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

Abstract

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 quantities.

Entropy, or Information, Unifies Ecology and Evolution and Beyond

“…Rarely used until recently [5]. Related measures are proposed as a primary measure of evolvability [12,13]. Commonly used for analyzing networks of physically linked or functionally interacting genes [5,[14][15][16][17].…”

“…For forecasts with other values of q, see Table 1b. [39] 'Directional' selection that favors a single variant ( [12,13] and Supp. S4, S5 of review [5])…”

“…However, there are now moves to make models that include adaptive differences between guilds of species [50]. Frank [13] and Day [12] have made a very clear case for assessing biological adaptation by entropic methods, which are a general method that allows us to connect underlying causes-such as adaptive differences of variants-to the resulting macropatterns, such as diversity within and between locations. For example, survival of individuals of a particular type (alleles, species) must often be combined over different life-stages such as:…”

“…is equivalent to addition of the logs of the survivals, and thus one often uses log fitnesses, e.g., log (p'/p), where p is the proportion of a particular type before selection and p' is its proportion after selection. Then the average of the log fitnesses is Average fitness = ∑ p p log p /p = −KL where KL is the classic expression for relative entropy (Kullback-Liebler) of the adult array of types relative to the initial newborn array [13]. This calculation provides immediate access to the maximum entropy production approach that is widely used throughout science for exploiting hypotheses about fundamental processes (e.g., inheritance mode and dispersal) to create forecasts of measurable patterns, including ecological adaptation and assemblages [51][52][53][54][55] (although some of those are not based on the four fundamental processes outlined above [51]).…”

The ‘algebra of evolution’: the Robertson–Price identity and viability selection for body mass in a wild bird population

“…In one of the best examples of its application, Price's theorem (or 'the Price equation'; [1]) provides a simple and yet comprehensive means of describing the change in a biological entity across a chosen time-step. The Price equation constitutes the 'algebra of evolution' [2] and arguably deserves to be known as the 'fundamental' theorem of evolution [3], as it holds true in all situations, for all biological entities, and can be used to describe any biological dynamics at any scale, from population genetics to population ecology [3,4]. As outlined in detail in other papers in this Special Issue, the Price equation is based on the straightforward observation that the mean value of any entity in a population at a given time point is determined by two components: the frequency of different classes of individuals in the population, and the value of the entity within each of those classes.…”