This paper deals with entire solutions (classical solutions defined globally in time and space) of a two-species strong competition model. For this system, it is well known that there exist two-front entire solutions which behave as two traveling fronts moving towards each other from both sides of the x-axis. In this paper, in terms of traveling fronts connecting two different constant states from the coexistence state and the two semi-trivial states, we build entire solutions originating from three and four fronts stuck between appropriate super and subsolutions. Moreover, the non-existence of entire solutions originating from more than seven traveling fronts is proved.
