  In this paper, we focus on studying the passivity properties of different
versions of replicator dynamics (RD). RD is an important class of evolutionary
dynamics in evolutionary game theory. Evolutionary dynamics describe how the
population composition changes in response to the fitness levels, resulting in
a closed-loop feedback system. RD is a deterministic monotone non-linear
dynamic that allows incorporation of the distribution of population types
through a fitness function. Here, in this paper, we use a tools for control
theory, in particular, the passivity theory, to study the stability of the RD
when it is in action with evolutionary games. The passivity theory allows us to
identify class of evolutionary games in which stability with RD is guaranteed.
We show that several variations of the first order RD satisfy the standard
loseless passivity property. In contrary, the second order RD do not satisfy
the standard passivity property, however, it satisfies a similar dissipativity
property known as negative imaginary property. The negative imaginary property
of the second order RD allows us to identify the class of games that converge
to a stable equilibrium with the second order RD.
