  A general model of catalytic branching process (CBP) with any finite number
of catalysis centers in a discrete space is studied. More exactly, it is
assumed that particles move in this space according to a specified Markov chain
and they may produce offspring only in the presence of catalysts located at
fixed points. The asymptotic (in time) behavior of the total number of
particles as well as the local particles numbers is investigated. The problems
of finding the global extinction probability and local extinction probability
are solved. Necessary and sufficient conditions are established for phase of
pure global survival and strong local survival. Under wide conditions the limit
theorems for the normalized total and local particles numbers in supercritical
CBP are proved in the sense of almost surely convergence as well as with
respect to convergence in distribution. Generalizations of a number of previous
results are obtained as well. In the proofs the main role is played by recent
results by the author devoted to classification of CBP and the moment analysis
of the total and local particles numbers in CBP.
  Keywords and phrases: catalytic branching process, extinction probability,
strong local survival, pure global phase, limit theorems, multi-type
Bellman-Harris process.
