2ND-ORDER, NUMERICAL-METHODS FOR THE SOLUTION OF EQUATIONS IN POPULATION MODELING

A second-order explicit method is developed for the numerical solution of the Ricatti (logistic) initial-value problem u' = du/dt = alpha-u(1 - u), t > 0, u(0) = U0, in which alpha not-equal 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. 1). 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-alpha, provided the denominator of the method does not vanish. Convergence is monotonic or oscillatory depending on the magnitude of the product-alpha-l, where l is the parameter in the discretization of the independent variable t. This dependence of the type of convergence on alpha-l 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 alpha-u(1 - 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.