This study investigates the evolutionary prisoner's dilemma game on a square lattice, where players can adopt one of two strategies: cooperate (C) or defect (D). Players update their strategies based on the payoff difference between themselves and their neighbors. Monte Carlo simulations and dynamical cluster techniques are used to analyze the density of cooperators in the stationary state. The system exhibits a continuous transition between two absorbing states as the temptation to defect varies. In the limits where the cooperator density approaches 0 or 1, the transitions belong to the universality class of directed percolation. The model considers two types of interactions: first-neighbor and second-neighbor. The payoff matrix is defined with R = 1, P = S = 0, and T = b. The evolutionary rule is based on the probability of adopting a neighbor's strategy, which depends on the payoff difference. The results show that the cooperator density decreases monotonically with increasing b until the second threshold, where cooperators vanish. The mean-field results agree with simulations for the 5-point approximation, while the pair approximation shows significant differences. The study finds that the critical behavior near the thresholds is consistent with the directed percolation universality class. The critical exponents derived from the simulations agree with those of directed percolation. The results also show that the active region is reduced by stochasticity, and the second threshold value of b is decreased by randomness. The generalized mean-field technique is effective for studying the coexistence region but is not applicable in the critical regions due to long-range correlations and fluctuations. The study concludes that the evolutionary prisoner's dilemma game on a square lattice exhibits critical behavior consistent with the directed percolation universality class, with two absorbing states (all cooperators or all defectors) separated by an active region. The results highlight the importance of spatial effects and stochasticity in promoting cooperation.This study investigates the evolutionary prisoner's dilemma game on a square lattice, where players can adopt one of two strategies: cooperate (C) or defect (D). Players update their strategies based on the payoff difference between themselves and their neighbors. Monte Carlo simulations and dynamical cluster techniques are used to analyze the density of cooperators in the stationary state. The system exhibits a continuous transition between two absorbing states as the temptation to defect varies. In the limits where the cooperator density approaches 0 or 1, the transitions belong to the universality class of directed percolation. The model considers two types of interactions: first-neighbor and second-neighbor. The payoff matrix is defined with R = 1, P = S = 0, and T = b. The evolutionary rule is based on the probability of adopting a neighbor's strategy, which depends on the payoff difference. The results show that the cooperator density decreases monotonically with increasing b until the second threshold, where cooperators vanish. The mean-field results agree with simulations for the 5-point approximation, while the pair approximation shows significant differences. The study finds that the critical behavior near the thresholds is consistent with the directed percolation universality class. The critical exponents derived from the simulations agree with those of directed percolation. The results also show that the active region is reduced by stochasticity, and the second threshold value of b is decreased by randomness. The generalized mean-field technique is effective for studying the coexistence region but is not applicable in the critical regions due to long-range correlations and fluctuations. The study concludes that the evolutionary prisoner's dilemma game on a square lattice exhibits critical behavior consistent with the directed percolation universality class, with two absorbing states (all cooperators or all defectors) separated by an active region. The results highlight the importance of spatial effects and stochasticity in promoting cooperation.