  In this manuscript, we are interested in the long-term behaviour of branching
processes with pairwise interactions (BPI-processes). A process in this class
behaves as a pure branching process with the difference that competition and
cooperation events between pairs of individuals are also allowed. Here, we
provide a series of integral tests that explain how competition and cooperation
regulate the long-term behaviour of BPI-processes. In particular, such integral
tests describe the events of explosion and extinction and provide conditions
under which the process comes down from infinity. Moreover, we also determine
whether the process admits, or not, a stationary distribution. Our arguments
use a random time change representation in terms of a modified branching
process with immigration and moment duality. The moment dual of BPI-processes
turns out to be a family of diffusions taking values on $[0,1]$ which are
interesting in their own right and that we introduce as generalised
Wright-Fisher diffusions.
