Here's some definitions. The first is taken from an interview with Michael J. Mauboussin by Tim Sullivan in the September 2011 edition of the Harvard Business Review, an issue that is dedicated to complexity:
A complex adaptive system (CAS) has three characteristics. The first is that the system consists of a number of heterogeneous agents, and each of those agents makes decisions about how to behave. The most important dimension here is that those decisions will evolve over time. The second characteristic is that the agents interact with one another. That interaction leads to the third—something that scientists call emergence: In a very real way, the whole becomes greater than the sum of the parts. The key issue is that you can’t really understand the whole system by simply looking at its individual parts.
And here's by Gökçe Sargut and Rita Gunther McGrath writing in an article in the same issue on the difference between complicated and complex:
Complicated systems, they say, have a lot of parts, but the parts interact in patterns of behavior we know and understand, and can reasonably predict.
Complex systems are versions of complicated systems wherein the patterns are there but difficult to know about (too many, too obscure, or outside our normal experience) and the interactions, though predictable, are too difficult to predict as a practical matter.
They observe:
Complex systems have always existed, of course—and business life has always featured the unpredictable, the surprising, and the unexpected. But complexity has gone from something found mainly in large systems, such as cities, to something that affects almost everything we touch: the products we design, the jobs we do every day, and the organizations we oversee.
Well, I buy the complex and complicated thing, but every example of CAS that anyone gives is more often biological than not. After all, the biological sciences has been the doman that has advanced the study of CAS the most.
What about agile?
Many say agile methods are themselves an example of CAS because of the property of emergence, and the myriad of agents (developers, testers, stakeholders, sponsors, customers, users et al) that are in constant interaction. Perhaps so. But I don't see agile projects and ant colonies acting the same way. There's simply too many intervening structures, inhibitions, rules, and constraints, to say nothing of project charters, vision, product managers, and market forces that focus the project.
As others have said, it's often nonsensical and many times misleading to cross domains too readily. For my money, I buy into emergence, and I buy into output effecting input (a necessary condition for adaption), but most of the other biological instinctive and survival behavior of ants is not what projects are about.
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