DOCSLIB.ORG

The Price Equation and the Mathematics of Selection∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Price Equation and the Mathematics of Selection∗ GENERAL ARTICLE The Price Equation and the Mathematics of Selection∗ Amitabh Joshi Fifty years ago, a small one and a half page paper without a single reference was published in the leading journal Na- ture. The paper laid out the most general mathematical for- mulation of natural selection that would work for all kinds of selection processes and under any form of inheritance (not just biological evolution and Mendelian genes), although the paper discussed the issue in a genetical framework. Writ- ten by a maverick American expatriate in England, with no prior background of studying evolution or genetics, the paper Amitabh Joshi studies and had initially been turned down by the editor of Nature as too teaches evolutionary biology and population ecology at ffi di cult to understand. Largely ignored by the evolutionary JNCASR, Bengaluru. He also biology community till the 1990s, the Price Equation is now has serious interests in poetry, widely recognized as an extremely useful conceptualization, history and philosophy, permitting the incorporation of non-genetic inheritance into especially the history and philosophy of genetics and evolutionary models, serving to clarify the relationship be- evolution. tween kin-selection and group-selection, unifying varied ap- proaches used in the past to model evolutionary change, and forming the foundation of multi-level selection theory. Background: Charles Darwin, Natural Selection, and Popu- lation Genetics To fully understand the significance of the Price Equation, we need to step back over a century and a half to Charles Darwin and the principle of natural selection. Darwin derived his concept of natural selection by analogy to plant and animal breeding, in which varieties of domesticated organisms with desired charac- teristics could be developed merely by choosing individuals with Keywords certain desired traits to breed from, generation after generation. Non-genic inheritance. Darwin realized that the ecological struggle for existence in na- ∗Vol.25, No.4, DOI: https://doi.org/10.1007/s12045-020-0967-1 RESONANCE | April 2020 495 GENERAL ARTICLE ture, as a result of competition for limiting resources, would act as a natural analogue of the breeder selecting which individuals Darwin saw that if got to reproduce, based on whether they had the desirable traits, offspring were relatively or not. Thus, individuals with traits that enabled them to func- more likely to carry trait tion well in their environment would tend to be more successful variations identical or ff very similar to their at surviving to breed, and eventually produce more o spring than parents, and if individuals that had traits less suited to the environment. The individuals with certain connection between greater reproductive success in the struggle trait variations were to for existence to longer term evolutionary change was provided routinely produce more ff offspring than others, by heredity, which implied that o spring would tend to resemble then eventually those their parents. Selection, therefore, involved both the differential ‘favourable’ trait survival and reproduction of certain individuals, and the similarity variations would become in traits between those individuals and their offspring. Of course, more common in the population, resulting in Darwin did not know the mechanisms of either the generation or adaptive evolutionary the inheritance of trait variations. Yet, he saw the important im- change. plication that, if offspring were relatively more likely to carry trait variations identical or very similar to their parents, and if individ- uals with certain trait variations were to routinely produce more offspring than others, then eventually those ‘favourable’ trait vari- ations would become more common in the population, resulting in adaptive evolutionary change. After the ‘rediscovery’ of Mendel’s Laws in 1900, a major fo- cus of theoretical research in evolutionary biology was to rec- oncile the mechanism of natural selection with the principles of Mendelian inheritance, especially since it had been widely be- lieved that natural selection would not be effective in mediating 1See Resonance, No.9, pp.3– change in the face of the conservative force of heredity1.This 5, 2000 and Resonance, No.6, naturally led to a focus on genic inheritance, leading to popula- pp.525–548, 2017; Boxes 1,2 tion genetic models of the process of adaptive changes in popula- After the ‘rediscovery’ tions under the influence of natural selection. These were clas- of Mendel’s Laws in sic models of dynamic systems, making a number of simplifying 1900, a major focus of assumptions, and focussing mechanistically on how genotype or theoretical research in evolutionary biology was allele frequencies would change over generations under the in- to reconcile the fluence of various evolutionary forces like mutation, migration, mechanism of natural selection, and random genetic drift. It was this blending of pop- selection with the ulation genetics with the principle of selection that largely made principles of Mendelian inheritance. 