High profile large scale public events are attractive targets for terrorist attacks. The recent Boston Marathon bombings on April 15, 2013 have further emphasized the importance of protecting public events. The security challenge is exacerbated by the dynamic nature of such events: e.g., the impact of an attack at different locations changes over time as the Boston marathon participants and spectators move along the race track. In addition, the defender can relocate security resources among potential attack targets at any time and the attacker may act at any time during the event. This paper focuses on developing efficient patrolling algorithms for such dynamic domains with continuous strategy spaces for both the defender and the attacker. We aim at computing optimal pure defender strategies, since an attacker does not have an opportunity to learn and respond to mixed strategies due to the relative infrequency of such events. We propose SCOUT-A, which makes assumptions on relocation cost, exploits payoff representation and computes optimal solutions efficiently. We also propose SCOUT-C to compute the exact optimal defender strategy for general cases despite the continuous strategy spaces. SCOUT-C computes the optimal defender strategy by constructing an equivalent game with discrete defender strategy space, then solving the constructed game. Experimental results show that both SCOUT-A and SCOUT-C significantly outperform other existing strategies.