Abstract: In this paper, we are concerned with the long-time behavior of competition systems between two species with ratio-dependent functional responses subject to the homogeneous Robin boundary condition. First, we establish the existence of positive solutions to these systems by using the fixed point index theory in coin and comparison principle. Second, we discuss the relationships between positive equi-libria and positive solutions of these systems over a large domain. Finally, we study the extinction and permanence of time-dependent positive solutions to these systems and obtain the conditions under which two species can coexist.

Key words: competition systems, existence, fixed point index theory, upper-lower solutions; extinction and permanence