Abstract: In this paper, the authors derive and study a delayed diffusion system, which models the interaction between the two species, the adult members of
which are in cooperation. By using the method of sub- and super-solutions due to Redlinger, we show that the diffusive delay model generates simple global dynamics, i.e., the zero steady state and the boundary equilibria are linear unstable and the unique positive steady state is globally asymptotically stable. We also establish the existence of traveling wave fronts connecting the zero solution of this equation with the unique positive steady state.
The approach used in this paper is the upper-lower solutions technique and the monotone iteration recently developed by Wang, Li and Ruan for reaction-diffusion systems with spatio-temporal delays.

Key words: Cooperation, time delay, traveling wave front, global stability, stage structure, reaction-diffusion equation