scholarly journals The extended Price equation: migration and sex

2021 ◽  
Author(s):  
Jake Brown ◽  
Jared M Field
Keyword(s):  
Natural Selection ◽  
Sexual Reproduction ◽  
Evolutionary Change ◽  
Price Equation ◽  
Hamilton’S Rule ◽  
Population Structures ◽  
Biological Interpretation ◽  
Hamilton's Rule

The Price equation provides a general partition of evolutionary change into two components. The first is usually thought to represent natural selection and the second, transmission bias. Here, we provide a new derivation of the generalised equation, which contains a largely ignored third term. Unlike the original Price equation, this extension can account for migration and mixed asexual and sexual reproduction. The derivation here expresses the generalised equation explicitly in terms of fitness, rendering this otherwise difficult third term more open to biological interpretation and use. This re-derivation also permits fundamental results, derived from the Price equation, to be more easily generalised. We take Hamilton's rule as a case study, and provide an exact, total expression that allows for population structures like haplodiploidy. Our analysis, more generally, makes clear the previously hidden assumptions in similar fundamental results, highlighting the caution that must be taken when interpreting them.

Inclusive Fitness and Hamilton’s Rule

2015 ◽  
Author(s):  
James A.R. Marshall
Keyword(s):  
Sex Allocation ◽  
Charles Darwin ◽  
Inclusive Fitness ◽  
Price Equation ◽  
Additive Effects ◽  
Eusocial Insect ◽  
Hamilton’S Rule ◽  
Biological Phenomena ◽  
Inclusive Fitness Theory ◽  
Hamilton's Rule

This chapter examines how the logic of inclusive fitness theory can be mathematically formalized using the Price equation, and how that formalization can be used to derive Hamilton's rule in its simplest form, as applied to unconditional behaviors having additive effects on fitness. Various biological phenomena, such as sex allocation and working policing within eusocial insect colonies, have been analyzed by considering what strategies maximize individuals' inclusive fitness, and how observed social behaviors should correlate with quantities such as relatedness. The chapter derives Hamilton's rule by introducing some notation for the effects of behaviors on fitnesses of individuals that interact socially, to make explicit precisely how genes (and later phenotypes) affect fitness, and to give a general form of Hamilton's rule that will apply to any (unconditional, additive) behavior regardless of its details. It shows that inclusive fitness is a genuinely novel extension of the classical fitness studied by Charles Darwin, R. A. Fisher, and others.

scholarly journals Using the Price equation to detect inclusive fitness in class-structured populations

2020 ◽  
Author(s):  
António M. M. Rodrigues
Keyword(s):  
Inclusive Fitness ◽  
Causal Structure ◽  
Regression Method ◽  
Structured Populations ◽  
Price Equation ◽  
Hamilton’S Rule ◽  
Inclusive Fitness Theory ◽  
Key Variables ◽  
Hamilton's Rule ◽  
Simple Regression

AbstractInclusive fitness theory has transformed the study of adaptive evolution since 1964, contributing to significant empirical findings. However, its status as a theory has been challenged by the proposals of several alternative frameworks. Those challenges have been countered by analyses that use the Price equation and the regression method. The Price equation is a universal description of evolutionary change, and the partitioning of the Price equation using the regression method immediately yields Hamilton’s rule, which embodies the main tenets of inclusive fitness. Hamilton’s rule captures the intensity and direction of selection acting on social behaviour and its underlying causal structure. Recent work, however, has suggested that there is an anomaly in this approach: in some cases, the regression method fails to estimate the correct values of the variables in Hamilton’s rule and the causal structure of the behaviour. Here, I address this apparent anomaly. I argue that the failure of the simple regression method occurs because social players vary in baseline fecundity. I reformulate the Price equation and regression method to recover Hamilton’s rule and I show that the method correctly estimates its key variables. I show that games where baseline fecundity varies among individuals represent a more general set of games that unfold in class-structured populations. This framework supports the robustness and validity of inclusive fitness.

