The Generalized Price Equation: Forces That Change Population Statistics

The generalized Price equation: forces that change population statistics

Steven A. Frank1 and William Godsoe2

The Price equation partitions the change in the expected value of a population measure. The first component describes the partial change caused by altered frequencies. The second component describes the partial change caused by altered measurements. In biology, frequency changes often associate with the direct effect of natural selection. Measure changes reflect processes during transmission that alter trait values. More broadly, the two components describe the direct forces that change population composition and the altered frame of reference that changes measured values. The classic Price equation is limited to population statistics that can expressed as the expected value of a measure. Many statistics cannot be expressed as expected values, such as the harmonic mean and the family of rescaled diversity measures. We generalize the Price equation to any population statistic that can be expressed as a function of frequencies and measurements. We obtain the generalized partition between the direct forces that cause frequency change and the altered frame of reference that changes measurements.

Introduction frame of reference are abstractions of the biologists’ components of natural selection and transmission. The Price equation has been used in many disci- A recent collection of articles celebrates the 50th plines to study the forces that cause change in pop- anniversary of the Price equation and summarizes ulations1. In evolutionary biology, the equation applications across a wide variety of disciplines8. separates the changes in traits caused by natural se- The classic Price equation describes change in lection from the changes in traits that arise during any population statistic that can be written in the transmission2. The abstract expression of change form of an expected value. Many population statis- often highlights general principles. For example, tics cannot be written in that form9. the Price equation shows that certain patterns of This article generalizes the Price equation to han- correlation between individuals provide sufficient dle most other forms of population statistics. We statistics for the evolutionary forces that influence end up with the same generic partitioning of popu- altruistic behaviors, the basis for kin selection the- lation change into two components: the forces that ory3,4. alter the frequencies of the entities that make up The Price equation has also been used in ecol- the population holding constant the measurements ogy to partition change in diversity indices into di- on each entity and the forces that alter the mea- 5 arXiv:2003.08976v1 [q-bio.PE] 19 Mar 2020 rect and indirect processes . Economists have used surements on each entity holding constant the fre- the equation to model the evolution of firms6,7. In quencies10. physics and other disciplines, the Price equation neatly separates change caused by directly acting forces from change that arises indirectly from an al- The Classic Price Equation tered frame of reference1. The direct forces and the Consider a population statistic that can be written 1 Department of Ecology and Evolutionary Biology, Uni- as the expected value of some measurement versity of California, Irvine, CA 92697–2525, USA

2 Bio-Protection Research Centre, Lincoln University, P.O. X z¯ = qizi = q · z. Box 85084, Lincoln 7647, New Zealand i

git • master @ arXiv-1.0-0::8db9321-2020-03-19 (2021-08-23 16:14Z) • safrank The index i denotes subsets of the population. The with the notation frequencies qi describe the weighting of each subset. The measurements zi are any value that may ∆¯z = ∆qz¯ + ∆zz,¯ (2) be associated with subsets. The statisticz ¯ is the expected value of the measurement. in which ∆q is the partial change with respect to q, Vector notation provides a more compact expres- and ∆z is the partial change with respect to z when 1 sion. The vectors q and z run over the indices evaluated in the altered population. See Frank for i = 1, 2, . . . , n. The dot product, q · z, yields the the abstract mathematical properties of the clas- summation that defines the expected value. sic Price equation and applications of that abstract For any population statistic that can be written interpretation. as an expected value, the classic form of the Price In biology, the partial frequency changes, ∆q, of- equation partitions the total change in the statistic ten associate with natural selection, which directly into two components. If we define the statistic in changes frequencies in proportion to Fisher’s aver- the altered population asz ¯0 = q0 · z0, then age excess in fitness. The partial measure changes, ∆z, often associate with changes in trait values dur- ∆¯z = q0 · z0 − q · z ing transmission. Thus, the Price equation provides a general separation between selection and trans- = q0 · z0 − q0 · z + q0 · z − q · z mission. = q0 · ∆z + ∆q · z.

