  This paper is concerned with front-like entire solutions for monostable
reactiondiffusion systems with cooperative and non-cooperative nonlinearities.
In the cooperative case, the existence and asymptotic behavior of spatially
independent solutions (SIS) are first proved. Combining a SIS and traveling
fronts with different wave speeds and directions, the existence and various
qualitative properties of entire solutions are then established using
comparison principle. In the non-cooperative case, we introduce two auxiliary
cooperative systems and establish some comparison arguments for the three
systems. The existence of entire solutions is then proved via the traveling
fronts and SIS of the auxiliary systems. Our results are applied to some
biological and epidemiological models. To the best of our knowledge, it is the
first work to study the entire solutions of non-cooperative reaction-diffusion
systems.
