Persistence and extinction in the anti-symmetric Lotka-Volterra systems

We investigate the global dynamics of the Lotka-Volterra systems with anti-symmetric interactions. We prove that the dynamics of system can be determined by some associated linear inequalities, which are of independent interest in linear programming. Necessary and sufficient conditions for the existence of solutions to the linear inequalities are established. When the solutions of the linear inequalities exist, the solutions of corresponding Lotka-Volterra systems are uniformly bounded, and the persistence and extinction of any species can be classified by an index set, which is uniquely determined by the linear inequalities. When the solutions of linear inequalities do not exist, the solutions of the Lotka-Volterra systems are unbounded. As an application, we analyze the three-species case and classify the long-time behaviors of the solutions to the system.