Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 88, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND STABILITY OF TRAVELING WAVES FOR A COMPETITIVE-COOPERATIVE RECURSION SYSTEM XIONGXIONG BAO, TING LI Abstract. This article concerns the existence and global stability of bistable traveling waves for a competitive-cooperative recursion system. We first show that the spatially homogeneous system associated with the competitive-cooperative recursion system admits a bistable structure. Then using the theory of bistable waves for monotone semiflows and a dynamical system approach, we prove that there exists an unique and global stable traveling wave solution connecting two stable equilibria for such recursion system under appropriate conditions. 1. Introduction In this article, we consider the existence and global stability of bistable traveling waves of the three-species competitive-cooperative recursion model un+1(x) = ∫ R (1 + r1)un(x− y) 1 + r1(a1un(x− y) + b1vn(x− y) + c1wn(x− y)) k1(y) dy vn+1(x) = ∫ R (1 + r2)vn(x− y) 1 + r2(a2vn(x− y)− b2wn(x− y) + c2un(x− y)) k2(y) dy, wn+1(x) = ∫ R (1 + r3)wn(x− y) 1 + r3(a3wn(x− y)− b3vn(x− y) + c3un(x− y)) k3(y) dy, (1.1) for n ≥ 0 and x ∈ R. Here un(x), vn(x) and wn(x) are the population densities of three species u, v and w, respectively, at time n and position x ∈ R; ri, ai, bi, ci (i = 1, 2, 3) are positive constants; ki(y) (i = 1, 2, 3) represents the dispersal kernel of three species. In (1.1), the variables v and w denote the densities of two species that work together in a mutualistic way, at the same time, the species v and w compete with u. Traveling wave solutions of recursion systems un+1 = Q[un] have been widely studied, see for example [2, 6, 7, 12, 13, 14, 15, 16, 17, 19, 20, 21, 24, 25, 26, 27, 30, 31] and references therein. Weinberger [24] studied the existence of asymptotic speeds for a scalar discrete-time recursion with that Q is a translation invariant order- preserving operator. Lui[19] extended the results in [24] to a multi-species verison of recursion system. Weinberger [25] also developed the theory in [19, 24] to the order-preserving operator with a periodic habitat. Weinberger et al. [26] further extended the results in [19, 24] so that they can be applied to invasion processes of 2010 Mathematics Subject Classification. 45G15, 45M10, 92D25. Key words and phrases. Existence; global stability; traveling waves; competitive-cooperative recursion system. c©2020 Texas State University. Submitted September 19, 2019. Published August 17, 2020. 1 2 X. BAO, T. LI EJDE-2020/88 cooperative or competitive models among multiple species. In fact, when specie v or w vanishes in (1.1), model (1.1) reduces to the classical two species competitive system. Lewis et al. [11] studied the linear determinacy of spreading speed for monotone discrete-time recursion system and applied their results to discrete-time two species competitive recursion model in the monostable case. Lin et al. [14] studied the spreading speed and traveling wave solutions of general discrete time recursion systems in the monostable case. Zhang and Zhao [29] also established the existence and global stability of bistable waves for discrete-time two species competition recursion systems with bistable structure. Recently, Wu and Zhao[28] studied the existence of spatially periodic traveling wave, single spreading speed and the linear determinacy for a class of intergrodifference competition models in a periodic habitat. For the competitive-cooperative reaction-diffusion system with nonlocal delays, ∂u ∂t = D1∆u+ r1u(1− a1u− b1g1 ∗ v − c1g2 ∗ w), ∂v ∂t = D2∆v + r2v(1− a2v + b2g3 ∗ w − c2g4 ∗ u), ∂w ∂t = D3∆w + r3w(1− a3w + b3g5 ∗ v − c3g6 ∗ u), (1.2) where the general kernel function gi(t, x) (i = 1, . . . , 6) satisfy gi ∗ z(t, x) = ∫ +∞ 0 ∫ +∞ −∞ gi(s, y)z(t− s, x− y) ds dy, ∀t > 0, x ∈ R. Tian and Zhao [23] have established the existence and global stability of bistable traveling wave for (1.2) with the infinite delay case by the finite-delay approximation