article: 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.
  A cooperator produces a common good (or benefit) available to all members of a population at a fitness cost to itself, and gains a net benefit if sufficiently reciprocated. However, cooperators are vulnerable to evolutionary displacement by defectors that exploit this benefit without paying the fitness cost of benefit production. Thus, the ubiquity of organisms displaying apparently cooperative behavior demands an explanation, and decades of experimental and theoretical research have been dedicated to unraveling this riddle. Previous research has uncovered multiple mechanisms that promote cooperation [1,2]. Recognition allows cooperators to direct benefits to other cooperators and exclude or punish defectors [3]. In the absence of recognition mechanisms, a spatially-structured environment can facilitate cooperation by promoting repeated interactions between individuals. For example, during repeated interactions, cooperators can persist in, and even invade, defector-dominated populations if an individual can adjust its cooperating strategy based on memory of past experience. Examples include “tit-for-tat” [4] and “win-stay-lose-shift” [5]. Even when organisms cannot adjust their cooperating strategy, limited migration in a spatially-structured environment (“population viscosity”) means that cooperators will preferentially interact with their genetic relatives who are also likely to be cooperators (also known as “kin selection”) [6,7]. However, increased population viscosity also increases local competition between cooperators for non-cooperative resources, and therefore migration to sites with additional resource is necessary for cooperation to survive [8–12]. A group of spatially-separated populations connected through migration is known as a metapopulation. For cooperation to persist in a metapopulation, cooperator frequency must vary greatly among different populations. This is because, when cooperator frequency is uniform across populations, the cooperator frequency of the metapopulation will mirror that of each population and decline over time due to the advantage of defectors over cooperators. In contrast, when cooperator frequency varies across populations, individuals in a population dominated by cooperators (which are likely cooperators) will grow faster than individuals in a population dominated by defectors (which are likely defectors). Thus, the global frequency of cooperators can increase despite the fact that defectors are more fit than cooperators in individual populations. This phenomenon is described by the Price equation [13,14] and is also known as “Simpson' s Paradox” [15,16]. A consequence of large variation in cooperator frequency is that population dynamics will be highly heterogeneous across the metapopulation: defector-dominated populations will go extinct while cooperator-dominated populations will thrive and migrate to populate vacant locations. One mechanism that generates large variation in cooperator frequency is small population size. In the most extreme case, if populations are founded by a single individual, then each population will have a cooperator frequency of 1 (if the founder is a cooperator) or 0 (if the founder is a defector). The possibility of small population size promoting cooperation has been widely observed in simulations [17–22], and was also demonstrated in an experimental system using Escherichia coli that was periodically diluted to very low population size [23]. Ecologically-relevant ways to achieve small population size include the migration of a small number of individuals to a vacant space or exposure to harsh environments that only allow survival of a small number of individuals. The idea of harsh environments and small population size supporting the evolution of cooperation [24] has received theoretical [21,25,26] and experimental support, both in animals [27] and microbes [28–30]. Surprisingly, one obvious way to achieve small population size has been largely overlooked: defectors themselves. The way in which defectors feed back onto population dynamics can be complex and produce surprising outcomes. Experimentally, defectors can increase the total biomass of a population under some circumstances [31]. Previous theoretical work has shown that cooperation can survive defector invasion if defector-induced environmental degradation promotes a small level of migration in cooperators and defectors [32,33]. Defectors can also, in theory, support the long-term success of cooperation by maintaining selective pressure for partner recognition [34]. We reasoned that the small population size known to promote cooperation could be created by defectors. To test this idea, we investigated a model of microbial cooperation based on the production of a common good. As microbial populations often violate common theoretical assumptions such as weak selection, fixed group size, and linear relationships between cooperator frequency and fitness [35,36], we developed a model based on processes that have been experimentally observed or are measurable. Similar to previous approaches used to simulate community growth [37–39], our model combined stochastic and deterministic components. The stochastic, agent-based component tracked the birth, death, mutation, and migration of all individuals in a metapopulation. This allowed us to capture stochastic events at small population size that would have been missed by deterministic models. The deterministic differential equation component allowed us to efficiently model the amount of common good available to a population without tracking each individual molecule of common good. We have found three ways in which defectors create conditions that rescue cooperation. First, during defector-induced population collapse, defectors occasionally go extinct before cooperators by chance, allowing cooperator-only but not defector-only populations to grow. Second, defectors cause population extinction, which creates empty locations that support cooperator but not defector migrants. This leads to the counterintuitive result that the initial presence of defectors in a population can sometimes improve the survival of cooperators when a subpopulation of cooperators mutate to defectors. Third, resource limitation, which can be precipitated by defectors, selects for rare cooperator and defector mutants adapted to this resource limitation. When defectors are initially present, this creates yet another population bottleneck beneficial to cooperation. These three factors act together to generate large variation in cooperator frequency across a metapopulation and therefore promote cooperation. Thus, by creating small population sizes, defectors provide a built-in mechanism that can rescue cooperation.
