Motion patterns and phase-transition of a defender–intruder problem and optimal interception strategy of the defender

• A novel type of conflicting interactions between agents provided and a defense–intrusion problem formulated.
• Interesting results of the optimal interception strategy of the defender provided that may intuitionally unexpected.
• The phase-transition for motion patterns of agents characterized.
• The optimal interception strategy of the defender provided with interpretations of its physical meaning.

In this paper, we consider a defense–intrusion interaction, in which an intruder is attracted by a protected stationary target but repulsed by a defender; while the defender tries to move towards an appropriate interception position (IP) between the intruder and the target in order to intercept the intruder and expel the intruder away from the target as maximum as possible. Intuitionally, to keep the intruder further away, one may wonder that: is it a better strategy for the defender trying to approach the intruder as near as possible? Unexpectedly and interestingly enough, this is not always the case. We first introduce the flexibility for IP selection, then investigate the system dynamics and the stable motion patterns, and characterize the phase-transition surface for the motion patterns. We show that, the phase-transition surface just defines the optimal interception strategy of the defender for IP selection; and from the perspective of mobility of agents, the optimal strategy just depends on relative mobility of the two agents.