  Environmental changes play a critical role in determining the evolution of
social dilemmas in many natural or social systems. Generally, the environmental
changes include two prominent aspects: the global time-dependent fluctuations
and the local strategy-dependent feedbacks. However, the impacts of these two
types of environmental changes have only been studied separately, a complete
picture of the environmental effects exerted by the combination of these two
aspects remains unclear. Here we develop a theoretical framework that
integrates group strategic behaviors with their general dynamic environments,
where the global environmental fluctuations are associated with a nonlinear
factor in public goods game and the local environmental feedbacks are described
by the `eco-evolutionary game'. We show how the coupled dynamics of local
game-environment evolution differs in static and dynamic global environments.
In particular, we find the emergence of cyclic evolutions of group cooperation
and local environment, which forms an interior irregular loop in the phase
plane, depending on the relative changing speed of both global and local
environments compared to the strategic change. Our results provide important
insights toward how diverse evolutionary outcomes could emerge from the
nonlinear interactions between strategies and the changing environments.
