  In this paper, we discuss a hierarchical model, based on a Ricker competition
model. The species considered are competing for resources and may be subject to
an Allee effect due to mate limitation, anti-predator vigilance or aggression,
cooperative predation or resource defense, or social thermoregulation. The
species may be classified into a more dominant species and less dominant or
``wimpy" species or just as a predator and prey. The model under consideration
also has components taking into account immigration in both species and a more
structured Holling type II functional response. Local and global stability
analyses are discussed and simulations are provided. We also consider
demographic stochasticity on the species due to environmental fluctuations in
the form of Wiener processes and we show that there are conditions under which
a global solution exists, stationary distributions exists, and strong
persistence in mean of the species is possible. We also use Wasserstein
distance to show empirically that stochasticity can acts as bifurcation
parameter.