  We modeled a social dilemma based on the production and consumption of common goods (Fig 1A). The cooperator type (C) generated publicly-available resource, which it also consumed in order to grow. The defector type (D) consumed this resource but did not produce it, and therefore enjoyed a fitness advantage which increased D' s birth rate and decreased its death rate relative to C (Fig 1B). Low-resource adapted mutant cooperators (C*) and defectors (D*), collectively referred to as “mutants”, grew much faster than their respective ancestor types in low-resource environments, and died more slowly in all environments (Fig 1B). To reflect experimental observations of fitness trade-offs [40–43], these mutants were poorly adapted to the resource-abundant environment of their ancestors (Fig 1B). This type of trade-off can arise, for example, when mutations increase surface expression of the transporter (s) responsible for resource uptake [42,44]. Increased transporter expression improves fitness in low-resource environments, but leads to excessive uptake and toxicity in high-resource environments [42,43]. Thus, the relative fitness of each type was ordered as C < D < C* < D* when resource was scarce and C* < D* < C < D when abundant (Fig 1B, see below for more details). When resource levels were intermediate, the fitness relationships changed at various line intersection points (Fig 1B); however, the fitness relationships always obeyed C < D and C* < D*. Because mutations that reduce or abolish cooperative benefit naturally occur [45–47], each cooperator type mutated to its corresponding defector type at rate uD (Fig 1A). We modeled cooperator to defector mutation but not the reverse, because the mutational target size for losing the ability to make common goods likely far exceeds that for regaining this ability. Mutation from ancestor to adaptive mutant occurred at rate u* (Fig 1A). In many simulations, we assumed that mutants adapted to low-resource environments and defectors were present at the beginning of simulations, because these variants are constantly generated from non-mutants and cooperators, respectively, during population growth [48]. We defined a population as a group of individuals residing in the same location, with each individual belonging to one of the four types (C, D, C*, or D*; Fig 1A). We defined a metapopulation as a set of populations that could interact through the migration of individuals (Fig 1A). Table 1 summarizes all parameter values used in this model. The birth rate of all types depended on the amount of resource available in their current location. We used a Holling Type II (or “Monod”) [49,50] function to relate resource concentration (S) to birth rate (b), because of its simplicity and empirical support. This function is defined by two parameters: V, the maximum birth rate, and K (sometimes referred to as the “Monod constant”), the concentration of S allowing half maximal birth rate. Let i = 0 or 1 denote cooperator or defector, respectively, and let j = 0 or 1 denote ancestor or mutant, respectively. Then the birth rate of a given type was described by bij (S) =VijSK/θj+S, (1) where θ (>1) denoted the mutant K advantage over the ancestor when resource became limited due to mutant' s higher affinity for resource. Death was modeled as a separate process with a rate dij independent of S, dij=dαiδj, (2) where α (>1) was the defector death advantage, and δ (>1) was the mutant death advantage. To enforce a strict social dilemma, we chose parameters such that defectors had a higher birth rate than their corresponding cooperators at all resource concentrations. In other words, for j = 0 or 1, b0j (S) < b1j (S) for all S. Specifically, we set the maximum birth rate of C and D to V00 = 0. 450 hr-1 and V10 = 0. 500 hr-1, respectively. To reflect the relative fitness deficit of the mutants at high resource concentrations, we set the maximum birth rate of C* and D* to V01 = 0. 293 hr-1 and V11 = 0. 333 hr-1, respectively. A visual representation of the net growth rate (i. e. , birth rate minus death rate) for each type as a function of S is plotted in Fig 1B.
