IUMJ

Title: Global-in-time behavior of Lotka-Volterra system with diffusion

Authors: Takashi Suzuki and Yoshio Yamada

Issue: Volume 64 (2015), Issue 1, 181-216

Abstract:

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 $n\leq2$, where $n$ denotes the space dimension. Under the presence of entropy, its $\omega$-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 $n\leq3$. Its $\omega$-limit set, furthermore, is contained in spatially homogeneous stationary states. In particular, no periodic-in-time solution is admitted.