  We describe the results of analytic calculations and computer simulations of
adaptive predictors (predictive agents) responding to an evolving chaotic
environment and to one another. Our simulations are designed to quantify
adaptation and to explore co-adaptation for a simple calculable model of a
complex adaptive system. We first consider the ability of a single agent,
exposed to a chaotic environment, to model, control, and predict the future
states of that environment. We then introduce a second agent which, in
attempting to model and control both the chaotic environment and the first
agent, modifies the extent to which that agent can identify patterns and
exercise control. We find that (i) optimal adaptive predictors have an optimal
memory and an optimal complexity, which are small for a rapidly changing map
dynamics and (ii) that the predictive power can be increased by imposing chaos
or random noise onto the map dynamics. The competition between the two
predictive agents can lead either to chaos, or to metastable emergent behavior,
best described as a leader-follower relationship. Our results suggest a
correlation between optimal adaptation, optimal complexity, and emergent
behavior, and provide preliminary support for the concept of optimal
co-adaptation near the edge of chaos.
