A novel second-order nonstandard finite difference method preserving dynamical properties of a general single-species model

Abstract

In this paper, we extend the Mickens' methodology to construct a second-order nonstandard finite difference (NSFD) method, which preserves dynamical properties including positivity, local asymptotic stability and especially, global asymptotic stability of a general single-species model. This NSFD method is based on a novel weighted non-local approximation of the right-hand side function in combination with the renormalization of the denominator function. The weight guarantees the dynamic consistency and the nonstandard denominator function ensures the convergence of order 2 of the NSFD method. The result is that we obtain a second-order and dynamically consistent NSFD method. It is proved that the NSFD method is simple and efficient and can be extended for solving a broad range of mathematical models arising in real-world applications. Also, we combine the constructed second-order NSFD method with Richardson's extrapolation technique to generate high-order numerical approximations. Finally, the theoretical findings are illustrated and supported by numerical experiments.

Keywords:

• NSFD method
• second-order
• single-species
• positivity
• global asymptotic stability

2020 Mathematics Subject Classifications:

• 34A45
• 65L05
• 65L12

Acknowledgments

We would like to thank the editor and anonymous referees for useful and valuable comments that led to a great improvement of the paper.

Authors' contributions

Manh Tuan Hoang: Conceptualization, Methodology, Software, Formal analysis, Writing- Original draft preparation, Methodology, Writing – Review & Editing, Supervision.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Availability of supporting data

The data supporting the findings of this study are available within the article [and/or] its supplementary materials.