# Stabilization of extensive fine-scale diversity by ecologically driven spatiotemporal chaos

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Contributed by Daniel S. Fisher, April 30, 2020 (sent for review September 4, 2019; reviewed by Oskar Hallatschek and Boris I. Shraiman)

## Significance

The diversity of living organisms is a fundamental feature of biology, yet still challenging to understand. Genomic data reveal that many closely related strains of bacteria can coexist for far longer than expected from “survival of the fittest.” To explain this, is it necessary to assume a multitude of specialist niches, or are other mechanisms possible? We investigate, from simple but general models, an alternate scenario: coexistence of a large number of closely related strains in a chaotic state driven by ecological interactions. The chaos is characterized by local blooms and busts that are out of sync at different spatial locations. A natural example is multiple nonspecialist strains of a pathogen species infecting multiple strains of a host species.

## Abstract

It has recently become apparent that the diversity of microbial life extends far below the species level to the finest scales of genetic differences. Remarkably, extensive fine-scale diversity can coexist spatially. How is this diversity stable on long timescales, despite selective or ecological differences and other evolutionary processes? Most work has focused on stable coexistence or assumed ecological neutrality. We present an alternative: extensive diversity maintained by ecologically driven spatiotemporal chaos, with no assumptions about niches or other specialist differences between strains. We study generalized Lotka–Volterra models with antisymmetric correlations in the interactions inspired by multiple pathogen strains infecting multiple host strains. Generally, these exhibit chaos with increasingly wild population fluctuations driving extinctions. But the simplest spatial structure, many identical islands with migration between them, stabilizes a diverse chaotic state. Some strains (subspecies) go globally extinct, but many persist for times exponentially long in the number of islands. All persistent strains have episodic local blooms to high abundance, crucial for their persistence as, for many, their average population growth rate is negative. Snapshots of the abundance distribution show a power law at intermediate abundances that is essentially indistinguishable from the neutral theory of ecology. But the dynamics of the large populations are much faster than birth–death fluctuations. We argue that this spatiotemporally chaotic “phase” should exist in a wide range of models, and that even in rapidly mixed systems, longer-lived spores could similarly stabilize a diverse chaotic phase.

Enormous diversity of species is one of the remarkable features of life on Earth. Once established by evolution, this diversity is traditionally explained in terms of niches and geographical separation. But spatial coexistence of a wide variety of species that seem to occupy similar niches is still a major puzzle (1).

Recent studies have found that even within individual microbial species [e.g., *Vibrio* (2) in the ocean, *Synechococcus* in hot springs (3, 4), *Staphylococcus epidermidis* on human skin (5, 6), *Neisseria* on the tongue (7), and *Bacteroides vulgatus* in guts (8)], many strains differing genetically on a broad spectrum of scales can coexist in nearby spatial locations. This fine-scale (or micro) diversity is especially surprising when strains mix together and are, hence, forced to compete. In some cases, the relevant mixing times are known: for the most abundant phytoplankton species, *Prochlorococcus*, which dominates tropical midoceans (9), a single sample contains hundreds of strains which diverged over timescales much longer than ocean-mixing times (10).

Why doesn’t survival of the fittest eliminate fine-scale diversity on timescales that are long compared to generation or spatial mixing times, but still short compared to the evolutionary timescales over which the diversity must have evolved and been maintained? To understand this, is it necessary to interpret the strains, substrains, and sub-substrains as “ecotypes” (11) adapted to microniches and differing phenotypically in essential ways? Or might there be more general explanations? Any satisfying theory should lead to understanding of how the statistical structure of diversity—not just its existence—arises and is maintained by evolution.

Microbial diversity is often characterized by abundance distributions. Abundances typically range over many orders of magnitude extending down to very rare species and are often fit by power laws (12, 13). However, data on abundance distributions of within-species fine-scale diversity are limited (3, 4, 7). Furthermore, dynamical data are crucial to distinguish scenarios—as we shall discuss—but are much harder to come by due to the need for deeply sequenced time-series data, as in, e.g., refs. 6 and 7. Is local fine-scale diversity relatively stable? Or are blooms from low to high abundance and back down—observed in a variety of contexts (14, 15)—more the norm?

