Journal of Mathematical Biology

, Volume 63, Issue 4, pp 601–635

Oligomorphic dynamics for analyzing the quantitative genetics of adaptive speciation

Article

DOI: 10.1007/s00285-010-0380-6

Cite this article as:
Sasaki, A. & Dieckmann, U. J. Math. Biol. (2011) 63: 601. doi:10.1007/s00285-010-0380-6

Abstract

Ecological interaction, including competition for resources, often causes frequency-dependent disruptive selection, which, when accompanied by reproductive isolation, may act as driving forces of adaptive speciation. While adaptive dynamics models have added new perspectives to our understanding of the ecological dimensions of speciation processes, it remains an open question how best to incorporate and analyze genetic detail in such models. Conventional approaches, based on quantitative genetics theory, typically assume a unimodal character distribution and examine how its moments change over time. Such approximations inevitably fail when a character distribution becomes multimodal. Here, we propose a new approximation, oligomorphic dynamics, to the quantitative genetics of populations that include several morphs and that therefore exhibit multiple peaks in their character distribution. To this end, we first decompose the character distribution into a sum of unimodal distributions corresponding to individual morphs. Characterizing these morphs by their frequency (fraction of individuals belonging to each morph), position (mean character of each morph), and width (standard deviation of each morph), we then derive the coupled eco-evolutionary dynamics of morphs through a double Taylor expansion. We show that the demographic, convergence, and evolutionary stability of a population’s character distribution correspond, respectively, to the asymptotic stability of frequencies, positions, and widths under the oligomorphic dynamics introduced here. As first applications of oligomorphic dynamics theory, we analytically derive the effects (a) of the strength of disruptive selection on waiting times until speciation, (b) of mutation on conditions for speciation, and (c) of the fourth moments of competition kernels on patterns of speciation.

Keywords

Adaptive dynamics Quantitative genetics theory Moment dynamics Adaptive speciation Evolutionarily stable strategy Convergence stability 

Mathematics Subject Classification (2000)

92D10 92D15 92D40 

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Evolutionary Studies of BiosystemsThe Graduate University for Advanced Studies (Sokendai)KanagawaJapan
  2. 2.Evolution and Ecology ProgramInternational Institute for Applied Systems AnalysisLaxenburgAustria
  3. 3.PRESTO, Japan Science and Technology AgencyKawaguchi, SaitamaJapan