Cheaters allow cooperators to prosper

Cooperation based on the production of costly common goods is observed throughout nature. This is puzzling, as cooperation is vulnerable to exploitation by defectors which enjoy a fitness advantage by consuming the common good without contributing fairly. Depletion of the common good can lead to population collapse and the destruction of cooperation. However, population collapse implies small population size, which, in a structured population, is known to favor cooperation. This happens because small population size increases variability in cooperator frequency across different locations. Since individuals in cooperator-dominated locations (which are most likely cooperators) will grow more than those in defector-dominated locations (which are most likely defectors), cooperators can outgrow defectors globally despite defectors outgrowing cooperators in each location. This raises the possibility that defectors can lead to conditions that sometimes rescue cooperation from defector-induced destruction. We demonstrate multiple mechanisms through which this can occur, using an individual-based approach to model stochastic birth, death, migration, and mutation events. First, during defector-induced population collapse, defectors occasionally go extinct before cooperators by chance, which allows cooperators to grow. Second, empty locations, either preexisting or created by defector-induced population extinction, can favor cooperation because they allow cooperator but not defector migrants to grow. These factors lead to the counterintuitive result that the initial presence of defectors sometimes allows better survival of cooperation compared to when defectors are initially absent. Finally, we find that resource limitation, inducible by defectors, can select for mutations adaptive to resource limitation. When these mutations are initially present at low levels or continuously generated at a moderate rate, they can favor cooperation by further reducing local population size. We predict that in a structured population, small population sizes precipitated by defectors provide a "built-in" mechanism for the persistence of cooperation

The initial presence of defectors can increase cooperator survival when empty locations are available.

<p><b>A-B</b>) 50% of locations were initially seeded with <i>C*</i> cells only and they mutated to <i>D*</i> at <i>u</i>_<i>D</i> = 10⁻⁷ hr^-1. <b>A</b>) All 144 locations on the same plot. <b>B</b>) Four plots of individual locations (resource in black). After the first 200 hours, initially-occupied (left) and initially unoccupied (right) locations showed the same behavior: resident or migrant <i>C*</i> cells grew to high density, and mutation to <i>D*</i> became appreciable. In all locations, although <i>D*</i> appeared at somewhat different times (mainly through mutation, since migration was insignificant at small population size), <i>D*</i> populations all slowly grew and caused local population collapse Consequently, most metapopulations went extinct. <b>C-D</b>) 50% of locations were initially seeded with <i>C*</i> and <i>D*</i> cells at 1:1. <b>C</b>) All 144 locations on the same plot. <b>D</b>) Four plots of individual locations (resource in black). While the initially-occupied locations crashed at about the same time (left), initially unoccupied locations showed different dynamics depending on whether <i>C*</i> or <i>D*</i> migrated into these locations (right). Thus, heterogeneity in dynamics promoted the survival of cooperation in these metapopulations because locations vacated by early population extinctions could be colonized by surviving <i>C*</i> from elsewhere.</p

Intermediate migration rates, initially empty locations, and intermediate release rates promote the survival of cooperation.

<p>Here, we consider metapopulations of <i>C</i> and <i>D</i>. <b>A</b>) Varying the fraction of the 144 locations initially occupied by populations of 10⁵ cells at a <i>C</i> to <i>D</i> ratio of 1:1. <b>B</b>) Initially fully-occupied metapopulations as in <b>A</b>, with different cooperator release rates. Error bars were calculated as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004645#pcbi.1004645.g003" target="_blank">Fig 3</a>. In all cases, <i>n</i> ≥ 30. Note the break on the X-axis. Points are shifted slightly about the x-axis to aid visualization.</p

Resource release rate determines multiple aspects of fitness.

