  In this paper, we introduce a two-sex controlled branching model to describe
the interaction between predator and prey populations with sexual reproduction.
This process is a two-type branching process, where the first type corresponds
to the predator population and the second one to the prey population. While
each population is described via a two-sex branching model, the interaction and
survival of both groups is modelled through control functions depending on the
current number of individuals of each type in the ecosystem. We provide
necessary and sufficient conditions for the ultimate extinction of both
species, the fixation of one of the species and the coexistence of both of
them. Moreover, the description of the present predator-prey two-sex branching
process on the fixation events can be performed in terms of the behaviour of a
one-type two-sex branching process with a random control on the number of
individuals, which is also introduced and analysed.