Nonadditive Interactions and Hamilton’s Rule

2015 ◽  
Author(s):  
James A.R. Marshall
Keyword(s):  
Replicator Dynamics ◽  
Structured Populations ◽  
The Other ◽  
Price Equation ◽  
Costs And Benefits ◽  
Partial Regression ◽  
Selective Pressures ◽  
Hamilton’S Rule ◽  
Hamilton's Rule ◽  
Synergistic Coefficient

This chapter examines what happens in nonadditive interactions when such interactions take place between relatives, and how Hamilton's rule can be extended in two different ways to accommodate such nonadditivity. It first considers the selective pressures on nonadditive behaviors directed towards relatives by making use of the replicator dynamics to capture interactions within structured populations, so that on average, interactions within the population occur between relatives. It then describes two extensions to Hamilton's rule to deal with nonadditive interactions. One approach takes deviations from additivity and accounts for them all in a single synergistic coefficient. The other approach applies partial regression to keep a version of Hamilton's rule with only three parameters, in which costs and benefits vary according to the frequency of social individuals in a population. The chapter also explains the use of the Price equation to study nonadditive social interactions between relatives.

scholarly journals Darwin's ‘one special difficulty’: celebrating Darwin 200

2009 ◽  
Vol 5 (2) ◽  
pp. 214-217 ◽  
Author(s):  
Joan M Herbers
Keyword(s):  
Natural Selection ◽  
Kin Selection ◽  
Inclusive Fitness ◽  
The Past ◽  
Hamilton’S Rule ◽  
Insect Societies ◽  
Hamilton's Rule

Darwin identified eusocial evolution, especially of complex insect societies, as a particular challenge to his theory of natural selection. A century later, Hamilton provided a framework for selection on inclusive fitness. Hamilton's rule is robust and fertile, having generated multiple subdisciplines over the past 45 years. His suggestion that eusociality can be explained via kin selection, however, remains contentious. I review the continuing debate on the role of kin selection in eusocial evolution and suggest some lines of research that should resolve that debate.

scholarly journals The general form of Hamilton’s rule makes no predictions and cannot be tested empirically

2017 ◽  
Vol 114 (22) ◽  
pp. 5665-5670 ◽  
Author(s):  
Martin A. Nowak ◽  
Alex McAvoy ◽  
Benjamin Allen ◽  
Edward O. Wilson
Keyword(s):  
Social Sciences ◽  
Natural Selection ◽  
Natural Law ◽  
Evolutionary Biology ◽  
Hamilton’S Rule ◽  
The Social ◽  
The Cost ◽  
Hamilton's Rule ◽  
Law Making

Hamilton’s rule asserts that a trait is favored by natural selection if the benefit to others, B, multiplied by relatedness, R, exceeds the cost to self, C. Specifically, Hamilton’s rule states that the change in average trait value in a population is proportional to BR−C. This rule is commonly believed to be a natural law making important predictions in biology, and its influence has spread from evolutionary biology to other fields including the social sciences. Whereas many feel that Hamilton’s rule provides valuable intuition, there is disagreement even among experts as to how the quantities B, R, and C should be defined for a given system. Here, we investigate a widely endorsed formulation of Hamilton’s rule, which is said to be as general as natural selection itself. We show that, in this formulation, Hamilton’s rule does not make predictions and cannot be tested empirically. It turns out that the parameters B and C depend on the change in average trait value and therefore cannot predict that change. In this formulation, which has been called “exact and general” by its proponents, Hamilton’s rule can “predict” only the data that have already been given.

scholarly journals The Price equation and the causal analysis of evolutionary change

2020 ◽  
Vol 375 (1797) ◽  
pp. 20190365 ◽  
Author(s):  
Samir Okasha ◽  
Jun Otsuka
Keyword(s):  
Natural Selection ◽  
Causal Model ◽  
Causal Structure ◽  
Causal Analysis ◽  
Evolutionary Change ◽  
Price Equation ◽  
Causal Interpretation ◽  
Theme Issue ◽  
Causal Processes ◽  
The Difference

Though the Price equation in itself is simply a statistical identity, biologists have often adopted a ‘causal interpretation’ of the equation, in the sense that its component terms have been supposed to correspond to distinct causal processes in evolution, such as natural selection and transmission bias. In this paper, we bring the issue of causal interpretation to the fore, by studying the conditions under which it is legitimate to read causal meaning into the Price equation. We argue that only if substantive assumptions about causal structure are made, which can be represented in the form of a causal model, can the component terms of the Price equation be interpreted as causally meaningful. We conclude with a reflection on the epistemic uses of the Price equation, emphasizing the difference between the description, explanation and prediction of evolutionary change. This article is part of the theme issue ‘Fifty years of the Price equation’.

