Professor Alan Grafen, FRS
The formal Darwinism project is an ambitious attempt to provide a mathematical population genetic justification for the concept of fitness optimisation, at a very high level of abstraction. As well as advancing the highly technical core of the project, I am also engaged in applications, including (i) inclusive fitness on networks, in relation to altruism and population viscosity (ii) reinterpreting bethedging in terms of fitness optimisation (iii) showing that group selection has no similar justification leading to a conclusion of optimisation of group fitness.
This work also has a historical interest. It is essentially capturing Darwin’s central claim in the Origin of Species that the mechanical processes of inheritance and reproduction (now represented by population genetics) give rise to the appearance of design (now represented by optimisation). By incorporating all the advances in evolutionary theory (ESS theory, inclusive fitness, optimality theory), the end result will be a grand overarching theory justifying fitness optimisation as a central concept in biology. It is a mathematical version of the synthesis of theory and its grounding in fundamental concepts to be found in Richard Dawkins’ “The Selfish Gene”. I also engage from time to time in other topics in evolutionary theory and in statistics. In February 2014 I completed "phyreg", an R package, available from CRAN, that implements my 1989 Phylogenetic Regression.
Additional Information
I am the coauthor with Rosie Hails of ‘Modern Statistics for the Life Sciences’ (2002, OUP), which introduces undergraduate students (and others!) to the powerful technique of General Linear Modelling. Online supplements show how to do all the exercises in Minitab, SPSS and SAS.

The Price equation and reproductive value
April 2020Journal articlePhilosophical Transactions of the Royal Society B: Biological Sciences 
A simple completion of Fisher's fundamental theorem of natural selection
January 2020Journal articleEcology and Evolution 
Should we ask for more than consistency of Darwinism with Mendelism?
December 2019Journal articleStudies in history and philosophy of biological and biomedical sciencesA nonmathematical exposition of the current status of the formal darwinism project is presented, linking it to the fundamental theorem of natural selection, which is regarded as Fisher's own 'formal darwinism project'. The purpose is to found organismlevel thinking about design and adaptation, in short Darwinism, on what is known about the mechanics of genetic inheritance, in short Mendelism, and the project is to do so in as general a biological setting as possible. This view also makes sense of the name 'fundamental theorem of natural selection'.Adaptation, Biological, Heredity, Models, Biological, Models, Genetic, Philosophy, Selection, Genetic, Biological Evolution 
Inclusive fitness is an indispensable approximation for understanding organismal design.
June 2019Journal articleEvolution; international journal of organic evolutionFor some decades most biologists interested in design have agreed that natural selection leads to organisms acting as if they are maximizing a quantity known as "inclusive fitness." This maximization principle has been criticized on the (uncontested) grounds that other quantities, such as offspring number, predict gene frequency changes accurately in a wider range of mathematical models. Here, we adopt a resolution offered by Birch, who accepts the technical difficulties of establishing inclusive fitness maximization in a fully general model, while concluding that inclusive fitness is still useful as an organizing framework. We set out in more detail why inclusive fitness is such a practical and powerful framework, and provide verbal and conceptual arguments for why social biology would be more or less impossible without it. We aim to help mathematicians understand why social biologists are content to use inclusive fitness despite its theoretical weaknesses. Here, we also offer biologists practical advice for avoiding potential pitfalls.Gene Frequency, Models, Biological, Models, Genetic, Selection, Genetic, Genetic Fitness 
The left hand side of the Fundamental Theorem of Natural Selection.
November 2018Journal articleJournal of theoretical biologyThe fundamental theorem of natural selection is explained here in very simple terms, suitable for students. The biological significance of the left hand side  the rate of change in mean fitness due to changes in gene frequencies, which is also described as the rate of change due to natural selection  has been regarded since 1972 as problematic, but here a simple graph is used to show that Fisher's poor explanation was of a robust and simple intuition. Simple numerical examples show the theorem at work with fixed genotypic fitness under two different mating systems, with bland density dependence, and also with fitnesses determined by an evolutionary game. The content of the theorem has long been taken for granted by wholeorganism evolutionary biologists, though in an imprecise way, even while mathematical population geneticists have been, in sequence, wrongly proving it false, wrongly proving it requires more assumptions than Fisher admitted, and accepting the truth of the theorem as Fisher proved it, but doubting its biological significance. An important emphasis on the instantaneous nature of natural selection, and of its measurement, emerges from the argument. Price's disappointments with the content of the theorem are directly confronted. The new explanation allows us to recognise the central place the theorem already occupies in evolutionary biology, and to begin to incorporate more fully the insights embedded in it.Animals, Gene Frequency, Quantitative Trait, Heritable, Models, Genetic, Selection, Genetic, Genetic Fitness 
The fundamental theorem of natural selection.
October 2018Journal articleJournal of animal breeding and genetics = Zeitschrift fur Tierzuchtung und Zuchtungsbiologie 
A general model of biological signals, from cues to handicaps.
June 2018Journal articleEvolution lettersOrganisms sometimes appear to use extravagant traits, or "handicaps", to signal their quality to an interested receiver. Before they were used as signals, many of these traits might have been selected to increase with individual quality for reasons apart from conveying information, allowing receivers to use the traits as "cues" of quality. However, current theory does not explain when and why cues of individual quality become exaggerated into costly handicaps. We address this here, using a gametheoretic model of adaptive signalling. Our model predicts that: (1) signals will honestly reflect signaler quality whenever there is a positive relationship between individual quality and the signalling trait's naturally selected, noninformational optimum; and (2) the slope of this relationship will determine the amount of costly signal exaggeration, with more exaggeration favored when the slope is more shallow. A shallow slope means that a lower quality male would pay only a small fitness cost to have the same trait value as a higher quality male, and this drives the exaggeration of signals as highquality signalers are selected to distinguish themselves. Our model reveals a simple and potentially widespread mechanism for ensuring signal honesty and predicts a natural continuum of signalling strategies, from costfree cues to costly handicaps. 
Defining fitness in an uncertain world.
April 2018Journal articleJournal of mathematical biologyThe recently elucidated definition of fitness employed by Fisher in his fundamental theorem of natural selection is combined with reproductive values as appropriately defined in the context of both random environments and continuing fluctuations in the distribution over classes in a classstructured population. We obtain astonishingly simple results, generalisations of the Price Equation and the fundamental theorem, that show natural selection acting only through the arithmetic expectation of fitness over all uncertainties, in contrast to previous studies with fluctuating demography, in which natural selection looks rather complicated. Furthermore, our setting permits each class to have its characteristic ploidy, thus covering haploidy, diploidy and haplodiploidy at the same time; and allows arbitrary classes, including continuous variables such as condition. The simplicity is achieved by focussing just on the effects of natural selection on genotype frequencies: while other causes are present in the model, and the effect of natural selection is assessed in their presence, these causes will have their own further effects on genoytpe frequencies that are not assessed here. Also, Fisher's uses of reproductive value are shown to have two ambivalences, and a new axiomatic foundation for reproductive value is endorsed. The results continue the formal darwinism project, and extend support for the individualasmaximisingagent analogy to finite populations with random environments and fluctuating classdistributions. The model may also lead to improved ways to measure fitness in real populations.Animals, Uncertainty, Stochastic Processes, Population Dynamics, Reproduction, Ploidies, Models, Biological, Female, Male, Mathematical Concepts, Selection, Genetic, Genetic Fitness, Biological Evolution
E:  alan.grafen@zoo.ox.ac.uk 
T:  01865 (2) 71211 
Personal website 