Journal of Mathematical Biology

, Volume 66, Issue 1–2, pp 225–279 | Cite as

What life cycle graphs can tell about the evolution of life histories

  • Claus RuefflerEmail author
  • Johan A. J. Metz
  • Tom J. M. Van Dooren


We analyze long-term evolutionary dynamics in a large class of life history models. The model family is characterized by discrete-time population dynamics and a finite number of individual states such that the life cycle can be described in terms of a population projection matrix. We allow an arbitrary number of demographic parameters to be subject to density-dependent population regulation and two or more demographic parameters to be subject to evolutionary change. Our aim is to identify structural features of life cycles and modes of population regulation that correspond to specific evolutionary dynamics. Our derivations are based on a fitness proxy that is an algebraically simple function of loops within the life cycle. This allows us to phrase the results in terms of properties of such loops which are readily interpreted biologically. The following results could be obtained. First, we give sufficient conditions for the existence of optimisation principles in models with an arbitrary number of evolving traits. These models are then classified with respect to their appropriate optimisation principle. Second, under the assumption of just two evolving traits we identify structural features of the life cycle that determine whether equilibria of the monomorphic adaptive dynamics (evolutionarily singular points) correspond to fitness minima or maxima. Third, for one class of frequency-dependent models, where optimisation is not possible, we present sufficient conditions that allow classifying singular points in terms of the curvature of the trade-off curve. Throughout the article we illustrate the utility of our framework with a variety of examples.


Adaptive dynamics Density dependence Frequency dependence Life history theory Matrix model Evolutionary optimisation 

Mathematics Subject Classification (2000)



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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Claus Rueffler
    • 1
    Email author
  • Johan A. J. Metz
    • 2
    • 3
    • 4
  • Tom J. M. Van Dooren
    • 4
    • 5
  1. 1.Mathematics and Biosciences Group, Department of MathematicsUniversity of ViennaViennaAustria
  2. 2.Mathematical Institute and Institute of BiologyLeiden UniversityLeidenThe Netherlands
  3. 3.Evolution and Ecology ProgramInternational Institute of Applied Systems AnalysisLaxenburgAustria
  4. 4.Netherlands Centre for Biodiversity, NaturalisLeidenThe Netherlands
  5. 5.UMR 7625 Ecology and Evolution, Eco-Evolutionary Mathematics, Ecole Normale SupérieureParis Cedex 05France

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