Evol Ecol Res 11: 191-208 (2009)     Full PDF if your library subscribes.

Multidimensional convergence stability

Olof Leimar

Department of Zoology, Stockholm University, Stockholm, Sweden

Correspondence: O. Leimar, Department of Zoology, Stockholm University, SE-10691 Stockholm, Sweden.
e-mail: olof.leimar@zoologi.su.se


Questions: Are there general stability conditions for the evolution of multidimensional traits, regardless of genetic correlations between traits? Can genetic correlations influence whether evolution converges to a stable trait vector?

Mathematical methods: Adaptive dynamics theory and the weak selection limit of quantitative genetics.

Key assumptions: Evolutionary change is represented as either (i) any gradualistic adaptive path in trait space, consisting of a sequence of small-effect mutant invasions, allowing for pleiotropic mutants, or (ii) a solution to the ‘canonical equation’ of adaptive dynamics with a gradually varying mutational covariance matrix. Assumption (ii) is a special case of (i).

Conclusions: It is possible to formulate robust stability conditions for multidimensional traits, but most evolutionary equilibria will not satisfy these conditions. Under the liberal assumption (i), there will in general be no ‘absolutely convergence stable’ equilibria in multidimensional trait spaces (except for simplified models). Under the more restrictive assumption (ii), a much larger proportion of evolutionary equilibria is ‘strongly convergence stable’, i.e. are stable irrespective of genetic correlations.

Keywords: adaptive dynamics, canonical equation, evolutionary stability, genetic correlations.

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