Evolutionary snowdrift game with disordered environments in mobile societies

We investigate an evolutionary snowdrift game on a square N=L\times L lattice with periodic boundary conditions, where a population of n_0 (n_0\leq N) players located on the sites of this lattice can either cooperate with or defect from their nearest neighbours. After each generation, every player moves with a certain probability p to one of the player's nearest empty sites. It is shown that, when p=0, the cooperative behaviour can be enhanced in disordered structures. When p>0, the effect of mobility on cooperation remarkably depends on the payoff parameter r and the density of individuals \rho (\rho=n_0/N). Compared with the results of p=0, for small r, the persistence of cooperation is enhanced at not too small values of \rho; whereas for large r, the introduction of mobility inhibits the emergence of cooperation at any \rho<1; for the intermediate value of r, the cooperative behaviour is sometimes enhanced and sometimes inhibited, depending on the values of p and \rho. In particular, the cooperator density can reach its maximum when the values of p and \rho reach their respective optimal values. In addition, two absorbing states of all cooperators and all defectors can emerge respectively for small and large r in the case of p>0.