### 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 language | English |
---|---|

Pages (from-to) | 1125-1152 |

Number of pages | 28 |

Journal | Journal of Mathematical Biology |

Volume | 72 |

Issue number | 4 |

DOIs | |

Publication status | Published - 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

*Journal of Mathematical Biology*,

*72*(4), 1125-1152. https://doi.org/10.1007/s00285-015-0938-4

}

*Journal of Mathematical Biology*, vol. 72, no. 4, pp. 1125-1152. https://doi.org/10.1007/s00285-015-0938-4

**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.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - The canonical equation of adaptive dynamics for life histories

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

AU - Metz, Johan A Jacob

AU - Staňková, Kateřina

AU - Johansson, Jacob

PY - 2016/3

Y1 - 2016/3

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.

KW - Animals

KW - Ecosystem

KW - Evolution, Molecular

KW - Female

KW - Genetic Fitness

KW - Humans

KW - Male

KW - Mathematical Concepts

KW - Models, Genetic

KW - Population Dynamics

KW - Selection, Genetic

KW - Journal Article

KW - Research Support, Non-U.S. Gov't

KW - Pontryagin's maximum principle

KW - STRUCTURED POPULATION-MODELS

KW - STRATEGIES

KW - Age-dependent resource allocation

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

U2 - 10.1007/s00285-015-0938-4

DO - 10.1007/s00285-015-0938-4

M3 - Article

C2 - 26586121

VL - 72

SP - 1125

EP - 1152

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 4

ER -