Evolutionary dynamics of continuous public goods games in structured populations.

Over the past few decades, many works have studied the evolutionary dynamics of continuous games. However, previous works have primarily focused on two-player games with pairwise interactions. Indeed, group interactions rather than pairwise interactions are usually found in real situations. The public goods game serves as a paradigm of multi-player interactions. Notably, various types of benefit functions are typically considered in public goods games, including linear, saturating, and sigmoid functions. Thus far, the evolutionary dynamics of cooperation in continuous public goods games with these benefit functions remain unknown in structured populations. In this paper, we consider the continuous public goods game in structured populations. By employing the pair approximation approach, we derive the analytical expressions for invasion fitness. Furthermore, we explore the adaptive dynamics of cooperative investments in the game with various benefit functions. First, for the linear public goods game, we find that there is no singular strategy, and the cooperative investments evolve to either the maximum or minimum depending on the benefit-to-cost ratio. Subsequently, we examine the game with saturating benefit functions and demonstrate the potential existence of an evolutionarily stable strategy (ESS). Additionally, for the game with the sigmoid benefit function, we observe that the evolutionary outcomes are closely related to the threshold value. When the threshold is small, a unique ESS emerges. For intermediate threshold values, both the ESS and repellor singular strategies can coexist. When the threshold value is large, a unique repellor displays. Finally, we perform individual-based simulations to validate our theoretical results.