For continuous-time nonlinear systems with inputs, we introduce the notion of an asymptotic ratio input-to-state stability (ISS) Lyapunov function. The derivative of such a function along solutions is upper-bounded by the difference of two terms whose ratio is asymptotically smaller than 1 for large states. This asymptotic ratio condition is sometimes more convenient to check than standard ISS Lyapunov function conditions. We show that the existence of an asymptotic ratio ISS Lyapunov function is equivalent to ISS. A related notion of ISS with nonuniform convergence rate is also explored.