# Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation models

## TL;DR

This paper develops Lyapunov-based methods to analyze the stability of nonstandard finite difference schemes for metapopulation models, ensuring the discrete models retain key properties of the continuous systems.

## Contribution

It introduces a Lyapunov function approach to verify stability of NSFD schemes, simplifying the process compared to previous methods.

## Key findings

- NSFD schemes preserve properties of metapopulation models
- Lyapunov method simplifies stability analysis
- Numerical examples confirm theoretical results

## Abstract

In this paper nonstandard finite difference (NSFD) schemes of two metapopulation models are constructed. The stability properties of the discrete models are investigated by the use of a generalization of Lyapunov stability theorem. Due to this result we have proved that the NSFD schemes preserve all properties of the metapopulation models. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes. The method of Lyapunov functions proves to be much simpler than the standard method for studying stability of the discrete metapopulation model in our very recent paper.