Natural populations can contain multiple types of coexisting individuals. How does natural selection maintain such diversity within and across populations? A popular theoretical basis for the maintenance of diversity is cyclic dominance, illustrated by the rock-paper-scissor game. However, it appears difficult to find cyclic dominance in nature. Why is this the case? Focusing on continuously produced novel mutations, we theoretically addressed the rareness of cyclic dominance. We developed a model of an evolving population and studied the formation of cyclic dominance. Our results showed that the chance for cyclic dominance to emerge is lower when the newly introduced type is similar to existing types compared to the introduction of an unrelated type. This suggests that cyclic dominance is more likely to evolve through the assembly of unrelated types whereas it rarely evolves within a community of similar types.

Natural populations ranging from microbial communities to animal societies consist of many different individuals. Some individuals compete with each other to exploit a shared resource (

In cyclic dominance, each type dominates one type and it is in turn dominated by another type. An arrow points from the dominated toward the dominant type. (

A famous example of this type of cyclic dominance in biology is toxin production in

However, traditional theoretical work assumes a set of predefined cyclic dominance types without asking how they developed or came together. In ecosystems, the introduction of a new species through migration can lead to such cyclic dominance. In this context, it is often termed intransitive competition (

Forming a cyclic dominance from a single type is challenging because as soon as the second type arises, the dominance type will take over the whole population, driving extinction of the other type. Therefore, a third type must arise before the population loses either of the two previous types. Such a precise timing of the arrival of a new type is critical for developing cyclic dominance and it can occur when new types arise at a high frequency, either through high mutation rates, recombination, or immigration (

Once we introduced our model in more detail, we will show that the interaction of ecology and evolution leads to increasing population size. This increases the chances that cyclic dominance arises, but it remains rare. Next, we rationalize this finding: While the lifespans of cyclic and non-cyclic dominance triplets are similar, it is more difficult to form cyclic dominance compared to non-cyclic dominance. The underlying reason is that similarity between parental and offspring payoffs suppresses the formation of cyclic dominance. Finally, we discuss which genealogical structure can promote or suppress cyclic dominance.

Interactions between individuals affect their death or birth. A traditional model for describing an interacting population is the generalized Lotka-Volterra equation (

We built the model based on individual reaction rules

Formulating the competition death rate

For a large population size, the abundance

Once a new mutant type arises during reproduction, new interactions occur. To describe these new interactions, we draw new payoff values from the parental payoff with Gaussian noise

(

To construct a pairwise interaction network, we used the stability between two types. Hence the term interaction refers to the pairwise relationship, considering the stability between two types. There are four possible scenarios for stability (see Appendix 1):

Dominance of type

represented by

Dominance of type

represented by

Bistability:

represented by

Coexistence:

represented by

Constructing the interaction network, we can examine the formation and the collapse of cyclic dominance, as shown in

The population dynamics described in

For small

(

The proportions of cyclic and non-cyclic dominance triplets increase in the early dynamics and quickly saturate. Whereas population dynamics illustrates the formation of both cyclic and non-cyclic dominance, cyclic dominance is much rarer than non-cyclic dominance. To quantify this rareness of cyclic dominance, we measured the fraction

The rareness of cyclic dominance triplets may be caused by their shorter lifespan compared with that of non-cyclic dominance triplets. Thus, we investigated the lifespan of triplets first to understand the rareness of cyclic dominance. Once triplets arise in populations, we can identify them, and trace how long they persist. Lifespan distributions in the steady state of both cyclic and non-cyclic dominance triplets decayed algebraically. We plotted the complementary cumulative distribution functions (CCDFs), clearly revealing a power law decay, as shown in

(

Why is it more difficult for cyclic dominance to emerge than for non-cyclic dominance? One factor is that the conditions needed for an interaction matrix to provide cyclic dominance are more restrictive than those for non-cyclic dominance. For a matrix to reveal cyclic dominance, it is necessary that in each of the three columns, the three payoffs

The fraction ^{5} samples), which is lower than that of the assembly of thee uncorrelated types. This is because the payoffs of a new offspring are similar to its parental payoffs: the offspring is likely to have the same relationships with other types in a population as the parental type and any type dominated by the parent will also be dominated by the offspring. The triplets including these offspring and parental types are more likely to form non-cyclic dominance than a cyclic one. Hence the correlation between payoffs affects the fractions by which cyclic dominance emerges compared with non-cyclic dominance.

