  We study the distribution of the maximal displacement of particles positions
for the whole time of the population existence in the model of critical and
subcritical catalytic branching random walk on Z. In particular, we prove that
in the case of simple symmetric random walk on Z, the distribution of the
maximal displacement has "a heavy tail" decreasing as a function of the power
1/2 or 1, when the branching process is critical or subcritical, respectively.
These statements describe new effects which do not arise in the corresponding
investigations of the maximal displacement of critical and subcritical
branching random walks on Z.
