The adaptive dynamics of life histories: From fitness-returns to selection gradients and Pontryagin's maximum principle

Metz, J.A.J. & Johansson, J. (2012). The adaptive dynamics of life histories: From fitness-returns to selection gradients and Pontryagin's maximum principle. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-12-054

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Abstract

Using a fitness-returns argument we derive an expression for the selection gradient for the age dependent allocation strategy in a common class of state variable based life-history models. By setting the selection gradient equal to zero as part of the calculation of the ESS-es for such models, we get a marginal value argument and through this recover the local version of Pontryagin's maximum principle. This fills in a minor gap in a recent paper by Parvinen, Heino and Dieckmann (2012; DOI10.1007/s00285-012-0549-2), who treated the calculation of the selection gradient and of the ESS-es as separate issues. As bonuses we (i) provide an extension of the framework of these authors that can handle also the more complicated evolutionary dynamics of the life histories that we consider, and (ii) derive also the full Pontryagin's maximum principle from a fitness-returns argument.

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