To check the effect of the correlation on emerging triplets, we measured the similarity between types as a proxy of the correlation between payoffs. We defined the trait vector

Between the last common ancestor and the present types, there are intermediary types accumulating mutations between them. Genealogies tell us who is whose parent, tracing back to the common ancestor of the observed types. From the genealogy, we can infer how many mutations were accumulated by each type and the time at which they diverged. If two types have only accumulated a few mutations from the most recent common ancestor, their payoffs are likely to be similar. Hence, the genealogy structure shapes the correlation between payoffs and affects the value of fraction _{1}, _{2},

(

We analyzed those mutational distances for cyclic and non-cyclic triplets found in the simulation at the steady-state. From the analysis, we inferred that the crucial parameter influencing the fraction

Payoff correlations must be developed before lineages become independent at _{1} and

In addition, numerically calculated minimizer genealogies almost completely suppressed the emergence of cyclic dominance

We also denoted three reference fractions: the maximizer genealogy (

Cyclic dominance is extremely interesting from a conceptual and theoretical perspective and it has thus been analyzed in great detail in mathematical biology (

We also examined the circumstances under which cyclic dominance can appear more frequently. While the probability of assembling such an interaction structure by chance in a payoff matrix with uncorrelated random entries is small, the probability to evolve such an interaction structure is even smaller. The inheritance of interactions from parent to offspring is a key mechanism shaping the correlations between payoffs and determines the formation of cyclic dominance. From our approach, we found that the introduction of an uncorrelated type is crucial for the formation of cyclic dominance triplets. Because the migration of new species can be interpreted as such an introduction, our results suggest that cyclic dominance might be more frequent on an inter-species basis than on an intra-species basis. As widespread intransitive competition is found in ecological systems (

Our approach, which reduces the complexity from continuous values to a categorical classification, may help to bridge the model dynamics and experimental data more easily. Experimental work has provided data regarding both the constituents of a microbial community but also the interactions between them. However for large communities, parameterizing all interactions in the model numerically makes it difficult to identify the fundamental factors shaping the dynamics. Reducing the complexity may permit study of the large scales of experimental data connecting the underlying model dynamics and large datasets.

An important limitation of our work is the assumption of global interactions. In our model, all individuals can interact with each other, ignoring the spatial population structure. A spatial model could localize the interactions and lead to the more frequent formation of cyclic dominance. Such a localization can foster cyclic dominance for a predefined cyclic set (

HJP, YP, and AT thank the Max Planck Society for generous funding. HJP was also supported by (1) an appointment to the JRG Program at the APCTP through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government and by (2) the Korean Local Governments - Gyeongsangbuk-do Province and Pohang city.

No competing interests declared

Conceptualization, Data curation, Software, Formal analysis, Visualization, Writing - original draft, Writing - review and editing

Formal analysis, Investigation, Writing - review and editing

Conceptualization, Supervision, Writing - review and editing

Our simulation code is available at

For large population sizes, the change of abundances of types can be described by deterministic equations. Linear stability analysis reveals which types will persist in equilibrium (

To determine the stability between two types, type 1 and type 2, we focus on the two equations_{ij} are all positive, the population would always go extinct for

The extinction point (0,0) is always unstable, because both eigenvalues are identical to

We classify the pairwise relationship based on these stabilities. Dominance relationships are given if only a single-type fixed point is stable while all other fixed points are unstable. When both single-type fixed points are stable, a bistability relationship is drawn. A coexistence relationship is achieved when only the coexistence 1xed point is stable. We summarized the stabilities of 1xed points at a given condition and named the pairwise relationship in

Stable fixed points are marked by S, while unstable fixed points are marke by U. Extinction is always unstable, while the stabilities of other fixed points depend on conditions. For the condition in the first row,

Conditions | Fixed points | Relationship | |||
---|---|---|---|---|---|

U | S | U | U | Dominance of type 1 | |

U | U | S | U | Dominance of type 2 | |

U | S | S | U | Bistability | |

U | U | U | S | Coexistence |

In this section, we show how the population size

Large population size reduces the time until new types occur in the population, and at the same time it takes longer to equilibrate. Thus, a large population size increases the number of different types in the population via two processes: (i) Even when the most competitive type is expected to fixate in the population, new types which are less fit can constantly emerge due to mutation. (ii) A second mechanism is ecological tunneling (

We also take a closer look at the average payoff values. Because of selection, the average payoff increases in time, but the increment becomes smaller as the average increases, see

The average population size _{0}. Particularly, the average increases logarithmically in the stationary regime. The left inset is the same data as the main panel on a linear-log scale and shows the logarithmic increase of the average payoffs. The right inset is the probability distribution function of all payoffs in the steady state for

From the payoff matrix, we can determine the pairwise relationship between types, constructing a network. We represent the relationship between two types

As a basic element of a network, we investigate the frequencies of link types first.

