中文摘要：

循环竞争博弈常被用来研究物种多样性。以前有关循环竞争博弈的研究工作所考虑的相互作用距离模式包括最近邻、取固定距离或一定距离以内的随机值，这与实际情况不相符。考虑到实际生物系统中物种个体做列维飞行与布朗运动的情况广泛存在，综合考虑了最近邻相互作用模式和列维飞行（布朗运动）长程相互作用模式，对循环竞争博弈及保持物种多样性的条件进行了研究。得到了最大飞行距离与选择概率的临界关系（包括Logistic式和指数式关系），进一步得到了幂指数与选择概率的临界关系，以及保持物种共存的条件。

英文摘要：

Cyclical game is often used to study the biodiversity in ecosystem. However, the interaction distance mode con-sidered in previous studies of cyclical game is only the interaction between nearest neighbors, a fixed distance, or a random value of fixed distance among the individuals of species. This is not consistent with the actual situation. In this paper, considering the fact that Levy flight and Brownian motion widespreadly exist in ecosystem, and comprehensively considering the nearest-neighbor-interaction and long-range-interaction given by Levy flight and Brownian motion, the cyclical game and conditions of maintaining biodiversity are investigated. The critical relation of maximal step length of flight versus choosing probability is presented, including Logistic and exponent relations. Further the critical relation between power-law exponent and choosing probability is found. The condition of maintaining species coexistence is also found.