<p>Individual locations (<i>n</i> = 1024) were seeded with a single <i>C</i> (red circles) or <i>C</i>* (magenta X's) individual, no initial resource, no cooperator to defector mutation, and no migration. Simulations were run until every location had a population size of 0 (i.e., extinction) or ≥10³ individuals. The maximal release rate <i>β</i>_<i>0j</i> = 2.4 units/cooperator/hr used in most simulations is marked by a solid black line. <b>A</b>) The relationship between survival probability and maximum release rate <i>β</i> was fit to the equation (<i>p</i>_<i>max</i>(<i>β</i> -<i>h</i>))/(<i>k+</i>(<i>β</i> -<i>h</i>)) using non-linear least squares regression (black dotted line). Here <i>p</i>_<i>max</i> was the maximum probability of survival, <i>h</i> was the smallest maximal release rate <i>β</i> allowing any surviving populations, and <i>k</i> was the <i>β</i> -<i>h</i> value allowing half-maximal probability of survival. Only rates giving survival probabilities > 0 were considered. For <i>C</i>, the parameter values were found to be <i>p</i>_<i>max</i> = 0.757 (0.754–0.760); <i>k</i> = 1.7 (1.6–1.8); <i>h</i> = 0.43 (0.40–0.46). For <i>C</i>*, <i>p</i>_<i>max</i> = 0.827 (0.824–0.829); <i>k</i> = 0.209 (0.201–0.217); <i>h</i> = 0.282 (0.276–0.288). We calculated the theoretical upper bound of <i>p</i>_<i>max</i> for <i>C</i> and found it to be <0.79 (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004645#pcbi.1004645.s008" target="_blank">S1 Appendix</a>), which was in agreement with our simulations. <b>B</b>) The net growth rates of surviving populations were determined by individually fitting the growth of five populations to the equation <i>A*</i>exp(<i>gt</i>) using non-linear least squares regression, where <i>A</i> was the initial number of cells after population growth rate had stabilized (>tens of cells, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004645#pcbi.1004645.s001" target="_blank">S1 Fig</a>), <i>g</i> was the net growth rate, and <i>t</i> was the time after the population had reached size <i>A</i>. In all cases, two standard errors were smaller than the plotting points, and were omitted. The net growth rate increased linearly as a function of maximum release rate until the amount released was greater than the amount that could be used by a population growing exponentially at its maximum net growth rate. For both <i>C</i> and <i>C</i>*, the slope of the linear portion (black dashed line) agreed with the theoretical value of 0.18 cooperators per unit of resource (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004645#pcbi.1004645.s009" target="_blank">S2 Appendix</a>).</p

The initial presence of defectors allows moderate mutation rates to improve cooperator survival.

<p><b>A</b>) Simulations were initialized with 10⁵ cells in all locations. The migration rate <i>m</i> and the mutation rate <i>u</i>_<i>D</i> were 10⁻⁷ hr^-1. Initial cooperator frequency was 50% (black circle) or 100% (orange triangle). <b>B-D</b>) One simulation initiated with 50% cooperators. The initial presence of defectors increased heterogeneity in dynamics due to the different timing of appearance of <i>C*</i> and <i>D*</i> in different locations. This heterogeneity allowed cooperation to persist over time. <b>E-G</b>) One simulation with 100% cooperators. In all locations, <i>C</i>* quickly outcompeted <i>C</i> and <i>D</i> and rose to high density. Mutation from <i>C</i>* to <i>D</i>* formed a “band” of <i>D*</i> growth across all locations (migration of <i>D*</i> was initially negligible due to the relatively small number of <i>D</i>* in each location). This resulted in a poor outcome for cooperation, since all populations eventually collapsed at about the same time, limiting the opportunity for cooperators to escape to empty locations. 50% initially-occupied metapopulations showed a similar phenomenon.</p

Migration, empty locations, and sometimes even the initial presence of defectors can increase the survival of cooperators.

<p>Each metapopulation was binned according to the frequency of cooperators averaged over the last 5 data points (800 hrs) of the entire simulation. <b>A-E</b>) Metapopulations with <i>C</i> and <i>D</i> only. <b>F-J</b>) Metapopulations with <i>C</i>* and <i>D</i>* only. Since the fitness of <i>C</i>* and <i>D</i>* differed from that of <i>C</i> and <i>D</i>, levels of cooperator survival differed in the two cases. <b>A</b>, <b>F</b>) In initially fully-occupied metapopulations without migration, cooperators went extinct in all metapopulations. <b>B</b>) Intermediate (10⁻⁷ hr^-1) migration could promote cooperator survival frequency (15/133 in <b>B</b> compared to 0/131 in <b>A</b>; <i>p</i> < 10⁻⁴, Fisher's Exact Test), demonstrating that empty space arising from defector-induced extinction can sometimes rescue cooperators. <b>C</b>, <b>H</b>) Increasing the initial cooperator frequency to 100% improved cooperator survival in an initially fully-occupied metapopulation (compare to <b>B</b>, <b>G</b>). <b>D</b>, <b>I</b>) The initial presence of empty locations improved the survival of cooperation (compare to <b>B</b>, <b>G</b>). <b>E, J</b>) With initially empty locations, the initial presence of defectors can promote cooperator survival (compare <b>D</b> with <b>E</b>; <b>I</b> with <b>J</b>). Error bars indicate an estimate of the 95% confidence interval (CI) based on the Wilson method [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004645#pcbi.1004645.ref055" target="_blank">55</a>,<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004645#pcbi.1004645.ref056" target="_blank">56</a>].</p

Multiple mechanisms allow cooperators to survive defector-induced population collapse.

