Journal of Mathematical Biology

, Volume 52, Issue 1, pp 1–26 | Cite as

Function-valued adaptive dynamics and the calculus of variations

  • Kalle ParvinenEmail author
  • Ulf Dieckmann
  • Mikko Heino


Adaptive dynamics has been widely used to study the evolution of scalar-valued, and occasionally vector-valued, strategies in ecologically realistic models. In many ecological situations, however, evolving strategies are best described as function-valued, and thus infinite-dimensional, traits. So far, such evolution has only been studied sporadically, mostly based on quantitative genetics models with limited ecological realism. In this article we show how to apply the calculus of variations to find evolutionarily singular strategies of function-valued adaptive dynamics: such a strategy has to satisfy Euler's equation with environmental feedback. We also demonstrate how second-order derivatives can be used to investigate whether or not a function-valued singular strategy is evolutionarily stable. We illustrate our approach by presenting several worked examples.

Mathematics Subject Classification (2000)

92D15 49-00 

Keywords or phrases

Adaptive dynamics Infinite-dimensional traits Reaction norms Calculus of variations Euler's equation 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Mathematics20014 University of TurkuFinland
  2. 2.Adaptive Dynamics NetworkInternational Institute for Applied Systems AnalysisLaxenburgAustria
  3. 3.Institute of Marine ResearchNordnesNorway

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