Skip to main content

Iteration of the Cycle II: Extension to the General Multilevel Hierarchy

  • 903 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2092)

Abstract

In the previous section we have followed phases 0–4 for up to 3 fitness levels and 2 spatial levels, i.e. we treated the transition \(E_{0} \rightarrow E_{1}\) and then \(E_{1} \rightarrow E_{2}\). The latter case of 3 fitness levels leading to fixation and neutral equilibrium on E 2 includes all the complexity of higher level systems, namely, the necessity to deal treating the McKean–Vlasov dynamic on types in E 2 the perturbations arising with the McKean–Vlasov dynamics at the next lower lower level in smaller space-time scales. The program can otherwise then be continued in essentially the same way to describe the emergence, fixation and neutral equilibrium at levels \(E_{3},E_{4},E_{5},\ldots\) in finite spatial systems with finitely many fitness levels. In the next subsection we briefly indicate how the general transition \(E_{j-1} \rightarrow E_{j}\) is carried out. In the second subsection we then explain how these transitions for the finite (spatial and number of fitness levels) systems is used to obtain the transitions for the original infinite systems.

Keywords

  • Fitness Level
  • Rare Mutation
  • Finite System
  • Founding Father
  • Spatial Level

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. D.A. Dawson, A. Greven, Hierarchically interacting Fleming-Viot processes with selection and mutation: Multiple space time scale analysis and quasi equilibria. Electron. J. Probab. 4, paper no. 4, 1–81 (1999)

    Google Scholar 

  2. D.A. Dawson, A. Greven, J. Vaillancourt, Equilibria and Quasi-equilibria for Infinite Collections of Interacting Fleming-Viot processes. Trans. Am. Math. Soc. 347(7), 2277–2360 (1995)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Dawson, D.A., Greven, A. (2014). Iteration of the Cycle II: Extension to the General Multilevel Hierarchy. In: Spatial Fleming-Viot Models with Selection and Mutation. Lecture Notes in Mathematics, vol 2092. Springer, Cham. https://doi.org/10.1007/978-3-319-02153-9_14

Download citation