  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.
