Fixation, transient landscape and diffusion's dilemma in stochastic evolutionary game dynamics

TL;DR

This paper explores the transient and fixation behaviors in stochastic evolutionary game dynamics using a landscape function, revealing complex phenomena like multiple time scales, discontinuous transitions, and the role of rare events in nonlinear biological systems.

Contribution

It introduces a landscape framework to unify various aspects of stochastic evolutionary dynamics, including fixation, transient behavior, and diffusion approximation challenges.

Findings

Landscape function captures stable and transient dynamics.

Multiple time scales are identified in intra- and inter-attractoral processes.

Rare, exponentially small probability events are crucial for understanding system complexity.

Abstract

Agent-based stochastic models for finite populations have recently received much attention in the game theory of evolutionary dynamics. Both the ultimate fixation and the pre-fixation transient behavior are important to a full understanding of the dynamics. In this paper, we study the transient dynamics of the well-mixed Moran process through constructing a landscape function. It is shown that the landscape playing a central theoretical "device" that integrates several lines of inquiries: the stable behavior of the replicator dynamics, the long-time fixation, and continuous diffusion approximation associated with asymptotically large population. Several issues relating to the transient dynamics are discussed: (i) multiple time scales phenomenon associated with intra- and inter-attractoral dynamics; (ii) discontinuous transition in stochastically stationary process akin to Maxwell construction in equilibrium statistical physics; and (iii) the dilemma diffusion approximation facing as a continuous approximation of the discrete evolutionary dynamics. It is found that rare events with exponentially small probabilities, corresponding to the uphill movements and barrier crossing in the landscape with multiple wells that are made possible by strong nonlinear dynamics, plays an important role in understanding the origin of the complexity in evolutionary, nonlinear biological systems.