approach and global convergence results for monotone semiflows. We refer the readers to [1, 4, 5, 8, 9] and references therein for the traveling waves and spreading speed of three-species competition system. To consider the internal interaction and propagation phenomenon of three-com- petitive and cooperative species in discrete-time case, we are interested in the study of traveling waves for competitive-cooperative system (1.1). Assume that the kernel function ki(y) (i = 1, 2, 3) is a continuous and nonnegative function satisfying (H1) ki(−y) = ki(y), ∫ R ki(y) dy = 1, ∫ R e αyki(y) dy < ∞ for all α ∈ R and i = 1, 2, 3. The symmetric property of the kernel functions ki(y) in (H1) implies that the dispersal of three species is isotropic and that the growth and dispersal properties are the same at each point. The spatially homogeneous system associated with (1.1) is un+1 = (1 + r1)un 1 + r1(a1un + b1vn + c1wn) , vn+1 = (1 + r2)vn 1 + r2(a2vn − b2wn + c2un) , wn+1 = (1 + r3)wn 1 + r3(a3wn − b3vn + c3un) , (1.3) for n ≥ 0. It is easy to see that (0, 0, 0), (1/a1, 0, 0), (0, 1/a2, 0) and (0, 0, 1/a3) are four boundary equilibria of (1.3). If a2a3 − b2b3 > 0, there is a nonnegative EJDE-2020/88 COMPETITIVE-COOPERATIVE RECURSION SYSTEMS 3 equilibrium (0, v+, w+) = ( 0, a3 + b2 a2a3 − b2b3 , a2 + b3 a2a3 − b2b3 ) . If a3−c1 a1a3−c1c3 > 0, a1−c3 a1a3−c1c3 > 0 and a1a3 66= c1c3, then there is a nonnegative equilibrium (û+, 0, ŵ+) = ( a3 − c1 a1a3 − c1c3 , 0, a1 − c3 a1a3 − c1c3 ) . If a2−b1 a1a2−b1c2 > 0, a1−c2 a1a2−b1c2 > 0 and a1a2 66= b1c2, then there is a nonnegative equilibrium (ŭ+, v̆+, 0) = ( a2 − b1 a1a2 − b1c2 , a1 − c2 a1a2 − b1c2 , 0 ) . If sign(1−b1 a3+b2 a2a3−b2b3−c1 a2+b3 a2a3−b2b3 ) = sign(|A|) and |A| 66= 0, then there is a positive equilibrium (u∗, v∗, w∗), where A = a1 b1 c1 c2 a2 −b2 c3 −b3 a3  . In this article, we study of the existence and stability of bistable traveling waves for system (1.1) in the case where the corresponding spatially homogeneous system admits a bistable structure. Though bistable waves in two species competition recursion system have been studied before (see Zhang and Zhao [29]), here we would like to emphasize that there is no result about the bistable waves of three species competitive-cooperative recursion model. We use the theory of monotone semiflows and squeezing technique to prove the existence and stability of bistable traveling waves. However, comparison to two species competition system, there exists eight equilibria for three species competitive-cooperative system, it is difficult to show the stability of these eight equilibria and the counter-propagation phenomenon between two different equilibria (see (A6) in Section 2). In this paper, we will show that the equilibria ( 1 a1 , 0, 0) and (0, v+, w+) are stable and the other equilibria are unstable. We first transfer system (1.1) into a cooperative system. By the changes of variables ũn = 1 a1 − un, ṽn = vn and w̃n = wn. (1.4) Dropping the tilde, we have un+1(x) = ∫ R 1 a1 r1(b1vn(x− y) + c1wn(x− y)) + un(x− y) 1 + r1(1− a1un(x− y) + b1vn(x− y) + c1wn(x− y)) k1(y) dy vn+1(x) = ∫ R (1 + r2)vn(x− y) 1 + r2(a2vn(x− y)− b2wn(x− y) + c2 a1 − c2un(x− y)) k2(y) dy, wn+1(x) = ∫ R (1 + r3)wn(x− y) 1 + r3(a3wn(x− y)− b3vn(x− y) + c3 a1 − c3un(x− y)) k3(y) dy. (1.5) By (1.4), the equilibria (0, 0, 0), (1/a1, 0, 0), (0, 1/a2, 0), (0, 0, 1/a3), (0, v+, w+), (û+, 0, ŵ+), (ŭ+, v̆+, 0) and (u∗, v∗, w∗) become (1/a1, 0, 0), (0, 0, 0), (1/a1, 1/a2, 0), 4 X. BAO, T. LI EJDE-2020/88 (1/a1, 0, 1/a3), (1/a1, v +, w+), ( 1 a1 −û+, 0, ŵ+), ( 1 a1 −ŭ+, v̆+, 0) and ( 1 a1 −u∗, v∗, w∗), respectively. Let 0 = (0, 0, 0), v1 := (v̂+ 1 , 0, v̂ + 3 ) = ( 1 a1 − û+, 0, ŵ+), v2 := (v̆+ 1 , v̆ + 2 , 0) = ( 1 a1 − ŭ+, v̆+, 0), v+ := (v+ 1 , v + 2 , v + 3 ) = ( 1 a1 , v+, w+), v∗ := (v∗1 , v ∗ 2 , v ∗ 3) = ( 1 a1 − u∗, v∗, w∗) and define the set E = {( 1 a1 , 0, 0 ) ,0, ( 1 a1 , 1 a2 , 0 ) , ( 1 a1 , 0, 1 a3 ) ,v1,v2,v +,v∗ } . Note that (1.5) is a cooperative system. To study the traveling wave solution of system (1.1) connecting ( 1 