Global compact attractors, in combination with Lyapunov functions and the Laplace transform, can be used to prove the global stability of extinction and persistence equilibria. A linearized stability analysis can be bypassed.  For illustration, two models are considered:
one for bacteria and phages in a chemostat;
and one for the spread of an infection in a spatially distributed population.


Hal L. Smith, Horst R. Thieme,
Dynamical Systems and Population Persistence, AMS, to appear 2011

Hal. L. Smith, Horst R. Thieme,
Persistence of  bacteria and phages in a chemostat, preprint

Horst R. Thieme,
Global stability of the endemic  equilibrium in
infinite dimension: Lyapunov functions and  positive operators, preprint