Theoretical ecologists have long endeavored to discover the ingredients necessary for diversity to persist in complex ecosystems. Most theoretical work has focused on “ecologically stable diversity” (reviewed in ref. 16). Common approaches include modeling interactions via competition for resources (17⇓⇓⇓–21) and approximating interspecies interactions as “random” (22⇓⇓⇓⇓–27). A standard assumption is that species, or strains, compete with themselves and, hence, suppress their own population growth, more strongly than they interact with other strains: this is equivalent to assuming that each strain has its own niche. Niches may be a good starting point for modeling interactions among different species. But for closely related strains, there is no obvious reason why the interactions between “siblings” should be much stronger than between distant “cousins.”

The opposite extreme to niche-based approaches is the “neutral theory of biodiversity” (28⇓–30), which posits that broad classes of species—such as all trees in some region—are ecologically equivalent. Species abundances fluctuate due to neutral birth, death, and migration processes instead of being stabilized by ecological interactions. The balance between these results in a broad distribution of abundances with a simple power-law tail (31) that compares well, at least semiquantitatively, to data on a variety of systems, such as marine plankton (13), diatoms of similar sizes (32), and trees in tropical forests (33).

But for microbial diversity, there is a major quantitative problem with neutral theory. The large sizes of microbial populations mean that birth–death fluctuations are far too slow to dominate population dynamics. And the short generation times mean that even tiny selective differences will be greatly amplified on modest timescales (34). To produce broad distributions from faster dynamics, recent work has generalized effective neutrality by considering species with different responses to a fluctuating environment, but still neutral time-averaged fitnesses (35, 36). But this does not solve the problem of fluctuating to extinction, or of how the average neutrality might arise.

A different approach, which we take here, is to ask whether ecological interactions and fitness differences can drive continually changing abundances in a state of “chaotic coexistence.” As recognized early by May (37), a range of simple deterministic ecological models exhibit abundances that vary chaotically (38). It has been proposed that chaotic coexistence is important for plankton biodiversity (39, 40), and chaos has been demonstrated in a long-term experiment of an isolated planktonic food web (41). In nature, large changes in the relative abundances of viral and microbial strains have been observed in controlled, aquatic environments (42), and cycling between being abundant and rare is commonly observed in oceanic bacterial taxa (43). Might chaos be a generic feature of complex ecosystems (44, 45)? Can chaos promote or stabilize extensive diversity among close relatives? General theoretical analyses have shown that spatiotemporal environmental fluctuations can prevent extinctions and lead to coexistence (46, 47). And particular studies have found that chaos reduces extinctions in spatially extended populations (48, 49), but the chaotic population dynamics were put in by hand for a single species rather than arising from interactions. It is unclear if these mechanisms can stabilize extensive diversity.

It is often said that pathogens promote diversity (50). Pathogens surely contribute to continual evolution: a ubiquitous driving force is the advantage of evolving to “kill-the-winner” via predation, pathogenicity, or evolutionary arms races with relatives such as among *Streptoomyces* bacteria (51). Yet, it is far from clear whether evolutionary kill-the-winner dynamics enhances or decreases diversity.

Population dynamics driven by the advantages of doing well against currently abundant strains also occurs on ecological timescales; this will be our focus here. For pathogens, increasing abundance of a host strain can result in increasing abundance of particular pathogen strains: this limits the successful host population, driving it down, thereby limiting those recently successful pathogen strains as well. The original predator–prey Lotka–Volterra (LV) model demonstrated that such dynamics can lead to coexistence with periodic variations of the predator and prey populations. What about host–pathogen dynamics with many interacting populations? Models with specialist pathogens can lead to stable static communities (52, 53), as indeed does slight modification of the original LV model. Coupled predator–prey cycles have also been studied (54, 55), but, generically, multiple coupled cycles tend to lead to chaos.

A major drawback of most models of host–pathogen dynamics is that they do not account for broadly nonspecialist pathogens. Assuming all are specialists is equivalent to giving each strain its own niche. This may be reasonable for multispecies communities, but it is not for understanding within-species fine-scale diversity. Indeed, phages are often found to infect related, but phenotypically distinct, bacterial strains (56, 57). A crucial need for understanding fine-scale diversity is consideration of models with many strains of a single host species and many strains of a single pathogen species with broad, but varying, infectivity.