a1 , 0, 0) and (0, v+, w+), it is equivalent to study the traveling wave solution of (1.5) connecting 0 and v+ = ( 1 a1 , v+, w+). We assume (H1) and that the parameters in (1.1) satisfy c2/a1 > 1, c3/a1 > 1, a1a3 < c1c3, a1a2 < b1c2 and 1− b1 a3 + b2 a2a3 − b2b3 − c1 a2 + b3 a2a3 − b2b3 < 0. Then we have the following results on traveling wave solution for system (1.5): • (Existence) There is c ∈ R such that system (1.5) admits a nondecreasing traveling wave solution Φ(x − cn) = (Φ1(x − cn),Φ2(x − cn),Φ3(x − cn)) with speed c satisfies Φ(−∞) = (0, 0, 0) and Φ(+∞) = (v+ 1 , v + 2 , v + 3 ) (see Theorem 3.5). • (Stability) If the initial value ψ(·) ∈ X[0,v+] satisfies one of the following two cases: Case (i) ψ(·) is nondecreasing and satisfies lim inf ξ→+∞ ψi(ξ) > v∗i > lim sup ξ→−∞ ψi(ξ) for i = 1, 2, 3; Case (ii) the kernel ki (i = 1, 2, 3) has a compact support and ψ(ξ) satisfies lim infξ→+∞ ψi(ξ) > v∗i > lim supξ→−∞ ψi(ξ) for i = 1, 2, 3, then there exists sψ such that limn→+∞ ‖Un(x,ψ)−Φ(x− cn+ sψ)‖ = 0 uniformly for x ∈ R (see Theorem 3.7). • (Uniqueness) Any monotone traveling wave solutions of (1.5) connecting 0 and v+ is a translation of Φ(·) (see Corollary 3.8). We end the introduction with the following remarks. By (1.4), for (1.2), there is an unique traveling wave solution Φ̂(x− cn) = (Φ̂1(x− cn), Φ̂2(x− cn), Φ̂3(x− cn)) connecting two stable points ( 1 a1 , 0, 0) and (0, v+, w+). Thus Φ̂(x−cn)→ ( 1 a1 , 0, 0) as n→∞ for c > 0, which implies that specie u will persistent and species v, w will extinct. If c < 0, Φ̂(x − cn) → (0, v+, w+) as n → ∞, which implies that species v and w are persistent and specie u will go to extinct. Hence, the traveling wave solution Φ̂(x − cn) can be used to determine the winner of such a competition- cooperative system in the presence of spatial diffusion and discrete time and the the sign of the wave speed c plays an important role, which will be considered in future. EJDE-2020/88 COMPETITIVE-COOPERATIVE RECURSION SYSTEMS 5 The rest of this paper is organized as follows. In Section 2, we will present some preliminaries for system (1.5). In Section 3, we will establish the existence and global stability of bistable traveling waves for system (1.5) by appealing the theory of bistable waves for monotone semiflows in [3] and a dynamical system approach. 2. Preliminary In this section, we introduce notation and show that system (1.3) admits a bistable structure. Let C := C(R,R3) be the set of all bounded and continuous functions from R to R3 equipped with the compact open topology. Let C+ = {(φ1, φ2, φ3) ∈ C : φi(x) ≥ 0,∀x ∈ R, i = 1, 2, 3}. Define Cr := {φ ∈ C : 0 ≤ φ ≤ r} and C[a,b] := {φ ∈ C : a ≤ φ ≤ b} for any a, b, r ∈ R3 with a ≤ b and r � 0. Define an operator Q = (Q1, Q2, Q3) on C by Q1[u, v, w](x) = ∫ R 1 a1 r1(b1v + c1w) + u 1 + r1(1− a1u+ b1v + c1w) k1(y) dy, Q2[u, v, w](x) = ∫ R (1 + r2)v 1 + r2(a2v(x)− b2w + c2 a1 − c2u) k2(y) dy, Q3[u, v, w](x) = ∫ R (1 + r3)w 1 + r3(a3w(x)− b3v + c3 a1 − c3u) k3(y) dy, Then system (1.5) can be expressed as Un+1(x) = Q[Un](x), Un := (un, vn, wn), n ≥ 0. In this article, we mainly consider the bistable structure of system (1.5). It is then needed to show that the fixed points 0 and v+ are stable and others are unstable. Let Q̂ be the spatially homogeneous operator of Q to [0,v+], where Q̂ = (Q̂1, Q̂2, Q̂3) and Q̂1[w1, w2, w3] = 1 a1 r1(b1w2 + c1w3) + w1 1 + r1(1− a1w1 + b1w2 + c1w3) , Q̂2[w1, w2, w3] = (1 + r2)w2 1 + r2(a2w2 − b2w3 + c2 a1 − c2w1) , Q̂3[w1, w2, w3] = (1 + r3)w3 1 + r3(a3w3 − b3w2 + c3 a1 − c3w1) , (2.1) To obtain the Jacobian matrices of Q̂[w1, w2, w3] at point (w1, w2, w3), we list the first row of Jacobian matrix as follows ∂Q̂1 ∂w1 = 1 + r1(1− a1w1 + b1w2 + c1w3) + r1a1[ r1a1 (b1w2 + c1w3) + w1] [1 + r1(1− a1w1 + b1w2 + c1w3)]2 , ∂Q̂1 ∂w2 = r1 a1 b1(1 + r1(1− a1w1 + b1w2 + c1w3))− [ r1a1 (b1w2 + c1w3) + w1]r1b1 [1 + r1(1− a1w1 + b1w2 + c1w3)]2 , ∂Q̂1 ∂w3 = r1 a1 c1(1 + r1(1− a1w1 + b1w2 + c1w3))− [ r1a1 (b1w2 + c1w3) + w1]r1c1 [1 + r1(1− a1w1 + b1w2 + c1w3)]2 ; the second row of Jacobian