It is convenient to switch the order of the terms in The Generalized Price Equation the last line Many population statistics cannot be expressed as ∆¯z = ∆q · z + q0 · ∆z. (1) the expected value of a particular measurement. Consider any population statistic that can be writ- This expression provides the most basic form of the ten as f(q, z), where f is a function that maps the 10 classic Price equation. Frank relates this vector population frequencies and measurements to some form to other notations, such as the covariance and value. For example, expectation used by Price11,12. Equation1 follows the strictly defined set map- 1 f(q, z) = P . (3) ping scheme for the relations between entities in the i qizi two populations given in Frank10. For example, q0 i Many population statistics depend on the frequen- is the frequency of entities in the altered population cies and values across population subsets9. So we derived from entities in the original population with should be able to write most population statistics index i. Our extension in the following section also as some function, f. The change in f is adheres to that strictly defined set mapping. See 10 Frank for further details. ∆f = f(q0, z0) − f(q, z). The Price equation is simply the chain rule for differences applied to a population statistic. The Applying the same approach as in the derivation first term on the right-hand side of eqn1 describes of the classic Price equation, we can add f(q0, z) − the change in frequencies when holding the mea- f(q0, z) = 0 to the expression above and then re- surements constant. The second term describes the arrange to obtain change in the measurements when holding the frequencies constant. In the second term, the con- ∆f = f(q0, z) − f(q, z) + f(q0, z0) − f(q0, z), stant frequencies come from the altered population, (4) because the consequences of the changed frame of which may be expressed in the partial difference reference depend on the frequencies in the altered notation as population. We can express these partial changes ∆f = ∆qf + ∆zf.

git • master @ arXiv-1.0-0::8db9321-2020-03-19 (2021-08-23 16:14Z) • safrank Example In the example of eqn5, the ∆ z term appears in the numerator of the final right-hand side term. Using the example function in eqn3 in the general That term can be expanded to evaluate hierarchi- form of eqn4, we obtain cally nested levels by the methods outlined here. 1 1 1 1 ∆f = − + − , q0 · z q · z q0 · z0 q0 · z Application from which we can derive an expression that con- Several widely used population statistics do not tains the standard partial change terms of the clas- match the standard expected value form of the classic Price equation, normalized by the average values sic Price equation9. Our generalization provides a of z, as way to partition changes in such statistics into fre- 1 ∆q · z q0 · ∆z quency changes and measure changes. We discuss ∆f = − + . (5) q0 · z z¯ z¯0 the harmonic mean and the numbers equivalent of a diversity index. Using the partial change notion in eqn2, we can The harmonic mean of a measure yi has the form write 1 ∆ z¯ ∆ z¯ of eqn3 when we define zi as 1/yi. The expression ∆f = − q + z . q0 · z z¯ z¯0 in eqn5 partitions changes in the harmonic mean into partial frequency change and partial measure This simple form emphasizes the partial change change components. expression. In many applications, these partial Harmonic means arise is various population biol- change terms associate with the direct effects of se- ogy applications. The carrying capacity of a poly- lection on frequency changes and the indirect effects morphic population can often be approximated by a of changes in measurement values during transmis- weighted harmonic mean of the carrying capacity of sion. each type15. Optimal residence times of consumers may vary with the harmonic mean of the quality of Recursive Multilevel Expansion resources in a given patch16. When mating occurs in local patches, the inbreeding level often follows The classic Price equation can be expanded recur- the harmonic mean number of females per patch17. 13 sively . In eqn1, the right-hand ∆ z term is the The numbers equivalent diversity index can be vector ∆zi over the index, i. Each ∆zi term may expressed in the functional form of f(q, z) by defin- itself be considered as the change in the average p−1 ing zi = qi and then writing value of z within the ith subset or group. Each of 1 those terms may be analyzed by the equation, in f(q, z) = (q · z) 1−p . which 0 This numbers equivalent form provides a general ∆¯zi = ∆qi · zi + qi · ∆zi. diversity expression that allows comparison of dif- Each group indexed by i contains the next ferent diversity indices when expressed in common level of nested subsets. For example, z = i units18–21. z , z , . . . , z . Multilevel expansions have pro- 1|i 2|i 1|mi Godsoe5 used the classic Price equation to an- vided insight into many applications4,13,14. alyze the simpler arithmetic mean diversity index We can hierarchically expand the generalized p−1 f = q · z for zi = q for p 6= 1. They demon- form in eqn4. In the third right-hand side term i strated the value of partitioning changes in diversity of that equation, we write into frequency change and measure change compo- f(q0, z0) = f(q0, z + ∆z), nents to evaluate the role of selection in the dynamics of diversity. However, they could not general- in which ∆z = z0 − z. Now we have a ∆z term, to ize their results to the broader numbers equivalent which we can apply the classic Price equation. form of diversity because the classic Price equation

