  Spontaneous formation of transverse patterns is ubiquitous in nonlinear
dynamical systems of all kinds. An aspect of particular interest is the active
control of such patterns. In nonlinear optical systems this can be used for
all-optical switching with transistor-like performance, for example realized
with polaritons in a planar quantum-well semiconductor microcavity. Here we
focus on a specific configuration which takes advantage of the intricate
polarization dependencies in the interacting optically driven polariton system.
Besides detailed numerical simulations of the coupled light-field exciton
dynamics, in the present paper we focus on the derivation of a simplified
population competition model giving detailed insight into the underlying
mechanisms from a nonlinear dynamical systems perspective. We show that such a
model takes the form of a generalized Lotka-Volterra system for two competing
populations explicitly including a source term that enables external control.
We present a comprehensive analysis both of the existence and stability of
stationary states in the parameter space spanned by spatial anisotropy and
external control strength. We also construct phase boundaries in non-trivial
regions and characterize emerging bifurcations. The population competition
model reproduces all key features of the switching observed in full numerical
simulations of the rather complex semiconductor system and at the same time is
simple enough for a fully analytical understanding of the system dynamics.
