The fundamental theorem of natural selection

John Baez

Suppose we have n different types of self-replicating entity, with the population P_i of the ith type changing at a rate equal to P_i times the fitness f_i of that type. Suppose the fitness f_i is any continuous function of all the populations P_1, \dots, P_n. Let p_i be the fraction of replicators that are of the ith type. Then p = (p_1, \dots, p_n) is a time-dependent probability distribution, and we prove that its speed as measured by the Fisher information metric equals the variance in fitness. In rough terms, this says that the speed at which information is updated through natural selection equals the variance in fitness. This result can be seen as a modified version of Fisher’s fundamental theorem of natural selection. We compare it to Fisher’s original result as interpreted by Price, Ewens and Edwards.