Figure 1.
There is one narrow peak of high fitness (peak 0), and one broader, flatter peak of lower fitness (peak 1).
Figure 2.
Verification of the method against analytical models for the error theshold.
Nowak and Schuster [23] present an analytical expression for the population size dependence of the error threshold (Equation 3). Ochoa et al. [50], [51] include a reformulation of the Nowak and Schuster analytical expression (Equation 4), in which they make explicit the reduction in the critical mutation rate when moving from infinite populations to those of size (see [50] section 3 for the detailed derivation). The observed consistency between our results and the analytical models provides verification for our results and the algorithmic method as a whole. It should be noted that the
axis represents the mutation rate by which 95% of runs have lost the lower, flatter peak (peak 1).
Figure 3.
The results of the simulation can be approximated by an exponential function.
This applies to both peak 0 (high, narrow peak) and peak 1 (lower, flatter peak). (with
being population size). The parameters (and their standard error in brackets) obtained by curve-fitting using a least squares method were, for the high, narrow peak (peak 0):
= 1.221% (0.0033%),
= 7.001% (1.4390%),
= 1.440 (0.1701),
= 0.3250 (0.02739), and for the lower, flatter peak (peak 1):
= 2.184% (0.0122%),
= 5.438% (1.0466%),
= 7.721 (0.2734),
= 0.3978 (0.0476).
Figure 4.
Transition from survival-of-the-fittest to survival-of-the-flattest and subsequently to the error catastrophe.
Each point represents the number of generations it took to lose the high, narrow peak (peak 0) and the number to lose the lower, flatter peak (peak 1), in a single run of the GA for population size 100, sequence length 30. Where a peak was not lost within 10,000 generations, a value of −1 was assigned for that particular run of the genetic algorithm: all points on the negative side of either axis should be taken to have a value greater than 10,000.
Figure 5.
The relationship between population size and critical mutation rate is consistent across haploids and diploids.
Here is the dominance parameter, as described in the section entitled Fitness Calculation. The simulation was run using the
values listed. The points show the results obtained, which can be approximated by exponential functions as shown by the lines (obtained by curve-fitting using a least squares method). The left graph shows the curve obtained for the critical mutation rate and the right graph shows the error threshold, both for a diploid population. Refer to Figure 3 for the equivalent curves for a haploid population.
Figure 6.
Percentage of runs losing the peaks at different mutation rates and population sizes.
The results shown are for the diploid method with , for peak 0 (a, left) and peak 1 (b, right). In the two lower projections the axis coming out of the page is the percentage of runs. The lower dashed line across these projections indicates, for population sizes of several hundred individuals, approximately where the percentage loss of peak 0 begins to rise steeply and that of peak 1 begins to fall steeply as mutation rate is increased: the transition from survival-of-the-fittest to survival-of-the-flattest. Likewise, the upper dashed line indicates approximately where the percentage loss of peak 0 has reached 100% and that of peak 1 has reached its minimum before rising back upward as mutation rate is increased further: the transition from survival-of-the-flattest to the error catastrophe.