Bi-level evolutionary graphs with multi-fitness

Evolutionary graphs (EGs) were initially introduced in 2005 and give an outlet for the evolutionary dynamic determined by the population structures, the fitness of the mutants and the number of the individuals. Bi-level EGs with different fitness are discussed. The corresponding fixation probability is given, especially when the lower level and the upper level are all isothermal. The optimisation fixation probabilities are also built if the lower and the upper graphs are all isothermal structures and the number of the total individuals is finite. At last, some applications to the system biology are involved.