Average payoff as a function of the parameters of the cost and benefit functions, spatial structure, and update rules.

<p>Each individual panel shows the average payoff of Mutualist A and Mutualist B, calculated as the arithmetic mean of their payoffs over the last generations, out of the total of generations, and averaged over five replicate model runs. The three parameters of the benefit and cost functions are varied as follows: and along the axes and between the upper () and lower () eight panels. The black line on white background indicates the threshold, below which no investments can evolve. Results for well-mixed populations are shown in the eight panels on the left, while results for spatially structured populations are shown in the eight panels on the right. Odd and even columns correspond to synchronous and asynchronous updating, respectively. Rows show results for a constant mutational standard deviation (first and third rows) and a constant mutational coefficient of variation (second and fourth rows). Other parameters: , , and .</p

Spatial bubble dynamics and appearance of the insulating boundary layer.

<p>(A) Typical snapshot of the spatial mosaic structure, indicating a high degree of polymorphism and spatial bubbles comprising different strategies. Each pixel represents an individual, rendered according to its payoff between zero (light gray) and the maximal value (black). (B) Enlargement of a bubble with its surrounding insulating boundary layer. Notice that individuals inside and outside the bubble both have higher payoffs than the individuals forming the boundary layer. This panel is obtained as an overlay of Mutualist A and B from the third column in D according to their average payoff values. (C) Shading of background strategies ranges from white to mid-gray, while shading for the focal bubble ranges from dark-gray to black, as the payoffs of individuals increase. (D) Time series of snapshots for a spatial bubble (black to dark-gray shading) that first expands and then vanishes, illustrating a spatial “boom and bust” cycle (snapshots are taken in generations 3013, 3040, 3260, 3399, 3493, 3625, 5165, 5620, 6400, and 6408). Parameters: in A and in D; , , , , , and .</p

Illustration of investment cycle and reciprocation threshold in well-mixed communities.

<p>(A) Best-response dynamics. Arrows indicate the succession of best responses, leading to in just four steps. (B) Evolutionary dynamics in a community with low degrees of polymorphism and “pairwise comparison” updating. Arrows indicate changes of the selection gradient along the investment cycle. In A and B, representative evolutionary trajectories are shown starting above the reciprocation threshold (thick gray lines). (C) Resultant changes of investment traits and payoffs along the investment cycle in B. Results in B and C are averaged over 15 replicate model runs for the same initial condition. Parameters: , , , , , , and .</p

Selection gradients along the investment cycle and resultant phase distribution in a well-mixed polymorphic community.

<p>(A) Arrows indicate the selection gradients on a random subsample of individuals of Mutualist A as that mutualist's trait distribution (gray dots) moves along the investment cycle (gray circular lines). Average long-term polymorphic distribution of (B) phases along the investment cycle and (C) corresponding phase asymmetries during the evolution of mutualistic investments, averaged over three replicate model runs and shown on logarithmic scales. The phase asymmetry in pairs of interacting individuals of Mutualist A and B is measured as the difference of their phases, . The peaks at and in B correspond, respectively, to the vertical and horizontal edges of the investment cycle. Parameters: in A and in B and C; in A and in B and C; , , , , and .</p

Possible replacement dynamics at the interface between two spatial bubbles.

<p>We presume that strategy pairs meet at the interface (white columns) of bubble 1 (dark-gray columns) and bubble 2 (light-gray columns). Here, the following cases can occur: (A) Unidirectional invasion: both mutualists from bubble 1 invade the other bubble, as both A1 and B1 have a higher payoff than A2 and B2. (B) Partner swapping: A1 has a higher payoff and outcompetes A2, but B2 has a higher payoff and outcompetes B1, hence A1 pairs up with B2. (C) Catalyzed invasion: only A1 is able to outcompete its competitor from bubble 2, but as it spreads, it makes it possible for B1 to follow. This is feasible because B2 fares worse with A1 than with A2, so as A1 spreads, the payoff of B2 decreases with its new partner, and hence B1 can now invade. (D) Insulating boundary layer: at the interface of two spatial bubbles, the originally competitively superior strategies A1 and B2 enter the interface, but as both then have a lower payoff than with their original partners, neither can spread further. Column heights depict the payoffs of strategies. For the described dynamics, the payoffs of a strategy with its two possible partners (i.e., from either bubble 1 or 2) at the interface must lie within the interval indicated by the two whiskers in the middle column. (E) Invasion dynamics depend on the strategy compositions of the mutualist pairs. Formation of an insulating boundary layer is the result of the encounter of two strategy pairs (A1&B1, A2&B2) that are mutually unable to invade each other (gray area). Otherwise, one bubble invades and replaces the other (in the white area, the strategy pair of bubble 1 wins, whereas in the black area, the strategy pair of bubble 2 wins). We evaluate these outcomes in the absence of evolution (no mutations) and for one strategy pair (A1&B1) initially occupying one half of the lattice and the other strategy pair (A2&B2) occupying the other half. Parameters: A1 and B1, and ; A2, and ; B2, and ; , , , , and .</p

Schematic representation of mutualistic interactions in (A) monomorphic and (C) polymorphic mutualist communities.

