Non-Oscillation Principle for Eventually Competitive and Cooperative Systems

A nonlinear dynamical system is called eventually competitive (or cooperative) provided that it preserves a partial order in backward (or forward) time only after some reasonable initial transient. We presented in this paper the Non-oscillation Principle for eventually competitive or cooperative systems, by which the non-ordering of (both $\omega$- and $\alpha$-) limit sets is obtained for such systems; and moreover, we established the Poincar\'{e}-Bendixson Theorem and structural stability for three-dimensional eventually competitive and cooperative systems.