Discrete and continuous fractional persistence problems - the positivity property and applications
This work provides theoretical foundations for preserving qualitative properties in fractional differential equations, benefiting researchers in fractional calculus and mathematical biology.
The authors study fractional persistence, proving conditions under which positivity, order preservation, equilibrium points, and stability are preserved when embedding classical differential equations into fractional Caputo systems. They also develop a nonstandard finite difference scheme using Grünwald-Letnikov derivatives that unconditionally preserves positivity, and apply the results to a fractional prey-predator model.
In this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical ($α=1$) differential equation in the fractional case (here using fractional Caputo's derivatives) and the numerical scheme which are associated (here with discrete Grünwald-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 Grünwald-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.