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Since almost twenty years, modified Patankar--Runge--Kutta (MPRK) methodshave proven to be efficient and robust numerical schemes that preservepositivity and conservativity of the production-destruction systemirrespectively of the time step size chosen. Due to these advantageousproperties they are used for a wide variety of applications. Nevertheless,until now, an analytic investigation of the stability of MPRK schemes is stillmissing, since the usual approach by means of Dahlquist's equation is notfeasible. Therefore, we consider a positive and conservative 2D test problemand provide statements usable for a stability analysis of general positive andconservative time integrator schemes based on the center manifold theory. Weuse this approach to investigate the Lyapunov stability of the second orderMPRK22($\alpha$) and MPRK22ncs($\alpha$) schemes. We prove thatMPRK22($\alpha$) schemes are unconditionally stable and derive the stabilityregions of MPRK22ncs($\alpha$) schemes. Finally, numerical experiments arepresented, which confirm the theoretical results.

On Lyapunov Stability of Positive and Conservative Time Integrators and Application to Second Order Modified Patankar--Runge--Kutta Schemes