In this paper we study the impact of tiny mutation on the evolutionary dynamics on one-dimensional cycle with shifting mechanism. The evolutionary success is evaluated by investigating the stationary distribution of the ergodic process with the idea of viscosity solutions. The cooperative behaviors in ecosystem and social system are briefly discussed by applying the results to the prisoner＇s dilemma game.
The weights of the drought risk index （DRI）, which linearly combines the reliability, resiliency, and vulnerability, are difficult to obtain due to complexities in water security during drought periods. Therefore, drought entropy was used to determine the weights of the three critical indices. Conventional simulation results regarding the risk load of water security during drought periods were often regarded as precise. However, neither the simulation process nor the DRI gives any consideration to uncertainties in drought events. Therefore, the Dempster-Shafer （D-S） evidence theory and the evidential reasoning algorithm were introduced, and the DRI values were calculated with consideration of uncertainties of the three indices. The drought entropy and evidential reasoning algorithm were used in a case study of the Haihe River Basin to assess water security risks during drought periods. The results of the new DRI values in two scenarios were compared and analyzed. It is shown that the values of the DRI in the D-S evidence algorithm increase slightly from the original results of Zhang et al. （2005）, and the results of risk assessment of water security during drought periods are reasonable according to the situation in the study area. This study can serve as a reference for further practical application and planning in the Haihe River Basin, and other relevant or similar studies.
