  In this monograph, we introduce a new model in population dynamics that
describes two species sharing the same environmental resources in a situation
of open hostility. The interactions among these populations are described not
in terms of random encounters, but via the strategic decisions of one
population that can attack the other according to different levels of
aggressiveness. This leads to a non-variational model for the two populations
at war, taking into account structural parameters such as the relative fit of
the two populations with respect to the available resources and the
effectiveness of the attack strikes of the aggressive population. The analysis
that we perform is rigorous and focuses on the dynamical properties of the
system, by detecting and describing all the possible equilibria and their
basins of attraction. Moreover, we will analyze the strategies that may lead to
the victory of the aggressive population, i.e. the choices of the
aggressiveness parameter, in dependence of the structural constants of the
system and possibly varying in time in order to optimize the efficacy of the
attacks, which take to the extinction in finite time of the defensive
population. The model that we present is flexible enough to include also
technological competition models of aggressive companies releasing computer
viruses to set a rival company out of the market.