matrix is ∂Q̂2 ∂w1 = (1 + r2)w2r2c2 [1 + r2(a2w2 − b2w3 + c2 a1 − c2w1)]2 , 6 X. BAO, T. LI EJDE-2020/88 ∂Q̂2 ∂w2 = (1 + r2)(1 + r2(a2w2 − b2w3 + c2 a1 − c2w1))− (1 + r2)w2r2a2 [1 + r2(a2w2 − b2w3 + c2 a1 − c2w1)]2 , ∂Q̂2 ∂w3 = (1 + r2)w2r2b2 [1 + r2(a2w2 − b2w3 + c2 a1 − c2w1)]2 ; and the third row is ∂Q̂3 ∂w1 = (1 + r3)w3r3c3 [1 + r3(a3w3 − b3w2 + c3 a1 − c3w1)]2 , ∂Q̂3 ∂w2 = (1 + r3)w3r3b3 [1 + r3(a3w3 − b3w2 + c3 a1 − c3w1)]2 , ∂Q̂3 ∂w3 = (1 + r3)(1 + r3(a3w3 − b3w2 + c3 a1 − c3w1))− (1 + r3)w3r3a3 [1 + r3(a3w3 − b3w2 + c3 a1 − c3w1)]2 . Thus the Jacobian matrix of Q̂ at 0 is J0 =  1 1+r1 r1 a1 b1 1+r1 r1 a1 c1 1+r1 0 1+r2 1+ r2c2 a1 0 0 0 1+r3 1+ r3c3 a1  and the characteristic equation of J0 is( λ− 1 1 + r1 )( λ− 1 + r2 1 + r2c2 a1 )( λ− 1 + r3 1 + r3c3 a1 ) = 0. It is obvious that J0 has three positive eigenvalues λ1 = 1 1 + r1 λ2 = 1 + r2 1 + r2c2 a1 , λ3 = 1 + r3 1 + r3c3 a1 . If c2/a1 > 1 and c3/a1 > 1, we obtain λ1 < 1 , λ2 < 1 and λ3 < 1. Then the fixed point 0 is stable (see [10, Chapter 1, Section 9]). Consider v+ = (1/a1, v + 2 , v + 3 ). Note that v+ is positive fixed point of (2.1) and 1 = 1 + r2 1 + r2(a2v + 2 − b2v + 3 ) and 1 = 1 + r3 1 + r3(a3v + 3 − b3v + 2 ) . In this case, it is easy to check that b1v + 2 + c1v + 3 6= 1. Thus the Jacobian matrix of Q̂ at v+ is Jv+ =  1+r1 1+r1(b1v + 2 +c1v + 3 ) 0 0 r2c2v + 2 1+r2 1− r2a2v + 2 1+r2 r2b2v + 2 1+r2 r3c3v + 3 1+r3 r3b3v + 3 1+r3 1− r3a3v + 3 1+r3  . Then the characteristic equation at v+ is( λ− 1 + r1 1 + r1(b1v + 2 + c1v + 3 ) )(( λ− 1 + r2a2v + 2 1 + r2 )( λ− 1 + r3a3v + 3 1 + r3 ) − r2b2v + 2 1 + r2 r3b3v + 3 1 + r3 ) = 0 (2.2) EJDE-2020/88 COMPETITIVE-COOPERATIVE RECURSION SYSTEMS 7 Recall that v+ = a3+b2 a2a3−b2b3 and w+ = a2+b3 a2a3−b2b3 . Thus, 1− b1 a3 + b2 a2a3 − b2b3 − c1 a2 + b3 a2a3 − b2b3 < 0 implies b1v + 2 + c1v + 3 > 1. It then follows that the Jacobian matrix Jv+ has three positive eigenvalues λi (i = 1, 2, 3) and λi < 1 for i = 1, 2, 3. It then follows that the fixed point v+ is stable. For v1 = (v1, v2, v3) = ( 1 a1 − û+, 0, ŵ+ ) , we have a1v1 = c1v3 and a3v3 + c3 a1 − c3v1 = 1. Then we have that the Jacobian matrixes of Q̂ at v1 is Jv1 =  1+r1c1v3 1+r1 r1b1 a1 (1−c1v3) 1+r1 r1a1 a1 (1−c1v3) 1+r1 0 1 0 r3c3v3 1+r3 r3b3v3 1+r3 1− r3a3v3 1+r3  . Assume a1a3 < c1c3. Since û+ = a3−c1 a1a3−c1c3 > 0, we obtain that a3 − c1 < 0 and c1v3 = c1 a1−c3 a1a3−c1c3 > 1. Thus it is easy to see that there is an eigenvalue λ1 > 1 for Jv1 . Hence v1 is unstable. Similarly, for v2 = (v̆1, v̆2, v̆3) = ( 1 a1 − ŭ+, v̆+, 0 ) , we have a1v̆1 = b1v̆2, a2v̆2 + c2 a1 − c2v̆1 = 1, and Jv2 =  1+r1b1v̆2 1+r1 r1b1 a1 (1−b1v̆2) 1+r1 r1c1 a1 (1−b1v̆2) 1+r1 r2c2v̆2 1+r2 1− r2b2v̆2 1+r2 r2a2v̆2 1+r2 0 0 1  . If a1a3 < b1c2, we have a2 < b1 and b1v̆2 > 1. Thus the point v2 is also unstable. For (1/a1, 0, 0), (1/a1, 1/a2, 0) and (1/a1, 0, 1/a3), the Jacobian matrices of Q at these three points are J( 1 a1 ,0,0) = 1 + r1 0 0 0 1 0 0 0 1  , J( 1 a1 , 1 a2 ,0) =  1+r1 1+r1 b1 a2 0 0 1 a2 r2c2 1+r2 1 1+r2 1 a1 r2b2 1+r2 0 0 1  J( 1 a1 ,0, 1 a3 ) =  1+r1 1+r1 c1 a3 0 0 0 1 0 1 a3 r3c3 1+r3 1 a3 r3b3 1+r3 1 1+r3  , respectively. Note that a1a3 < c1c3 and a1a3 < b1c2 imply a3 < c1 and a2 < b1. Hence ( 1 a1 , 0, 0), ( 1 a1 , 1 a2 , 0) and ( 1 a1 , 0, 1 a3 ) are unstable. Next, we consider v∗ = (v∗1 , v ∗ 2 , v ∗ 3) = ( 1 a1 − u∗, v∗, w∗). In this case, we have b1v ∗ 2 + c1v ∗ 3 = a1v ∗ 1 , a2v ∗ 2 − b2v∗3 + c2 a1 − c2v∗1 = 1, a3v ∗ 3 − b3v∗3 + c3 a1 − c3v∗1 = 1. (2.3) 8 X. BAO, T. LI EJDE-2020/88 The Jacobian matrix of Q at v∗ is Jv∗ =  1+r1a1v ∗ 1 1+r1 b1r1 1+r1 ( 1 a1 − v∗1) c1r1 1+r1 ( 1 a1 − v∗1) r2c2v ∗ 2 1+r2 1− r2a2v ∗ 2 1+r2 r2b2v ∗ 2 1+r2 r3c3v ∗ 3 1+r3 r3b3v ∗ 3 1+r3 1− r3a3v ∗ 3 1+r3  . Then F (λ) = |λI − Jv∗ | = ∣∣∣∣∣∣∣ λ− 1+r1a1v ∗ 1 1+r1 − b1r1 1+r1 ( 1 a1 − v∗1) − c1r1 1+r1 ( 1 a1 − v∗1) − r2c2v ∗ 2 1+r2 λ− 1 + r2a2v ∗ 2 1+r2 − r2b2v ∗ 2 1+r2 − r3c3v ∗ 3 1+r3 − r3b3v ∗ 3 1+r3 λ− 1 + r3a3v ∗ 3 1+r3 ∣∣∣∣∣∣∣ . Note that F (+∞) = +∞, and F (1) = ∣∣∣∣∣∣∣ 1− 1+r1a1v ∗ 1 1+r1 − b1r1 1+r1 ( 1 a1 − v∗1) − c1r1 1+r1 ( 1 a1 − v∗1) − r2c2v ∗ 