As a reference, we consider a random payoff matrix model wherein all payoffs are randomly drawn from the standard normal distribution. In this case, the probability that one payoff is larger than another is 0.5 because all payoffs are independently sampled from the same distribution. Thus each condition in

With population dynamics, the behavior of link frequencies differs for different

Time series of the average link proportions in the stationary regime for small baseline death rates (

(

Dashed boxes marks sets of triplets, which have the same link composition, but a different structure.

Triad ID | Degeneracy | Proportions |
---|---|---|

1 | 1 | 4/108 |

2 | 3 | 3/108 |

3 | 3 | 3/108 |

4 | 1 | 4/108 |

5 | 6 | 12/108 |

6 | 6 | 3/108 |

7 | 6 | 12/108 |

8 | 6 | 12/108 |

9 | 3 | 12/108 |

10 | 6 | 6/108 |

11 | 3 | 3/108 |

12 | 3 | 3/108 |

13 | 6 | 6/108 |

14 | 3 | 12/108 |

15 | 2 | 1/108 |

16 | 6 | 12/108 |

Let us now consider triplets that are constructed by three links. With four kinds of link relationships, in total we can find 64 possible triplets. However, if we take into account symmetries, this reduces to 16 different triplet structure, see

Again we use a random payoff matrix to attain the reference triplet proportions first. For three types

All elements are independently drawn from the standard normal distribution

To determine the type of triplet structure, we focus on one type of individual

where

Thus we calculate the probability to find a certain set of subs for a type. The probability

In the same way,

With those probabilities and the degeneracies, we can calculate the triplet proportions expected in a random payoff matrix. The results are summarized in

Now we move to triplet proportions in population dynamics, see

The emergence of a new mutant type

(_{1}; _{2};

Genealogies can be schematically depicted with time and mutational distance axes, see

We begin with a payoff matrix of types

For a pair of types

Since the random variable

Thus, multiple mutation events can be written as a single mutation event with larger variance. For any other order of the mutation events, the final expression is the same even though all intermediate terms will be different.

In the same way, we can proceed with the other three payoffs. Consequently, all four payoffs can be written as

Self-interactions,

Next, we consider type _{2} is the number of mutations accumulated in the lineage of type

Hence, all nine payoffs describing interactions between types _{1} is the number of mutations accumulated from the type

The same calculations apply to any other genealogy, for example ones presented in

However, due to the genealogy structure, some payoffs have additional common ancestors later than

In this section, we numerically identify those genealogies which produce the largest or smallest proportions of certain triplets. First of all, we note that the fractions of triplets do not depend on scales of the set of mutational distances

We search for extreme genealogies by numerical optimization, implemented by a hill climbing algorithm. The optimization starts from a random set of _{m} and identifies the local optimum. Since our method can find only local optima, we use multiple initial values to find optimal set. After we identified the mutational distance sets which give local optima, we cluster them using principle component analysis (PCA). We denote the

First, we ask which genealogies maximize the number of cyclic dominance triplets. Performing PCA analysis, we found five different classes of such genealogies, see _{15} and non-cyclic dominance ones as _{16}. Classes

Median values of each of five mutational distances _{15}.

Class | Counts | Proportion of _{15} | _{1} | _{2} | |||
---|---|---|---|---|---|---|---|

5044 | 0.020 | 0.0020 | 0.98 | 0 | 0 | 0.010 | |

2012 | 0.019 | 0 | 0 | 0 | 0 | 0.98 | |

683 | 0.016 | 0 | 0.0021 | 0.0017 | 0.37 | 0.59 | |

2701 | 0.014 | 0.0058 | 0.32 | 0 | 0.23 | 0.41 | |

2160 | 0.016 | 0.0046 | 0.52 | 0 | 0 | 0.44 |

Next, we ask which genealogies instead minimize the number of non-cyclic dominance triplets, with the expectation that they will be similar to the ones above. Performing PCA analysis, we again find five different classes, see _{15} have the same characteristics of mutational distances with the genealogies maximizing _{16}, see section F. The genealogy class _{1} and _{16}.

