The replicator equation is the cornerstone of evolutionary game dynamics that shows the growth rate of the proportion of agents using a certain strategy and whose rate is equal to the difference between the average payoff of that strategy and the average payoff of the agent population as a whole. The general idea is that replicators whose fitness is larger (smaller) than the average fitness of population will increase (decrease) in numbers. The static approach of evolutionary game theory has been complemented by a dynamic stability analysis of stationary (rest) solutions of the replicator equations. In this work, we apply the replicator dynamics of evolutionary game theory to the study of optimal transport networks. We discuss the Wardrop optimal networks in which Nash equilibria and the global optima coincide and whose price of anarchy is 1, present a discrete-time replicator dynamical systems on Wardrop optimal networks based on similar-order preserving mappings, and propose a vast class of discrete-time replicator dynamics generated by Schur-convex potential functions. We discuss Nash equilibria, convergence to fixed points, and asymptotic stability conditions of the replicator equation dynamics.