Evolution kinetics and phase transitions of complex adaptive systems.
Motivated by the fact that a population of competing agents never set up a true stationary distribution, we propose a theory of evolution kinetics for complex adaptive systems. The formula derived for the survival probability is used to describe different phases in the population evolution when the prize-to-fine ratio as well as time changes. A kinetic phase diagram is obtained to show the phase boundaries for the self-segregating region, the intermediate region, and the clustering region. The kinetic evolutionary equations of strategy distribution are also established and used to give the root-mean-square separation of strategy distribution. It is revealed that the rootstock of the phase transitions is ascribed to the cooperation and competition among agents with different gene values for a limited resource.