Ecological diversification in rapidly evolving populations

Abstract

Microbial populations have an enormous capacity for rapid evolutionary change. Some mutations increase the fitness of their lineage and compete with each other in a process known as clonal interference. Other mutations can evade competitive exclusion by diversifying into distinct ecological niches. Both processes are frequently observed in natural and experimental settings, yet little is known about how they interact in the parameter regimes most relevant for microbes. Here we address this gap by analyzing the dynamics of ecological diversification in a simple class of resource competition models, where individuals acquire mutations that alter their resource uptake rates. We focus on large adapting populations, where mutations occur so frequently that their ecological and evolutionary timescales overlap. In this regime, we show that the competition between linked mutations causes the population to self-organize into a smaller number of distinct ecotypes, driven by an emergent priority effect that favors the resident strains. We demonstrate that these priority effects bias the long-term metabolic structure of the population, producing qualitative departures from existing ecological theory. We argue that similar dynamics should arise for other rapidly evolving ecosystems, where adaptive mutations accumulate at many linked genetic loci.

Introduction

Microbial populations can adapt to their local environment by rapidly acquiring new mutations. Many such mutations are subject to competitive exclusion, and will either take over the population or go extinct. However, some mutations can evade competitive exclusion by expanding into a different ecological niche. Striking examples of this behavior have been observed in microbial evolution experiments, where initially clonal populations spontaneously diversify into multiple coexisting ecotypes (1–12). The mechanism of coexistence can often be traced to subtle differences in resource use (7, 13), but similar effects can also arise through differences in predation (14–16), antagonism (16–18), and immunological memory (19, 20). While these ecotypes coexist with each other in the short-term, their relative balance can shift over longer evolutionary timescales as their descendants acquire further mutations. Evidence for such quasi-stable coexistence has been observed in a variety of natural (21–27) and experimental (11, 12, 28–32) settings. However, our theoretical understanding of this diversification process – and its impact on microbial evolution – remain poorly understood in comparison.

Most existing approaches for modeling ecological diversification assume that evolution is much slower than ecology (33–40). Mutations arise one-by-one, and their fates are determined by an ecological competition against the other resident strains in the population. While this weak-mutation limit has shed light on the conditions where ecological diversification is likely to occur, it is known to break down in microbial populations where ecotypes are commonly observed. In many cases of interest, from laboratory evolution experiments (28, 41, 42) to natural populations of viruses (43–45), bacteria (24, 46) and certain cancers (47), the size of the population is large enough that multiple new mutations will arise and compete with each other at the same time. When recombination is limited, the competition between these linked mutations (“clonal interference”; 48) ensures that the ecological fate of a new mutation will rarely be decided before the next mutation occurs. This leads to a more complex set of dynamics, in which the ecological and evolutionary timescales are inherently intertwined.

Clonal interference has been extensively studied in the absence of ecological interactions (48–56), but much less is understood in the more general case where ecotype structure can emerge. Several recent studies have started to explore these effects using computer simulations (18, 34, 57, 58). However, our limited theoretical understanding of these overlapping timescales leaves many basic questions unresolved: Does clonal interference tend to promote ecological diversification by producing a wider range of new mutations? Or does it constrain coexistence by allowing the resident strains to constantly encroach on each other’s niches? What signatures do these dynamics leave in genomic data? And how do they impact the ecological or metabolic phenotypes that emerge in the population in the long run?

Here we address these questions by developing a population genetic theory of ecological diversification that is valid for rapidly evolving populations. We focus on a simple class of resource competition models, where cells compete for different metabolites in a well-mixed environment, while acquiring heritable mutations in their genomes. Focusing on the two resource case, we show that rapidly evolving populations naturally cluster into a smaller number of distinct ecotypes, even when their underlying strain diversity is very large. By extending traveling wave models of clonal interference, we show that this self-organization is driven by an emergent priority effect that hinders ecological diversification by favoring the resident ecotypes. We show how these priority effects bias the long-term ecological structure of the community, and discuss the implications of these results in the context of recent microbial evolution experiments. Together, these results provide a quantitative framework for modeling ecological diversification in settings where ecology and evolution are both widespread.

