Effect of the fixed cost per individual on the fixation probability under the weak selection.

Abstract

<p>Four heterogeneous graphs, SW-I (the Small-World graph generated according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560-Watts2" target="_blank">[27]</a> with rewiring probability 0.1), SW-II (the Small-World graph generated according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560-Watts2" target="_blank">[27]</a> with rewiring probability 1), RD (random graph generated according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560-Nobari1" target="_blank">[62]</a>) and SF (scale-free graph generated according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560-Barabsi1" target="_blank">[28]</a>) are used to test the theoretical predictions. Simulation results for the fixation probability of a single cooperator with neighbors in SW-I, SW-II and RD are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g004" target="_blank">Figure 4A</a>. For , the harmonic means of the degree distribution are , , in SW-I, , , in SW-II, and , , in RD, respectively. The vertical dash line represents the harmonic mean of degree distribution, , for each of , where the red, green and blue vertical dash lines correspond to SW-I, SW-II and RD, respectively. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g004" target="_blank">Figure 4B</a> shows the simulation results for the fixation probability of a single cooperator with neighbors in SF. For , the harmonic means of the degree distribution are , , , respectively. The vertical dash line represents the harmonic mean of degree distribution, , where the blue, green and red vertical dash lines correspond to , respectively. In both <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g004" target="_blank">Figures 4A and 4B</a>, the -axis denotes the benefit-to-cost ratio, , the -axis the fixation probability, and the horizontal dash-point line denotes the fixation probability of a single cooperator under neutral selection (i.e. ) which is . The fixation probability of a single cooperator is measured using the fraction of runs where cooperators reached fixation out of runs (based on graphs and runs per graph). Both <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g004" target="_blank">Figures 4A and 4B</a> show that the theoretical predictions present a good approximation to the numerical results.</p