This article proposes Persistence Administered Collective Navigation (PACNav) as an approach for achieving decentralized collective navigation of Unmanned Aerial Vehicle (UAV) swarms. The technique is based on the flocking and collective navigation behavior observed in natural swarms, such as cattle herds, bird flocks, and even large groups of humans. As global and concurrent information of all swarm members is not available in natural swarms, these systems use local observations to achieve the desired behavior. Similarly, PACNav relies only on local observations of relative positions of UAVs, making it suitable for large swarms deprived of communication capabilities and external localization systems. We introduce the novel concepts of path persistence and path similarity that allow each swarm member to analyze the motion of other members in order to determine its own future motion. PACNav is based on two main principles: (1) UAVs with little variation in motion direction have high path persistence, and are considered by other UAVs to be reliable leaders; (2) groups of UAVs that move in a similar direction have high path similarity, and such groups are assumed to contain a reliable leader. The proposed approach also embeds a reactive collision avoidance mechanism to avoid collisions with swarm members and environmental obstacles. This collision avoidance ensures safety while reducing deviations from the assigned path. Along with several simulated experiments, we present a real-world experiment in a natural forest, showcasing the validity and effectiveness of the proposed collective navigation approach in challenging environments. The source code is released as open-source, making it possible to replicate the obtained results and facilitate the continuation of research by the community.