Hamilton’s Rule as an Organizing Framework

2017 ◽  
Author(s):  
Jonathan Birch
Keyword(s):  
Social Evolution ◽  
Natural World ◽  
Breeding Value ◽  
Price Equation ◽  
Causal Explanations ◽  
Hamilton’S Rule ◽  
Indirect Fitness ◽  
The Mean ◽  
The Cost ◽  
Hamilton's Rule

Queller’s version of Hamilton’s rule (HRG), derived from the Price equation, states that the mean breeding value for a social character increases if and only if rb > c, where r is the coefficient of relatedness between social partners, b is the benefit conferred on recipients, and c is the cost incurred by actors. The value of HRG lies in its ability to provide an organizing framework for social evolution theory, helping us to interpret, classify, and compare more detailed models of particular scenarios. HRG does this by allowing us to classify causal explanations of positive change by their commitments regarding the sign of rb and c. This leads to a four-part taxonomy of explanations, comprising indirect fitness explanations, direct fitness explanations, hybrid explanations, and wholly or partially non-selective explanations. There are plausible instances of all four categories in the natural world.

Social Evolution, Hamilton’s Rule, and Inclusive Fitness

2018 ◽  
Author(s):  
Samir Okasha
Keyword(s):  
Social Behaviour ◽  
Social Evolution ◽  
Inclusive Fitness ◽  
Popular Approach ◽  
Hamilton’S Rule ◽  
Inclusive Fitness Theory ◽  
The Social ◽  
Social Encounters ◽  
The Individual ◽  
Hamilton's Rule

Inclusive fitness theory, originally due to W. D. Hamilton, is a popular approach to the study of social evolution, but shrouded in controversy. The theory contains two distinct aspects: Hamilton’s rule (rB > C); and the idea that individuals will behave as if trying to maximize their inclusive fitness in social encounters. These two aspects of the theory are logically separable but often run together. A generalized version of Hamilton’s rule can be formulated that is always true, though whether it is causally meaningful is debatable. However, the individual maximization claim only holds true if the payoffs from the social encounter are additive. The notion that inclusive fitness is the ‘goal’ of individuals’ social behaviour is less robust than some of its advocates acknowledge.

scholarly journals Evolution and evolvability: celebrating Darwin 200

2008 ◽  
Vol 5 (1) ◽  
pp. 44-46 ◽  
Author(s):  
John F.Y Brookfield
Keyword(s):  
Genetic Variation ◽  
Natural Selection ◽  
Time Scales ◽  
Evolutionary Change ◽  
Adaptive Mutation ◽  
Genetic Covariances ◽  
The Relationship ◽  
The Way

The concept of ‘evolvability’ is increasingly coming to dominate considerations of evolutionary change. There are, however, a number of different interpretations that have been put on the idea of evolvability, differing in the time scales over which the concept is applied. For some, evolvability characterizes the potential for future adaptive mutation and evolution. Others use evolvability to capture the nature of genetic variation as it exists in populations, particularly in terms of the genetic covariances between traits. In the latter use of the term, the applicability of the idea of evolvability as a measure of population's capacity to respond to natural selection rests on one, but not the only, view of the way in which we should envisage the process of natural selection. Perhaps the most potentially confusing aspects of the concept of evolvability are seen in the relationship between evolvability and robustness.

Hamilton’s rule and kin competition in a finite kin population

2021 ◽  
pp. 110862
Author(s):  
Qiao-Qiao He ◽  
Xiu-Deng Zheng ◽  
Ruth Mace ◽  
Yi Tao ◽  
Ting Ji
Keyword(s):  
Kin Competition ◽  
Hamilton’S Rule ◽  
Hamilton's Rule
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