Universal expressions of population change by the Price equation: natural selection, information, and maximum entropy production

The Price equation shows the unity between the fundamental expressions of change in biology, in information and entropy descriptions of populations, and in aspects of thermodynamics. The Price equation partitions the change in the average value of a metric between two populations. A population may be composed of organisms or particles or any members of a set to which we can assign probabilities. A metric may be biological fitness or physical energy or the output of an arbitrarily complicated function that assigns quantitative values to members of the population. The first part of the Price equation describes how directly applied forces change the probabilities assigned to members of the population when holding constant the metrical values of the members---a fixed metrical frame of reference. The second part describes how the metrical values change, altering the metrical frame of reference. In canonical examples, the direct forces balance the changing metrical frame of reference, leaving the average or total metrical values unchanged. In biology, relative reproductive success (fitness) remains invariant as a simple consequence of the conservation of total probability. In physics, systems often conserve total energy. Nonconservative metrics can be described by starting with conserved metrics, and then studying how coordinate transformations between conserved and nonconserved metrics alter the geometry of the dynamics and the aggregate values of populations. From this abstract perspective, key results from different subjects appear more simply as universal geometric principles for the dynamics of populations subject to the constraints of particular conserved quantitiesComment: v2: Complete rewrite, new title and abstract. Changed focus to Price equation as basis for universal expression of changes in populations. v3: Cleaned up usage of terms virtual and reversible displacements and virtual work and usage of d'Alembert's principle. v4: minor editing and correction

Simple unity among the fundamental equations of science

The Price equation describes the change in populations. Change concerns some value, such as biological fitness, information or physical work. The Price equation reveals universal aspects for the nature of change, independently of the meaning ascribed to values. By understanding those universal aspects, we can see more clearly why fundamental mathematical results in different disciplines often share a common form. We can also interpret more clearly the meaning of key results within each discipline. For example, the mathematics of natural selection in biology has a form closely related to information theory and physical entropy. Does that mean that natural selection is about information or entropy? Or do natural selection, information and entropy arise as interpretations of a common underlying abstraction? The Price equation suggests the latter. The Price equation achieves its abstract generality by partitioning change into two terms. The first term naturally associates with the direct forces that cause change. The second term naturally associates with the changing frame of reference. In the Price equation's canonical form, total change remains zero because the conservation of total probability requires that all probabilities invariantly sum to one. Much of the shared common form for the mathematics of different disciplines may arise from that seemingly trivial invariance of total probability, which leads to the partitioning of total change into equal and opposite components of the direct forces and the changing frame of reference.Comment: arXiv admin note: text overlap with arXiv:1810.0926

Natural selection. III. Selection versus transmission and the levels of selection

George Williams defined an evolutionary unit as hereditary information for which the selection bias between competing units dominates the informational decay caused by imperfect transmission. In this article, I extend Williams' approach to show that the ratio of selection bias to transmission bias provides a unifying framework for diverse biological problems. Specific examples include Haldane and Lande's mutation-selection balance, Eigen's error threshold and quasispecies, Van Valen's clade selection, Price's multilevel formulation of group selection, Szathmary and Demeter's evolutionary origin of primitive cells, Levin and Bull's short-sighted evolution of HIV virulence, Frank's timescale analysis of microbial metabolism, and Maynard Smith and Szathmary's major transitions in evolution. The insights from these diverse applications lead to a deeper understanding of kin selection, group selection, multilevel evolutionary analysis, and the philosophical problems of evolutionary units and individuality

Natural selection. IV. The Price equation

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