Journal of Mathematical Biology

, Volume 73, Issue 4, pp 903–917 | Cite as

On fitness in metapopulations that are both size- and stage-structured

  • Kalle ParvinenEmail author
  • Anne Seppänen


A proxy for the invasion fitness in structured metapopulation models has been defined as a metapopulation reproduction ratio, which is the expected number of surviving dispersers produced by a mutant immigrant and a colony of its descendants. When a size-structured metapopulation model involves also individual stages (such as juveniles and adults), there exists a generalized definition for the invasion fitness proxy. The idea is to calculate the expected numbers of dispersers of all different possible types produced by a mutant clan initiated with a single mutant, and to collect these values into a matrix. The metapopulation reproduction ratio is then the dominant eigenvalue of this matrix. The calculation method has been published in detail in the case of small local populations. However, in case of large patches the previously published numerical calculation method to obtain the expected number of dispersers does not generalize as such, which gives us one aim of this article. Here, we thus derive a generalized method to calculate the invasion fitness in a metapopulation, which consists of large local populations, and is both size- and stage-structured. We also prove that the metapopulation reproduction ratio is well-defined, i.e., it is equal to 1 for a mutant with a strategy equal to the strategy of a resident. Such a proof has not been previously published even for the case with only one type of individuals.


Adaptive dynamics Invasion fitness Structured metapopulation Numerical method 

Mathematics Subject Classification

92D15 37N25 



The authors wish to thank Avidan Neumann for valuable discussions on viral evolution which brought up the need for the methods developed in this article. This study was funded by the Academy of Finland, project number 128323 to K.P.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

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