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

Table of Contents

Notation

  • crewe17_defin_fitnes_uncer_world_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png denotes individual
  • crewe17_defin_fitnes_uncer_world_e9f4d218474bc10aa94958ec30139aee865c0173.png the class of individual crewe17_defin_fitnes_uncer_world_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png (this year)
  • crewe17_defin_fitnes_uncer_world_31d78e3dc24ad8fd3cc1059bd6a51f0efa2d5a32.png denotes the reproductive value of the class crewe17_defin_fitnes_uncer_world_e9f4d218474bc10aa94958ec30139aee865c0173.png, the class crewe17_defin_fitnes_uncer_world_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png is assigned to (this year)
  • crewe17_defin_fitnes_uncer_world_ab46261adcaca18ea00f0604f3ca4aa48b4feb58.png 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):
  • crewe17_defin_fitnes_uncer_world_e3c2acd8fda16fe4a737b744feab5cbfe8ecaf48.png fitness of an individual crewe17_defin_fitnes_uncer_world_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png:

    crewe17_defin_fitnes_uncer_world_a990c709cd3c6224e9819aff31549746bd1da0f5.png

  • crewe17_defin_fitnes_uncer_world_0207be880056b9a69e22e729dd37bced29cd174a.png (possibly infinite) measure space of classes
  • crewe17_defin_fitnes_uncer_world_968aae4e9a48f70992d7a484a93d5dda62f8b266.png finite set of genotypes
  • crewe17_defin_fitnes_uncer_world_ed39d9a397196f8f0ce6388b0ea4e0c1dd8becee.png denotes the class of each member of the population belongs to crewe17_defin_fitnes_uncer_world_8588ee7f792521e7c0b61148f08ebb89b205c995.png if the population is of size crewe17_defin_fitnes_uncer_world_094b02afce734f4ce51933d0093ef3d2da9f8123.png
  • crewe17_defin_fitnes_uncer_world_4e6445b713672ad576b57903394f79d1d559739f.png is the union of countable number of measure spaces, and so itself forms a measure space with appropriate sigma-algebra
  • crewe17_defin_fitnes_uncer_world_de5529d5a48369da6cb8aa79853c98453e81ddfc.png denotes the genotype of each member of the population, which belongs to the measure space crewe17_defin_fitnes_uncer_world_25b587d51bd3a148dae320d4ec29e86932c6660b.png
  • Population at year crewe17_defin_fitnes_uncer_world_eb5d809ed7c492fae7d4927a6fc9a5e22f9b3831.png is described by arrays

    crewe17_defin_fitnes_uncer_world_7682d18772b5a863b8c8bbd8f782b105a27c64c5.png

    representing the class and genotype of each individual, with typical elements crewe17_defin_fitnes_uncer_world_90579f152672127cff797cbd95e8a5778a2e93b6.png and crewe17_defin_fitnes_uncer_world_7ef5369cf9050b63bdcf1a8bc7f918b25d2cd128.png.

  • Environment crewe17_defin_fitnes_uncer_world_b897eeb8fb0a355796e406854d6a6df63f7fd82e.png (measure space) in the next year, according to the measure crewe17_defin_fitnes_uncer_world_7e2fb1e7336af2a0d933bd85846eae177789e5b2.png where crewe17_defin_fitnes_uncer_world_eaac7646cb022aa963b396c0e936a271d8f5e34f.png is the current environment
  • crewe17_defin_fitnes_uncer_world_5afc1df35dd9c66f2ccc62dab3b3ba45b0fdbce4.png denotes environment in year crewe17_defin_fitnes_uncer_world_eb5d809ed7c492fae7d4927a6fc9a5e22f9b3831.png
  • crewe17_defin_fitnes_uncer_world_6e2ff9643a68a33815b244e99fc81ba6facb32aa.png and crewe17_defin_fitnes_uncer_world_5970a08ca4e9daf54cc95ab35e2a8fec8f4beb78.png indexes sets crewe17_defin_fitnes_uncer_world_0207be880056b9a69e22e729dd37bced29cd174a.png and crewe17_defin_fitnes_uncer_world_968aae4e9a48f70992d7a484a93d5dda62f8b266.png when individuals reerred to are in the offspring year
  • crewe17_defin_fitnes_uncer_world_3b5cae0fcd9240a12d412906021e5c72a9b60550.png denotes reproductive output of an individual crewe17_defin_fitnes_uncer_world_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png, where crewe17_defin_fitnes_uncer_world_35025b9001a8d2a9c3d5059d93789e0522dcae09.png is the number of whole offspring of crewe17_defin_fitnes_uncer_world_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png in class crewe17_defin_fitnes_uncer_world_6e2ff9643a68a33815b244e99fc81ba6facb32aa.png with genotype crewe17_defin_fitnes_uncer_world_5970a08ca4e9daf54cc95ab35e2a8fec8f4beb78.png, per habloid set of the parent, weighted by the parent's genetic contribution
  • crewe17_defin_fitnes_uncer_world_23796db635a31d9c3fe202779289c1ad33e299ad.png is the ploidy of crewe17_defin_fitnes_uncer_world_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png

