We  construct a  continuous state  branching process  with immigration
  (CBI) whose  immigration depends  on the CBI  itself and we  recover a
  continuous branching  process (CB). This provides  a dual construction
  of the pruning at nodes of  CB introduced by the authors in a previous
  paper. This construction  is a natural way to  model neutral mutation.
  Using exponential formula, we compute the probability of extinction of
  the original  type population in  a critical or  sub-critical quadratic
  branching,  conditionally   on  the   non  extinction  of   the  total
  population.