Results

Modeling ecological diversification in rapidly evolving populations

To understand how rapid evolution influences ecological diversification, we focus on a simple class of resource competition models that have been studied in a number of previous works (34–37, 59–65). These models aim to capture several key empirical features of microbial evolution experiments (3, 5, 28), while remaining as analytically tractable as possible. In the simplest version of the model, we assume that individuals compete for $ℛ$ substitutable resources in a chemostat-like environment (Fig. 1A). Each resource is externally supplied at a constant rate ${\beta }_i$, which represents the fraction of available biomass that is supplied through resource $i$. Each strain $\mu$ in the population is then characterized by a resource-utilization vector ${\overrightarrow{r}}_{\mu }=\left(r_{\mu ,1},\dots ,r_{\mu ,ℛ}\right)$, which describes how rapidly it consumes each of the available resources. Following previous work (34, 59), it will be useful to decompose this phenotype into an overall magnitude (or “budget”) $X_{\mu }\equiv \mathrm{l}\mathrm{o}\mathrm{g}\sum _i{r}_{\mu ,i}$, and a normalized “strategy” ${\overrightarrow{\alpha }}_{\mu }={\overrightarrow{r}}_{\mu }/\sum _i{r}_{\mu ,i}$, which reflects the relative effort that strain $\mu$ devotes to importing each of the available resources.

Fig 1. Modeling ecological diversification in rapidly evolving asexual populations.

(a) Individuals compete for $ℛ$ substitutable resources in a chemostat-like environment. (b) Individuals acquire mutations that alter their resource uptake rates ($\overrightarrow{r}$). Mutations that impact the normalized resource uptake strategy ($\overrightarrow{\alpha }=\overrightarrow{r}/\sum _i{r}_i$) occur at rate $U_{\alpha }$, while mutations that only increase the overall magnitude $X=\mathrm{l}\mathrm{o}\mathrm{g}\sum _i{r}_i$ occur at rate $U_b$. (c, d) Simulated dynamics for $ℛ=2$ resources at small ($\mathrm{c},N={10}^6$) and large ($\mathrm{d},N={10}^9$) population sizes, where strategy mutations are sampled uniformly from the discrete strategies (or “ecotypes”) depicted at top. Muller plots show the relative abundances of each strain that reaches at least 5% frequency in the population. Strains are colored by their resource uptake strategies (top), with shading denoting different genotypes. Other parameters are $U_b={10}^{-8},s_b=0.02,U_{\alpha }={10}^{-7}$. (e) Total relative abundances of the ecotypes in panel d. (f) Shannon diversity (base e) of the individual strains in panel d (black) as well as the coarse-grained ecotypes in panel e (green). The dashed line depicts the maximum stable diversity possible in the rare mutation regime in panel c, when two coexisting strains are at equal frequencies. These data show that large populations cluster into a few coarse-grained ecotypes, even when their underlying strain diversity exceeds the competitive exclusion bound.

When resource uptake is fast compared to dilution, the population rapidly reaches a fixed carrying capacity $N$, and the state of the ecosystem can be described by the collection of relative strain frequencies $f_{\mu }$. Under suitable assumptions (SI Section 1), these strain frequencies can be approximated by the stochastic differential equation,

$$\frac{\partial f_{\mu }}{\partial t}\approx [X_{\mu }+\sum _{i=1}^{ℛ}{\alpha }_{\mu ,i}\cdot g_i(t)-\mathrm{{\rm Y}}(t)]f_{\mu }+\sqrt{\frac{f_{\mu }}{N}}{\eta }_{\mu }(t),$$ (1)

where ${\eta }_{\mu }(t)$ is a Brownian noise term (66) that captures the stochastic effects of genetic drift, $g_i(t)\approx ({\beta }_i-\sum _{\mu }{\alpha }_{\mu ,i}{f}_{\mu })/{\beta }_i$ is an intensive variable that measures the local availability of resource $i$, and $\mathrm{{\rm Y}}(t)\approx \sum _{\mu }{X}_{\mu }{f}_{\mu }+\sum _{\mu ,i}{\alpha }_{\mu ,i}{g}_i{f}_{\mu }$ is an additional gauge term that ensures that the relative frequencies remain normalized. These Lotka-Volterra-like dynamics allow for stable coexistence of up to $ℛ$ competing strains that partition the resources in different ways (34, 59, 67).