<p>Spheres depict strategies, and the links between spheres represent the interactions between interacting strategies from the two mutualist guilds. (B) According to its own and its partner's strategy, an individual receives a payoff (schematically illustrated by two triangles that become darker and wider as the received payoff increases). The comparison of payoffs between partners shows whether their interaction is more mutualistic (middle) or more exploitative (bottom and top). (D) Average distribution of interaction types in our model, showing that small relative differences between the payoffs of interacting individuals are more common or longer-lasting than extreme exploitations. (E) Average distribution of the payoff sums and relative payoff differences for interacting individuals of Mutualist A and Mutualist B, demonstrating that, on average, payoffs in asymmetric, or exploitative, interactions are lower than in symmetric, or more mutualistic, interactions. The distributions in D and E are based on sampling all individuals in every generation for generations and for five replicate model runs. The relative difference between the payoffs and of individuals and is given by , and is given by . Parameters: , , , , , , and .</p

Evolution and stability of mutualistic investments in communities with higher degrees of polymorphism.

<p>(A) Diversity thresholds revealed by the effect of mutational variability on the average payoff in the community. For lower mutational standard deviations , there is no mutualism (left-hand side), while stable community-level mutualism evolves abruptly once mutational variability is high enough (right-hand side). Results are averaged over the two mutualists and 15 replicate model runs. Payoffs can range between 0 and the maximal potential payoff . (B, C, D) Polymorphic spread of strategies in well-mixed communities, and their evolution along the investment cycle, with low, medium, or high mutational standard deviations: in B, in C, and in D. As the averages of the traits and move along the investment cycle, they trace out the shown circular lines, corresponding to cyclic oscillations whose amplitudes decrease as increases. Other parameters: in A and in B, C, and D; lower benefit-to-cost ratio of in A: , , ; higher benefit-to-cost ratio of in A, B, C, and D: , , ; , .</p

The evolution of division of labor between minus () and plus () strands.

<p>(<b>A</b>) A representative example of simulations resulting in asymmetric strand separation averaged over the population of vesicles (: red; : orange; : dark blue; : light blue). Starting from an initially symmetric state, i.e. all strand types are represented in equal numbers (), and of equal replication rates () (<i>J</i> denotes the mutation class with trait ). The trade-off in this case is assumed to be strong between the replication affinity and the catalytic activity. Hence the trait of the minus strand (<b>B</b>) gradually evolves towards lower replication rates () in order to achieve higher metabolic activity (). During trait evolution the ratio of minus (dark shadings) and plus (light shadings) strands changes, and the minuses significantly increase in numbers. At stable equilibrium, for the very extreme cases, only 4–8% of the macromolecules, on average 2 or 3 per vesicle, are plus strands. Other parameters: , , , , , , , , , , , and .</p

The effect of degradation rate of macromolecules on strand asymmetry.

<p>The equilibrium ratio of the minus and plus strands (indicated by the heights as well as the colors of the bars; red: 0.9→yellow: 0.5) is not affected significantly by the rate of degradation, however increasing the degradation rate above a threshold results in the extinction of the replicators (notice the flat grey area on the right hand side of the graph). For strong trade-off (), this threshold is at a lower rate of degradation, whereas higher degradation rates are tolerated as the strength of trade-off decreases (). The results are averaged over 3 replicate model runs. Other parameters: , , , , , , , , , and .</p

Schematic representation of the main reactions and components of vesicles with complementary replicating strands.

<p>Vesicles are composed of two types of macromolecules (type 1 as red, and type 2 as blue), and with two strand types (plus () strands with light, and minus () strands with dark shading). The minus () strands (molecules colored dark red) serve both as enzymes (enzymatic activity indicated with asterisk) for producing monomers (molecule colored green) from source material, and as templates for producing plus () strands (molecules colored orange). The monomers are used as the building blocks (green arrow) for the productions of replicators (replication complexes are indicated in curly brackets). The plus strand only serves as template for producing minus strands. For molecule type 2, the metabolic and replication processes are similar to those of molecule type 1 described above, except that the minus () strand catalyzes a different chemical reaction.</p