496 RESONANCE | April 2020 GENERAL ARTICLE up the so-called Neo-Darwinian Synthesis. The Price Equation took a very different approach, and sought to ask why various systems under selection changed in the manner they did, over The Price Equation generations, rather than focussing on how they changed. Thanks sought to ask why to this shift in focus, and its avoidance of traditional mechanistic various systems under selection changed in the modelling aimed at prediction, the Price Equation threw light on manner they did, over the underlying mathematical structure of selection itself, yielding generations, rather than insights of great originality and profundity into the evolutionary focussing on how they process. changed. Thanks to this shift in focus, and its avoidance of traditional mechanistic modelling What is the Price Equation? aimed at prediction, the Price Equation threw The Price Equation is simple and elegant, constituting a theorem light on the underlying encapsulating evolutionary change. In one of its commonly en- mathematical structure countered forms, it is presented as of selection itself. 1 Δφ¯ = cov(W,φ) + E(Wδ¯) W¯ Here, in the familiar context of biological evolution by natural selection, W is Darwinian fitness (the number of offspring an in- dividual having a particular phenotype leaves in the next time step), W¯ is the population mean fitness i.e., the mean number of offspring produced per individual in the population, φ is the phenotype, δ¯ is the population mean of the difference between an individual’s phenotype and the mean phenotype of its offspring, and Δφ¯ is the change in the mean phenotype in the population after one round of selection. Before we get into examining what exactly this equation means, and why, let us see where it is com- ing from. There are several different ways in which one can de- rive the Price Equation. We will follow the derivation in Chapter 6 of Rice (2004). Therefore, the notations used here are differ- ent from those originally used by Price, which are shown in the accompanying Article-in-a-box. Consider a set of N individuals in the parental generation, and let φi denote the phenotype of the ith individual (0 < i ≤ N). Then, φ¯ denotes the population mean phenotype in the parental gener- RESONANCE | April 2020 497 GENERAL ARTICLE ation. Let δi, j denote the difference in phenotype between the parental individual i and its jth offspring. Then, δ¯i is the differ- ence between the phenotype of parental individual i and the mean phenotype of all its offspring. Finally, let Wi denote the number of offspring of parental individual i,andW¯ the mean number of off- spring produced by individuals of the parental generation. Given the above, the phenotype of the jth offspring of the ith parent is given by φi + δi, j. The mean phenotype of all individuals in the offspring generation, φ¯, is thus given by N Wi = = φi + δi, j φ¯ = i 1 j 1 N i=1 Wi The numerator in this equation adds up the phenotype of all off- spring of all N parents, and the denominator is the total number of all offspring. Let us next examine the various summations in the above expres- sion. Since the individual phenotype of the ith parent, φi,isfirst being summed over its own Wi number of offspring, rather than over the i = 1...N parents, we can simply write this sum as the product of Wi and φi. Wi φi = Wiφi . j=1 Similarly, the summation of the difference in phenotype between each of the Wi offspring of the ith parent with the parental pheno- type (δi, j) can be written as the product of the number of offspring of the ith parent (Wi)andthedifference between their mean phe- notype and that of the parent (δ¯i), as Wi δi, j = Wiδ¯i . j=1 498 RESONANCE | April 2020 GENERAL ARTICLE And, finally, the summation of offspring number per parent over all N parents in the denominator can be written as N Wi = NW¯ . i=1 Substituting these three values into the equation for φ¯,weget ⎡ ⎤ ⎢N N ⎥ 1 ⎢ ⎥ φ¯ = ⎣⎢ Wiφi + Wiδ¯i⎦⎥ NW¯ i=1 i=1 1 = E(Wφ) + E(Wδ¯) . W¯ Let us briefly examine what these terms on the right hand side are. Within the square brackets are two expectations. If you are not quite familiar with the concept of an ‘expectation’ of a random variable, it is essentially very similar to the arithmetic mean (for details, see Resonance, February 1996 pp.55–68). For example, the first term in the square brackets is the expectation of the product of number of offspring (W) and phenotype (φ), which is equivalent to the mean of the offspring number × pheno- type (product of fitness and phenotype) in the parental generation. Similarly, the second expectation is essentially the mean, in the parental generation, of the product of fitness and the difference between an individual’s own phenotype and the mean phenotype of all its offspring.
