The Fundamental Theorem of Natural Selection

Suppose we have n different types of self-replicating entity, with the population [Formula: see text] of the ith type changing at a rate equal to [Formula: see text] times the fitness [Formula: see text] of that type. Suppose the fitness [Formula: see text] is any continuous function of all the populations [Formula: see text]. Let [Formula: see text] be the fraction of replicators that are of the ith type. Then [Formula: see text] 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.