Mean level of altruism during the 500 generations of selection.

<p>Lines represent the mean level of altruism, defined as the proportion of food items shared with other group members, for each of the 25 treatments (five relatedness values × five <i>c/b</i> ratios, 20 replicates per treatment), with yellow dashed lines representing treatments where <i>r  =  c/b</i>, red lines treatments where <i>r</i> < <i>c/b</i>, and blue dotted lines treatments where <i>r</i> > <i>c/b</i>.</p

Mean group foraging efficiency during the 500 generations of selection.

<p>Lines represent the mean number of items transported successfully for each of the 25 treatments (five relatedness values × five <i>c/b</i> ratios, 20 replicates per treatment), with yellow dashed lines representing treatments where <i>r  =  c/b</i>, red lines treatments where <i>r</i> < <i>c/b</i>, and blue dotted lines treatments where <i>r</i> > <i>c/b</i>.</p

Mean level of altruism at the end of the 500 generations of selection.

<p>The mean (±SD) level of altruism as a function of the <i>c/b</i> ratio and level of relatedness (20 replicates per treatment) for <i>r  =  c/b</i> (yellow lines), <i>r</i> < <i>c/b</i> (red lines), and <i>r</i> > <i>c/b</i> (blue lines). Dashed lines are fits based on a model by McNamara et al. <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1000615#pbio.1000615-McNamara1" target="_blank">[44]</a>–<a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1000615#pbio.1000615-Gardner3" target="_blank">[46]</a>.</p

Effects of a single mutation on performance, altruism, and both.

<p>Number of individuals for whom a single mutation of moderate effect significantly altered only performance, only the level of altruism, and both performance and the level of altruism (i.e., pleiotropic effects). Experiments were performed on 4,000 individuals in the two treatments with intermediate <i>r</i> and <i>c/b</i> values (see <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1000615#s4" target="_blank">Materials and Methods</a>).</p

The Viability Evolution (ViE) algorithm.

<p>The population under evolution is shown in a two-dimensional objective space, defined by the and objective functions in this example. (A) Individuals of the initial population (black circles) are randomly generated. The region enclosed by the target viability boundaries (gray stripes) is extremely unlikely to contain any of the randomly generated individuals in the initial population. (B) The initial viability boundaries are set by the algorithm in terms of inequalities on the objectives to encompass all individuals in the initial population. (C) Viability boundaries are modified to approach the target boundaries ; as a result, a fraction of the population becomes non-viable (gray shaded circles) and is marked for elimination. The way in which the boundaries are modified depends on the specific viability boundary update procedure implemented by the user. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086831#pone-0086831-g002" target="_blank">Figure 2</a> for further details on the update mechanism used in this paper. (D) All viable individuals are allowed to reproduce by making one mutated copy at each iteration of the algorithm. Mutated copies that fall within the viability boundaries are allowed to stay along with the parent. Mutated copies that fall outside the viability boundaries are marked for elimination. (E) Non viable individuals are eliminated from the population. (F) The process described in (C–E) is repeated for many iterations until the viability boundaries reach the target values or the maximum number of evaluations is exhausted. (G) The algorithmic description of Viability Evolution. Pseudo-code for the <i>relaxBoundaries</i> and <i>updateBoundaries</i> procedures is shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086831#pone.0086831.s008" target="_blank">Figure S8</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086831#pone.0086831.s009" target="_blank">S9</a>.</p

Efficiency of SSGA and Viability Evolution.

<p>Efficiency is measured as number of evaluations used per target area discovered over 50 repetitions of the experiment (, otherwise , Wilcoxon rank-sum test; N.S. not significant). A repetition of the evolutionary experiment lasts a higher number of evaluations in Viability Evolution. However, Viability Evolution is able to discover more target areas per repetition than SSGA. Its efficiency is significantly better than SSGA (, Wilcoxon rank sum test). To enhance readability of the box plots, we removed two outlier data points: Griewangk SSGA (500), Value 8676 and Griewangk SSGA (750), Value 12984. The computation of efficiency was performed only on Griewangk and Shubert, since the target areas in these benchmarks are regularly distributed in the search space and therefore have the same probability of being discovered.</p

Number of successful repetitions for SSGA and ViE.

<p>Results for SSGA and ViE on single-objective, multi-modal problems out of a total of 50 repetitions.</p

Example of viability boundaries definition for a filter design problem.

<p>A candidate filter design being optimized with Viability Evolution must satisfy certain requirements defined by the user as viability boundaries. Here, the filter gain-bandwidth product (GBW, computed at the cutoff frequency ) must satisfy the viability boundary . The stop-band attenuation (SBA) of the filter is also constrained by the viability boundary . Finally, a filter must also satisfy a requirement on the pass-band flatness (PBF), i.e. the deviation of amplitude gain from the gain at cut-off frequency, such that . The response for two different filters is depicted in figure. The first filter (solid line) is viable as it satisfies all viability boundaries, while the second filter (dashed line response) is non-viable, as it violates the viability boundaries expressed on pass-band flatness.</p

Boundary update mechanism in the Viability Evolution (ViE) algorithm.

<p>(A) Let us assume, without loss of generality, that the problem to be solved is defined by two objectives, and . The target regions of the given problem are defined by the <i>target</i> viability boundaries - [, ] for each objective function respectively. Thus, the goal is to find solutions which have values between and for each objective function respectively. (B) Individuals of the initial population are randomly generated. Each individual is represented using a circle on the axis of each objective function. The position of a circle on the axis of an objective function indicates the value of the corresponding individual for that particular objective. In this example, each individual is represented using 2 circles - one on each axis of the two objective functions. (C) The <i>initial</i> viability boundaries are set for each objective by identifying the extreme values [, ] on either side of the corresponding target viability boundaries [, ]. The initial viability boundaries thus encompass all individuals in the initial population. (D) The viability boundaries are then tightened such that at least a minimum fraction of individuals become non-viable. To illustrate this clearly, the intervals - [, ], and [, ] are both rescaled to [, ] here. The new values for the viability boundaries [, ] (shown as dotted lines) for each objective function are computed such that at least a minimum fraction of individuals in the population become non-viable (shown as light gray circles). (E) Non viable individuals are eliminated from the population. The population continues to evolve with the new viability boundaries until the next boundary update. Pseudo-code for the boundary update procedure illustrated in figure is shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086831#pone.0086831.s009" target="_blank">Figure S9</a>.</p

Genetic diversity maintained by SSGA and ViE.

<p>Average population genetic diversity (and confidence intervals) maintained during evolution for the 50 repetitions of each experiment.</p