  Global dynamical behaviors of the competitive Lotka-Volterra system even in
3-dimension are not fully understood. The Lyapunov function can provide us such
knowledge once it is constructed. In this paper, we construct explicitly the
Lyapunov function in three examples of the competitive Lotka-Volterra system
for the whole state space: (1) the general 2-dimensional case; (2) a
3-dimensional model; (3) the model of May-Leonard. The dynamics of these
examples include bistable case and cyclical behavior. The first two examples
are the generalized gradient system defined in the Appendixes, while the model
of May-Leonard is not. Our method is helpful to understand the limit cycle
problems in general 3-dimensional case.