  Each simulation tracked one metapopulation comprising 144 locations. All locations initially contained no resource. Metapopulation replicates were performed by choosing different seeds for the random number generators used in the simulation. Simulations terminated upon extinction of cooperators, or after 20,000 hrs. We determined that, in simulations where cooperators survived longer than 10,000 hours, only simulations with very low u* (<10−10 hr-1) or very high migration rates (>10−6 hr-1) occasionally resulted in cooperator extinction by 20,000 hrs. Thus, some simulations outside of this parameter range were run for 10,000 hrs.
  We first modeled a simple scenario unfavorable to cooperation and without C* or D* mutants. Each location of the metapopulation was occupied by a large population (105 total C and D individuals) and all populations shared the same initial defector frequency of 50%. As per the Price equation or Simpson' s paradox, this lack of variation in cooperator frequency among populations meant that all populations would suffer a similar fate of defector-induced population extinction. Indeed, due to the fitness advantage of D over C under all resource concentrations (Fig 1B), the frequency of C, as well as the amount of resource, decreased over time (S2A and S2B Fig). In most metapopulations, cooperators would go extinct before defectors in all locations, and this led to metapopulation extinction after ~200 hrs (S2C Fig). However, occasionally in one of the locations, defectors would by chance go extinct before cooperators (similar to Fig 4B). This allowed rapid growth of the resident cooperators in these defector-free locations (similar to Fig 4B), temporarily rescuing cooperation. These events would not have been observed in deterministic models, which would predict deterministic and synchronous extinction of cooperation in all locations. In the absence of migration, even populations of pure cooperators were short-lived: defectors would eventually rise from cooperators through mutation, and re-induce population collapse. Thus, in the absence of migration, defectors drove all metapopulations extinct (Figs 3A and 5). However, at intermediate migration rates (e. g. , 10−7 hr-1, corresponding to a maximum of ~0. 1 individuals migrating from each location per hr, with a generation time of ≥ 2 hrs), a fraction of initially fully-occupied metapopulations became dominated by C (Fig 3B and black in Fig 5A). This is because cooperators (e. g. , from those locations where they happened to outlast defectors, Fig 4A and 4B) could migrate and grow in locations vacated by defector-induced extinctions (Fig 4A, 4B and 4C). Since defector migrants could not grow in empty locations, cooperators were favored over defectors in these metapopulations. Lower migration rates (<10−9 hr-1) approximated no migration, while very high migration rates (>10−5 hr-1) approximated a single, well-mixed population (Fig 5A). Thus, migration rates too low or too high were unfavorable to cooperation, as observed previously [20,33,54]. In initially fully-occupied metapopulations with an intermediate migration rate, cooperator survival improved when starting with 100% C (Fig 3C) instead of 50% C and 50% D (Fig 3B). Unlike the 50% C-50% D case where D caused population collapse with very similar dynamics in all locations (Fig 4A), in the 100% C case, D arose at somewhat different times in different locations. The latter occurred because mutation rate uD was small (10−7 hr-1) compared to the local population size (~106 individuals/location) (S3 Fig), and the small size of D meant that migration of D was negligible. This heterogeneity in dynamics across locations increased the chance of cooperator survival because locations vacated by early population extinctions could be colonized by surviving C from other locations. The presence of initially empty locations dramatically improved cooperator survival (compare Fig 3D to 3B and Fig 5A), as also observed in other models [19,33]. In contrast to initially fully-occupied metapopulations where all empty locations used by cooperator migrants resulted from defector-induced extinction events (Fig 4A, 4B and 4C), pre-existing empty locations gave cooperators another way to escape defectors (Fig 4D). In summary, when a small number of individuals migrated, empty locations (either vacated by defector-induced population extinction or preexisting) aided the survival of cooperation. All these trends held for metapopulations comprising C* and D* only (Fig 3F, 3G, 3H and 3I), despite the fact that cooperator survival frequency differed between C*/D* metapopulations and C/D metapopulations. Strikingly, at 50% initial occupancy, the initial presence of defectors increased cooperator survival (Fig 3D and 3I) compared to metapopulations starting with pure cooperators (Fig 3E and 3J). This is because, when starting with pure cooperators, cooperators in each occupied location quickly grew to saturation (Fig 6B, left), and sent migrants to other locations, which also quickly reached saturation (Fig 6B, right). In all locations, high population density led to low resource supply. Thus, defector populations, arising mainly from mutation, increased slowly (Fig 6B). This meant that cooperator