Despite the widespread adoption of open source software (OSS), its sustainability remains a critical concern, particularly in light of security vulnerabilities and the often inadequate end-of-service (EoS) processes for OSS projects as they decline. Existing models of OSS community participation, like the Onion model and the episodic contribution model, offer valuable insights but are fundamentally incompatible and fail to provide a comprehensive picture of contributor engagement with OSS projects. This paper addresses these gaps by proposing the CROSS model, a novel contributor-project interaction lifecycle model for open source, which delineates the various lifecycle stages of contributor-project interaction, along with the driving and retaining forces pertinent to each stage. By synthesizing existing research on OSS communities, organizational behavior, and human resource development, it explains a range of archetypal cases of contributor engagement and highlights research gaps, especially in EoS/offboarding scenarios. The CROSS model provides a foundation for understanding and enhancing the sustainability of OSS projects, offering a robust foundation for future research and practical application.