In the present paper, we show that an analogous N-barrier maximum principle (see [1]) remains true for lattice systems. This extends the results in [1] from continuous equations to discrete equations. In order to overcome the difficulty induced by a discretized version of the classical diffusion in the lattice systems, we propose a more delicate construction of the N-barrier which is appropriate for the proof of the N-barrier maximum principle for lattice systems. As an application of the discrete N-barrier maximum principle, we study a coexistence problem of three species arising from biology, and show that the three species cannot coexist under certain conditions.

 

[1] C.-C. Chen and L.-C. Hung, A maximum principle for diffusive lotka-volterra systems of two competing species, JDE