<p><b>A</b> and <b>D</b> each shows one example simulation of one metapopulation of <i>C</i> and <i>D</i> only, with 144 locations and a migration rate and a mutation rate <i>u_D</i> of 10⁻⁷ hr^-1. Each red or blue curve depicts the trajectory of <i>C</i> or <i>D</i>, respectively, in one location within the metapopulation. Small open triangles at the beginning of a population trajectory indicate the appearance of a new subpopulation via immigration or mutation. While we do not visually distinguish between these two types of events, one can infer, for example, that the appearance of an individual in an initially empty location must have occurred due to migration. (<b>A</b>-<b>C</b>) In an initially fully-occupied metapopulation, all empty locations must have been created by defector-induced extinction. All 144 locations of a metapopulation were initialized with 10⁵ individuals at a 1:1 <i>C</i> to <i>D</i> ratio. In an example metapopulation where cooperators achieved a high final frequency (<b>A</b>), <i>D</i> drove <i>C</i> extinct (X's at 1 on the ordinate) before suffering self-extinction in 143 out of 144 locations. However, in one location (black arrowhead, details in <b>B</b>), <i>D</i> went extinct before <i>C</i>, allowing the increase of <i>C</i> and the spread of <i>C</i> by migration to locations where <i>D</i> had driven <i>C</i> extinct (example shown by white arrowhead, details in <b>C</b>). In <b>B</b> and <b>C</b>, resource levels (black) were plotted. Downward-facing arrowhead on resource indicates that the resource concentration went below the plotted scale. (<b>D</b>) In a metapopulation with 90% locations initially occupied, <i>C</i> individuals were also able to migrate into and grow in initially empty locations (gray arrowhead).</p

A social dilemma defined by resource use and fitness trade-offs.

<p><b>A</b>) Model overview. Ancestor types mutate to evolved types (denoted by a '*') at rate <i>u</i>*. Cooperators (<i>C</i> and <i>C</i>*), which produce resource (green triangles) mutate at rate <i>u</i>_<i>D</i> to defectors (<i>D</i> and <i>D</i>*), which produce nothing. All types migrate randomly to new locations at rate <i>m</i>. <b>B</b>) When resource is abundant, <i>C</i> (red) and <i>D</i> (blue) have faster net growth rates than <i>C</i>* (magenta) and <i>D</i>* (cyan). When resource is low, <i>C</i>* and <i>D</i>* grow faster than <i>C</i> and <i>D</i>. However, the net growth rate of <i>D</i> is always greater than <i>C</i>, and the net growth rate of <i>D</i>* is always greater than <i>C</i>*, resulting in a social dilemma. (Inset) The fitness functions at low resource levels.</p

High-fitness mutants can improve the survival of cooperation.

<p><b>A-B</b>) Non-empty locations in each simulation were initialized with a total of 10⁵ cells at a cooperator to defector ratio of 1:1. Each of these populations were seeded with a total of 0 (black circle), 2 (orange triangle), 20 (green plus), or 200 (blue x) initial mutants, also at a <i>C</i>* to <i>D</i>* ratio of 1:1. <b>A</b>) 100% of locations initially occupied. <b>B</b>) 50% of locations initially occupied. <b>C-D</b>) Instead of seeding the initial mutants, initial <i>C</i> and <i>D</i> populations were given an ancestor to evolved mutation rate <i>u*</i> of 0 (black circle), 10⁻⁷ (orange triangle), 10⁻⁶ (green plus), or 10⁻⁵ (blue x) hr^-1. At an extremely high mutation rate, cooperator survival should approximate that in <i>C*</i>/<i>D*</i> metapopulations, plotted here for comparison (purple diamond). Trends in <b>C</b> and <b>D</b> are qualitatively similar to those in <b>A</b> and <b>B</b>. Data is slightly shifted about the x-axis to aid visualization of overlapping points.</p