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We describe a general mechanism for adaptation in multi-agent systems in which agents modify their behavior in response to changes in the environment or actions of other agents. The agent use memory to estimate the global state of the system from individual observations and adjust their actions accordingly. We present a mathematical model of the dynamics of collective behavior in such systems and apply it to study adaptive task allocation in mobile robots. In this application, the robots task is to forage for red or green pucks. As it travels around the arena, a robot records observations of puck and other robots, and uses these observations to compute the estimated density of each. If it finds there are not enough robots of a specific type, it may switch its foraging state to fill a gap. After a transient, we expect the number of robots in each foraging state to reflect the prevalence of each puck type in the environment. We modelled adaptive task allocation and studied the dynamics of the system for different transition rates between states. We find that for some rates lead to fast convergence times and a steady state solution.