  We construct a continuous state branching process with immigration (CBI)
whose immigration depends on the CBI itself and we recover a continuous state
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.
