  In numerous papers, the behaviour of stochastic population models is
investigated through the sign of a real quantity which is the growth rate of
the population near the extinction set. In many cases, it is proven that when
this growth rate is positive, the process is persistent in the long run, while
if it is negative, the process converges to extinction. However, the critical
case when the growth rate is null is rarely treated. The aim of this paper is
to provide a method that can be applied in many situations to prove that in the
critical case, the process congerves in temporal average to extinction. A
number of applications are given, for Stochastic Differential Equations and
Piecewise Deterministic Markov Processes modelling prey-predator,
epidemilogical or structured population dynamics.
