A second-order explicit method is developed for the numerical solution of the Ricatti (logistic) initial- value problem u′ ≡ du/dt = αu(l - u), t > 0, u (0) = U0, in which α ≠ 0 is a real parameter. The method is based on two first-order methods which appeared in an earlier paper by the authors (Twizell et al. '). In addition to being chaos-free and of higher order, the novel method is seen to converge to the correct, stable, steady-state solution for any value of the parameter α, provided the denominator of the method does not vanish. Convergence is monotonic or oscillatory depending on the magnitude of the product αl, where l is the parameter in the discretization of the independent variable t. This dependence of the type of convergence on αl is likened to the behaviour of the well known Crank-Nicolson method for solving the simple heat equation. Conversion of the numerical method to a reliable, empirical model for predicting the limited growth of successive generations of a population is given. When extended to the numerical solution of Fisher's equation, in which the quadratic polynomial αu(l - u) appears as the reaction term, the numerical solution is found by solving a linear algebraic system at each time step, as opposed to solving a non-linear system, which often happens when solving non-linear partial differential equations. [ABSTRACT FROM AUTHOR]