We investigate global behavior of $x_{n+1} = T(x_{n}),\quad n=0,1,2,...$ (E) where $T:\mathcal{ R}\rightarrow \mathcal{ R}$ is a competitive (monotone with respect to the south-east ordering) map on a set $\mathcal{R}\subset \mathbb{R}^2$ with nonempty interior. We assume the existence of a unique fixed point $\overline{e}$ in the interior of $\mathcal{ R}$. We give very general conditions which are easily verifiable for (E) to exhibit either competitive-exclusion or competitive-coexistence. More specifically, we obtain sufficient conditions for the interior fixed point $\overline{ e}$ to be a global attractor when $\mathcal{ R}$ is a rectangular region. We also show that when $T$ is strongly monotone in $\mathcal{ R}^{\circ}$ (interior of $\mathcal{ R}$), $\mathcal{ R}$ is convex, the unique interior equilibrium $\overline{ e}$ is a saddle, and a technical condition is satisfied, the corresponding global stable and unstable manifolds are the graphs of monotonic functions, and the global stable manifold splits the domain into two connected regions, which under additional conditions on $\mathcal{R}$ and on $T$ are shown to be basins of attraction of fixed points on the boundary of $\mathcal{R}$. Applications of the main results to specific difference equations are given.

This paper discusses the use of observational studies of human-robot social interaction in open human-inhabited environments as a method for improving on the design and evaluating the interactive capabilities of social robots. First, we discuss issues that have surfaced in attempts to evaluate social interactions between humans and robots. Next, we review two observational studies involving robots interacting socially with humans and discuss how the results can be applied to improving robot design. The first is an analysis of a mobile conference-attending robot that performed a search task by augmenting its perception through social interaction with human attendees. The second is an analysis of a stationary robotic receptionist that provides information to visitors and enhances interaction through story-telling. Through these examples, we show how observational studies can be applied to human-robot social interactions in varying contexts and with differing tasks to quantitatively and qualitatively evaluate (and discover unanticipated aspects of) the social interaction. Finally, we discuss design recommendations suggested by insights gained through these analyses