Load more

Recommended publications
  • The Mirror of Physics: on How the Price Equation Can Unify Evolutionary Biology
    Synthese https://doi.org/10.1007/s11229-021-03339-6 ORIGINAL RESEARCH The mirror of physics: on how the Price equation can unify evolutionary biology Victor J. Luque1 · Lorenzo Baravalle2 Received: 30 September 2020 / Accepted: 26 July 2021 © The Author(s) 2021 Abstract Due to its high degree of complexity and its historical nature, evolutionary biol‑ ogy has been traditionally portrayed as a messy science. According to the supporters of such a view, evolutionary biology would be unable to formulate laws and robust theories, instead just delivering coherent narratives and local models. In this article, our aim is to challenge this view by showing how the Price equation can work as the core of a general theoretical framework for evolutionary phenomena. To support this claim, we outline some unnoticed structural similarities between physical theo‑ ries (in particular, classical mechanics) and evolutionary biology. More specifcally, we shall argue that the Price equation, in the same way as fundamental formalisms in physics, can serve as a heuristic principle to formulate and systematise diferent theories and models in evolutionary biology. Keywords Evolutionary biology · Price equation · Complexity · Metatheoretical structuralism · Guiding principle · Classical mechanics This article belongs to the topical collection “Simplicity out of Complexity? Physics and the Aims of Science,” Edited by Florian J. Boge, Martin King, Paul Grünke and Miguel Ángel Carretero Sahuquillo. * Victor J. Luque [email protected] Lorenzo Baravalle [email protected]
    [Show full text]
  • The 'Algebra of Evolution': the Robertson–Price Identity and Viability Selection for Body Mass in a Wild Bird Population
    Edinburgh Research Explorer The ‘algebra of evolution’: the Robertson–Price identity and viability selection for body mass in a wild bird population Citation for published version: Hajduk, GK, Walling, CA, Cockburn, A & Kruuk, LEB 2020, 'The ‘algebra of evolution’: the Robertson–Price identity and viability selection for body mass in a wild bird population', Philosophical Transactions of the Royal Society B: Biological Sciences, vol. 375, no. 1797, 20190359. https://doi.org/10.1098/rstb.2019.0359 Digital Object Identifier (DOI): 10.1098/rstb.2019.0359 Link: Link to publication record in Edinburgh Research Explorer Document Version: Publisher's PDF, also known as Version of record Published In: Philosophical Transactions of the Royal Society B: Biological Sciences General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 26. Sep. 2021 The ‘algebra of evolution’: the Robertson–Price identity and royalsocietypublishing.org/journal/rstb viability selection for body mass in a wild bird population Research G. K. Hajduk1, C. A. Walling1, A. Cockburn2 and L. E. B. Kruuk2 1 Cite this article: Hajduk GK, Walling CA, Institute of Evolutionary Biology, University of Edinburgh, Edinburgh EH9 3FL, UK 2Research School of Biology, Australian National University, Canberra, ACT 2601, Australia Cockburn A, Kruuk LEB.
    [Show full text]
  • Multilevel Selection and Population Genetics in Structured
    Multilevel Selection, Population Genetics and Cooperation in Structured Populations NIMBioS Tutorial: Game Theoretical Modeling of Evolution in Structured Populations Jeremy Van Cleve University of Kentucky 26 April 2016 Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY How do cooperative behaviors evolve? Cooperation occurs when: focal: cost to improve state of its partner partner: beneft from improved state Food improves state ⟶ higher ftness Multilevel selection, pop gen and games in structured pops UNIVERSITY OF KENTUCKY trate into a tightWhen form they known cease as moving,and the a when cells Mexican there ofenvironment hat. is the lacks Then, either slug electrolytes, concen- directional when light it orimmune is system, no before they very are light. shed at moist, thefunctioning rear of simultaneously the slug. asthe liver, slug from kidney, front and toered back innate class picking up of toxins cells and calledand bacteria, ultimately sentinel become cells the stalk. that There is sweep athe through recently discov- constituent cells. Those atthe slug the lacks a front nervous system, direct there are movement differencesmove: among an important advantage to themore quickly social and farther stage. than Though any individual amoebaeffectively could recovering the solitaryrear, stage. and these The cells slug can feed moves sheath on any largely bacteria made they discover, upfrom of it cellulose, it in drops some cells important at ways. the Astoward light and heat it and away from ammonia crawls ( throughwhich a then elongatesAfter slightly a few and hours, beginsa this center to center concentrates into crawl in aconcentrate around a mound, in great process streams of calledhighest dicty cells, aggregation concentration.
