Dynamics and Evolution: Evolutionarily Stable Attractors, Invasion Exponents and Phenotype Dynamics

D. A. Rand, H. B. Wilson, J. M. McGlade

Abstract

We extend the ideas of evolutionary dynamics and stability to a very broad class of biological and other dynamical systems. We simultaneously develop the general mathematical theory and a discussion of some illustrative examples. After developing an appropriate formulation for the dynamics, we define the notion of an evolutionary stable attractor (ESA) and give some samples of ESAS with simple and complex dynamics. We discuss the relationship between our theory and that for ESSS in classical linear evolutionary game theory by considering some dynamical extensions. We then introduce and develop our main mathematical tool, the invasion exponent. This allows analytical and numerical analysis of relatively complex situations, such as the coevolution of multiple species with chaotic population dynamics. Using this, we introduce the notion 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 phenotype dynamics which describe how a population with a distribution of 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 form of evolution in which various learning rules compete and evolve. We give a very tentative economic application which has interesting ESAS and phenotype dynamics.

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