# Notes on: Crewe, P., Gratwick, R., & Grafen, A. (2017): Defining fitness in an uncertain world

## Table of Contents

## Notation

- denotes individual
- the class of individual (this year)
- denotes the
*reproductive value*of the class , the class is assigned to (this year) - is the "Willias reproductive value", which is the sum over its offspring and its own surviving self of its share of reproductive value from these descendants (surviving self devalued by chance of survival):
fitness of an individual :

- (possibly infinite) measure space of classes
- finite set of genotypes
- denotes the class of each member of the population belongs to if the population is of size
- is the union of countable number of measure spaces, and so itself forms a measure space with appropriate sigma-algebra
- denotes the genotype of each member of the population, which belongs to the measure space
Population at

*year*is described by arraysrepresenting the class and genotype of each individual, with typical elements and .

- Environment (measure space) in the
*next*year, according to the measure where is the*current*environment - denotes environment in year
- and indexes sets and when individuals reerred to are in the
*offspring*year - denotes reproductive
*output*of an individual , where is the number of whole offspring of in class with genotype ,*per habloid set of the parent*, weighted by the parent's genetic contribution - is the
*ploidy*of

## Terminology

- Phenotype
- elating to the observable characteristics of an individual resulting from the interaction of its genotype with the environment
- Allele
- a variant form of a gene
- ESS
- Evolutionary Stable Strategies (ESS)
- Ploidy
- the number of sets of chromosomes in a cell, or in the cells of an organism.
- Neutral stochasticity process
- assumes phenotypic uniformity
- Taylor year
- the first year, which is initialized by ergodic distribution of the
*neutral process*, allowing genotype and class to affect phenotypes (after this year, revert to*neutral process*)

## Introduction

- Structured populations iwth stochastic demography, i.e. with continuing fluctuations in class distribution
- Assume populations are
*fitness-maximizing* **Reproductive value**can be viewed as the fraction of long distant gene pol whose ancestral paths pass through an individual or a class of individuals today- Fitness does not measure an absolute contribution to future gene pools (reproductive value does measure an absolute
*expected*contribution), but rather how well an individual performs relative to what is expected for its class.

## Preliminaries

- Basic model of a class-structured population with environemental and class-distribution stochasticity
- "neutral stochastic process" assumes phenotypic uniformity
- "Taylor's swtich stochastic process"
- Takes its initial conditions from ergodic distribution of the eutral process
- Allows genotype and class to affect phenotypes in the first year
- After first year, reverts to the same uniform phenotype as the first process

### Basic model

#### Assumptions

- Finite population in discrete time with overlapping generations
- Each individual belongs to a single class from a (possibly infinite) measure space of classes , and also possesses a genotype from a finite set
- Environment modeled as a Markov process
- environment in the next year will take values in the measure space , according to the measure where is the
*current*environment

- environment in the next year will take values in the measure space , according to the measure where is the
- Reproductive
*output*is a set of random variable, whose distributions depends on on a probability space - To understand evolution of class and genotype we need to understand the joint distribution of the offspring functions as the population and environemt fluctuate
- Probability that the value of implies a zero-size population in the next is year is zero
- Entries of are uniformly bounded
Require that the term "phenotype" is being used in line with standard usage

- Restrictions on are:
- Possbility of geograhical, group or family structure is elimanted by the permutation assumption, except where that structure can be created by classes alone
- Prevents extinction from occurring
- Insists that all effects of genotype, except on the actual genotypes of the offspring, must act through phenotype

In addition to for the class, and for the genotype, of individual , we will write the phenotype for one individual and for the whole population as

We suppose that the reproduction of the population between each year is enapsulated by a single random variable , the **population reproduction map**, on the probability space . that takes an environment and class, genotype and phenotype arrays , , and and returns an array for the number of offspring-equivalents (per parental haploid set) of individual in class and with genotype . Our notational conventions already imply that a full notation for would be

which establishes the joint distribution of in terms of our single random function .