We provide a procedure which generates a rational control Lyapunov function and a polynomial stabilizer for nonlinear systems described by analytic functions satisfying some regularity conditions. Furthermore, an improved estimate of the domain of attraction of the closed-loop system can be computed by means of previously introduced rational Lypunov functions. For polynomial systems, we indicate that the existence of a polynomial feedback stabilizer is guaranteed by the existence of a rational control Lyapunov function. We illustrate the proposed procedure for the stabilization of the population co-existence equilibrium of a predator-prey model describing tumor dynamics.