2 1+r2 r2a2v ∗ 2 1+r2 − r2b2v ∗ 2 1+r2 − r3c3v ∗ 3 1+r3 − r3b3v ∗ 3 1+r3 r3a3v ∗ 3 1+r3 ∣∣∣∣∣∣∣ = r1u ∗ 1 + r1 r2v ∗ 1 + r2 r3w ∗ 1 + r3 ∣∣∣∣∣∣ a1 b1 c1 c2 a2 −b2 c3 −b3 a3 ∣∣∣∣∣∣ . Since sign(1− b1 a3 + b2 a2a3 − b2b3 − c1 a2 + b3 a2a3 − b2b3 ) = sign(|A|) and 1− b1 a3+b2 a2a3−b2b3 − c1 a2+b3 a2a3−b2b3 < 0, we have F (1) < 0. Then there is λ′ > 1 such that F (λ′) = 0, which implies that v∗ is unstable. From above all calculations, we have the following lemma. Lemma 2.1. The following statements are valid. (1) If c2/a1 > 1, c3/a1 > 1, then 0 is stable. (2) If 1− b1 a3 + b2 a2a3 − b2b3 − c1 a2 + b3 a2a3 − b2b3 < 0, (2.4) then v+ is stable. (3) If (2.4) hold, then v∗ is unstable. (4) If a1a3 < c1c3, then v1 is unstable. (5) If a1a3 < b1c2, then v2 is unstable. By Lemma 2.1, if the parameters satisfy c2/a1 > 1, c3/a1 > 1, a1a3 < c1c3, a1a2 < b1c2 and 1 − b1 a3+b2 a2a3−b2b3 − c1 a2+b3 a2a3−b2b3 < 0, then system (1.5) is of the bistable structure. According to [3, Theorem 3.1], if the operator Q satisfies the following conditions: (A1) (Translation invariance) Ty ◦ Q[Φ] = Q ◦ Ty[Φ], for all Φ ∈ Cv+ , y ∈ R, where Ty[Φ](x) = Φ(x− y). (A2) (Continuity) Q : Cv+ → Cv+ is continuous with respect to the compact open topology. (A3) (Monotonicity) Q is order preserving in the sense that Q[Φ] ≥ Q[Ψ] when- ever Φ ≥ Ψ in Cv+ . (A4) (Compactness) Q : Cv+ → Cv+ is compact with respect to the compact open topology. EJDE-2020/88 COMPETITIVE-COOPERATIVE RECURSION SYSTEMS 9 (A5) (Bistability) Two fixed points 0 and v+ are strongly stable from above and below, respectively, for the map Q : [0,v+]→ [0,v+], that is, there exist a number δ > 0 and unit vectors e1 and e2 ∈ Int(R3) such that Q[ηe1]� ηe1, Q[v+ τ − ηe2]� v+ − ηe2, ∀η ∈ (0, δ) and the set E \ {0,v+} is totally unordered. (A6) (Counter-propagation) For each α ∈ E \ {0,v+}, c∗−(α,v+) + c∗+(0, α) > 0, where c∗−(α,v+) and c∗+(0, α) represent the leftward and rightward spread- ing speeds of monotone subsystem {Qn}n≥0 restricted on [α,v+] and [0, α], respectively. then there exists a nondecreasing traveling wave solution Φ(x − cn) = (Φ1(x − cn),Φ2(x− cn),Φ3(x− cn)) with speed c ∈ R and connecting two bistable points 0 and v+. Hence, in section 3, we will verify that operator Q given by system (1.5) satisfies assumptions (A1)–(A6). 3. Existence and globally stability of traveling wave In this section, we establish the existence and stability of bistable traveling waves for system (1.5). Since (1.5) is cooperative, it is easy to verify that the map Q satisfies (A1)–(A4). In the following, we show that (A5) and (A6) also hold. Lemma 3.1. Assume that c2/a1 > 1, c3/a1 > 1, a1a3 < c1c3, a1a2 < b1c2 and (2.4) holds. Then Q satisfies (A5). Proof. From Lemma 2.1, we know that 0 and v+ are stable. We now prove that 0 is strongly stable from above and v+ is strongly stable from below. Since c2/a1 > 1 and c3/a1 > 1, then the Jacobian matrix J0 has three eigenvalues λi, i = 1, 2, 3. If 1 > max{λ2, λ3} > λ1, then J0 has a unit eigenvector e0 > 0 associated with max{λ2, λ3} such that J0(e0) = max{λ2, λ3}e0 � e0. If 1 > λ1 > max{λ2, λ3}, take k ∈ (λ1, 1), ε0 ∈ ( 0, a1(1+r1) r1b1 ) , η0 ∈ ( 0, a1(1+r1) r1c1 ) and unit vector e0 = ( ε0√ 1 + ε2 0 + η2 0 , η0√ 1 + ε2 0 + η2 0 , 1√ 1 + ε2 0 + η2 0 ) such that J0(e0)� ke0 � e0. By the continuous differentiality of Q̂, there exists δ > 0 such that Q̂(ηe0) = Q̂(0) + ∫ 1 0 DQ̂(tηe0)ηe0 dt = η ∫ 1 0 DQ̂(tηe0)e0 dt ≤ ηke0 � ηe0 for all η ∈ (0, δ] and hence 0 is strongly stable from above for the map Q̂. By similar argument, we also have that v+ is strongly stable form below. From above arguments, we have that 0 is strongly stable from above and v+ is strongly stable from below. Next, we mainly show that E \ {0,v+} are totally unordered. We first show v̂+ 3 > v∗3 if a1a3 < c1c3. From (2.1), we have v̂+ 1 = 1 a1 r1(0 + c1v̂ + 3 ) + v̂+ 1 1 + r1(1− a1v̂ + 1 + c1v̂ + 3 ) , 10 X. BAO, T. LI EJDE-2020/88 v̂+ 3 = (1 + r3)v̂+ 3 1 + r3(a3v̂ + 3 + c3 a1 − c3v̂+ 1 ) , that is, a1v̂ + 1 − c1v̂ + 3 = 0 and a3v̂ + 3 + c3 a1 − c3v̂+ 1 = 1. Then v̂+ 3 = 1− c3 a1 a1 − c1c3 a1 = a1 − c3 a1a3 − c1c3 . On the other hand, a1v ∗ 1 − c1v∗3 = b2v ∗ 3 > 0, 1− a3v ∗ 3 − c3 a1 + c3v ∗ 1 = −b3v∗2 < 0, which implies that v∗3 < a1 − c3 a1a3 − c1c3 for a1a3 < c1c3. Thus v∗3 < v̂+ 3 if a1a3 < c1c3. By the similar way, we have v̆+ 2 > v∗2 if a1a2 < b1c2. It follows that the set E \ {0,v+} are totally unordered and Q satisfies (A5). � Lemma 3.2. c∗(0,v∗) + c∗(v∗,v+) > 0. Proof. Recall that v∗ = (v∗1 , v ∗ 2 , v ∗ 3) satisfies (2.3). To consider c∗(v∗,v+), let ũn = un − v∗1 , ṽn = vn − v∗2 and w̃n = wn − v∗3 . Then system (1.5) becomes ũn+1(x) = −v∗1 + ∫ R 1 a1 r1(b1ṽn(x− y) + c1w̃n(x− y)) + ũn(x− y) + (1 + r1)v∗1 1 + r1(1− a1ũn(x− y) + b1ṽn(x− y) + c1w̃n(x− y)) k1(y) dy, ṽn+1(x) = −v∗2 + ∫ R (1 + r2)(ṽn(x− y) + v∗2) 1 + r2(1 + a2ṽn(x− y)− b2w̃n(x− y)− c2ũn(x− y)) k2(y) dy, w̃n+1(x) = −v∗3 + ∫ R (1 + r3)(w̃n(x− y) + v∗3) 1 + r3(1 + a3w̃n(x− y)− b3ṽn(x− y)− c3ũn(x− y)) k3(y) dy. (3.1) It is easy to verify that system (3.1) is cooperative and positively invariant in C[0,β] = {ψ ∈ C : 0 ≤ ψ ≤ β}, where β = v+−v∗ � 0. The spatially homogeneous system ũn+1 = −v∗1 + 1 a1 r1(b1ṽn + c1w̃n) + ũn + (1 + r1)v∗1 1 + r1(1− a1ũn + b1ṽn + c1w̃n) ṽn+1 = −v∗2 + (1 + r2)(ṽn + v∗2) 1 + r2(1 + a2ṽn − b2w̃n − c2ũn) , w̃n+1 = −v∗3 + (1 + r3)(w̃n + v∗3) 1 + r3(1 + a3w̃n − b3ṽn − c3ũn) (3.2) has stable equilibrium β and unstable one 0, and there are no other equilibria between these two equilibria in [0, β] ∈ R3. EJDE-2020/88 COMPETITIVE-COOPERATIVE RECURSION SYSTEMS 11 Now we consider the linearization of (3.1) at 0, ũn+1(x) = ∫ R ( (1 + r1a1v ∗ 1) 1 + r1 ũn + ( 1 a1 − v∗1)r1b1 1 + r1 ṽn + ( 1 a1 − v∗1)r1c1 1 + r1 w̃n ) k1(y) dy ṽn+1(x) = ∫ R (r2c2v ∗ 2 1 + r2 ũn + (1 + r1)− r2a2v ∗ 2 1 + r2 ṽn + r2b2v ∗ 3 1 + r2 w̃n ) k2(y) dy, w̃n+1(x) = ∫ R (r3c3v ∗ 3 1 + r3 ũn + r3b3v ∗ 2 1 + r3 ṽn + (1 + r3)− r3a3v ∗ 3 1 + r3 w̃n ) k3(y) dy. (3.3) For any µ ∈ R+, let ûn(x) = e−µxαn, v̂n(x) = e−µxβn and ŵn(x) = e−µxγn for n ≥ 0. Then αn, βn and γn satisfies αn+1 = (1 + r1a1v ∗ 1) 1 + r1 K1(µ)αn + ( 1 a1 − v∗1)r1b1 1 + r1 K1βn + ( 1 a1 − v∗1)r1c1 1 + r1 K1(µ)γn, βn+1 = r2c2v ∗ 2 1 + r2 K2(µ)αn + (1 + r1)− r2a2v ∗ 2 1 + r2 K2(µ)βn + r2b2v ∗ 3 1 + r2 K2(µ)γn, γn+1 = r3c3v ∗ 3 1 + r3 K3(µ)αn + r3b3v ∗ 2 1 + r3 K2(µ)βn + (1 + r3)− r3a3v ∗ 3 1 + r3 K2(µ)γn, (3.4) where Ki(µ) := ∫ ∞ −∞ eµyki(y) dy > ∫ ∞ −∞ ki(y) dy = 1, i = 1, 2, 3. (3.5) We define the matrix Bµ :=  (1+r1a1v ∗ 1 ) 1+r1 K1(µ) ( 1 a1 −v∗1 )r1b1 1+r1 K1(µ) ( 1 a1 −v∗1 )r1c1 1+r1 K1(µ) r2c2v ∗ 2 1+r2 K2(µ) (1+r1)−r2a2v∗2 1+r2 K2(µ) r2b2v ∗ 3 1+r2 K2(µ) r3c3v ∗ 3 1+r3 K3(µ) r3b3v ∗ 2 1+r3 K2(µ) (1+r3)−r3a3v∗3 1+r3 K2(µ)  . Note that each entry of Bµ is positive for any µ ≥ 0, then Bµ is positive. Let λ(µ) be the principal eigenvalue of Bµ. By [22, Theorem A4], λ(µ) is positive with a strongly positive eigenvector. Let λ(0) be the principle eigenvalue of Bµ with µ = 0, we have B0 :=  (1+r1a1v ∗ 1 ) 1+r1 ( 1 a1 −v∗1 )r1b1 1+r1 ( 1 a1 −v∗1 )r1c1 1+r1 r2c2v ∗ 2 1+r2 (1+r1)−r2a2v∗2 1+r2 r2b2v ∗ 3 1+r2 r3c3v ∗ 3 1+r3 r3b3v ∗ 2 1+r3 (1+r3)−r3a3v∗3 1+r3  . Since 0 is unstable, then λ(0) > 1. By Ki(µ) > 1, ∀µ > 0, i = 1, 2, 3, we have Bµ > B0 for any µ > 0. The monotonicity of the principle eigenvalue with respect to the positive matrix implies that λ(µ) > λ(0) > 1 for any µ > 0. Let Ψ(µ) := lnλ(µ) µ , then Ψ(µ) > 0 for any µ > 0 and limµ→0+ Ψ(µ) = +∞, we also have lim inf µ→∞ Ψ(µ) = lim inf µ→∞ lnλ(µ) µ ≥ lim inf µ→ ln (trBµ) µ ≥ lim inf µ→ ln (1+r1a1v ∗ 1 ) 1+r1 eµy0 ∫ +∞ y0 k1(y) dy µ = y0 > 0. 