  Feedback loops between population dynamics of individuals and their
ecological environment are ubiquitously found in nature, and have shown
profound effects on the resulting eco-evolutionary dynamics. Incorporating
linear environmental feedback law into replicator dynamics of two-player games,
recent theoretical studies shed light on understanding the oscillating dynamics
of social dilemma. However, detailed effects of more general nonlinear feedback
loops in multi-player games, which is more common especially in microbial
systems, remain unclear. Here, we focus on ecological public goods games with
environmental feedbacks driven by nonlinear selection gradient. Unlike previous
models, multiple segments of stable and unstable equilibrium manifolds can
emerge from the population dynamical systems. We find that a larger relative
asymmetrical feedback speed for group interactions centered on cooperators not
only accelerates the convergence of stable manifolds, but also increases the
attraction basin of these stable manifolds. Furthermore, our work offers an
innovative manifold control approach: by designing appropriate switching
control laws, we are able to steer the eco-evolutionary dynamics to any desired
population states. Our mathematical framework is an important generalization
and complement to coevolutionary game dynamics, and also fills the theoretical
gap in guiding the widespread problem of population state control in microbial
experiments.
