Discrete and continuous fractional persistence problems - the positivity property and applications

TL;DR

This paper investigates the positivity, order preservation, and stability in continuous and discrete fractional differential equations, providing conditions and schemes that maintain these properties, with applications to a fractional prey-predator model.

Contribution

It introduces explicit conditions for positivity and order preservation in fractional systems and develops a non-standard finite difference scheme that unconditionally preserves positivity.

Findings

Positivity and stability are preserved under certain fractional conditions.

A discrete scheme based on Gr"unwald-Letnikov derivatives preserves positivity unconditionally.

Application to a fractional prey-predator model demonstrates practical relevance.

Abstract

In this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical ($\alpha=1$) differential equation in the fractional case (here using fractional Caputo's derivatives) and the numerical scheme which are associated (here with discrete Gr\"unwald-Letnikov derivatives). Our main concerns are positivity, order preserving ,equilibrium points and stability of these points. We formulate explicit conditions under which a fractional system preserves positivity. We deduce also sufficient conditions to ensure order preserving. We deduce from these results a fractional persistence theorem which ensures that positivity, order preserving, equilibrium points and stability is preserved under a Caputo fractional embedding of a given differential equation. At the discrete level, the problem is more complicated. Following a strategy initiated by R. Mickens dealing with non local approximations, we define a non standard finite difference scheme for fractional differential equations based on discrete Gr\"unwald-Letnikov derivatives, which preserves positivity unconditionally on the discretization increment. We deduce a discrete version of the fractional persistence theorem for what concerns positivity and equilibrium points. We then apply our results to study a fractional prey-predator model introduced by Javidi and al.