Two slightly different test problems have been used to examine nonlinear stability behaviour of numerical methods for solving systems of ordinary differential equations. These test problems, for BN-stability and for monotonicity, both lead to the concept of algebraic stability. In this paper, it is shown that these two problems may be combined to yield weaker stability conditions. Nonautonomous systems give a more restrictive stability condition than autonomous systems. Irreducible methods with some zero quadrature weights can be stable in this weak sense.