    [Show full text]
  • The Purpose of Adaptation
    Downloaded from http://rsfs.royalsocietypublishing.org/ on February 1, 2018 The purpose of adaptation Andy Gardner rsfs.royalsocietypublishing.org School of Biology, University of St Andrews, Dyers Brae, St Andrews KY16 9TH, UK AG, 0000-0002-1304-3734 A central feature of Darwin’s theory of natural selection is that it explains the purpose of biological adaptation. Here, I: emphasize the scientific impor- tance of understanding what adaptations are for, in terms of facilitating Research the derivation of empirically testable predictions; discuss the population genetical basis for Darwin’s theory of the purpose of adaptation, with refer- Cite this article: Gardner A. 2017 The ence to Fisher’s ‘fundamental theorem of natural selection’; and show that a purpose of adaptation. Interface Focus 7: deeper understanding of the purpose of adaptation is achieved in the context 20170005. of social evolution, with reference to inclusive fitness and superorganisms. http://dx.doi.org/10.1098/rsfs.2017.0005 One contribution of 20 to a theme issue ‘New 1. The purpose of adaptation trends in evolutionary biology: biological, Darwinism is a theory of the process of adaptation, i.e. the appearance of design philosophical and social science perspectives’. in the biological world. The problem of how to explain adaptation is an ancient one, and it famously provided the basis for William Paley’s [1] argument for Subject Areas: the existence of an intelligent, divine designer. This problem was decisively solved by Charles Darwin [2], whose theory of evolution by natural selec- biomathematics tion—in which heritable variations associated with greater survival and reproductive success are identified as being more likely to accumulate in natural Keywords: populations—explained the adaptation of organisms in purely naturalistic, natural selection, fundamental theorem, mechanical terms.
    [Show full text]
  • Struggle for Existence: the Models for Darwinian and Non-Darwinian Selection
    Struggle for Existence: the models for Darwinian and non-Darwinian selection G. Karev1, F. Berezovskaya2 1National Center for Biotechnology Information, Bethesda, MD 20894, USA. Email: [email protected] 2 Howard University, Washington, DC 20059, USA e-mail: [email protected] Abstract Classical understanding of the outcome of the struggle for existence results in the Darwinian “survival of the fittest”. Here we show that the situation may be different, more complex and arguably more interesting. Specifically, we show that different versions of inhomogeneous logistic-like models with a distributed Malthusian parameter imply non-Darwinian “survival of everybody”. In contrast, the inhomogeneous logistic equation with distributed carrying capacity shows Darwinian “survival of the fittest”. We also consider an inhomogeneous birth-and-death equation and give a simple proof that this equation results in the “survival of the fittest”. In addition to this known result, we find an exact limit distribution of the parameters of this equation. We also consider “frequency-dependent” inhomogeneous models and show that although some of these models show Darwinian “survival of the fittest”, there is not enough time for selection of the fittest species. We discuss the well-known Gauze’s Competitive exclusion principle that states that “Complete competitors cannot coexist”. While this principle is often considered as a direct consequence of the Darwinian “survival of the fittest”, we show that from the point of view of developed mathematical theory complete competitors can in fact coexist indefinitely. 1. Introduction. Darwinian evolution and Competitive Exclusion principle In his 1934 book, G.F. Gause described his experiments where he cultivated two types of yeast, S.
    [Show full text]
  • George Price's Contributions to Evolutionary Genetics
    J. theor. Biol. (1995) 175, 373–388 George Price’s Contributions to Evolutionary Genetics S A. F Department of Ecology and Evolutionary Biology, University of California, Irvine, CA 92717, U.S.A. (Received on 9 November 1994, Accepted in revised form on 14 February 1995) George Price studied evolutionary genetics for approximately seven years between 1967 and 1974. During that brief period Price made three lasting contributions to evolutionary theory; these were: (i) the Price Equation, a profound insight into the nature of selection and the basis for the modern theories of kin and group selection; (ii) the theory of games and animal behavior, based on the concept of the evolutionarily stable strategy; and (iii) the modern interpretation of Fisher’s fundamental theorem of natural selection, Fisher’s theorem being perhaps the most cited and least understood idea in the history of evolutionary genetics. This paper summarizes Price’s contributions and briefly outlines why, toward the end of his painful intellectual journey, he chose to focus his deep humanistic feelings and sharp, analytical mind on abstract problems in evolutionary theory. 7 1995 Academic Press Limited Introduction (iii) Price (1972b) was the first to crack the most obscure and misunderstood of R. A. Fisher’s many At the age of 44, George Price quit his engineering job oracles: the fundamental theorem of natural selection. at IBM and took up evolutionary genetics. Price This theorem is perhaps the most widely quoted result moved to the Galton Laboratories at London in 1967, in evolutionary theory, yet Price demonstrated where C. A. B. Smith, Weldon Professor of Biometry, convincingly that the consensus interpretation of the provided him with space and encouragement.
