  In this paper we investigate the stability properties of the so-called gBBKS
and GeCo methods, which belong to the class of nonstandard schemes and preserve
the positivity as well as all linear invariants of the underlying system of
ordinary differential equations for any step size. A stability investigation
for these methods, which are outside the class of general linear methods, is
challenging since the iterates are always generated by a nonlinear map even for
linear problems. Recently, a stability theorem was derived presenting criteria
for understanding such schemes.
  For the analysis, the schemes are applied to general linear equations and
proven to be generated by $\mathcal C^1$-maps with locally Lipschitz continuous
first derivatives. As a result, the above mentioned stability theorem can be
applied to investigate the Lyapunov stability of non-hyperbolic fixed points of
the numerical method by analyzing the spectrum of the corresponding Jacobian of
the generating map. In addition, if a fixed point is proven to be stable, the
theorem guarantees the local convergence of the iterates towards it.
  In the case of first and second order gBBKS schemes the stability domain
coincides with that of the underlying Runge--Kutta method. Furthermore, while
the first order GeCo scheme converts steady states to stable fixed points for
all step sizes and all linear test problems of finite size, the second order
GeCo scheme has a bounded stability region for the considered test problems.
Finally, all theoretical predictions from the stability analysis are validated
numerically.
