  The next-iterate operators and corresponding next-iterate root-sets and
root-curves associated with the nullclines of a planar discrete map are
introduced. How to augment standard phase portraits that include the nullclines
and the direction field, by including the signs of the root-operators
associated with their nullclines, thus producing an augmented phase portrait,
is described. The sign of a next-iterate operator associated with a nullcline
determines whether a point is mapped above or below the corresponding nullcline
and can, for example, identify positively invariant regions. Using a
Lotka-Volterra type competition model, we demonstrate how to construct the
augmented phase portrait. We show that the augmented phase portrait provides an
elementary, alternative approach for determining the complete global dynamics
of this model. We further explore the limitations and potential of the
augmented phase portrait by considering a Ricker competition model, a model
involving mutualism, and a predator-prey model.