git • master @ arXiv-1.0-0::8db9321-2020-03-19 (2021-08-23 16:14Z) • safrank is limited to expected value statistics. We can now equation and the analysis of economic evolu- use the generalized Price equation to partition the tion. Evolutionary and Institutional Economics change in the numbers equivalent diversity index. Review 1, 127–148 (2004). 8. Lehtonen, J., Okasha, S. & Helanterä, Conclusion H. Fifty years of the Price equation. Philosophical Transactions of the Royal The classic Price equation has been applied to a Society B 375, 20190350 (2020). URL wide variety of problems8. Our results generalize https://royalsocietypublishing. the equation to population statistics that can be org/doi/abs/10.1098/rstb.2019.0350. expressed as a function of frequencies and measure- https://royalsocietypublishing.org/ ments. That generalization opens new problems for doi/pdf/10.1098/rstb.2019.0350. study. 9. Bullen, P. S. Handbook of Means and Their In- equalities (Kluwer Academic Publishers, Dor- drecht, Netherlands, 2003). Acknowledgments 10. Frank, S. A. Natural selection. IV. The Price equation. Journal of Evolutionary Biology 25, The project was supported by the Donald Bren 1002–1019 (2012). Foundation (SAF) and the New Zealand Tertiary 11. Price, G. R. Selection and covariance. Nature Education Commission CoRE grant to the Bio- 227, 520–521 (1970). Protection Research Centre (WG). 12. Price, G. R. Extension of covariance selection mathematics. Annals of Human Genetics 35, References 485–490 (1972). 13. Hamilton, W. D. Innate social aptitudes of 1. Frank, S. A. The Price equation program: man: an approach from evolutionary genetics. simple invariances unify population dynamics, In Fox, R. (ed.) Biosocial Anthropology, 133– thermodynamics, probability, information and 155 (Wiley, New York, 1975). inference. Entropy 20, 978 (2018). 14. Hilbert, M. Linking information, knowledge 2. Gardner, A. Price’s equation made and evolutionary growth: A multilevel inter- clear. Philosophical Transactions of the play between natural selection and informed Royal Society B 375, 20190361 (2020). intervention (SSRN Scholarly Paper No. ID URL https://royalsocietypublishing. 2397751) (2013). org/doi/abs/10.1098/rstb.2019.0361. 15. Anderson, W. W. Genetic equilibrium and pop- https://royalsocietypublishing.org/ ulation growth under density-regulated selec- doi/pdf/10.1098/rstb.2019.0361. tion. American Naturalist 105, 489–498 (1971). 3. Hamilton, W. D. Selfish and spiteful behaviour 16. Calcagno, V., Grognard, F., Hamelin, F. M., in an evolutionary model. Nature 228, 1218– Wajnberg, E. & Mailleret, L. The functional 1220 (1970). response predicts the effect of resource distri- 4. Frank, S. A. Foundations of Social Evolution bution on the optimal movement rate of con- (Princeton University Press, Princeton, New sumers. Ecology Letters 17, 1570–1579 (2014). Jersey, 1998). 17. Herre, E. A. Sex ratio adjustment in fig wasps. 5. Godsoe, W., Eisen, K. & Stanton, D. Trans- Science 228, 896–898 (1985). mission bias’s fundamental role in biodiversity 18. MacArthur, R. H. Patterns of species diversity. change. bioRxiv 527028 (2019). Biological Reviews 40, 510–533 (1965). 6. Metcalfe, J. S. Evolutionary Economics 19. Hill, M. O. Diversity and evenness: a unify- and Creative Destruction (Routledge, London, ing notation and its consequences. Ecology 54, 1998). 427–432 (1973). 7. Andersen, E. S. Population thinking, Price’s

git • master @ arXiv-1.0-0::8db9321-2020-03-19 (2021-08-23 16:14Z) • safrank 20. Jost, L. Entropy and diversity. Oikos 113, 363–375 (2006). 21. Jost, L. Partitioning diversity into independent alpha and beta components. Ecology 88, 2427– 2439 (2007).

git • master @ arXiv-1.0-0::8db9321-2020-03-19 (2021-08-23 16:14Z) • safrank