Median values of _{m} are given. We use 20000 independent simulations.

Class | Counts | Proportion of _{16} | _{1} | _{2} | |||
---|---|---|---|---|---|---|---|

1576 | 0.11 | 0.0045 | 0.97 | 0 | 0 | 0.012 | |

697 | 0.11 | 0.0012 | 0.0021 | 0 | 0.011 | 0.96 | |

5883 | 0.11 | 0 | 0.29 | 0.019 | 0.22 | 0.45 | |

598 | 0.11 | 0.019 | 0.29 | 0.014 | 0.23 | 0.44 | |

11246 | 0.11 | 0.034 | 0.31 | 0 | 0.21 | 0.43 |

Alternatively, we can also ask for which genealogies it is hardest to obtain cyclic dominances. Performing PCA analysis, we find three different classes, see

Median values of _{m} are given. We use 20000 independent simulations.

Class | Counts | Fraction of _{15} | _{1} | _{2} | |||
---|---|---|---|---|---|---|---|

4995 | 0.97 | 0 | 0.018 | 0 | 0.0051 | ||

5011 | 0.014 | 0 | 0.97 | 0 | 0.0057 | ||

9994 | 0.00012 | 0.48 | 0 | 0.49 | 0.0 | 0.014 |

Now we ask for which genealogies it is easiest to obtain non-cyclic dominances. Performing PCA analysis, we find again three different classes of geneaolgies, see _{15}: The class

Median values of _{m} are given.

Class | Counts | Proportion of _{16} | _{1} | _{2} | |||
---|---|---|---|---|---|---|---|

^{16}-1 | 4772 | 0.24 | 0.97 | 0 | 0.025 | 0 | 0.0019 |

^{16}-2 | 11017 | 0.24 | 0.019 | 0 | 0.98 | 0 | 0.0023 |

^{16}-3 | 4211 | 0.24 | 0.45 | 0 | 0.53 | 0.0 | 0.0037 |

From all optimization results we find two extreme genealogies for maximizing and minimizing the fraction

The fraction

Matrix generation | _{15} | _{16} | |
---|---|---|---|

Random matrix | 0.0093 | 0.11 | 0.077 |

Maximizer | 0.02 | 0.11 | 0.154 |

Minimizer | < 10^{-4} | 0.24 | 0.000 |

We calculate the proportions of cyclic and non-cyclic dominance triplets at maximizer genealogies. In that case, the cyclic dominance triplets occur

Altogether, this results in _{16} has degeneracy six, and they occur

Altogether, this results in

(_{1} originates from the type _{2} mutated from _{1} has variance _{1}, and _{2}. Small _{1} and _{2} become uncorrelated.

Besides genealogies, there is also another way to shape the payoff correlation. Mutational distances are controlled by not only how many steps proceed but also how big the jump is related variance of noise. In a minimal model, we can directly control the closeness of new-born type by using a different variance _{1} from the original type _{1}. Then, we assume that _{1} is the parental type of type _{2}, and the variance

We investigated the correlation between diversity and the probability of having cyclic and non-cyclic dominance triplets in the system at the steady-state. We measured the Pearson correlation coefficient

We used three different diversity indices: (1) richness

In total 1944 realizations are used for the calculations and the average is across 500 time steps. We find a weak anti-correlation between the fraction of cyclic dominance and diversity while the fraction of non-cyclic dominance has stronger anti-correlation with diversity. Despite these differences, almost no correlation between the fraction and diversity is found.

Richness index | Shannon index | Simpson index | |
---|---|---|---|

-0.05 | -0.12 | -0.11 | |

-0.15 | -0.35 | -0.34 | |

-0.04 | -0.08 | -0.07 | |

-0.12 | -0.35 | -0.34 | |

0.01 | -0.01 | -0.004 |

Anti-correlation between the fraction of non-cyclic dominance

In the first and the second rows, we plot

(

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

The reviewers and I found your article to be an insightful perspective regarding why non-transitive (e.g. rock-paper-scissors) interactions might be rare in natural communities, despite the potential stabilizing effects that non-transitive interactions could have in sustaining diversity. In particular, your approach of analyzing the evolutionary origin of different interactions provides fascinating insight to an important problem.

