Appendix A. The existence and stability of equilibria.

The existence and stability of equilibria

The vertices and degrees in a heterogeneous graph.

<p>The vertices are labeled vertex 1, vertex 2, , vertex . The degree of vertex is , and where is the number of -neighbors and the number of -neighbors.</p

Effect of graph heterogeneity on the fixation probability under weak selection and fixed cost per game.

<p>Simulation results for the fixation probability in random graphs and scale-free graphs with different heterogeneities are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3A and 3B</a>, respectively (see the generation of these graphs in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560.s001" target="_blank">Supporting Information S1</a>). In <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3A</a>, and . The benefit-to-cost ratio is taken as 16, 20, 24, 28 and 32, respectively, and the selection intensity is . In <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3B</a>, and . The benefit-to-cost ratio is taken as 5, 10, 15, 20 and 25, respectively, and the selection intensity is . In both <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figures 3A and 3B</a>, the -axis denotes and the -axis the fixation probability of a single cooperator with neighbors. The fixation probability is measured using the fraction of runs where cooperators reached fixation out of runs. The dash line in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3A</a> and the dash line in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figure 3B</a> denote respectively the fixation probability under neutral selection (). Both <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g003" target="_blank">Figures 3A and 3B</a> show the tendency that for all different values of , fixation probability approaches that of neutral selection as increases.</p

Evolution of Cooperation in a Heterogeneous Graph: Fixation Probabilities under Weak Selection

<div><p>It has been shown that natural selection favors cooperation in a homogenous graph if the benefit-to-cost ratio exceeds the degree of the graph. However, most graphs related to interactions in real populations are heterogeneous, in which some individuals have many more neighbors than others. In this paper, we introduce a new state variable to measure the time evolution of cooperation in a heterogeneous graph. Based on the diffusion approximation, we find that the fixation probability of a single cooperator depends crucially on the number of its neighbors. Under weak selection, a cooperator with more neighbors has a larger probability of fixation in the population. We then investigate the average fixation probability of a randomly chosen cooperator. If a cooperator pays a cost for each of its neighbors (the so called fixed cost per game case), natural selection favors cooperation if the benefit-to-cost ratio is larger than the average degree. In contrast, if a cooperator pays a fixed cost and all its neighbors share the benefit (the fixed cost per individual case), cooperation is favored if the benefit-to-cost ratio is larger than the harmonic mean of the degree distribution. Moreover, increasing the graph heterogeneity will reduce the effect of natural selection.</p></div

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

<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

Qualitative outcomes of the optimal foraging strategy (13) and (14) for increasing recognition time .

<p>Panel (a) assumes for which and . The optimal foraging strategy is at (i.e. always pay the cost of recognition and then never attack the less profitable prey type) and the NE component (shown as the gray line segment) (corresponding to the NE outcome of attacking immediately) is suboptimal. In each of the other three panels, the (union of the) thick edges forms a strict equilibrium set (SES, for definition see section Zero-one rule and the Nash equilibrium of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0088773#pone.0088773.s001" target="_blank">Appendix S1</a>) that is the globally stable evolutionary outcome. Panel (b) assumes , and . The union of the two edges and forms one NE component corresponding to optimal foraging behavior. Panel (c) assumes , and . The edge forms a NE component corresponding to optimal foraging behavior. Panel (d) assumes for which and The edge forms a NE component corresponding to optimal foraging behavior. The arrows in each panel indicate the direction of increasing energy intake per unit time at points in the unit square. Other parameters , , , and .</p

The decision tree for two prey types.

<p>The first level gives the prey encounter distribution. The second level gives the predator activity distribution. The final row of the diagram gives the probability of each predator activity event and so sum to . Since each entry here is simply the product of the probabilities along the path leading to this endpoint, we do not provide them in the decision trees from now on. With random prey distribution and large, and . If prey is the more profitable type, the edge in the decision tree corresponding to not attacking this type of prey is never followed at optimal foraging (indicated by the dotted edge in the tree). The reduced tree is then the resulting diagram with this edge removed.</p

Game-Theoretic Methods for Functional Response and Optimal Foraging Behavior

<div><p>We develop a decision tree based game-theoretical approach for constructing functional responses in multi-prey/multi-patch environments and for finding the corresponding optimal foraging strategies. Decision trees provide a way to describe details of predator foraging behavior, based on the predator's sequence of choices at different decision points, that facilitates writing down the corresponding functional response. It is shown that the optimal foraging behavior that maximizes predator energy intake per unit time is a Nash equilibrium of the underlying optimal foraging game. We apply these game-theoretical methods to three scenarios: the classical diet choice model with two types of prey and sequential prey encounters, the diet choice model with simultaneous prey encounters, and a model in which the predator requires a positive recognition time to identify the type of prey encountered. For both diet choice models, it is shown that every Nash equilibrium yields optimal foraging behavior. Although suboptimal Nash equilibrium outcomes may exist when prey recognition time is included, only optimal foraging behavior is stable under evolutionary learning processes.</p></div

Decision tree for prey recognition game.

<p>In the reduced tree, the dotted edges are deleted.</p

Effect of an individual's degree on the fixation probability under neutral selection.

<p>The scale-free network (generated according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone.0066560-Barabsi1" target="_blank">[28]</a>) and the 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>) are used to test the effect of graph heterogeneity on neutral selection (). For each of these two graphs, the total population size is and 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). The simulation results are plotted in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g002" target="_blank">Figure 2A</a> for the scale-free network and in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g002" target="_blank">Figure 2B</a> for the random graph. For both <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g002" target="_blank">Figures 2A and 2B</a>, the -axis denotes the number of a single cooperator's neighbors and the -axis the fixation probability of cooperation. In each of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g002" target="_blank">Figures 2A and 2B</a>, the three solid lines represent the theoretical predictions of fixation probabilities for three average degrees, where in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g002" target="_blank">Figure 2A</a> and in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0066560#pone-0066560-g002" target="_blank">Figure 2B</a>, and diamonds, squares and triangles denote the simulation results. It is clear that the simulation results match the theoretical prediction well.</p