?, n ∈ N, obtained in Theorem 3.5. Then,are uniformly bounded and equicontinuous.

The proof is easy, so we omit it here.

Theorem 3.7Suppose that (H) holds. Then, system (1.1) has a positive traveling wave(uj(t),vj(t)) = (φ1(j +ct),φ2(j +ct)) with speed c = c?, which connects the equilibria (1,0)and (k,k).

ProofLet cn? (c?,c?+1) be a decreasing sequence such thatFollowing Theorem 3.5,for each cn,there exists a traveling wave solution(φ1n,φ2n)∈of (1.1)satisfying(1.2) with (1.3).

For c > c?, as φi≥ 0 for ξ ∈ R, it is not hard to prove that φi(ξ) > 0 for ξ ∈ R, i = 1,2.Let

Then, we can derive from the second equation in (1.2) that

which implies that there exists M >0 sufficiently large such thatwhen ξ < ?M.

Because (φ1n(·+a),φ2n(·+a)), for any a ∈ R, are also solutions of (1.2) with (1.3), from the above proof, we can assume that

where δ is small enough satisfying 0< δ

We now prove that (φ1n,φ2n) has a subsequence converging point-wise to (φ1,φ2), which satisfies(1.2)for c=c?. By using Lemma 3.6,the Arzela-Ascoli theorem and a diagonalization argument, we can find a subsequence of (φ1n,φ2n), again denoted by (φ1n,φ2n), such that(φ1n,φ2n) andconverge uniformly on every bounded interval (and so point-wise on R) to functions (φ1,φ2) andrespectively. Therefore, it follows immediately that(φ1,φ2) satisfies system (1.2).

Because (φ1n,φ2n)∈, we have(φ1,φ2)∈. Hence, by a similar argument as to that for the proof of Theorem 3.5, we may confirm that

Then, we divide our proof into the following two cases:

Case 1φ1(?∞) exists. In this case, we first prove that φ2(?∞) exists and is equal to 0,and then we get φ1(?∞)=1. If we suppose thatthen there exist sequences{xn}and{yn} satisfying xn,yn→ ?∞ as n → ∞ such that

Because φ1(?∞) exists, by the Barbalat Lemma, we get thatIt is clear thatHence, by taking ξ = xn, ynin the first equation of (1.2), and by letting n → ∞, we get

We shall show that φ1(?∞) = 0 is not possible. By contradiction, we assume that φ1(?∞)=0. Then, for any ε ∈ (0,1/2), there exists M1? 1 such that for any ξ ≤ ?M1, one has

Because φ1n(ξ)→ φ1(ξ) as n → ∞ for any ξ ∈ R, there exists N >0 such that for any n ≥ N,φ1(ξ) ≥ φ1n(ξ)? ε for any ξ ∈ R. In particular,

By Theorem 3.5, φ1N(?∞)=1. Hence, for the above ε,there exists M2? 1 such that for any ξ ≤ ?M2, we obtain

Combining (3.23) and (3.24), for any ξ ≤ ?M2, one has

Set M =max{M1,M2}. Then, it follows from (3.25) that

which contradicts (3.22), due to the choice of ε. Hence, we obtain

which implies that φ2(?∞) exists. Then, it follows from (1.2) that

From(3.26),it is not hard to see that φ2(?∞)=0 and φ1(?∞)=1,or φ2(?∞)= φ1(?∞)=k.Due to (3.21), it is clear that the case φ2(?∞) = φ1(?∞) = k cannot occur. Therefore,φ2(?∞)=0, φ1(?∞)=1.

Case 2We are going to prove that= 0. By (3.21), we suppose that there exists ? ∈ (0,δ]such that

Then, there exists a sequence {ξi}i∈N, ξi→ ?∞ as i → ∞ such that

Furthermore, it follows from the continuity of φ2(ξ) that there exists ε1>0 such that

Consider the initial value problem

By (3.27), we can take subsequences ξi1nand ξi2nof {ξi}i∈Nsatisfying i1n

Furthermore, by (3.32),

where j0satisfies that |j0| < c1t. Let j0= 0, t = (ξi1n? ξi2n)/c?in (3.33). It is clear that by(3.31), t > T andBecausefor t ≥ T by (3.30), we further get that φ2(ξi1n)>2δ > ?. Hence, we obtain

which contradicts (3.27). Thus,

It is easy to see that

Taking ξ =xn, ynin the first equation of (1.2), and letting n → ∞, we obtain

Proof of Theorem 1.2If the statement is false,then there exists c0satisfying 0

By the continuity of φ1(ξ) and (3.34), there exists δ >0 such that φ1(ξ)> δ for ξ ∈ R. Hence,it can be seen from the second equation of (1.1) that

By Lemmas 3.3 and 3.4, for any ε ∈ (0,c?), we have

Choose ρ ∈ (0,1)such that c0< ρc?. Furthermore,we let j = ?[ρc?t]and ε=(1 ?ρ)c?. Then,|j|=[ρc?t]<(c?? ε)t for t>0. Then, by (3.35), we have

On the other hand,j+c0t=c0t?[ρc?t]