  For over fifty years, the dynamical systems perspective has had a prominent
role in evolutionary biology and economics, through the lens of game theory. In
particular, the study of replicator differential equations on the standard
(probability) simplex, specified by fitness maps or payoff functions, has
yielded insights into the temporal behavior of such systems. However behavior
is influenced by context and environmental factors with a game-changing quality
(i.e., fitness maps are manipulated). This paper develops a principled
geometric approach to model and understand such influences by incorporating
replicator dynamics into a broader control-theoretic framework. Central to our
approach is the construction of a Lie algebra structure on the space of fitness
maps, mapping homomorphically to the Lie algebra of replicator vector fields.
This is akin to classical mechanics, where the Poisson bracket Lie algebra of
functions maps to associated Hamiltonian vector fields. We show, extending the
work of Svirezhev in 1972, that a trajectory of a replicator vector field is
the base integral curve of a solution to a Hamiltonian system defined on the
cotangent bundle of the simplex. Further, we exploit the Lie algebraic
structure of fitness maps to determine controllability properties of a class of
replicator systems.
