  The paradox of cooperation among selfish individuals still puzzles scientific
communities. Although a large amount of evidence has demonstrated that
cooperator clusters in spatial games are effective to protect cooperators
against the invasion of defectors, we continue to lack the condition for the
formation of a giant cooperator cluster that assures the prevalence of
cooperation in a system. Here, we study the dynamical organization of
cooperator clusters in spatial prisoner's dilemma game to offer the condition
for the dominance of cooperation, finding that a phase transition characterized
by the emergence of a large spanning cooperator cluster occurs when the initial
fraction of cooperators exceeds a certain threshold. Interestingly, the phase
transition belongs to different universality classes of percolation determined
by the temptation to defect $b$. Specifically, on square lattices, $1<b<4/3$
leads to a phase transition pertaining to the class of regular site
percolation, whereas $3/2<b<2$ gives rise to a phase transition subject to
invasion percolation with trapping. Our findings offer deeper understanding of
the cooperative behaviors in nature and society.
