  Considering feedback of collective actions of cooperation on common resources
has vital importance to reach sustainability. But such efforts may have not
immediate consequence on the state of environment and it is unclear how they
influence the strategic and environmental dynamics with feedbacks. To address
this issue, we construct a feedback-evolving game model in which we consider
the growth capacity of resources and the punishment efficiency on defectors who
do not provide returns to the environment. Importantly, we further assume a
delay in adopting the contribution of cooperative individuals to environmental
change in our model. We find that when this contribution amount from
cooperators' endowment is fixed, the time delay has no particular consequence
on the coevolutionary dynamics. However, when the return is proportional to
their endowment, then the time delay can induce periodic oscillatory dynamics
of cooperation level and environment. Our work reveals the potential effects of
time delay of cooperative actions on the coevolutionary dynamics in strategic
interactions with environmental feedback.
