  Modified Patankar (MP) schemes are conservative, linear implicit and
unconditionally positivity preserving time-integration schemes constructed for
production-destruction systems. For such schemes, a classical stability
analysis does not yield any information about the performance. Recently, two
different techniques have been proposed to investigate the properties of MP
schemes. In Izgin et al. [ESAIM: M2AN, 56 (2022)], inspired from dynamical
systems, the Lyapunov stability properties of such schemes have been
investigated, while in Torlo et al. [Appl. Numer. Math., 182 (2022)] their
oscillatory behaviour has been studied. In this work, we investigate the
connection between the oscillatory behaviour and the Lyapunov stability and we
prove that a condition on the Lyapunov stability function is necessary to avoid
oscillations. We verify our theoretical result on several numerical tests.