    [Show full text]
  • The Place of Development in Mathematical Evolutionary Theory SEAN H
    COMMENTARY AND PERSPECTIVE The Place of Development in Mathematical Evolutionary Theory SEAN H. RICEÃ Department of Biological Sciences, Texas Tech University, Lubbock, Texas ABSTRACT Development plays a critical role in structuring the joint offspring–parent phenotype distribution. It thus must be part of any truly general evolutionary theory. Historically, the offspring–parent distribution has often been treated in such a way as to bury the contribution of development, by distilling from it a single term, either heritability or additive genetic variance, and then working only with this term. I discuss two reasons why this approach is no longer satisfactory. First, the regression of expected offspring phenotype on parent phenotype can easily be nonlinear, and this nonlinearity can have a pronounced impact on the response to selection. Second, even when the offspring–parent regression is linear, it is nearly always a function of the environment, and the precise way that heritability covaries with the environment can have a substantial effect on adaptive evolution. Understanding these complexities of the offspring–parent distribution will require understanding of the developmental processes underlying the traits of interest. I briefly discuss how we can incorporate such complexity into formal evolutionary theory, and why it is likely to be important even for traits that are not traditionally the focus of evo–devo research. Finally, I briefly discuss a topic that is widely seen as being squarely in the domain of evo–devo: novelty. I argue that the same conceptual and mathematical framework that allows us to incorporate developmental complexity into simple models of trait evolution also yields insight into the evolution of novel traits.
    [Show full text]
  • The Price Equation, Fisher's Fundamental Theorem, Kin Selection, and Causal Analysis
    Evolution, 51(6), 1997, pp. 1712-1729 THE PRICE EQUATION, FISHER'S FUNDAMENTAL THEOREM, KIN SELECTION, AND CAUSAL ANALYSIS STEVEN A. FRANK Department of Ecology and Evolutionary Biology, University of California, Irvine, California 92697-2525 E-mail: [email protected] Abstract.--A general framework is presented to unify diverse models of natural selection. This framework is based on the Price Equation, with two additional steps. First, characters are described by their multiple regression on a set of predictor variables. The most common predictors in genetics are alleles and their interactions, but any predictor may be used. The second step is to describe fitness by multiple regression on characters. Once again, characters may be chosen arbitrarily. This expanded Price Equation provides an exact description of total evolutionary change under all conditions, and for all systems of inheritance and selection. The model is first used for a new proof of Fisher's fundamental theorem of natural selection. The relations are then made clear among Fisher's theorem, Robertson's covariance theorem for quantitative genetics, the Lande-Arnold model for the causal analysis of natural selection, and Hamilton's rule for kin selection. Each of these models is a partial analysis of total evolutionary change. The Price Equation extends each model to an exact, total analysis of evolutionary change for any system of inheritance and selection. This exact analysis is used to develop an expanded Hamilton's rule for total change. The expanded rule clarifies the distinction between two types of kin selection coefficients. The first measures components of selection caused by correlated phenotypes of social partners.
    [Show full text]
  • Joint Phenotypes, Evolutionary Conflict and the Fundamental Theorem of Natural Selection David C
    Washington University in St. Louis Washington University Open Scholarship Biology Faculty Publications & Presentations Biology 5-19-2014 Joint phenotypes, evolutionary conflict and the fundamental theorem of natural selection David C. Queller Washington University in St Louis, [email protected] Follow this and additional works at: https://openscholarship.wustl.edu/bio_facpubs Part of the Biology Commons, Evolution Commons, and the Population Biology Commons Recommended Citation Queller, David C., "Joint phenotypes, evolutionary conflict and the fundamental theorem of natural selection" (2014). Biology Faculty Publications & Presentations. 57. https://openscholarship.wustl.edu/bio_facpubs/57 This Article is brought to you for free and open access by the Biology at Washington University Open Scholarship. It has been accepted for inclusion in Biology Faculty Publications & Presentations by an authorized administrator of Washington University Open Scholarship. For more information, please contact [email protected]. Joint phenotypes, evolutionary conflict, and the fundamental theorem of natural selection David C. Queller Department of Biology Washington University in St. Louis St. Louis, MO 63130 USA Email: [email protected] Prepublication version. Pulbished in Phil. Trans. Roy. Soc. B 369:20130423 doi:10.1098/rstb.2013.0423. 1 Abstract Multiple organisms can sometimes affect a common phenotype. For example the portion of a leaf eaten by an insect is a joint phenotype of the plant and the insect and the amount of food obtained by an offspring can be joint trait with its mother. Here I describe the evolution of joint phenotypes in quantitative genetic terms. A joint phenotype for multiple species evolves as the sum of additive genetic variances in each species, weighted by the selection on each species.
