Cyclical game coupling with Levy flight and Brownian motion and stable coexistence conditions of species

Abstract

Cyclical game is often used to study the biodiversity in ecosystem. However, the interaction distance mode considered 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.