populations in all locations, whether initially occupied or not, more or less synchronously reached saturation and generated slowly-rising defector populations (a band of cooperator increase, followed by a band of defector increase, followed by extinctions of cooperators and then defectors in Fig 6A). In contrast, at 50% cooperator and 50% defector, even though population dynamics initially looked homogeneous across occupied locations (Fig 6D, left), cooperator and defector migration to empty locations led to heterogeneous dynamics (Fig 6D, right). This heterogeneity across locations (Fig 6C, lack of cooperator or defector bands) favored cooperation. All surviving metapopulations we examined displayed oscillations in cooperator frequency that dampened with time to a stable amplitude range, lasting the extent of simulations (S4 Fig). This dampening occurred as the initially chaotic population crash and stochastic rescue gave way to a more stable “chasing” dynamic (S1 Movie, S2 Movie). The fact that oscillations between cooperators and defectors were also seen in models with different assumptions and initial conditions [17,53,57–59] suggests that this type of behavior might be relatively easy to achieve. The level of cooperative benefit production is a double-edged sword for the survival of cooperation. On the one hand, producing too little benefit is similar to defecting. On the other hand, excess benefit production can support defectors [26,31,46,60–64]. We found that reducing resource release rate β from 2. 4 units hr-1 to 1. 2 units hr-1 (limiting to C, Fig 2B) greatly improved the survival of cooperation (Fig 5B). Further reduction to 0. 6 units hr-1 showed a less significant improvement than 1. 2 units hr-1. This also shifted the rates of migration that supported the survival of cooperation to higher values. This is presumably because a higher migration rate would allow multiple cooperator immigrants to arrive at the same location, which can facilitate the survival of cooperators with lower release rates. As expected, reducing the release rate to a value where a single C had no chance to grow into a successful colony (0. 3 units hr-1) abolished the survival of cooperation. We used 2. 4 units hr-1 a condition suboptimal for cooperation—in all other simulations. Resource limitation, imposed by defectors or encountered in empty or densely-populated locations, can select for mutants adapted to resource limitation. Because of the fitness trade-off (Fig 1B), these mutants will be kept at low frequencies when resource is abundant (S5 Fig). We reasoned that the initial rarity of these mutants could help rescue cooperation from defectors, similar to the effect of small population size caused by defector-induced population collapse or individual migration into empty locations. To gain intuition about how adaptive mutants might help rescue cooperation from defectors, we initiated metapopulations with a mixture of C and D (initially 100% occupied, 105 total cells/population, 1: 1 C: D) so that defectors could induce resource limitation. When we added a small number (Fig 7A, orange triangles) but not a large number (green pluses and blue crosses) of C* and D* to every population, cooperator survival improved compared to the no-mutant scenarios (black circles). This appeared to be general, holding regardless of initial occupancy (Fig 7B) or details of the fitness relationships between the four types (S6A and S6B Fig). The only exception was that rare initial mutants might not improve cooperation when cooperator survival was already high (S6C Fig). Rare initial mutants promoted cooperation because, under resource limitation precipitated by defectors, the population dynamics of a metapopulation were dominated by the dynamics of mutants due to their high resource affinity (low K) compared to ancestors (Fig 8). When the initial C* and D* were rare but not when they were abundant, it was almost guaranteed that in at least one population, D* would go extinct before C* by chance. The remaining C* would then rapidly increase into a large population (Fig 8B) and, at intermediate migration rates, spread to locations emptied by D*-induced extinction (Fig 8C). In these metapopulations, oscillations between C* and D* were also observed (S4C and S4D Fig). To test whether continuous mutation from C and D to C* and D* could support cooperator survival, we varied u* in metapopulations initiated with 1: 1 C: D. As expected, at very high mutation rates (e. g. , Fig 7C and 7D, blue), cooperator survival frequency resembled that of metapopulations initiated with C* and D*, which in turn differed greatly depending on the initial occupancy of metapopulation (Fig 7C and 7D, purple). Otherwise, except when u* was unrealistically low (S7A and S7B Fig, orange) or zero (Fig 7C and 7D, black), cooperator survival appeared to be largely independent of the initial occupancy of the metapopulation (S7A and S7B Fig, blue and green; Fig 7C and 7D, orange and green). Overall, unless cooperator survival was already very high without mutants (S7B Fig), moderate u* improved cooperator survival (Fig 7C and 7D, orange and green; S7A Fig, grey, blue and green; all compared to black). The initial presence of defectors allowed moderate u* to promote