Evolutionary dynamics of [formula omitted]-person snowdrift game with two thresholds in well-mixed and structured populations.

In this paper, to state the necessary and the critical number of cooperators for completing the snow shoveling, we investigate the evolutionary dynamics of an N -person snowdrift game with two thresholds in well-mixed populations and structured populations. Firstly, we discuss the effects of these thresholds on equilibria in infinite well-mixed populations by applying the replicator equation, and observe that there exist three types of equilibria with the variations of parameters. Secondly, we study the game in finite well-mixed populations by means of the Markov process, and find that the evolutionary results converge to those in infinite population as the finite population size increases. Finally, we explore such a game in structured populations by using the pair approximation approach, and derive the dynamical equation under weak selection, which is ignored in the previous work. Results present that structured populations can create more complicated behaviors of the cooperative dynamics, and to some extent facilitate cooperation compared with well-mixed populations. Our work helpfully provides a theoretical approximation for the N -person snowdrift game in more complicated scenarios. • Some concise critical conditions of equilibria are obtained in infinite and well-mixed populations. • The relationship between equilibria in finite and infinite well-mixed populations is discussed. • The dynamical equation of the double threshold game in structured populations is derived. • Structure populations to some extent facilitate cooperation compared with well-mixed populations. [ABSTRACT FROM AUTHOR]