12 X. BAO, T. LI EJDE-2020/88 It then follows that c := infµ>0 Ψ(µ) > 0. By [16, Theorem 3.10], we have c∗(0, β) ≥ c. Then c∗(v∗,v+) = c∗(0, β) ≥ c > 0. To compute c∗(0,v∗), let ũn = −un + v∗1 , ṽn = −vn + v∗2 and w̃n = −wn + v∗3 . Then system (1.5) becomes ũn+1(x) = v∗1 − ∫ R − 1 a1 r1(b1ṽn(x− y) + c1w̃n(x− y))− ũn(x− y) + (1 + r1)v∗1 1 + r1(1 + a1ũn(x− y) + b1ṽn(x− y) + c1w̃n(x− y)) k1(y) dy ṽn+1(x) = v∗2 − ∫ R (1 + r2)(−ṽn(x− y) + v∗2) 1 + r2(1− a2ṽn(x− y) + b2w̃n(x− y) + c2ũn(x− y)) k2(y) dy, w̃n+1(x) = v∗3 − ∫ R (1 + r3)(−w̃n(x− y) + v∗3) 1 + r3(1− a3w̃n(x− y) + b3ṽn(x− y) + c3ũn(x− y)) k3(y) dy. (3.6) Note that this system is cooperative and the spatially homogeneous system of (3.6) has unstable equilibrium 0 and stable equilibrium v∗ � 0 in [0,v∗] ⊂ R3. By the similar arguments as for system (3.1), we have c∗(0,v∗) > 0. Therefore, c∗(0,v∗)+ c∗(v∗,v+) > 0. � Lemma 3.3. Let v0 = (1/a1, 0, 0). Then c∗(0,v0) + c∗(v0,v +) > 0. Proof. To calculate the speed c∗(0,v0), we only need to consider the following one-dimensional monotone subsystem of (1.1) un+1(x) = ∫ R (1 + r1)un(x− y) 1 + r1a1un(x− y) k1(y) dy, n ≥ 0. (3.7) By [7, Theorem 2.1], (3.7) has a monotone traveling wave connecting 0 and 1 with the minimal wave c∗, where c∗ = inf µ>0 { 1 µ ln ( 1 + r1 1 + r1a1 ∫ R eµyk1(y) dy )} is the spreading speed and c∗(0,v0) = c∗ (also see [26, Lemma 2.3]. As the proof of [29, Lemma 2.1], we have c∗(0,v0) > 0. Next, we consider c∗(v0,v +). We consider the two-dimensional monotone system vn+1(x) = ∫ R (1 + r2)vn 1 + r2(a2vn(x− y)− b2wn(x− y)) k2(y) dy, wn+1(x) = ∫ R (1 + r3)wn 1 + r3(a3wn(x− y)− b3vn(x− y)) k3(y) dy. (3.8) Note that vn+1(x) ≥ ∫ R (1 + r2)vn 1 + r2a2vn(x− y) k2(y) dy, wn+1(x) ≥ ∫ R (1 + r3)wn 1 + r3a3wn(x− y) k3(y) dy. Thus, c∗(v0,v +) ≥ min{c∗v, c∗w}, where c∗v = inf µ>0 { 1 µ ln ( 1 + r2 1 + r2a2 ∫ R eµyk1(y) dy )} > 0, EJDE-2020/88 COMPETITIVE-COOPERATIVE RECURSION SYSTEMS 13 c∗w = inf µ>0 { 1 µ ln ( 1 + r3 1 + r3a3 ∫ R eµyk1(y) dy )} > 0. It follows that c∗(v0,v +) > 0. Therefore, c∗(0,v0)+c∗(v0,v +) > 0. This completes the proof. � As in Lemmas 3.2 and 3.3, for the other equilibria in E \ {0,v+}, we have the following results. Lemma 3.4. Assume that c2/a1 > 1, c3/a1 > 1, a1a3 < c1c3, a1a2 < b1c2 and (2.4) hold. Then {Qn}∞n=0 satisfies (A6). As a consequence of Lemmas 3.1–3.4 and [3, Theorem 3.1], we have the following results. Theorem 3.5. Assume that c2/a1 > 1, c3/a1 > 1, a1a3 < c1c3, a1a2 < b1c2 and (2.4) hold. Then system (1.5) admits a nondecreasing traveling wave solution Φ(x− cn) = (Φ1(x− cn),Φ2(x− cn),Φ3(x− cn)) with speed c ∈ R and connecting two stable equilibria 0 and v+ = (v+ 1 , v + 2 , v + 3 ). Next, we study the global stability and uniqueness of the bistable traveling waves Φ(x−cn) = (Φ1(x−cn),Φ2(x−cn),Φ3(x−cn)) for system (1.5). Let z = x−c(n+1). Thus (1.5) can be transformed into the system Un+1(z) = Q[Un](z + c), n ≥ 0. (3.9) In the following, Un(z,ψ) denotes the solution of (3.9) with the initial value U0(z,ψ) = ψ := (ψ1, ψ2, ψ3). Then Φ(z) is an equilibrium solution of system (3.9), that is, Φ(z) = Q[Φ](z + c), ∀z ∈ R. Clearly, the solution Un(x,ψ) of (1.5) with initial value is given by Un(x,ψ) = Un(x− cn,ψ). We choose 0 < δ̂i < δ̃ < 1 (i = 1, 2) with δ̂2/δ̂1 and δ̂3/δ̂1 sufficiently small such that b1 a1 δ̂2 δ̂1 + c1 a1 δ̂3 δ̂1 < 1. (3.10) We choose δ̆2, δ̆3 < 1 such that b3/a3 < breveδ3/δ̆2 < a2/b2 and choose δ̆1 < δ̃ small enough to satisfy c2 δ̆1 δ̆2 − a2 + b2 δ̆3 δ̆2 < 0 and c3 δ̆1 δ̆3 − a3 + b3 δ̆2 δ̆3 < 0. (3.11) Define a continuous function δ(ξ) = (δ1(ξ), δ2(ξ), δ3(ξ)) by δi(ξ) =  δ̂i, ξ < ξ̂ < 0, nondecreasing, ξ̂ ≤ ξ ≤ ξ̆, δ̆i, ξ > ξ̆ > 0, for i = 1, 2, 3. Moreover, we define W± n (z) = (U±n , V ± n ,W ± n ) as follows U±n (z) = Φ1 ( z ± ẑ ± ε(1− e−σn) ) ± εδ1(z ± ẑ)e−σn, V ±n (z) = Φ2 ( z ± ẑ ± ε(1− e−σn) ) ± εδ2(z ± ẑ)e−σn, W±n (z) = Φ3 ( z ± ẑ ± ε(1− e−σn) ) ± εδ3(z ± ẑ)e−σn. 14 X. BAO, T. LI EJDE-2020/88 The following Lemma shows that W± n (z) = (U±n , V ± n ,W ± n ) is the upper and lower solution of (3.9). Lemma 3.6. Assume that c2/a1 > 1, c3/a1 > 1, a1a3 < c1c3, a1a2 < b1c2 and (2.4) hold. There exist positive number σ and ε0 ∈ (0, 1) such that for any ẑ and ε ∈ (0, ε0), (U±n , V ± n ,W ± n ) are upper solution and lower solution of system (3.9) for n ≥ 0 and z ∈ R, respectively. The proof of the above lemma is similar to that of [29, Lemma 3.2]; and we omit it here. With the help of Lemma 3.6, the bistable traveling waves Φ(z) is a Lyapunov stable equilibrium of system (3.9). Let X = BUC(R,R3) be the Banach space of all bounded and uniformly continuous functions from R to R3 with the usual