cooperator survival (Fig 9, black). In contrast, metapopulations initiated with 100% C did not benefit from moderate u* (Fig 9, orange). This is because in the absence of initial D, C mutated to C* which quickly migrated and dominated all locations (Fig 9E, 9F and 9G). This in turn triggered a synchronous rise of D* in all locations (Fig 9E, 9F and 9G), which was detrimental to cooperator survival compared to when defectors were initially present (where D* could contribute to de-synchronization, Fig 9B, 9C and 9D). Thus, the initial presence of D in conjunction with moderate u* can promote the survival of cooperation. In natural populations, we expect defectors to be present due to mutation from cooperators. We also expect that there will be many types of mutants, each conferring a different fitness improvement over the ancestor in resource-limited environments. The fittest C* and D* that survive genetic drift will dominate a metapopulation. Regardless of the average fitness of the population, the fittest mutants are expected to be relatively rare (by definition), but exist (since being the fittest is relative to other mutants) [65–67]. Thus, this mechanism of reducing population size could promote cooperation even as the population evolves.
  Defectors, by degrading the environment through resource exploitation, create small population sizes that can help rescue cooperation from defectors. This happens via three mechanisms. First, through resource depletion, defectors directly reduce population size and, occasionally, cooperators outlive defectors by chance (Fig 4A and 4B). Second, while pre-existing empty locations aid the survival of cooperation (Figs 3 and 5A, [19,33]), locations emptied by defector-induced extinction can also promote the survival of cooperation by supporting the growth of migrant cooperators but not defectors (Figs 4C and 8C). In fact, the initial presence of defectors often increases cooperator survival (Figs 3,6 and 9). This is because, relative to metapopulations starting with pure cooperators, defectors can increase the heterogeneity in population collapse dynamics, reducing the chances that all populations go extinct simultaneously. Third, selection for rare mutants adapted to resource limitation, which can be induced by defectors, introduces another population bottleneck (Fig 8). This facilitates the survival of cooperation (Fig 7) when defectors are initially present (Fig 9). Thus, in structured populations with intermediate migration rates, defectors themselves can help rescue cooperation. These relatively rare events (e. g. , mutation, migration, chance survival of cooperators) captured by our stochastic, agent-based modeling approach may be an important contributor to the success of cooperation over evolutionary time. Defector-induced local population extinction has been observed in microbes. For example, when the social bacterium Myxococcus xanthus encounters starvation, individuals aggregate into fruiting bodies. In a fruiting body, a small fraction of cells become stress-resistant spores while the rest undergo autolysis or form the peripheral shell of the fruiting body. When mixed with cooperators, obligate defectors incapable of developing into fruiting bodies on their own can exploit cooperators and have a disproportionately large contribution to spore formation. Some of these obligate defector types can drive the entire mixed population to extinction during cycles of famine and feast [68]. A collapsing population can be rescued when an obligate defector mutates into a super-cooperator [69]. We have observed that too much or too little resource production is detrimental to cooperator survival (Fig 5B). Experimentally, siderophore-producing bacteria readily evolve to reduce siderophore production if exposed to extremely iron-limited conditions, but complete loss of siderophore production was rarely observed [46]. The authors of that study considered reduction in resource production a type of defecting. However, if cooperation is defined as the production of a common good—and not the amount of common good produced—our results (Fig 5B) suggest that evolution to a lower resource release rate might be an effective way for cooperators to survive non-producing defectors. Intriguingly, they also found that reduced investment in siderophore production was not associated with reduced population growth rates, suggesting that the evolved populations may have also adapted to their nutrient-limited environment. Populations with high defector load show decreased growth rates and eventually go extinct because defectors impose severe resource limitation. In microbial populations, resource limitation will rapidly select for mutants adapted to this new environment [42,44,70]. Despite this fact, most models have only considered migration rate [53] and investment in resource production [20,25,71] as evolvable parameters. One group modeled what would happen if cooperators and defectors had to adapt to an evolutionary pressure unrelated to cooperation, such as a predator or parasite [72]. They found positive frequency-dependent effects: if cooperators were initially dominant, it was harder for defectors to invade, and vice versa. They speculated that this occurred because the