  We establish the relation between local stability of equilibria and slopes of
critical curves for a specific class of difference equations. We then use this
result to give global behavior results for nonnegative solutions of the system
of difference equations \begin{equation*} %\tag{LGIN} \begin{array}{rcl}
x_{n+1} & = & \displaystyle \frac{b_1 x_n}{1+x_n+c_1 y_{n}} +h_1 y_{n+1} & = &
\displaystyle \frac{b_2 y_n}{1+y_n+c_2 x_{n}} +h_2 \end{array} \quad n=0,1,...,
\quad (x_0,y_0) \in [0,\infty)\times [0,\infty) \end{equation*} with positive
parameters. In particular, we show that the system has between one and three
equilibria, and that the number of equilibria determines global behavior as
follows: if there is only one equilibrium, then it is globally asymptotically
stable. If there are two equilibria, then one is a local attractor and the
other one is nonhyperbolic. If there are three equilibria, then they are
linearly ordered in the south-east ordering of the plane, and consist of a
local attractor, a saddle point, and another local attractor. Finally, we give
sufficient conditions for having a unique equilibrium.
