  Modified Patankar-Runge-Kutta (MPRK) methods preserve the positivity as well
as conservativity of a production-destruction system (PDS) of ordinary
differential equations for all time step sizes. As a result, higher order MPRK
schemes do not belong to the class of general linear methods, i.e. the iterates
are generated by a nonlinear map $\mathbf g$ even when the PDS is linear.
Moreover, due to the conservativity of the method, the map $\mathbf g$
possesses non-hyperbolic fixed points.
  Recently, a new theorem for the investigation of stability properties of
non-hyperbolic fixed points of a nonlinear iteration map was developed. We
apply this theorem to understand the stability properties of a family of second
order MPRK methods when applied to a nonlinear PDS of ordinary differential
equations. It is shown that the fixed points are stable for all time step sizes
and members of the MPRK family. Finally, experiments are presented to
numerically support the theoretical claims.