Thank you for submitting your article "Why is cyclic dominance so rare?" for consideration by

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

We would like to draw your attention to changes in our revision policy that we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, we are asking editors to accept without delay manuscripts, like yours, that they judge can stand as

Summary:

Cyclic dominance has been proposed as a possible stabilizing mechanism for diversity in communities. However, empirical evidence has suggested that cyclic dominance appears to be rare. Why might this be? The authors' explanation for this observation, based on a dynamical eco-evo model and various theoretical arguments, can be summed up as follows:

– When drawing a network of qualitative dominance relations, it may seem that cyclic dominance could happen often (1/4th of possible triplets if relations are drawn at random).

– But when the interactions arise from an underlying matrix of payoffs, the conditions on payoffs that create cyclical dominance are actually quite constraining, happening at random only in 1/13th of triplets of dominance links (which themselves are only 1/10th of triplets, since dominance is not the only outcome: one can also observe coexistence or bistability).

– And when the payoffs themselves arise from a continuous evolutionary process (where new types are introduced with small changes in payoffs from their ancestors), the probability is even lower. The authors' simulations show that it occurs only in 1/30th of dominance triplets.

The authors provide arguments for why this probability is even lower: it is due to correlations between parents and offsprings. They show in their simulation results that cyclical dominance is associated with more difference between types (measured as an overall difference in their payoffs) than non-cyclical dominance. They also run a simple genealogy process for a 3-type subcommunity (where, instead of each mutation leading to a new type that competes with all the existing types, mutations are now accumulated along genealogical branches as if, for every mutation, the parent type is automatically replaced) to show that cyclical dominance can be favored when many mutations occur after the branching out of the 3 types, thus decorrelating their properties.

Essential revisions:

1) The authors argue the importance of studying cyclic dominance by noting its potential to increase diversity. This paper would, therefore, be strengthened by including an analysis of whether diversity increased in simulations in which cyclic dominance emerged (or at time points during which it was present) as compared to simulations or time points in/at which there was no cyclic dominance. If there is no correlation between the emergence of cyclic dominance and increases in diversity, the authors should address this in the Discussion section. Relatedly, the authors claim in the Discussion "our results indicate cyclic dominance can support diversity over long time scales". We do not believe the first half of this claim (that cyclic dominance can support diversity) is currently supported by their results (it is rather from other literature).

2) For readers to make sense of the triplet analysis it is necessary to know the link frequencies (e.g. between dominance, coexistence, and bistability). This can be brief (no need for a full supplementary figure in main text).

3) The authors argue that it is difficult to evolve cyclic dominance due to mutants being similar to parents. The paper would therefore be strengthened by including an analysis of how the mutation size sigma influences the frequency of cyclic dominance. This is shown for the toy model in Appendix 6, but not for the main model. Many readers would likely appreciate the results of varying sigma within the main model.

4) On a related note, the authors could directly measure and report on the time passed and/or number of mutations accrued since species in cyclic dominance triplets diverged from their last common ancestor and make appropriate comparisons. The current section "Genealogical structure can promote and can suppress cyclic dominance" could be moved to the Appendix if a direct tracing of the genealogy of triplets is added to the paper. (We found the genealogy section to be a bit difficult to parse).

5) The authors could include a brief analysis of how often the species in cyclic dominance triplets gain large population fractions after emerging. This question is relevant because if the three species in a triplet never reach large population fractions then their effects on each other would presumably be small compared to the effects from other members of the population (which would not affect any of the conclusions of this paper but may affect some readers' interpretations of those conclusions).

6) We believe that an outline of the whole argument (a bit like the summary that we give above) should be introduced early in the article, so that the structure is clear: currently, the text reads like a journey where we discover things on the way, but it is not always clear at first why we are doing one thing or another. In particular, something that may not be an issue for theoretical readers, but would be for a more general audience, is that it is not obvious at first which aspects of the model are going to be important or not for the argument, and therefore, whether the results are specific to the model assumptions. In summarizing the results above, we tried to present them in a way that clarifies that a large part of the argument could be made without invoking any specific simulation model. Readers may worry about the interpretation or lack of realism of some assumptions of the dynamics, when they do not really matter for the argument. This is an issue at different points in the paper: for the general eco-evo model, but also for the genealogy "mini-model", where a reader could find it concerning that the process is now different from the main simulation (the text makes it sound as if a type can accumulate mutations without branching out). It would be helpful for the genealogy section to introduce a quick outline of the argument it will make. We also suggest showing the Appendix 5—figure 1A in the main text, maybe as a panel in Figure 5. It would help understand what this mini-model assumes compared to the main simulation model (that all branches except for 3 have disappeared or can be ignored, thus explaining what it means to "accumulate mutations along a branch").

