We introduce nonlinear attractive effects into a spatial Prisoner's Dilemma game where the players located on a square lattice can either cooperate with their nearest neighbors or defect. In every generation, each player updates its strategy by firstly choosing one of the neighbors with a probability proportional to A(alpha) denoting the attractiveness of the neighbor, where A is the payoff collected by it and alpha (>= 0) is a free parameter characterizing the extent of the nonlinear effect; and then adopting its strategy with a probability dependent on their payoff difference. Using Monte Carlo simulations, we investigate the density rho(c) of cooperators in the stationary state for different values of alpha. It is shown that the introduction of such attractive effect remarkably promotes the emergence and persistence of cooperation over a wide range of the temptation to defect. In particular, for large values of alpha, i.e., strong nonlinear attractive effects, the system exhibits two absorbing states (all cooperators or all defectors) separated by an active state (coexistence of cooperators and defectors) when varying the temptation to defect. In the critical region where rho(c) goes to zero, the extinction behavior is power-law-like rho(C) similar to (b(c) - b)(beta), where the exponent beta accords approximatively with the critical exponent (beta approximate to 0.584) of the two-dimensional directed percolation and depends weakly on the value of alpha.
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