  The dynamics of a prey-predator system with foraging facilitation among
predators are investigated. The analysis involves the computation of many
semi-algebraic systems of large degrees. We apply the pseudo-division
reduction, real-root isolation technique and complete discrimination system of
polynomial to obtain parameter conditions for the exact number of equilibria
and their qualitative properties as well as a complete investigation of
bifurcations including saddle-node, transcritical, pitchfork, Hopf and
Bogdanov-Takens bifurcations. Moreover, numerical simulations are presented to
support our theoretical results.
