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.
authors.affiliation: Mathematical Institute and Institute of Biology, Leiden University, 2333 CA, Leiden, The Netherlands.|Evolution and Ecology Program, International Institute for Applied Systems Analysis, 2361, Laxenburg, Austria.|Department of Marine Zoology, Naturalis Biodiversity Center, 2333 CR, Leiden, The Netherlands.
authors.forename: Johan A Jacob
authors.initials: JAJ
authors.lastname: Metz
authors.affiliation: Department of Knowledge Engineering, Maastricht University, 6211 LH, Maastricht, The Netherlands. k.stankova@maastrichtuniversity.nl.|Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD, Delft, The Netherlands. k.stankova@maastrichtuniversity.nl.
authors.forename: Kateřina
authors.initials: K
authors.lastname: Staňková
authors.affiliation: Evolution and Ecology Program, International Institute for Applied Systems Analysis, 2361, Laxenburg, Austria.|Theoretical Population Ecology and Evolution Group, Department of Biology, Lund University, 22362, Lund, Sweden.
authors.forename: Jacob
authors.initials: J
authors.lastname: Johansson
mesh_terms: D000818:Animals; D017753:Ecosystem; D019143:Evolution, Molecular; D005260:Female; D056084:Genetic Fitness; D006801:Humans; D008297:Male; D055641:Mathematical Concepts; D008957:Models, Genetic; D011157:Population Dynamics; D012641:Selection, Genetic
keywords: Age-dependent resource allocation; Canonical equation of adaptive dynamics; Evolution in periodic environments; Function valued traits; Mendelian take on life history theory; Pontryagin’s maximum principle