Donations triggered by inequality tolerance affect the evolution of cooperation in spatial public goods game

Generosity through donation plays a crucial role in reducing inequality and influencing human behavior. However, previous research on donation has overlooked individuals' acceptance of the extent of inequality, which acts as a trigger for donation. To address this gap, this paper systematically explores the impact of donation based on inequality tolerance on the evolution of cooperation in spatial public goods game. Specifically, donation occurs only when an individual's payoff advantage exceeds her inequality tolerance. The results show that donation patterns are crucial for the emergence and stability of cooperation. In the enduring period, the defector-to-cooperator donation pattern helps to form cooperative clusters. In the expanding period, cooperator-to-cooperator, defector-to-defector, and defector-to-cooperator donation patterns create a stable three-layer structure through self-organization, providing a payoff advantage to boundary cooperators. As donation ratio increases, the three-layer structure provides a greater payoff advantage to boundary cooperators, leading to an increase in cooperation. As inequality tolerance increases, changes in donation patterns weaken the three-layer structure, causing cooperation to decrease or disappear through discontinuous phase transitions. Subsequently, all critical points of discontinuous phase transitions are identified by specific spatial configurations. In addition, the influence of donation patterns on the evolution of cooperation is robust, even in heterogeneous small-world networks. This paper offers valuable insights into the dynamics of cooperation evolution and the role of donation in shaping behavior.