  From critical infrastructure, to physiology and the human brain, complex
systems rarely occur in isolation. Instead, the functioning of nodes in one
system often promotes or suppresses the functioning of nodes in another.
Despite advances in structural interdependence, modeling interdependence and
other interactions between dynamic systems has proven elusive. Here we define a
broadly applicable dynamic dependency link and develop a general framework for
interdependent and competitive interactions between general dynamic systems. We
apply our framework to studying interdependent and competitive synchronization
in multi-layer oscillator networks and cooperative/competitive contagions in an
epidemic model. Using a mean-field theory which we verify numerically, we find
explosive transitions and rich behavior which is absent in percolation models
including hysteresis, multi-stability and chaos. The framework presented here
provides a powerful new way to model and understand many of the interacting
complex systems which surround us.
