DYNAMICS AND EVOLUTION - EVOLUTIONARILY STABLE ATTRACTORS, INVASION EXPONENTS AND PHENOTYPE DYNAMICS

We extend the ideas of evolutionary dynamics and stability to a very b road class of biological and other dynamical systems. We simultaneousl y develop the general mathematical theory and a discussion of some ill ustrative examples. After developing an appropriate formulation for th e dynamics, we define the notion of an evolutionary stable attractor ( ESA) and give some samples of ESAS With Simple and complex dynamics. W e discuss the relationship between our theory and that for ESSS in cla ssical linear evolutionary game theory by considering some dynamical e xtensions. We then introduce and develop our main mathematical tool, t he invasion exponent. This allows analytical and numerical analysis of relatively complex situations, such as the coevolution of multiple sp ecies with chaotic population dynamics. Using this, we introduce the n otion of differential selective pressure which for generic systems is nonlinear and characterizes internal ESAS. We use this to analytically determine the ESAS in our previous examples. Then we introduce the ph enotype dynamics which describe how a population with a distribution o f phenotypes changes in time with or without mutations. We discuss the relation between the asymptotic states of this and the ESAS. Finally, we use our mathematical formulation to analyse a non-reproductive for m of evolution in which various learning rules compete and evolve. We give a very tentative economic application which has interesting ESAS and phenotype dynamics.