    [Show full text]
  • One Equation to Rule Them All
    One Equation To Rule Them All: A philosophical analysis of the Price equation Victor J. Luque ABSTRACT This paper provides a philosophical analysis of the Price equation and its role in evolutionary theory. Traditional models in population genetics postulate simplifying assumptions in order to make the models mathematically tractable. On the contrary, the Price equation implies a very specific way of theorizing, starting with assumptions that we think are true and then deriving from them the mathematical rules of the system. I argue that the Price equation is a generalization-sketch, whose main purpose is to provide a unifying framework for researchers, helping them to develop specific models. The Price equation plays this role because, like other scientific principles, shows features as abstractness, unification and invariance. By underwriting this special role for the Price equation some recent disputes about it and other evolutionary equations could be diverted. This move also links Price’s equation with Newton’s second law of motion, and therefore vindicates the Newtonian analogy. KEYWORDS: Price equation, generalization-sketch, abstractness, unification, Newtonian analogy. 1. INTRODUCTION The Price equation, first presented by George Price at the beginning of the 1970s, is one of the key equations in evolutionary theory. Price believed that he had found an equation so special that it could describe any evolutionary situation and any evolutionary problem –in other words, Price developed an abstract way of theorizing and thinking about evolution. Nevertheless, this equation has been involved in a great dispute the last decade due to its special nature, after a long period of oblivion when it was used by very few researchers.
    [Show full text]
  • Arxiv:1808.05643V1 [Q-Bio.PE] 16 Aug 2018 Subject to Surprisingly Little Conceptual Or Theoretical Challenge to Its Underlying Premises and Methods
    Not even wrong: The spurious link between biodiversity and ecosystem functioning Pradeep Pillaia1 , Tarik C. Gouhier1 1Marine Science Center, Northeastern University, 30 Nahant Rd, Nahant, MA 01908 a To whom correspondence should be addressed. E-mail: [email protected] Abstract Resolving the relationship between biodiversity and ecosystem functioning has been one of the central goals of modern ecology. Early debates about the relationship were finally resolved with the advent of a statistical partitioning scheme that decomposed the biodiversity effect into a \selection" effect and a \complementarity" effect. We prove that both the biodiversity effect and its statistical decomposition into selection and complementarity are fundamentally flawed because these methods use a na¨ıve null expectation based on neutrality, likely leading to an overestimate of the net biodiver- sity effect, and they fail to account for the nonlinear abundance-ecosystem functioning relationships observed in nature. Furthermore, under such nonlinearity no statistical scheme can be devised to partition the biodiversity effects. We also present an alter- native metric providing a more reasonable estimate of biodiversity effect. Our results suggest that all studies conducted since the early 1990s likely overestimated the positive effects of biodiversity on ecosystem functioning. Keywords: biodiversity; ecosystem functioning; complementarity effect; selection effect; Jensen's inequality; Price equation; species coexistence Introduction The notion that increasing biodiversity
    [Show full text]
  • The Price Equation As a Bridge Between Animal Breeding and Evolutionary Biology Philosophical Transactions of the Royal Society B
    The Price equation as a bridge between animal breeding and evolutionary biology Philosophical Transactions of the Royal Society B. Biological sciences Bijma, P. https://doi.org/10.1098/rstb.2019.0360 This article is made publicly available in the institutional repository of Wageningen University and Research, under the terms of article 25fa of the Dutch Copyright Act, also known as the Amendment Taverne. This has been done with explicit consent by the author. Article 25fa states that the author of a short scientific work funded either wholly or partially by Dutch public funds is entitled to make that work publicly available for no consideration following a reasonable period of time after the work was first published, provided that clear reference is made to the source of the first publication of the work. This publication is distributed under The Association of Universities in the Netherlands (VSNU) 'Article 25fa implementation' project. In this project research outputs of researchers employed by Dutch Universities that comply with the legal requirements of Article 25fa of the Dutch Copyright Act are distributed online and free of cost or other barriers in institutional repositories. Research outputs are distributed six months after their first online publication in the original published version and with proper attribution to the source of the original publication. You are permitted to download and use the publication for personal purposes. All rights remain with the author(s) and / or copyright owner(s) of this work. Any use of the publication or parts of it other than authorised under article 25fa of the Dutch Copyright act is prohibited.
    [Show full text]