Stochastically stable equilibria in the nonlinear public goods game

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In this paper, we study the stochastically stable equilibria in a nonlinear public goods game. In the nonlinear public goods game, we consider that only when the number of cooperators in a group exceeds a threshold, group members can obtain the benefit of cooperation, otherwise they obtain nothing. By studying the stochastic replicator equation, we obtain a sufficient and necessary condition in which there is a unique stochastically stable equilibrium. And cooperators can coexist with defectors at this stochastically stable equilibrium. Furthermore, numerical calculations show that the frequency of cooperators at the stochastically stable equilibrium decreases with increasing the group size, increases with increasing the synergy factor, and increases with increasing the threshold value in the nonlinear public goods game.