For microbial populations, evolutionary and ecological timescales overlap, and diversity is always being produced. However, there is an important question of principle—as well as, quantitatively, in practice: Does evolution generally lead to extensive spatially coexisting fine-scale diversity that would persist—in the absence of further evolution—for much longer than ecological timescales? A first step is to understand whether and how—without special assumptions—such diversity could persist. Can host–pathogen, or more general kill-the-winner, ecological dynamics stabilize fine-scale diversity on long timescales?

We analyze a general class of LV models and begin by investigating the nature of a diverse chaotic state that exists in a special case (58). However, in the generic case, chaos drives a cascade of extinctions that destroys the diversity. We then add the simplest form of spatial structure—identical islands with migration between them—and show that this leads to a robustly stable spatiotemporally chaotic “phase” that we argue should occur far more generally.

## Models of Complex Diverse Ecosystems

We focus on generalized LV models widely used to model ecological interactions between species. For competing species, the dynamics of the population size,

This implicitly assumes that each species has its own “niche” and treats the interactions between species differently than interactions within species.

For closely related substrains, our focus, there is no a priori reason that interactions with the same strain are particularly different from with other strains. It is useful to separate out the overall constraints from use of the same resources, which will keep the total population of the K strains, *s*elective differences—assumed much smaller than the average growth rate,

Competition for resources will result in positive correlations between how strains interact with each other—e.g., between *SI Appendix*, section 4 discusses how the results derived here can be generalized to the bacteria–phage model.

For simplicity, we focus on interactions between strains of a single species. Being sums and differences between similar interactions and similar growth rates, it is then natural to treat the

### Niche-Like Interactions and Large Stable Communities.

If, in spite of the similarities between strains, competition with others of the same strain are stronger—niche-like interactions—the diagonal terms would be negative on average, and we thus define

What happens when there is no large stable community? Our focus will primarily be on

## Chaotic Ecological Dynamics

### Perfectly Antisymmetric LV.

The idealized model of perfectly antisymmetric interactions (*SI Appendix*, section 3.A, which greatly facilitate its analysis. We here summarize its key features—see Figs. 1 and 4.

The antisymmetric model (ASM) has a family of stable chaotic states in which a fraction of the initial K strains—a unique set we call the “persistent strains”—coexist in a chaotic steady state, while the others have gone extinct and cannot reinvade. In steady state, each strain is characterized by its invasion growth rate,

The effects of the interactions make the growth rates vary with time, so the logarithms of the fractional abundances *B*. This “kill-the-winner” dynamics causes the peaks to typically last only for a short time,

The wildly fluctuating growth rates for large Θ cause the log-abundances of the persistent strains to undergo superdiffusive random walks, seen in Fig. 1*A*. The statistical properties of these random walks turns out to be crucial and much more general than the ASM. Eventually, on a long timescale—of order the equilibration time of the chaotic state,

The essential nature of the ASM is only revealed by its dynamics, but a snapshot of the distribution of abundances of the persistent strains shows their spread over a logarithmically wide range. As shown in *SI Appendix*, section 3.A, the full joint distribution of abundances can be found exactly. For large K, the abundances are essentially independent, and the probability density of the abundance of strain i has the simple form:

Intriguingly, the form of Eq. **4**, as well as the joint distribution of all of the *Dynamical Mean-Field Theory*, with details relegated to *SI Appendix*, section 3.

### Diverging Fluctuations for Near-Antisymmetric Interactions.

When *SI Appendix*, section 3.F, continues to amplify the fluctuations, eventually driving extinctions, as shown in *SI Appendix*, Fig. S3.

Our analyses have focused on γ substantially negative. But in parallel work, Roy et al. (62) have studied the case

### Chaotic Coexistence Stabilized by Migration.

In a well-mixed population, ecological chaos for

Instead of a mainland, we consider the simplest spatial model: a large set of I islands (or “demes”) with migration between all of them. For each strain, this presents a large pool of potential immigrants, but allows for global extinction of its whole pool. The islands are identical (strains interact via the same matrix V on each), and there is no additional spatial structure: migration rates between any pair of islands are all of the same with rate m per individual out of each island. The dynamics of the abundance, **2**, with an additional term for the net migration of individuals,

Local extinctions occur when the population of a strain on an island decreases below one individual, i.e., when its fractional abundance

