Evolutionary dynamics: How payoffs and global feedback control the stability

Biological as well as socio-economic populations can exhibit oscillatory dynamics. In the simplest case this can be described by oscillations around a neutral fixed point as in the classical Lotka-Volterra system. In reality, populations are always finite, which can be discussed in a general framework of a finite-size expansion which allows to derive stochastic differential equations of Fokker-Planck type as macroscopic evolutionary dynamics. Important applications of this concept are economic cycles for “cooperate—defect—tit for tat” strategies, mating behavior of lizards, and bacterial population dynamics which can all be described by cyclic games of rock-scissors-paper dynamics. Here one can study explicitly how the stability of coexistence is controlled by payoffs, the specific behavioral model and the population size. Finally, in socio-economic systems one is often interested in the stabilization of coexistence solutions to sustain diversity in an ecosystem or society. Utilizing a diversity measure as dynamical observable, a feedback into the payoff matrix is discussed which stabilizes the steady state of coexistence.