Conditions for the existence of a stable equilibrium and for the existence of an asymptotically stable equilibrium for a strongly
order preserving semiflow are presented. Analyticity of the semiflow and the compactness of certain subsets of the set of
equilibria are required for the latter and yield finiteness of the equilibrium set. Our results are applied to semilinear
parabolic partial differential equations and to the classical Kolmogorov competition system with diffusion.