Even with migration, some strains go extinct globally, as shown in Fig. 2*A*. But for a wide range of m, a majority of strains persist for long times in spite of some local extinctions; Fig. 3. The dynamics of the abundances of a single persistent strain across all of the islands is shown in Fig. 2*B*. While much of the time, the population on an island is near the migration floor, as indicated, on each island there are local blooms up to high abundance which produce enough emigrants to other islands to avoid local extinctions and enable later blooms on the other islands. Indeed, even a small initial population on a single island can bloom and seed other islands, leading to long term persistence, as shown in Fig. 2*C*. Global extinctions only occur when such blooms are too rare.

A crucial feature that enables global persistence is that the chaos on different islands is desynchronized. This occurs if the migration rate is less than the largest Lyapunov exponent on a single island (64):

The key feature that makes coexistence of a majority of the strains possible is that even strains with negative

A snapshot of the abundances of all of the strains on a single island, Fig. 4, shows the broad spread of the log-abundances **4**), but here over a range *Inset*).

Much of our theoretical understanding of the spatiotemporal chaos is obtained from the limit of many strains and many islands, for which the dynamical mean-field analysis developed in the next section is asymptotically exact. But understanding the statistical dynamics of the blooms in this limit is the key to understanding how and for how long strains with negative *Global Extinctions*. Strains can thus persist for exponentially long times, as shown in Fig. 5. Quantitatively,**14**.

### Dynamical Mean-Field Theory.

The dynamics of the multistrain models are intractable, even for the special single-island ASM. However, we can take advantage of the large number of strains and the random nature of the interactions among them and use an approach developed for other random systems: dynamical mean-field theory (DMFT) (65). This is valid when **3**). The DMFT approach approximates the population dynamics of a strain of interest, i, which is deterministically driven by its interaction with many others, by an effective stochastic integro-differential equation for that strain alone on a single island:*Discussion* and *SI Appendix*, section 3.B.

The crucial features are the effects on strain i of interactions with the other strains: these have two contributions. The first, and intuitive, part, is an effectively random time-dependent drive—i.e., instantaneous growth rate—*when strain* i *is absent*,

The second effect of interactions occurs when

### DMFT for Antisymmetric Single-Island Model.

The mathematical analysis of *SI Appendix*, sections 3.D and 3.E reveals the full quantitative properties of the self-consistent solution of the DMFT for the single-island ASM. But the key features of the strongly chaotic steady state for large Θ are relatively intuitive once one knows it exists: this relies on the special properties elucidated in *SI Appendix*, section 3.A.

Each persistent strain has a—distinct—nonzero average abundance,

Only strains with positive bias persist. Thus, since the **6** shows that the average abundance of the persistent strains is simply proportional to their bias*SI Appendix*, section 3.B.

Kill-the-winner feedback is represented by (for **6**, which prevents each *SI Appendix*, section 3.E). Far away from its own peaks, when

Since abundances are distributed broadly on a log scale, the sums over strains determining *SI Appendix*, section 3.D, we show that the self-consistency condition for

On longer timescales, **4**).

With *SI Appendix*, section 3.F) using the DMFT solution for the ASM to describe how the temperature scale Θ gradually increases with time (*SI Appendix*, Fig. S3).

## Analysis of Spatiotemporal Chaos

We now analyze the spatiotemporally chaotic behavior that occurs in the island model for *Diverging Fluctuations for Near-Antisymmetric Interactions*. Since the distribution of the average drives,

Strains with large positive biases have dynamics that are minimally affected by the small rates of migration and are quantitatively similar to the ASM. In particular, the statistics of the peaks will be similar and produce correlation,

The magnitude of the chaos on each island can, as for the ASM, be characterized by a logarithmic scale, Θ. However, in contrast to the ASM, Θ, has a unique value in the spatiotemporally chaotic state. This can be estimated from marginal strains with biases near zero. With no bias, their log abundances will spread out roughly uniformly on a logarithmic scale from the peaks down to the migration floor: thus, over a range

### Persistence and Averages with Many Islands.

In the limit of infinitely many islands, in steady state, the island average

The dependence of the island average on the bias

For strains with large negative biases, abundances typically hover around the migration floor at *SI Appendix*, section 5.A. The power-law correlations of the drive

It would appear that the integral in Eq. **11** diverges at large

The negative biases are typically of order

An analysis of very negative biases in *SI Appendix*, section 5.A shows that for M large, in principle, only a very small fraction of order

We next analyze the collective fluctuations that can lead to global extinctions when the number of islands is finite.

