  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.
