Evol Ecol Res 1: 423-441 (1999)     Full PDF if your library subscribes.

Allocation of energy between growth and reproduction: The Pontryagin Maximum Principle solution for the case of age- and season-dependent mortality

Jan Kozłowski1 and Anatoly T. Teriokhin2

1Institute of Environmental Sciences, Jagiellonian University, Oleandry 2A, 30-063 Kraków, Poland and 2Department of Biology, Moscow State University, Moscow 119899, Russia

Author to whom all correspondence should be addressed.
e-mail: kozlo@eko.uj.edu.pl


Optimal allocation of energy to growth and reproduction was studied with a dynamic optimization method for long-lived animals inhabiting a seasonal environment. We found that, after maturation, it is optimal to grow either over several years (if the animals age in the sense that their mortality increases with age) or asymptotically (if there is no ageing). Asymptotic growth can be well approximated with Bertalanffy’s equation, even if the animals are able to grow to infinity. This is because it is optimal to allocate an increasing proportion of surplus energy to reproduction year after year following maturation. Age at first reproduction depends on mortality in both favourable and unfavourable seasons, and also depends on the favourable season length – it is optimal to mature early when mortality is high in either season or the season is short. Size at maturity depends additionally on the parameters of the equation describing the size-dependence of the production rate. When mortality in the unfavourable season is high compared to mortality in the favourable season, a large part of growth is realized after maturation. The model qualitatively explains growth patterns and life histories in fish, reptiles, amphibians and some long-lived invertebrates which grow after maturation. To understand better the diversity of life histories, field biologists working on such animals should focus on differences in the ageing rate, favourable season length, mortality rates in both seasons and the size-dependence of production rate. Optimization models based on the allocation principle are promising tools to integrate these kinds of data.

Keywords: age at maturity, Bertalanffy’s growth curve, life-history evolution, optimal resource allocation, Pontryagin Maximum Principle, seasonality.

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        © 1999 Jan Kozłowski. All EER articles are copyrighted by their authors. All authors endorse, permit and license Evolutionary Ecology Ltd. to grant its subscribing institutions/libraries the copying privileges specified below without additional consideration or payment to them or to Evolutionary Ecology, Ltd. These endorsements, in writing, are on file in the office of Evolutionary Ecology, Ltd. Consult authors for permission to use any portion of their work in derivative works, compilations or to distribute their work in any commercial manner.

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