### Global Extinctions.

With a finite number of islands, the island average, *SI Appendix*, section 5.B and discussed in parallel work (62). In simulations, for 10 islands,

When the population size on each island is finite, the migration floor,

The extinction probability for a strain can be estimated heuristically from its bloom probability. An unlucky fluctuation of the migration floor from *SI Appendix*, section 5.C.) Blooms of sufficient size for recovery (**12**. Given that blooms on different islands, or at well-separated times, are roughly independent, the probability of extinction happening when no large blooms occur for a time *Inset*).

## Discussion

We have shown that antisymmetric correlations in the LV interaction matrix, together with simple spatial structure, are sufficient to stabilize extensive diversity of an assembled community. No niche-like assumptions or special properties are needed. This spatiotemporally chaotic “phase” is very robust; the key ingredient is the negative feedback induced by the antisymmetric correlations and sufficient—albeit very small—migration. While some fraction of the strains go deterministically extinct, a majority persist for very long times. This includes—crucially—strains with substantially negative average growth rate, which, in a well-mixed population, would guarantee their extinction. The key to their persistence with spatial structure is that each strain occasionally has a local bloom to high abundance which provides migrants to the other islands. As these blooms are nearly independent from island to island, global extinctions occur only if blooms do not happen on any of the islands for a sufficiently long period. This makes the spatiotemporally chaotic phase very stable, even for modest numbers of islands.

Much of our analysis has focused on the asymptotic, but unrealistic, regime when logarithmic functions of the population size and the migration rate are large. This has enabled us to obtain many results in a general framework based on DMFT. Yet our simulations show that the predicted behaviors are correct, even semiquantitatively, for modest sizes of parameters. A crucial prediction is the exponential scaling of survival times with the number of islands, illustrated in Fig. 5, for realistic parameters: total population per island,

### Generalizations.

The diverse spatiotemporally chaotic phase should exist far beyond the models we have analyzed. In *SI Appendix*, section 6, we discuss the behavior of the general random LV model with niche-like interactions (parameterized by *SI Appendix*, section 3.B. But the qualitative behavior is unchanged.

More generally, the spatiotemporally chaotic phase should exist with sparse interactions, broad distributions of interaction strengths, correlations due to phylogenetic relatedness of strains, or with some variations between islands. Indeed, even overall antisymmetric correlations of the interactions are not essential. The key is the absence of strains that always outcompete most other strains, and that, for each strain, there are some ecological interactions that provide negative feedback, preventing it from persisting at high abundance. We showed how this occurs for antisymmetrically correlated interactions, but this can also occur with moderately strong niche-like competition with its own strain (

#### Extrinsically driven chaos.

Dynamics with large swings of local abundances are crucial for stabilizing the spatiotemporally chaotic phase. But these need not be driven primarily by complex ecology. Extrinsic spatiotemporal environmental fluctuations, to which different strains respond differently, can cause blooms on different islands roughly independently. Blooms will be limited by the overall carrying capacity on each island, but no strain-specific interactions are needed. Migration can then stabilize the coexistence of diverse strains, even if they do badly on average, i.e., with time-averaged fitness, analogous to their *global*-invasion eigenvalue (as analyzed for two islands in *SI Appendix*, section 5.B). When its global-invasion eigenvalue is positive, a strain can invade from low abundance and persist (47).

Although much of the behavior is similar, there is a crucial distinction between chaotic dynamics driven by intrinsic ecological interactions between multiple strains and that driven by extrinsic spatiotemporal variations. For ecologically driven chaos, the magnitude of the dynamical variations in growth rates relative to their averages “self-tunes” the statistics of blooms to yield stable coexistence. Such a balance will generally not occur for extrinsically driven spatiotemporal chaos: extensive coexistence will depend sensitively on parameters and is by no means assured. [Indeed, it has been shown that the storage effect’s contribution to the global invasion eigenvalue decreases with the number of strains (47).]

