Typical trajectories (10 independent runs) for a population size of N = 50 for mutants with a twofold fitness advantage, r = 2. In the well-mixed population, the mutant goes to fixation 50% of the time. This is identical for the ring-structured population, but it rises to 75% in the star. The time to fixation, on the other hand, is substantially longer for structured populations: for the star, it takes approximately N times as long for either fixation or extinction (filled squares represent the times where a trajectory has reached either extinction or fixation of the mutants, filled circles show the points where new mutants arise).
The probability (top) and the time (bottom) to fixation of a single mutant in different population structures depending on the fitness of the mutant (left) and the population size (right). Lines represent analytical results, symbols are computer simulations. The star increases the fixation probability (top) and also the fixation time (bottom) compared with the well-mixed population. The one-dimensional lattice (or ring) and the two-dimensional lattice lead to the same fixation probabilities as the well-mixed population, but they both increase the fixation time.
(a) The probability that the process starting from a single mutant ever reaches a certain number of mutants (state). Symbols are simulations, lines represent analytical results. For the star, different approximations are used for K ≤ 3 and K ≥ 3. On a star, there will be a second mutant with a very high probability, but the probability to reach three mutants drops substantially. (b) Average unconditional sojourn in a state with a certain number of mutants, i.e. the average time spend in this state before the mutant is either lost or takes over (r = 1.1, N = 50, simulation averages over 100 000 realizations).