Coevolutionary Dynamics of Stochastic Replicator Systems

In this thesis we establish a theory of evolutionary dynamics that accounts for the following requirements. 1. The evolutionary process is considered in a coevolutionary context. 2. The theory describes the full dynamics of the coevolutionary process. 3. The coevolutionary dynamics are derived from the underlying population dynamics. 4. The theory accounts for the stochastic aspects of the evolutionary process. To our knowledge the mathematical framework advanced here is the first to simultaneously combine these four key features of evolution. We present a hierarchy of three dynamical models for the investigation of coevolutionary systems; each of these models stands for a different balance between descriptive capacity and corresponding analytic tractability. Deductions are given to clarify the interconnections between the models; from the assumptions necessary for these derivations we infer their domains of validity. Equations central to the fields of evolutionary game theory, replicator dynamics and adaptive dynamics are recovered as specialized cases from our mathematical framework. In particular, the canonical equation of adaptive dynamics, which so far has been used on the grounds of plausibility arguments, is underpinned by a formal derivation.