Neimark-Sacker bifurcation in a discrete dynamical model of population genetics

A. Dénes, Bolyai Institute, University of Szeged, Hungary

E. J. Qualitative Theory of Diff. Equ., Proc. 8'th Coll. Qualitative Theory of Diff. Equ., No. 6. (2008), pp. 1-10.

Abstract: In this paper we investigate a genetic model initiated by G\'abor Tusn\'ady. This model is a four-dimensional system of difference equations which describes the change of distribution of genotypes from generation to generation in the case of one locus and four alleles considering selection and mutation. During computer experiments Tusn\'ady discovered cyclic behaviour in the evolution of genotypes. Using bifurcation theory we prove that the system indeed can have cycles and this occurrence is caused by a Neimark-Sacker bifurcation.

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