  We study effects of strategy-dependent time delays on equilibria of evolving
populations. It is well known that time delays may cause oscillations in
dynamical systems. Here we report a novel behavior. We show that microscopic
models of evolutionary games with strategy-dependent time delays lead to a new
type of replicator dynamics. It describes the time evolution of fractions of
the population playing given strategies and the size of the population. Unlike
in all previous models, stationary states of such dynamics depend continuously
on time delays. We show that in games with an interior stationary state (a
globally asymptotically stable equilibrium in the standard replicator
dynamics), at certain time delays, it may disappear or there may appear another
interior stationary state. In the Prisoner's Dilemma game, for time delays of
cooperation smaller than time delays of defection, there appears an ustable
interior equilibrium and therefore for some initial conditions, the population
converges to the homogeneous state with just cooperators.
