A symmetry of fixation times in evoultionary dynamics

Abstract

In this paper, we show that for evolutionary dynamics between two types that can be described by a Moran process, the conditional fixation time of either type is the same irrespective of the selective scenario. With frequency dependent selection between two strategies A and B of an evolutionary game, regardless of whether A dominates B, A and B are best replies to themselves, or A and B are best replies to each other, the conditional fixation times of a single A and a single B mutant are identical. This does not hold for Wright-Fisher models, nor when the mutants start from multiple copies.

Figure 1

Conditional mean fixation time of a single mutant of strategy A and B as a function of selection strenght w for different payoff matrices. N = 5. Top figure: Moran process; the conditional mean fixation times of a single A mutant is the same as that of a single B mutant for all w. Bottom figure: Wright-Fisher process; the conditional mean fixation time of a single A mutant (solid line) and that of a single B mutant (dashed line) are the same for small w and different for large w.

Figure 2

Conditional mean fixation time of 2 mutants of strategy A and B as a function of selection strenght w for different payoff matrices. N = 5. Top figure: Moran process; the conditional mean fixation times of 2 A mutants (solid line) and that of 2 B mutants (dashed line) are different for all w ≠ 0. Bottom figure: Wright-Fisher process; the conditional mean fixation time of 2 A mutants (solid line) and that of 2 B mutants (dashed line) are different, however they are very close for small w.