Journal of Mathematical Biology

, Volume 72, Issue 4, pp 1081–1099 | Cite as

Mutual invadability near evolutionarily singular strategies for multivariate traits, with special reference to the strongly convergence stable case

  • Stefan A. H. Geritz
  • Johan A. J. Metz
  • Claus RuefflerEmail author


Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces is sadly lacking. This note aims at getting a research program of the ground leading to such an understanding. In particular, we show that, as long as the evolutionary trajectory stays within the reign of the local quadratic approximation of the fitness function, any initial small scale polymorphism around an attracting invadable evolutionarily singular strategy (ess) will evolve towards a dimorphism. That is, provided the trajectory does not pass the boundary of the domain of dimorphic coexistence and falls back to monomorphism (after which it moves again towards the singular strategy and from there on to a small scale polymorphism, etc.). To reach these results we analyze in some detail the behavior of the solutions of the coupled Lande-equations purportedly satisfied by the phenotypic clusters of a quasi-n-morphism, and give a precise characterisation of the local geometry of the set \(\mathcal D\) in trait space squared harbouring protected dimorphisms. Intriguingly, in higher dimensional trait spaces an attracting invadable ess needs not connect to \(\mathcal D\). However, for the practically important subset of strongly attracting ess-es (i.e., ess-es that robustly locally attract the monomorphic evolutionary dynamics for all possible non-degenerate mutational or genetic covariance matrices) invadability implies that the ess does connect to \(\mathcal D\), just as in 1-dimensional trait spaces. Another matter is that in principle there exists the possibility that the dimorphic evolutionary trajectory reverts to monomorphism still within the reign of the local quadratic approximation for the invasion fitnesses. Such locally unsustainable branching cannot occur in 1- and 2-dimensional trait spaces, but can do so in higher dimensional ones. For the latter trait spaces we give a condition excluding locally unsustainable branching which is far stricter than the one of strong convergence, yet holds good for a relevant collection of published models. It remains an open problem whether locally unsustainable branching can occur around general strongly attracting invadable ess-es.


Adaptive dynamics Evolutionary branching Multi-dimensional trait space Mutual invadability Strong attractivity Local dimorphic divergence 

Mathematics Subject Classification

92D15 92D25 



The authors gratefully acknowledge the help of Mattias Siljestam with the numerics (including a number of stochastic simulations the results of which did not end up in the paper, but which were quite informative as they eventually suggested that locally unsustainable branching cannot occur in two-dimensional trait spaces). This work benefitted from the support from the “Chair Modélisation Mathématique et Biodiversité of Veolia Environnement-Ecole Polytechnique-Museum National d’Histoire Naturelle-Fondation X”.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Stefan A. H. Geritz
    • 1
  • Johan A. J. Metz
    • 2
    • 3
    • 4
  • Claus Rueffler
    • 5
    Email author
  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  2. 2.Mathematical Institute and Institute of BiologyLeiden UniversityLeidenThe Netherlands
  3. 3.Netherlands Centre for BiodiversityNaturalisLeidenThe Netherlands
  4. 4.Evolution and Ecology ProgramInternational Institute of Applied Systems AnalysisLaxenburgAustria
  5. 5.Animal Ecology, Department of Ecology and GeneticsUppsala UniversityUppsalaSweden

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