The basic reproduction number R0 in structured population dynamics is defined as the spectral radius of the generation evolution operator induced by the integral kernel of the renewal integral equation. If the basic population dynamics is described by the evolutionary system associated with a nonautonomous differential equation, the generation evolution operator is calculated from the infinitesimal generator of the evolution semigroup induced from the evolutionary system. Using the basic reproduction number defined by the generation evolution operator, we can examine existence and stability of total orbits of nonlinear nonautonomous system, in which the total orbit is given as a fixed point of the evolution semigroup. Thus the idea of R0 is uniquely extended to the threshold value for extinction and persistence of population in time-heterogeneous environments.
