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

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.

Adaptive Dynamics with Interaction Structure

“…We depart from most prior work on adaptive dynamics in structured populations (van Baalen and Rand, 1998;Ferrière and Le Galliard, 2001;Metz and Gyllenberg, 2001;Durinx et al, 2008) in our use of fixation probabilities in place of invasion fitness-defined as the exponential growth rate of a rare mutant subject to the current environment (Metz et al, 1992;Rand et al, 1994;Dieckmann and Law, 1996;Metz et al, 1996;Heino et al, 1998).…”

Evolution of flowering strategies inOenothera glazioviana: an integral projection model approach

“…Sensitivities estimate the effect on l of absolute changes in parameter values. They quantify the intensity and direction of selection, and may be interpreted as selection pressures (Metz et al 1992;Rand et al 1994;Caswell 2001 Kachi & Hirose (1985).…”

Surprising evolutionary predictions from enhanced ecological realism

“…The second idea facilitating a merger of game theoretical and optimization methods is to derive the fitness functions to be analyzed from an individual-based view of the underlying population dynamics-in earlier work, such functions had often been instead assumed a priori. Taken together, these ideas motivated the introduction of the so-called invasion fitness of a given phenotype in a given environment, defined and derived-analytically, numerically, or empirically-as the phenotype's asymptotic exponential growth rate in the considered environment Rand et al, 1994).…”