7) Regarding the main simulation model, there is in fact a concern that does not impact the results. A common issue in eco-evo modela is that the evolving traits will keep changing in the same direction, which is generally avoided by imposing some trade-off.

As the authors note, the average payoff here tends to increase indefinitely, which means that all competition coefficients d_{ij} eventually converge to just α. In ecological terms, we would call that going toward neutral dynamics (where all individuals have the same competition strength, both within and between types, see S.P. Hubbell's 2001 book). This issue is not deadly because, as differences of d_{ij} between types become smaller, all that really happens is that the time it takes for a type to exclude another diverges. But the qualitative nature of the relationships is not changed: there is still dominance or coexistence or bistability, even if by a very small margin. This is simply impractical because one must wait longer and longer for extinctions to actually happen (this is actually theorized in ecology: types could coexist by becoming so similar than one winning over the other takes forever, see Scheffer and van Nes, 2006). So it seems to me that the simulation would have been easier if the payoffs were relative, e.g. constantly setting the average at zero, so that interactions do not converge to neutrality.

In the same spirit:

"For small α values (rich environments) in particular, the population size N at the steady state becomes large, containing many different types " the value of α should change nothing except the biomass scale (population size N ~ M lambda/alpha – incidentally, why introduce parameter M at all? it is never explained). The fact that more types exist for lower α seems like an artefact, perhaps because it takes longer for interactions to tend toward neutrality.

8) Perhaps one could directly show how adding correlations in the matrix diminishes the occurrence of cyclical dominance? This would directly show that nothing else is needed (i.e. that the eco-evo process does nothing more than add these correlations).

9) It is worth noting that there is a large ecological literature about extensions of cyclic dominance to more than triplets, called intransitive competition. Some of that literature has claimed (although we are not necessarily convinced) that intransitive competition is actually common and important. We would suggest reading these articles and figuring out why their claims would be so different – and if that difference is important, then discussing it in the article.

See for instance as starting points: Soliveres et al., 2015; Gallien et al., 2017.

Essential revisions:

1) The authors argue the importance of studying cyclic dominance by noting its potential to increase diversity. This paper would, therefore, be strengthened by including an analysis of whether diversity increased in simulations in which cyclic dominance emerged (or at time points during which it was present) as compared to simulations or time points in/at which there was no cyclic dominance. If there is no correlation between the emergence of cyclic dominance and increases in diversity, the authors should address this in the Discussion section. Relatedly, the authors claim in the Discussion "our results indicate cyclic dominance can support diversity over long time scales". We do not believe the first half of this claim (that cyclic dominance can support diversity) is currently supported by their results (it is rather from other literature).

Thank you for this comment. Indeed, we have started out from the concept of diversity as it has often been stated as a crucial consequence of cyclic dominance. However, we have only shown the rareness of cyclic dominance and discussed mechanisms leading to it. As you pointed out, we did not explicitly show that cyclic dominance increases diversity. We briefly mentioned the lifetime of both cyclic and non-cyclic dominance triplets but we did not explicitly analyze correlations between triplets with diversity. We have now explored these correlations and added the detailed results in Appendix 7 and argued them in the Discussion. In a nutshell, we found that the fraction of cyclic dominance \chi has almost no correlation with diversity. This means – in contrast to general believe – that diversity is not mainly supported by the emergence of cyclic dominance in our model. However, the long time to equilibration with the removal of all non-competitive types and a short time to the emergence of new types are main drivers of diversity.

2) For readers to make sense of the triplet analysis it is necessary to know the link frequencies (e.g. between dominance, coexistence, and bistability). This can be brief (no need for a full supplementary figure in main text).

Agreed, we have added an explanation for link frequencies in the main text before the Results section. In addition, we have distinguished the terms “frequency” and “fraction”

preventing confusion. Frequency is only used when the population frequency is considered. In all other cases, we have used “proportion” or “fraction”.

3) The authors argue that it is difficult to evolve cyclic dominance due to mutants being similar to parents. The paper would therefore be strengthened by including an analysis of how the mutation size sigma influences the frequency of cyclic dominance. This is shown for the toy model in Appendix 6, but not for the main model. Many readers would likely appreciate the results of varying sigma within the main model.

