  Recently, the eco-evolutionary game theory which describes the coupled
dynamics of strategies and environment have attracted great attention. At the
same time, most of the current work is focused on the classic two-player
two-strategy game. In this work, we study multi-strategy eco-evolutionary game
theory which is an extension of the framework. For simplicity, we'll focus on
the voluntary participation Prisoner's dilemma game. For the general class of
payoff-dependent feedback dynamics, we show the conditions for the existence
and stability of internal equilibrium by using the replicator dynamics,
respectively. Where internal equilibrium points, such as, two-strategy
coexistence states, three-strategy coexistence states, persistent oscillation
states and interior saddle points. These states are determined by the relative
feedback strength and payoff matrix, and are independent of the relative
feedback speed and initial state. In particular, the three-strategy coexistence
provides a new mechanism for maintaining biodiversity in biology, ecology, and
sociology. Besides, we find that this three-strategy model return to the
persistent oscillation state of the two-strategy model when there is no
defective strategy at the initial moment.