supreme norm. Let X+ = {(ψ1, ψ2, ψ3) ∈ X : ψi(x) ≥ 0,∀x ∈ R, i = 1, 2, 3}. Now we are ready to prove the global stability of the bistable wave Φ(z). Theorem 3.7. Let Φ(x− cn) be a monotone traveling wave solutions of (1.5) and Un(x,ψ) be the solution of (1.5) with U0(·,ψ) = ψ(·) := (ψ1(·), ψ2(·), ψ3(·)) ∈ X[0,v+]. Let the initial value ψ satisfy one of the following statements: (i) ψ ∈ X[0,v+] is nondecreasing and satisfies lim inf ξ→+∞ ψi(ξ) > v∗i > lim sup ξ→−∞ ψi(ξ) for i = 1, 2, 3; (ii) the kernel ki (i = 1, 2, 3) has a compact support and ψ(ξ) satisfies lim inf ξ→+∞ ψi(ξ) > v∗i > lim sup ξ→−∞ ψi(ξ) for i = 1, 2, 3, then there exists sψ ∈ R such that lim n→+∞ ‖Un(x,ψ)−Φ(x− cn+ sψ)‖ = 0 (3.12) uniformly for x ∈ R. Proof. Let ε ∈ (0, ε0) small. For any given nondecreasing initial value ψ ∈ X[0,v+] satisfying lim infξ→+∞ ψi(ξ) > v∗i and v∗i > lim supξ→−∞ ψi(ξ) for i = 1, 2, 3, we can show that there exist ẑ = ẑ(ε,ψ) > 0 such that Φ(z − ẑ)− εe ≤ U0(z,ψ) ≤ Φ(z + ẑ) + εe, ∀z ∈ R. (3.13) By [32, Theorem 2.2.4] and the same strategy as in [29, Theorem 3.1], we have that (3.12) holds true. In fact, by the comparison principle and the definition of upper and lower solutions W± n (z) in Lemma 3.6, we have W− n (z) ≤ Un(z,U0(·)) ≤W+ n (z), ∀z ∈ R, n ∈ N, which implies Φ(z− ẑ− ε0)− εδ(z− ẑ)e−σn ≤ Un(z,ψ) ≤ Φ(z+ ẑ+ ε0) + εδ(z+ ẑ)e−σn. (3.14) We define the semiflow solution of (3.9) as Ψn(ψ) := Un(z,ψ), ∀ψ ∈ X , n ∈ N+. Since limz→−∞Φ(z) = 0 and limz→+∞Φ(z) = v+, by (3.14), we have γ+(ψ) := {Ψn(ψ) : n ≥ 0} is bounded in X . The Ascoli-Arzelà theorem implies that γ+(ψ) is precompact in X and the omega limit set ω(ψ) is nonempty, compact and invariant. Let z0 = ẑ + ε0 and n → ∞ in (3.14), we have the omega limit set ω(ψ) ⊂ I := EJDE-2020/88 COMPETITIVE-COOPERATIVE RECURSION SYSTEMS 15 [Φ(·−z0),Φ(·+z0)]X . Let h(s) = Φ(·+s), for all s ∈ [−z0, z0]. Then h is a monotone homeomorphism from [−z0, z0] onto a subset Î ⊂ I. Then Ψn : X[0,v+] → X[0,v+] is a monotone autonomous semiflow and each h(s) is stable equilibrium for Ψn. Clearly, each ψ ∈ Î satisfies lim infξ→+∞ ψi(ξ) > v∗i > lim supξ→−∞ ψi(ξ) and then γ+(ψ) is precompact. By the similar process in [29, Theorem 3.1] and [32, Theorem 2.2.4], there is sψ ∈ [−z0, z0] such that ω(ψ) = h(sψ) = Φ(·+ sψ). Then limn→∞Ψn(ψ) = Φ(· + sψ). Since Un(x,ψ) = Un(x − cn,ψ) = Ψn(ψ)(x − cn), we obtain lim n→+∞ ‖Un(x,ψ)−Φ(x− cn+ sψ)‖ = 0 uniformly for x ∈ R. If the kernel ki (i = 1, 2, 3) has a compact support, then for any ε > 0 and any ψ(ξ) satisfies lim infξ→+∞ ψi(ξ) > v∗i > lim supξ→−∞ ψi(ξ), then there exist ẑ = ẑ(ε,ψ) > 0 and a large time n0 ∈ N+ such that Φ(z − ẑ)− εe ≤ Un0 (z,ψ) ≤ Φ(z + ẑ) + εe. Then there are n0 and ẑ such that for any z ∈ R, Un0 (z,ψ) ≤ Φ(z + ẑ) + εδ̂ ≤ Φ(z + ẑ) + εδ̂(z + ẑ) = W+ 0 (z + ẑ), Un0 (z,ψ) ≥ Φ(z − ẑ)− εδ̂ ≥ Φ(z − ẑ)− εδ̂(z − ẑ) = W− 0 (z + ẑ) It follows that W− n (z) ≤ Un(z,Un0 (·)) ≤W+ n (z) for z ∈ R and n ∈ N. Note that Un(z,Un0 (·)) = Un+n0 (z,ψ) for any z ∈ R and n ∈ N. 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Differential Equations, 264 (2018), 6507-6534. [29] Y. Zhang, X.-Q. Zhao; Bistable traveling waves in competitive recursion systems, J. Differ- ential Equations, 252 (2012), 2630-2647. [30] L. Zhang, S. Pan; Entire solutions of integrodifference equations, Journal of Difference Equa- tions and Applications, 25 (2019), 504-515. [31] L. Zhang, W.-T. Li, Z.-C. Wang, Y.-J. Sun; Entire solutions for nonlocal dispersal equations with bistable nonlinearity: asymmetric case, Acta Mathematica Sinica, English Series, 35 (2019), 1771-1794. [32] X.-Q. Zhao; Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. Xiongxiong Bao (corresponding author) School of Science, Chang’an University, Xi’an, Shaanxi 710064, China Email address: baoxx2016@chd.edu.cn EJDE-2020/88 COMPETITIVE-COOPERATIVE RECURSION SYSTEMS 17 Ting Li School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China Email address: liting2017@lzu.edu.cn 1. Introduction 2. Preliminary 3. Existence and globally stability of traveling wave Acknowledgments References