In our toy model, the mutational distances, between o and M_{1} and between M_{1} and M_{2}, are controlled by different variances while they are controlled by the number of mutations in the main model. In both cases, the absolute scale does not affect the results but the ratio of mutational distances is important. Hence, changing sigma in the main model affects the overall mutational distance, but the ratio remains unchanged, keeping the fraction of cyclic is the same. We have clarified this by discussing two explicit sigma values in the toy model.

4) On a related note, the authors could directly measure and report on the time passed and/or number of mutations accrued since species in cyclic dominance triplets diverged from their last common ancestor and make appropriate comparisons. The current section "Genealogical structure can promote and can suppress cyclic dominance" could be moved to the Appendix if a direct tracing of the genealogy of triplets is added to the paper. (We found the genealogy section to be a bit difficult to parse).

Thank you. Following this suggestion, we have moved the previous genealogy analysis into the Appendix and instead mentioned the direct measure of genealogical properties found in the simulations. The key variable to find cyclic or non-cyclic triplets is the fraction of accumulated mutational distances before and after the payoff correlation appears during diverging process, which agrees well with the toy model result. We have included a new Figure 5 to explain it and have Introduced F_{l} defined by the fraction of mutational distances. Addressing major comment 6, in the new figure we first explained the genealogy. Then we show an example of genealogy for three types and the relationship between the fraction \chi and F__{l}. As many mutations are accumulated after each type evolves independently from others, the chance to emerge the cyclic dominance systematically increases.

5) The authors could include a brief analysis of how often the species in cyclic dominance triplets gain large population fractions after emerging. This question is relevant because if the three species in a triplet never reach large population fractions then their effects on each other would presumably be small compared to the effects from other members of the population (which would not affect any of the conclusions of this paper but may affect some readers' interpretations of those conclusions).

Thank you for this very interesting suggestion. We have measured the population frequencies which belong to cyclic and non-cyclic dominance, f_{cyc} and f_{ncyc}. On average, cyclic dominance takes around 33% of the population, while non-cyclic dominance takes around 83%. To check how often cyclic dominance can take more than a half population, we have checked the probability distribution function (PDF) of f_{cyc} and f_{ncyc} at the steady-state. Cyclic dominance usually takes very low population frequencies or rarely take the majority, showing a bimodal distribution with two peaks around 0 and 1. The peak near 0 is higher than the other, implying the cyclic dominance usually emerges in small groups. Also, we found that a chance that cyclic dominance takes more than a half of the population is around 30%. On the other hand, the non-cyclic dominance usually takes the majority and 85% chance to take over the half population. These results have addressed in a new Appendix 7 and have shortly mentioned it in the Discussion.

6) We believe that an outline of the whole argument (a bit like the summary that we give above) should be introduced early in the article, so that the structure is clear: currently, the text reads like a journey where we discover things on the way, but it is not always clear at first why we are doing one thing or another. In particular, something that may not be an issue for theoretical readers, but would be for a more general audience, is that it is not obvious at first which aspects of the model are going to be important or not for the argument, and therefore, whether the results are specific to the model assumptions. In summarizing the results above, we tried to present them in a way that clarifies that a large part of the argument could be made without invoking any specific simulation model. Readers may worry about the interpretation or lack of realism of some assumptions of the dynamics, when they do not really matter for the argument. This is an issue at different points in the paper: for the general eco-evo model, but also for the genealogy "mini-model", where a reader could find it concerning that the process is now different from the main simulation (the text makes it sound as if a type can accumulate mutations without branching out). It would be helpful for the genealogy section to introduce a quick outline of the argument it will make. We also suggest showing the Appendix 5—figure 1A in the main text, maybe as a panel in Figure 5. It would help understand what this mini-model assumes compared to the main simulation model (that all branches except for 3 have disappeared or can be ignored, thus explaining what it means to "accumulate mutations along a branch").

Thank you for this very helpful suggestion. Based on the summary provided by the reviewers, we have now added an outline after making a point of the rareness of cyclic dominance. Also, taking a suggestion, we have moved Appendix 5—figure 1A to a new Figure 5A in the main text.

