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Debates persist regarding how different forms of rewards influence public cooperation. In many real-world scenarios, individuals who consistently honor multiple commitments receive higher credit scores than those who fulfill only one commitment. Inspired by this, herein dynamic reward is introduced into the spatial public goods game, wherein the probability of receiving a reward increases with the number of consecutive cooperations by individuals. To illustrate the role of the dynamic reward mechanism, fixed reward is used as a comparison to investigate the impact of dynamic reward on the evolution of cooperation. Through extensive simulations, we find that the phases presented by dynamic reward are more than those by fixed reward. Moreover, it is more conducive to cooperation than fixed reward, indicating that dynamic reward can more effectively resolve social dilemmas. Additionally, the higher initial probability of reward promotes the earlier emergence of cooperation. This research broadens the form of reward, enhancing our understanding of dynamic reward in promoting prosocial behaviors in the spatial public goods game. • Instead of traditional reward, consider dynamic reward that are based on the number of consecutive cooperations of players in the Public Goods Game. • In the dynamic reward, a stable coexistence of cooperators and rewarders emerges. • Fixed reward and dynamic reward both have a positive impact on cooperation, whereas dynamic reward is more conducive to promoting cooperation. • In both models, a counterintuitive decline appeared in the overall trend of cooperation levels. • The higher initial reward probability can lower the threshold for the emergence of cooperation. [ABSTRACT FROM AUTHOR]

In ecosystems, cooperative behavior is universal and can dramatically improve a species' chances of survival. Nevertheless, the situations that can occur when different species with cooperative tendencies interact are veiled. To explore such a situation, in this paper, we investigate how cooperative behavior can affect biodiversity in the population system. Based on the spatial rock–paper–scissors (RPS) game, which incorporates the relative power between predator and prey species, we redefine the competition rate to facilitate cooperative behavior. Competition rates are modulated by the sensitivity parameter, which regulates alterations in competition rates stemming from variations in predator–prey population disparities. Through comprehensive numerical analysis, we have demonstrated compelling evidence confirming the nature of cooperative behavior in maintaining biodiversity. The sensitivity parameter acts as a double-edged sword; it hampers biodiversity when it falls below a certain level. Conversely, when it exceeds the threshold, it supports the maintenance of biodiversity. From snapshots and the coefficient analysis based on spatial autocorrelation, we found that empty sites are essential to promote coexistence as resource nodes. Compared with previous studies in spatial RPS games, our findings suggest that simple modification of a competition rate rather than exploiting cooperative games can realize the cooperative behavior of cyclically competing populations, and biodiversity is sensitively affected by cooperation. • We define cooperative behavior between species by redefining competition rates. • Cooperation affects competition and exchange sensitively, affecting biodiversity. • Empty spaces play an important role and are crucial in promoting coexistence. [ABSTRACT FROM AUTHOR]