The rapid development of generative artificial intelligence (GAI) has been effective in accelerating the formation of new quality productivity and other aspects, but the governance challenges it faces in terms of data, information and content need to be solved urgently, and there is an urgent need to find a balance point between technological development and security. Based on information ecosystem theory, a collaborative governance game model involving three stakeholders (managers, providers, and users of GAI services) and numerical simulation analysis were conducted. It is found that GAI safety governance is a process of multi-object collaborative participation, which is affected by ecological factors, and increasing the co-governance gains of each subject and the losses brought by negative governance behaviors are conducive to promoting the information ecosystem to be stable in the ideal state of {strict regulation, active governance, and compliant use}. Regarding image factors, increasing the regulatory image gain of managers and the reputation image gain of providers can promote the information ecosystem to evolve to the ideal state. Regarding rewards and penalties, increasing the rewards from managers to providers and from providers to users can promote the evolution of the entire information ecosystem to a desirable state, even though it may cause a slight loss in the interests of the party who imposes the rewards, where the rewards given by the managers are within an effective threshold; and increasing the penalties imposed by the managers and providers can promote the synergistic governance of the three parties. With regard to the factor of excess earnings, compressing the space for excess earnings of providers and users can promote the synergistic governance of the three parties. The research conclusions presented herein provide theoretical support for formulating security governance strategies for generative artificial intelligence technologies, thereby promoting the development of a healthy and orderly online ecosystem.