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A mutation-selection-recombination model in population genetics. (English) Zbl 1176.92039
Summary: We construct a new continuous time selection-mutation-recombination model for population dynamics, which describes the development of the distribution of the different gametes in the population. We show that cyclic mutation rates can result in stable and unstable limit cycles due to Hopf bifurcations. In addition, we give a qualitative characterization of the whole dynamics in the simplex, which is the phase space of the system. If only selection acts, then Fisher’s Fundamental Law is valid: the mean fitness is a Lyapunov function and every orbit converges to some rest point.
MSC:
92D15Problems related to evolution
92D10Genetics
37G15Bifurcations of limit cycles and periodic orbits
92D25Population dynamics (general)
37N25Dynamical systems in biology