Global-in-time behavior of lotka-volterra system with diffusion: Skew-symmetric case

We study the global-in-time behavior of the Lotka- Volterra system with diffusion. In the first category, the interaction matrix is skew-symmetric and the linear terms are non-increasing. There, the solution exists globally in time with compact orbit, provided that n ≥ 2, where n denotes the space dimension. Under the presence of entropy, its Ö-limit set is composed of a spatially homogeneous orbit. Furthermore, any spatially homogeneous solution is periodic in time, provided with constant entropy. In the second category, the interaction matrix exhibits a dissipative profile. There, the solution exists globally in time with compact orbit if n ≥ 3. Its ω-limit set, furthermore, is contained in spatially homogeneous stationary states. In particular, no periodic-in-time solution is admitted.