The canonical equation of adaptive dynamics for life histories: from fitness-returns to selection gradients and Pontryagin's maximum principle

Johan A Jacob Metz, Kateřina Staňková, Jacob Johansson

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)

Abstract

This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013) are chosen such that they avoid a number of the problems that one gets in this most relevant of applications, (2) derive the fitness gradient occurring in the CE from simple fitness return arguments, (3) show explicitly that setting said fitness gradient equal to zero results in the classical marginal value principle from evolutionary ecology, (4) show that the latter in turn is equivalent to Pontryagin's maximum principle, a well known equivalence that however in the literature is given either ex cathedra or is proven with more advanced tools, (5) connect the classical optimisation arguments of life history theory a little better to real biology (Mendelian populations with separate sexes subject to an environmental feedback loop), (6) make a minor improvement to the form of the CE for the examples in Dieckmann et al. and Parvinen et al.

Original languageEnglish
Pages (from-to)1125-1152
Number of pages28
JournalJournal of Mathematical Biology
Volume72
Issue number4
DOIs
Publication statusPublished - Mar 2016

Keywords

  • Animals
  • Ecosystem
  • Evolution, Molecular
  • Female
  • Genetic Fitness
  • Humans
  • Male
  • Mathematical Concepts
  • Models, Genetic
  • Population Dynamics
  • Selection, Genetic
  • Journal Article
  • Research Support, Non-U.S. Gov't
  • Pontryagin's maximum principle
  • STRUCTURED POPULATION-MODELS
  • STRATEGIES
  • Age-dependent resource allocation
  • Canonical equation of adaptive dynamics
  • Evolution in periodic environments
  • EVOLUTION
  • Mendelian take on life history theory
  • Function valued traits
  • TRAITS

Cite this

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title = "The canonical equation of adaptive dynamics for life histories: from fitness-returns to selection gradients and Pontryagin's maximum principle",
abstract = "This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013) are chosen such that they avoid a number of the problems that one gets in this most relevant of applications, (2) derive the fitness gradient occurring in the CE from simple fitness return arguments, (3) show explicitly that setting said fitness gradient equal to zero results in the classical marginal value principle from evolutionary ecology, (4) show that the latter in turn is equivalent to Pontryagin's maximum principle, a well known equivalence that however in the literature is given either ex cathedra or is proven with more advanced tools, (5) connect the classical optimisation arguments of life history theory a little better to real biology (Mendelian populations with separate sexes subject to an environmental feedback loop), (6) make a minor improvement to the form of the CE for the examples in Dieckmann et al. and Parvinen et al.",
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author = "Metz, {Johan A Jacob} and Kateřina Staňkov{\'a} and Jacob Johansson",
year = "2016",
month = "3",
doi = "10.1007/s00285-015-0938-4",
language = "English",
volume = "72",
pages = "1125--1152",
journal = "Journal of Mathematical Biology",
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The canonical equation of adaptive dynamics for life histories : from fitness-returns to selection gradients and Pontryagin's maximum principle. / Metz, Johan A Jacob; Staňková, Kateřina; Johansson, Jacob.

In: Journal of Mathematical Biology, Vol. 72, No. 4, 03.2016, p. 1125-1152.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

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T2 - from fitness-returns to selection gradients and Pontryagin's maximum principle

AU - Metz, Johan A Jacob

AU - Staňková, Kateřina

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N2 - This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013) are chosen such that they avoid a number of the problems that one gets in this most relevant of applications, (2) derive the fitness gradient occurring in the CE from simple fitness return arguments, (3) show explicitly that setting said fitness gradient equal to zero results in the classical marginal value principle from evolutionary ecology, (4) show that the latter in turn is equivalent to Pontryagin's maximum principle, a well known equivalence that however in the literature is given either ex cathedra or is proven with more advanced tools, (5) connect the classical optimisation arguments of life history theory a little better to real biology (Mendelian populations with separate sexes subject to an environmental feedback loop), (6) make a minor improvement to the form of the CE for the examples in Dieckmann et al. and Parvinen et al.

AB - This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013) are chosen such that they avoid a number of the problems that one gets in this most relevant of applications, (2) derive the fitness gradient occurring in the CE from simple fitness return arguments, (3) show explicitly that setting said fitness gradient equal to zero results in the classical marginal value principle from evolutionary ecology, (4) show that the latter in turn is equivalent to Pontryagin's maximum principle, a well known equivalence that however in the literature is given either ex cathedra or is proven with more advanced tools, (5) connect the classical optimisation arguments of life history theory a little better to real biology (Mendelian populations with separate sexes subject to an environmental feedback loop), (6) make a minor improvement to the form of the CE for the examples in Dieckmann et al. and Parvinen et al.

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KW - Ecosystem

KW - Evolution, Molecular

KW - Female

KW - Genetic Fitness

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KW - Male

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KW - Selection, Genetic

KW - Journal Article

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KW - Pontryagin's maximum principle

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KW - STRATEGIES

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KW - Canonical equation of adaptive dynamics

KW - Evolution in periodic environments

KW - EVOLUTION

KW - Mendelian take on life history theory

KW - Function valued traits

KW - TRAITS

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EP - 1152

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

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