7) Regarding the main simulation model, there is in fact a concern that does not impact the results. A common issue in eco-evo model is that the evolving traits will keep changing in the same direction, which is generally avoided by imposing some trade-off. As the authors note, the average payoff here tends to increase indefinitely, which means that all competition coefficients d_{ij} eventually converge to just α. In ecological terms, we would call that going toward neutral dynamics (where all individuals have the same competition strength, both within and between types, see S.P. Hubbell's 2001 book). This issue is not deadly because, as differences of d_{ij} between types become smaller, all that really happens is that the time it takes for a type to exclude another diverges. But the qualitative nature of the relationships is not changed: there is still dominance or coexistence or bistability, even if by a very small margin. This is simply impractical because one must wait longer and longer for extinctions to actually happen (this is actually theorized in ecology: types could coexist by becoming so similar than one winning over the other takes forever, see Scheffer and van Nes, 2006. So it seems to me that the simulation would have been easier if the payoffs were relative, e.g. constantly setting the average at zero, so that interactions do not converge to neutrality.

We agree with your point. In our model, species or types can coexist because the system is close to the neutral regime. However, our finding (rareness of cyclic dominance) is robust even with rescaling of payoffs. This is because the main driver of the diversity in that case is also not the formation of cyclic dominance. Cyclic dominance cannot be stably sustained, but appears because the system stays in non-equilibrium. Hence, we observe the emergence of cyclic dominance as a chance event only. We have also simulated a model with rescaling of payoff so that the mean payoff becomes zero. In this case the overall population size is controlled by the control parameter M with \alpha=0. Diversity is only observed for large population sizes (large M values), implying that the diversity originated from a large mutation supply. Cyclic dominance does not contribute to diversity, and thus the underlying mechanism found based on the matrix approach also holds. We have cited the paper Scheffer and van Nes, 2006, and have discussed the results with the rescaling method in the Discussion.

The payoffs are rescaled every mutation and extinction events so that the average of payoffs becomes zero. The population starts from a single-type population with the payoff zero. 2000 realizations are used for obtaining averages. At the steady-state, the average fractions are smaller than 0.02, which are smaller than that of our model.

In the same spirit:

"For small α values (rich environments) in particular, the population size N at the steady state becomes large, containing many different types" the value of α should change nothing except the biomass scale (population size N ~ M lambda/alpha – incidentally, why introduce parameter M at all? it is never explained). The fact that more types exist for lower α seems like an artefact, perhaps because it takes longer for interactions to tend toward neutrality.

We agree that this deserves a better justification. We originally introduced the two parameters α and M for two different reasons: When the interaction parameters A_{ij} become large, α ensures that the population size remains restricted – and the magnitude α controls the importance of competition in the absence of any game theoretical interactions. On the other hand, M gives us the possibility to control the impact of the overall competition term. However, since we only focused a single M value, we have now dropped M by rescaling α and shifting A_{ij} by _{lnM}. We now have used this alternative notion which hopefully reduces confusion, and thus parameters are recalculated while the results are robust.

8) Perhaps one could directly show how adding correlations in the matrix diminishes the occurrence of cyclical dominance? This would directly show that nothing else is needed (i.e. that the eco-evo process does nothing more than add these correlations).

In our model, the payoff correlations are an emergent property. The payoff matrix changes according to the simple rule with the correlation between parental and offspring types. But we do not impose any genealogies. Because the genealogy is an outcome of the eco-evo process, the whole structure of payoff correlations is also an outcome of the eco-evo process determining the fraction of emerging cyclic and non-cyclic dominance. Thus, introducing external correlations seems to come with a risk of introducing artefacts.

9) It is worth noting that there is a large ecological literature about extensions of cyclic dominance to more than triplets, called intransitive competition. Some of that literature has claimed (although we are not necessarily convinced) that intransitive competition is actually common and important. We would suggest reading these articles and figuring out why their claims would be so different – and if that difference is important, then discussing it in the article.

See for instance as starting points: Soliveres et al., 2015; Gallien et al., 2017.

Thank you for these pointers. First of all, we would like to say that the findings in those papers and in our manuscript are not mutually exclusive. In our model, the intransitive links always appear but it does not always connect to the emergence of cyclic triplets due to other link types such as coexistence and bistability. This was also observed in a soil bacterial community (Kotil and Vetsigian, 2018); the structure is hierarchical with some intransitive relationships, which do not lead to the cyclic dominance. Second, we considered non-equilibrium steady state while others have focused on equilibrium states. The cyclic dominance we studied stands out from other forms of intransitive competition since it is virtually unable to evolve from the equilibrium state. This a major difference may be that ecologists typically think about the interaction between different species and evolutionary biologists more in terms of the origin of the interacting partners. Reconciling both aspects, our manuscript supports the basic idea that assembly of unrelated types is more likely to lead to cyclic triplets than evolution, in which emerging types are closely related. We have addressed this literature in the Discussion.