In general, environmental variations will occur alongside complex ecological interactions. And the resulting behavior can be complicated; see, for example, ref. 68. But the analytical methods developed in this paper for treating the dynamics of multiple strains should be useful for developing more general understanding of the complex interplay of predation, competition, and environmental fluctuations.

#### Abundance distributions.

A key quantitative characteristic of the spatiotemporal chaos is a general consequence of the dramatic fluctuations of local populations. “Snapshots” of the local abundances will be broadly distributed on a logarithmic scale (Fig. 4). How universal are such abundance distributions? At low abundance, they will be affected by details of migration, while at high abundance, different mechanisms for limiting blooms (*SI Appendix*, section 6), such as strong niche-like interactions, or some strains doing atypically well on average (*SI Appendix*, section 3.B), will change the abundance distribution. But the wide intermediate range of abundances will be far more universal—with dynamics like the seemingly special perfectly antisymmetric model! The lack of universality at both high and low abundance means that summary statistics like the “Shannon diversity” (entropy of the distribution) or “species richness” (total number of strains observed) are poor characterizations of the abundance distributions: the former is dominated by high-abundance strains and the latter by very-low-abundance strains.

The more universal intermediate part of the abundance distribution will be approximately a power law with exponent one. This is the same as from the neutral theory of ecology (31). In fact, a neutral model with immigration from a mainland at strain-dependent rates

#### Bacteria-phage strain-level diversity.

The most natural context for antisymmetric correlations in the interactions between multiple types is diverse strains of both a host species and a generalist pathogen species, especially bacteria and phage. Models of such systems with perfectly antisymmetric (*SI Appendix*, section 4 enables understanding of large numbers of strains with broad, randomly varying infectivity. The primary additional parameter is the ratio of timescales over which the differences between strains result in substantial abundance changes for the bacterial strains vs. that for the phage strains. If these ecological timescales are identical, the mean-field dynamics of both the bacterial and phage strains is identical to the ASM. If the timescales differ substantially, we expect quantitative, but not qualitative, differences. Spatial structure obviates the need for perfect antisymmetry—unrealistic in any case—and our scenario and analysis of a spatiotemporally chaotic phase should hold for generalist bacteria-phage models.

Dynamical diversity from bacteria–phage interactions has recently been studied by other approaches. For example, some explicit kill-the-winner models treat phage predation as stochastic events leading to bacterial population collapse (73). While these exhibit persistent diversity, it is unclear whether there is a reasonable underlying population dynamics that could give rise to the caricature used. The advantage of models with explicit population dynamics for bacteria and phage is that avoidance of extinctions cannot be put in “by hand.” Furthermore, models of the strain interactions as we have studied naturally allow for nonspecific interactions, instead of the one-bacteria-one-phage scenario of the original kill-the-winner models (52, 53, 74), which is destabilized by demographic stochasticity. Indeed, nonspecific interactions may well be needed to stabilize a chaotic phase (75).

### Extensions.

Our conceptual and analytical frameworks should enable biologically relevant extensions in various directions.

#### Realistic spatial structure.

The effects of real spatial structure surely merit exploration. With conditions being the same everywhere, but transport either by local diffusion or occasional long-distance dispersal by wind, ocean currents, or hitchhiking on animals (76), the process of recovering from local extinctions is more complex, as it must involve spatial propagation of repopulation “fronts.” With a small number of strains having cyclic “rock–paper–scissors” interactions, spatiotemporally chaotic coexistence has been shown with repopulation fronts taking the form of spiral waves. With many similar strains, can a substantial fraction survive globally in a chaotic state, even in a system of infinite spatial extent?

#### Microbial spores or “seedbanks.”

When faced with unfavorable environmental conditions, many microbes enter a reversible state of dormancy by forming spores or other resting structures (77). The formation of long-lived—but not immortal—spores is another mechanism that can stabilize a diverse ecologically chaotic phase—indeed, even in a well-mixed system with no spatial structure. If some fraction of each strain forms spores that die off slowly, but occasionally germinate to produce actively dividing cells, the spore population effectively averages the chaotic dynamics of the active cell populations over time. This is analogous to averaging over a set of islands. For populations with large fluctuations driven by extrinsic environmental variations, this is the temporal “storage effect” (46). The product of the average spore lifetime,