numerically dominant population was more likely to attain a better fitness-enhancing mutation than the minority population [72]. We found that externally-generated selective pressures were not necessary to help the survival of cooperation. Instead, the rise of defectors could create an environment favoring mutants adapted to defector-induced resource limitation. Mutants adapted to resource limitation probably exist in any microbial population of sufficient size [42,65–67]. We did not model multiple types of mutants with fitness differences because mutants with smaller fitness advantages would be displaced by mutants with larger fitness advantages. We propose that only the population size of the “best” type of mutant will determine the survival of cooperation. We therefore predict that the population size of the mutant conferring the largest fitness advantage will determine the overall frequency of the survival of cooperation. If these mutants pre-exist at a low level (Fig 7A and 7B), or occur at a moderate rate (Fig 7C and 7D and S7A Fig), they can promote cooperator survival. The mutation scenarios assumed in our models are likely to occur. Theoretical [65] work and experiments with bacteria [66] and viruses [73] suggest that the fitness distribution of adaptive mutations may be exponentially distributed and invariant to the mean fitness of the population regardless of biological details. Recent work suggests that the fitness distribution of adaptive mutations is exponential for modest fitness gains, but exhibits a flat tail for highly adaptive mutants [67]. Thus, high-fitness mutants are likely to pre-exist [48] regardless of the mean fitness of the population, but at low levels due to their rarity and the high probability that they will be less fit than the ancestor in the ancestral environment. The large range (several orders of magnitude) of moderate u* capable of promoting cooperation is also likely to be met in reality. Thus, in sufficiently large populations, selection for high-fitness mutants could in principle help the survival of cooperation through repeated rounds of defector-induced collapses, even as the population evolves improved starvation tolerance. We join others in advocating for modeling techniques that narrow the gap between theory and experiment by basing parameters on physically measurable quantities [36,74]. Our model assumptions were based on measurable physical parameters, such as birth, death, and release and consumption rates. While we chose specific parameters to model mutants with fitness trade-offs, our modeling framework is general, and is designed to handle arbitrary numbers of types with arbitrary fitness functions, and can therefore be used to model many types of ecological interactions.
  The modeling framework was written in Java and is available on GitHub at https: //github. com/nodice73/metapop. All data was analyzed with code written in the R programming language [75] and is available in the file' metapop. r' , which is included in the source tree for the simulation framework. Each simulation was assigned a “universally unique identifier” (UUID) by placing the output of the simulation into a folder named with this UUID. A separate file produced by the script running the simulation recorded the arguments used to run the simulation. In addition to the parameters used for the simulation, these arguments also included the set of random numbers used to seed the appropriate random number generators, as well as the location of the simulation (which included the UUID). The UUID of a simulation was used to look up the set of seeds which initialized its RNGs (Random Number Generators). These seeds were used to exactly replicate the output of the original run while, for example, reducing the amount of time it was run or increasing the frequency of output data. These re-runs also received their own UUIDs.
summary: When defectors appear in a population of cooperators due to mutation or migration, defectors should become more common because they exploit the resource provided by cooperators without making a fair contribution to its production. The environmental degradation caused by defectors eventually leads to population collapse, and should result in the extinction of cooperators and defectors alike. By explicitly tracking all individuals in a spatially-structured environment where they can migrate to different locations, we find that cooperation can survive due to under-appreciated consequences of defector-induced population collapse. Population collapse creates vacant territories which can support the growth of cooperator but not defector migrants. This leads to the counterintuitive result that cooperation sometimes survives better when defectors are initially present compared to when they are initially absent. When defectors are initially present, the emergence of mutants best adapted to defector-degraded environments can also promote cooperation. Thus, by reducing population size through physiological and evolutionary mechanisms, defectors create “built in” mechanisms that protect cooperation from defectors. This may help explain the near ubiquity of cooperation despite its inherent vulnerability.
title: Defectors Can Create Conditions That Rescue Cooperation