This work presents a cyclic pursuit model based on nonlinear inter-agent interactions driven by double-integrator dynamics. Cyclic pursuit models, beginning with the famous Three-Bug problem, have spurred a strong interest in the research community, encompassing linear and non-linear control models, synchronous and asynchronous pursuit and other similar variants and extensions. Much of the work, however, has been towards the development of kinematic pursuit models. Doubleintegrator models with directed interactions such as in cyclic pursuit models present strong challenges in the evaluation of the system stability and the emergence of global dynamic attributes based on local non-linear agent interactions. This work proposes and evaluates a control model based on a particular form of agent interactions involving linear attractive and non-linear repulsive forcing functions, directed from each agent to its leading agent along the pursuit curve. Specifically, the non-linear controller enforces a stricter implementation of collision avoidance among agents during pursuit, allowing for a reversal in pursuit direction when inter-agent separations drop to low values. Starting from arbitrary initial conditions for the multi-agent system, the emergence of stable cyclic pursuit configurations purely from local agent interactions is demonstrated. The proposed work has strong potential in co-operative perimeter-tracking applications such as wildfire monitoring and border patrol, where the need for efficient spatial distribution of agents around perimeters while in pursuit is essential for efficient data gathering and developing improved situational awareness for dynamic events.