  This paper investigates the dynamical properties of the Leslie-Gower model
with Alle effects, cooperative hunting, and constant placement rates. The
conditions for the existence of the triple equilibrium point of the model are
first analyzed. Subsequently, the canonical type theory and the qualitative
theory of planar systems are applied to obtain that the triple equilibrium
point can be a node with a residual dimension of 2 and an equilibrium point
with a residual dimension of 3 under different parameter conditions. Finally,
it is proved that the system bifurcates with a residual dimension of 2 in the
vicinity of the node with cooperative hunting and placement rate as branching
parameters.
