  In a famous paper, Bezuidenhout and Grimmett demonstrated that the contact
process dies out at the critical point.Their proof technique has often been
used to study the growth of population patterns. The present text is intended
as an introduction to their ideas, with examples of minimal technicality. In
particular, we recover the basic theorem about Galton-Watson chains: except in
a degenerate case, survival is possible only if the fertility rate exceeds 1.
The classical proof that is taught in classrooms is essentially analytic, based
on generating functions and convexity arguments. Following the
Bezuidenhout-Grimmett way, we propose a proof that is more consistent with
probabilistic intuition. We also study the survival problem for an original
model, mixing sexual and asexual reproduction.