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 crewe17_defin_fitnes_uncer_world_0207be880056b9a69e22e729dd37bced29cd174a.png, and also possesses a genotype from a finite set crewe17_defin_fitnes_uncer_world_968aae4e9a48f70992d7a484a93d5dda62f8b266.png
  • Environment modeled as a Markov process
    • environment crewe17_defin_fitnes_uncer_world_78125f57e5239da97499020997238375c59b154a.png in the next year will take values in the measure space crewe17_defin_fitnes_uncer_world_15291eee4cb1177a9c3c351646ab7084a3558131.png, according to the measure crewe17_defin_fitnes_uncer_world_7e2fb1e7336af2a0d933bd85846eae177789e5b2.png where crewe17_defin_fitnes_uncer_world_eaac7646cb022aa963b396c0e936a271d8f5e34f.png is the current environment
  • Reproductive output crewe17_defin_fitnes_uncer_world_83bcd3069ffa513feddf14b92192d8d0b2f5e285.png is a set of random variable, whose distributions depends on crewe17_defin_fitnes_uncer_world_d76ba657fc0c8c889f7a06450543776c50a209c3.png on a probability space crewe17_defin_fitnes_uncer_world_4dcdbc57e21a4143bfe27bedc7fc956fe69279a3.png
  • To understand evolution of class and genotype crewe17_defin_fitnes_uncer_world_df9ab4bdb40f2d6489da496f7c01a10635560167.png we need to understand the joint distribution of the offspring functions as the population and environemt fluctuate
  • Probability that the value of crewe17_defin_fitnes_uncer_world_4016a81e52b1f12c3e1d907335d404f43d7cf519.png implies a zero-size population in the next is year is zero
  • Entries of crewe17_defin_fitnes_uncer_world_4016a81e52b1f12c3e1d907335d404f43d7cf519.png are uniformly bounded
  • Require that the term "phenotype" is being used in line with standard usage

    crewe17_defin_fitnes_uncer_world_6202f1476d549c54a63336ec8eea19403c79b31a.png

  • Restrictions on crewe17_defin_fitnes_uncer_world_4016a81e52b1f12c3e1d907335d404f43d7cf519.png 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 crewe17_defin_fitnes_uncer_world_e9f4d218474bc10aa94958ec30139aee865c0173.png for the class, and crewe17_defin_fitnes_uncer_world_5b75907baddc1f20479fae69727ff4819121a7ed.png for the genotype, of individual crewe17_defin_fitnes_uncer_world_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png, we will write the phenotype crewe17_defin_fitnes_uncer_world_8d44cd5b3aaa1d39a7bc0113d3715143f965f94b.png for one individual and for the whole population as

crewe17_defin_fitnes_uncer_world_d7c0f5154e380855300ad983b2812567ed6b6cdd.png

We suppose that the reproduction of the population between each year is enapsulated by a single random variable crewe17_defin_fitnes_uncer_world_4016a81e52b1f12c3e1d907335d404f43d7cf519.png, the population reproduction map, on the probability space crewe17_defin_fitnes_uncer_world_4dcdbc57e21a4143bfe27bedc7fc956fe69279a3.png. that takes an environment crewe17_defin_fitnes_uncer_world_eaac7646cb022aa963b396c0e936a271d8f5e34f.png and class, genotype and phenotype arrays crewe17_defin_fitnes_uncer_world_ed39d9a397196f8f0ce6388b0ea4e0c1dd8becee.png, crewe17_defin_fitnes_uncer_world_de5529d5a48369da6cb8aa79853c98453e81ddfc.png, and crewe17_defin_fitnes_uncer_world_9a580382f022b555ca2ca1a422438494b77efc53.png and returns an crewe17_defin_fitnes_uncer_world_2a16e094e679ff6a0eab12ef8104f639a17021fe.png array for the number of offspring-equivalents (per parental haploid set) of individual crewe17_defin_fitnes_uncer_world_97eb714dfbd8abb06c6ee1fb2cb049cdaa7defd1.png in class crewe17_defin_fitnes_uncer_world_6e2ff9643a68a33815b244e99fc81ba6facb32aa.png and with genotype crewe17_defin_fitnes_uncer_world_5970a08ca4e9daf54cc95ab35e2a8fec8f4beb78.png. Our notational conventions already imply that a full notation for crewe17_defin_fitnes_uncer_world_b5c83d93d47000416160b06ed52d0fb50c1165ee.png would be

crewe17_defin_fitnes_uncer_world_c85e5e147a31a203fbf7334f6f7f68832cfc5717.png

which establishes the joint distribution of crewe17_defin_fitnes_uncer_world_b5c83d93d47000416160b06ed52d0fb50c1165ee.png in terms of our single random function crewe17_defin_fitnes_uncer_world_4016a81e52b1f12c3e1d907335d404f43d7cf519.png.

Neutral Stochastic Process