  We discuss a discrete-time model for motion of substance in a channel of a
network. For the case of stationary motion of the substance and for the case of
time-independent values of the parameters of the model we obtain a new class of
statistical distributions that describe the distribution of the substance along
the nodes of the channel. The case of interaction between a kind of substance
specific for a node of the network and another kind of substance that is leaked
from the channel is studied in presence of possibility for conversion between
the two substances. Several scenarios connected to the dynamics of the two
kinds of substances are described. The studied models: (i) model of motion of
substance through a channel of a network, and (ii) model of interaction between
two kinds of substances in a network node connected to the channel, are
discussed from the point of view of human migration dynamics and interaction
between the population of migrants and the native population of a country.
