Natural selection. IV. The Price equation

Abstract

The Price equation partitions total evolutionary change into two components. The first component provides an abstract expression of natural selection. The second component subsumes all other evolutionary processes, including changes during transmission. The natural selection component is often used in applications. Those applications attract widespread interest for their simplicity of expression and ease of interpretation. Those same applications attract widespread criticism by dropping the second component of evolutionary change and by leaving unspecified the detailed assumptions needed for a complete study of dynamics. Controversies over approximation and dynamics have nothing to do with the Price equation itself, which is simply a mathematical equivalence relation for total evolutionary change expressed in an alternative form. Disagreements about approach have to do with the tension between the relative valuation of abstract versus concrete analyses. The Price equation's greatest value has been on the abstract side, particularly the invariance relations that illuminate the understanding of natural selection. Those abstract insights lay the foundation for applications in terms of kin selection, information theory interpretations of natural selection and partitions of causes by path analysis. I discuss recent critiques of the Price equation by Nowak and van Veelen.

Fig. 1

Geometric expression of selection. The plots show the equivalence of the dot product, the geometric expression and the covariance, as given in eqn 5. For both plots, z = (1, 4) and ƶ = z/||z|| = (0.24, 0.97). The dashed line shows the perpendicular between the pattern of frequency changes derived from fitnesses, Δq, and the phenotypic pattern, ƶ. The vertex of the two vectors is at the origin (0, 0). The distance from the origin to the intersection of the perpendicular along ƶ is the total amount of selection, ||Δq||cos φ. (a) The vector of frequency changes that summarize fitness is Δq = (−0.4, 0.4). The angle between the vector of frequency changes and the phenotypes is φ = arccos [(Δq · ƶ)/||Δq||] which, in this example, is 1.03 radians or 59°. In this case, the total selection is ||Δq||cos φ = 0.29. (b) In this plot, Δq = (0.4, −0.4), yielding an angle φ of 121°. The perpendicular intersects the negative projection of the phenotype vector, shown as a dashed line, associated with the negative change by selection of ||Δq||cos φ = −0.29.