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The genealogical decomposition of a matrix population model with applications to the aggregation of stages

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  • Bienvenu, François
  • Akçay, Erol
  • Legendre, Stéphane
  • McCandlish, David M.

Abstract

Matrix projection models are a central tool in many areas of population biology. In most applications, one starts from the projection matrix to quantify the asymptotic growth rate of the population (the dominant eigenvalue), the stable stage distribution, and the reproductive values (the dominant right and left eigenvectors, respectively). Any primitive projection matrix also has an associated ergodic Markov chain that contains information about the genealogy of the population. In this paper, we show that these facts can be used to specify any matrix population model as a triple consisting of the ergodic Markov matrix, the dominant eigenvalue and one of the corresponding eigenvectors. This decomposition of the projection matrix separates properties associated with lineages from those associated with individuals. It also clarifies the relationships between many quantities commonly used to describe such models, including the relationship between eigenvalue sensitivities and elasticities. We illustrate the utility of such a decomposition by introducing a new method for aggregating classes in a matrix population model to produce a simpler model with a smaller number of classes. Unlike the standard method, our method has the advantage of preserving reproductive values and elasticities. It also has conceptually satisfying properties such as commuting with changes of units.

Suggested Citation

  • Bienvenu, François & Akçay, Erol & Legendre, Stéphane & McCandlish, David M., 2017. "The genealogical decomposition of a matrix population model with applications to the aggregation of stages," Theoretical Population Biology, Elsevier, vol. 115(C), pages 69-80.
    • Handle: RePEc:eee:thpobi:v:115:y:2017:i:c:p:69-80
    DOI: 10.1016/j.tpb.2017.04.002
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    References listed on IDEAS

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    1. Benjamin Allen & Christine Sample & Yulia Dementieva & Ruben C Medeiros & Christopher Paoletti & Martin A Nowak, 2015. "The Molecular Clock of Neutral Evolution Can Be Accelerated or Slowed by Asymmetric Spatial Structure," PLOS Computational Biology, Public Library of Science, vol. 11(2), pages 1-32, February.
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    Cited by:

    1. Giaimo, Stefano, 2022. "Selection on age-specific survival: Constant versus fluctuating environment," Theoretical Population Biology, Elsevier, vol. 145(C), pages 136-149.
    2. Smith, Phoebe & Guiver, Chris & Adams, Ben, 2022. "Quantifying the per-capita contribution of all components of a migratory cycle: A modelling framework," Ecological Modelling, Elsevier, vol. 471(C).
    3. Coste, Christophe F.D. & Austerlitz, Frédéric & Pavard, Samuel, 2017. "Trait level analysis of multitrait population projection matrices," Theoretical Population Biology, Elsevier, vol. 116(C), pages 47-58.

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