  A class of models of biological population and communities with a singular
equilibrium at the origin is analyzed; it is shown that these models can
possess a dynamical regime of deterministic extinction, which is crucially
important from the biological standpoint. This regime corresponds to the
presence of a family of homoclinics to the origin, so-called elliptic sector.
The complete analysis of possible topological structures in a neighborhood of
the origin, as well as asymptotics to orbits tending to this point, is given.
An algorithmic approach to analyze system behavior with parameter changes is
presented. The developed methods and algorithm are applied to existing
mathematical models of biological systems. In particular, we analyze a model of
anticancer treatment with oncolytic viruses, a parasite-host interaction model,
and a model of Chagas' disease.
