Beyond payoff neutrality: How generalized subpopulation interactions drive cooperation in structured populations

Understanding how cooperation evolves in multi-subpopulation is crucial for addressing social challenges. While previous studies show that payoff-neutral subpopulations in structured populations can enhance cooperation, the role of broader inter-subpopulation relationships remains unclear. We extend this framework to include generalized relationships-competition, mutualism, and parasitism-modeled by inter-subpopulation payoffs α and β. Within subpopulations, individuals play the prisoner's dilemma, while inter-subpopulation interactions yield payoffs based on α and β. Evolutionary analysis and simulations reveal that, in fully connected networks, generalized relationships yield outcomes almost indistinguishable from the payoff-neutral scenario (α=0,β=0). However, in structured populations, these relationships introduce additional pathways for sustaining cooperation beyond those observed under payoff neutrality. When the network structure alone can support cooperation, only mutualistic relationships (α>0,β>0) enable the full dominance of cooperative strategies. Conversely, when the network structure alone cannot maintain cooperation, competitive (α<0,β<0) or parasitic (α>0,β<0) relationships allow cooperation to persist or even achieve complete dominance, whereas mutualism offers only limited support. These findings provide deeper insights into how diverse inter-subpopulation relationships shape the evolution of cooperation in multi-subpopulation social systems.