Positivity and boundedness preserving nonstandard finite difference schemes for solving Volterra’s population growth model

Abstract

In this work, we extend the Mickens’ methodology to construct nonstandard finite difference (NSFD) schemes preserving positivity and boundedness of the nonlinear Volterra integro-differential population growth model. A rigorously mathematical study for the positivity, boundedness, convergence and error bounds of the proposed NSFD schemes is provided. It is proved that the NSFD schemes not only converge, but also preserve the positivity and boundedness of the integro-differential model for all finite step sizes. Furthermore, the constructed NSFD schemes can be extended to formulate nonstandard discretization schemes for general Volterra integral equations and fractional-order Volterra integro-differential population growth models. Finally, the theoretical results and advantages of the NSFD schemes over standard ones are supported and illustrated by a set of numerical experiments. The numerical experiments show that some typical standard numerical schemes such as the Euler scheme, the second order and classical fourth order Runge–Kutta schemes fail to correctly preserve the positivity and boundedness for some given step sizes. As a result, they can generate numerical approximations that are completely different from the solutions of the integro-differential model. However, these properties are preserved by the NSFD schemes when using the same step sizes.