#### Phenotype models.

An unsatisfactory feature of LV models of many interacting strains is that the organisms do not interact via their phenotypic properties: their phenotype is defined by their interactions with others. There is a long history of more explicit phenotype modeling for species interacting via consumption of common resources—MacArthur consumer resource models and generalizations (18⇓⇓–21, 78). More generally, one can consider interactions via many chemicals in the environment, with the dynamics of the populations set by the chemical concentrations and the dynamics of the chemical concentrations set by the population sizes of the various strains. Our analysis should be readily extendable to at least simple versions of such models, most simply when they reduce to LV models, with interactions determined by the effect of one strain on the environment and the response of the other strain to this. Under what conditions will a spatiotemporally chaotic phase exist in ecological models with interactions only via modifications of the environment? And how much diversity can be stabilized by the interplay between chaos and migration? This is a productive avenue for future research. A more general question is how systems will behave with direct interaction between pairs of individuals—and thus, again, LV structure—but with the interactions determined by phenotypic properties of the two organisms (rather than an abstract

#### Evolution and ecology.

We have studied how a large number of closely related strains can coexist when assembled together, but can such communities of strains be evolved in the first place? Although for microbial populations, there is usually no clear separation between ecological and evolutionary timescales, coexisting strains are found to have a wide spectrum of genetic differences and, hence, times to their common ancestors. The most basic case—and the hardest scenario for extensive diversity to exist—is when evolution is the slowest process. In simple models of assembled communities, mutations to new strains can lead to collapse (78) or to buildup of diversity (79, 80), depending on how the new mutants’ interactions are modeled. Eco–evo models should characterize the strains by phenotypic properties, which are what evolves, and the interactions be determined by these, without strict tradeoffs [which are often assumed (20, 81)]. In an already-evolved population, there are likely few generalist mutations (higher

Can a highly diverse chaotic community evolve in general phenotype-based models? If so, will the system undergo continual Red Queen evolution with no systematic “improvement”? Or will the evolution tend to get slower and slower? The statistical properties conditioned on evolution will likely be quite different from those in an assembled community, described in ref. 26. What general features might emerge? And how might different scenarios—and different possible “phases” of the eco–evo dynamics—potentially be distinguished by data from real microbial populations?

### Data Availability Statement.

This is a purely theoretical paper; there are no data associated with it.

## Acknowledgments

A.A. was supported by a Bowes BioX fellowship and Stanford’s Center for Computational, Evolutionary, and Human Genomics. M.T.P. was supported by William R. and Sara Hart Kimball as a Stanford Graduate Fellow and by NSF Graduate Research Fellowship. DGE-114747. All authors were supported by NSF Grant PHY-1607606. We thank Giulio Biroli, Felix Roy, and Guy Bunin for valuable discussions and sharing the results of their parallel work. Computer resources were provided by the Stanford Research Computing Center’s Sherlock cluster.

## Footnotes

↵

^{1}M.T.P. and A.A. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: dsfisher{at}stanford.edu.

Author contributions: M.T.P., A.A., and D.S.F. designed research, performed research, and wrote the paper.

Reviewers: O.H., University of California Berkeley; and B.I.S., University of California Santa Barbara.

The authors declare no competing interest.

See online for related content such as Commentaries.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1915313117/-/DCSupplemental.

Published under the PNAS license.

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- T. Chawanya,
- K. Tokita

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- M. Vallade,
- B. Houchmandzadeh

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- F. Roy,
- G. Biroli,
- G. Bunin,
- C. Cammarota

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- R. H. MacArthur,
- E. O. Wilson

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- A. Pikovsky,
- M. Rosenblum,
- J. Kurths

- ↵
- H. Sompolinsky,
- A. Zippelius

- ↵
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- F. Roy,
- M. Barbier,
- G. Biroli,
- G. Bunin

- ↵
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- J. S. Weitz,
- H. Hartman,
- S. A. Levin

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- T. F. Thingstad,
- S. Våge,
- J. E. Storesund,
- R. A. Sandaa,
- J. Giske

- ↵
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- O. Hallatschek,
- D. S. Fisher

- ↵
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- N. Shoresh,
- M. Hegreness,
- R. Kishony

- ↵
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- E. T. Miller,
- C. A. Klausmeier

- ↵

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