General nonstandard finite-difference schemes for differential equations with three fixed-points

We construct general nonstandard finite-difference (NSFD) schemes that provide explicit methods for first-order differential equations with three fixed-points y'(t) = +/- y(y alpha)(y - 1) where 0 <= alpha <= 1. For y >= 0, these methods, regardless of the step-size chosen, are stable with respect to the monotonicity of solutions and are elementary stable. That is, they preserve the critical properties of the original differential equation such as the positivity of the solutions, the stability behavior of all fixed-points, and the monotonicity of solutions within each subinterval (0, alpha), (alpha, 1), and (1, infinity). (C) 2008 Elsevier Ltd. All rights reserved.