The authors study wave propagation for the system
where and are periodic in . The traveling waves for the one-dimensional spatial discrete version of (1) is considered in [J.-S. Guo and F. Hamel, Math. Ann. 335, No. 3, 489–525 (2006; Zbl 1116.35063)], the results of which are extended to the two dimensional spatial discrete version in the present article. The authors investigate the following system:
for some positive integer . Here, is the wave speed, with . Let be the wave profile. Under some assumptions, the authors show the following conclusions:
there is a such that (2) admits a traveling wave solution for ;
if (2) admits a traveling wave solution for , then ;
if is a traveling wave solution of (2) with , then is strictly increasing in ;
by constructing a discrete approximate system for (1) in the two dimensional case, the minimal wave speed of the approximate system converges to the minimal wave speed of (1).