DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2010.14.111 DYNAMICAL SYSTEMS SERIES B Volume 14, Number 1, July 2010 pp. 111–128 TRAVELING PLANE WAVE SOLUTIONS OF DELAYED LATTICE DIFFERENTIAL SYSTEMS IN COMPETITIVE LOTKA-VOLTERRA TYPE Cheng-Hsiung Hsu Department of Mathematics National Central University Chung-Li 32001, Taiwan Ting-Hui Yang Department of Mathematics Tamkang University Tamsui, Taipei County 25137, Taiwan (Communicated by Yuan Lou) Abstract. In this work we consider the existence of traveling plane wave solutions of systems of delayed lattice differential equations in competitive Lotka-Volterra type. Employing iterative method coupled with the explicit construction of upper and lower solutions in the theory of weak quasi-monotone dynamical systems, we obtain a speed, c * , and show the existence of traveling plane wave solutions connecting two different equilibria when the wave speeds are large than c * . 1. Introduction. The purpose of this work is to investigate the existence of trav- eling plane wave solutions of systems of N delayed 2-dimensional lattice differential equations (2D-LDEs) in competitive Lotka-Volterra type. The nth 2D-LDE in the systems is of the form d dt u n;i,j (t)= L n [u n;i,j ](t)+ u n;i,j (t)f n ( u i,j (t), (u i,j )  n t ) , (1) for (i,j ) ∈ Z 2 and 1 ≤ n ≤ N , where u i,j (t) := (u 1;i,j (t), ··· ,u N;i,j (t)), (u i,j )  n t (τ 1 , ··· ,τ n−1 ,τ n+1 , ··· ,τ N ) :=(u 1;i,j (t − τ 1 ), ··· ,u n−1;i,j (t − τ n−1 ),u n+1;i,j (t − τ n+1 ), ··· ,u N;i,j (t − τ N )), and L n [u n;i,j ](t)= d n,1 u n;i+1,j (t)+d n,2 u n;i,j+1 (t) + d n,3 u n;i−1,j (t)+ d n,4 u n;i,j−1 (t) − d n,0 u n;i,j (t). Here τ i and d i,j are positive real constants which represent the time delays and coupling coefficients respectively. Let τ := max{τ 1 , ··· ,τ n−1 ,τ n+1 , ··· ,τ N }. All f n are C 1 functions from R N × C 1 ([−τ, 0], R) N−1 to R where C 1 ([−τ, 0], R) N−1 is 2000 Mathematics Subject Classification. Primary: 34A33, 34C37, 34K10, 35C07; Secondary: 92B20. Key words and phrases. Traveling Wave Solution, Delayed Lattice Differential Equations, Up- per And Lower Solutions, Heteroclinic Solutions. 111