To incorporate evolution, we assume that the individuals in the population can also acquire mutations that alter their resource consumption phenotypes. As above, it will be convenient to classify these mutations by their effects on the overall magnitude ($X_{\mu }$) and resource uptake strategy (${\overrightarrow{\alpha }}_{\mu }$) of the mutant strain (Fig. 1B). For most of this work, we will consider two broad classes of mutations. Strategy mutations, which occur at a total per capita rate $U_{\alpha }$, alter the resource uptake strategy of an individual and potentially its overall magnitude as well. We also allow for a second class of “constitutively beneficial” mutations, which arise at rate $U_b$ and increase the total magnitude of ${\overrightarrow{r}}_{\mu }$ by a characteristic amount $\mathrm{\Delta }X\approx s_b$, while leaving the relative rates intact ($\mathrm{\Delta }\overrightarrow{\alpha }\approx 0$). This second class of mutations models changes to cellular functions that are only tangentially related to resource consumption (e.g. eliminating a pathway that is not necessary in the current environment). Such mutations are routinely observed in microbial evolution experiments (28, 41, 68), and have previously been shown to have large impacts on the emergent eco-evolutionary dynamics (34). While the assumptions of a fixed effect size ($\mathrm{\Delta }X\approx s_b$) and a perfect tradeoff ($\mathrm{\Delta }\overrightarrow{\alpha }\approx 0$) may initially seem to be artificial, we will show below that both assumptions can be relaxed without influencing our main results.

Together, these assumptions define a simple evolutionary model, similar to Refs. (34, 58), which allows us to explore the dynamics of ecological diversification across a range of population sizes and mutation rates. The simulations in Fig. 1C,D show an example of this diversification process for $ℛ=2$ resources and two different simulated population sizes: a smaller population where mutations occur infrequently ($NU\approx 0.1$; Fig. 1C), and a larger population where multiple new mutations are produced in each generation ($NU\approx 100$; Fig. 1D). These additional mutations lead to qualitatively different behavior, which we explore in more detail below.

Rapidly evolving populations cluster into a few distinct ecotypes

In small populations ($NU\lesssim 1$), the diversification dynamics in Fig. 1C are well-described by the strong-selection, weak-mutation (SSWM) framework in Ref. (34). The salient features are present even for $ℛ=2$ resources, which we focus on for most of this work. In this case, both the resource supply rates and uptake vectors can be characterized by scalar values, $\beta$ and ${\alpha }_{\mu }$ (Fig. 1C, top) corresponding to the first resource dimension, with the remaining components fixed from normalization. When mutations occur infrequently, the population reaches its stable equilibrium long before the next mutation occurs. This separation of timescales leads to the punctuated dynamics in Fig. 1C, where extended periods of stasis are interspersed with rapid switches in composition. When a new mutation arises in such a setting, its initial growth rate (or invasion fitness) is given by

$$s_{\mathrm{i}\mathrm{n}\mathrm{v}}\approx \mathrm{\Delta }X+\sum _{i=1}^{ℛ}\mathrm{\Delta }{\alpha }_i\cdot g_i\left(0\right)\equiv \mathrm{\Delta }X+s_{\mathrm{e}\mathrm{c}\mathrm{o}},$$ (2)

where $\mathrm{\Delta }X=X_2-X_1$ and $\mathrm{\Delta }\overrightarrow{\alpha }={\overrightarrow{\alpha }}_2-{\overrightarrow{\alpha }}_1$ denote the phenotypic effects of the mutation, and the resource availabilities $g_i(0)$ are set by the previous ecological equilibrium (34). The first term in this expression acts like a constitutive cost or benefit, while the ecological component ($s_{\text{eco}}\equiv \sum _i\mathrm{\Delta }{\alpha }_i{g}_i$) is a variable quantity that depends on the current resource utilization of the population.

Most such mutations will drift to extinction even when they are deterministically favored to invade ($s_{\text{inv}}>0$). However, a lucky fraction ($p_{\text{est}}\approx 2s_{\text{inv}}$) will establish in the population and eventually grow to larger frequencies, producing a new ecological equilibrium. The properties of this state will strongly depend on the phenotype of the invading strain (34, 35, 60). Some mutants will outcompete the current residents and fix ($f^{\mathrm{*}}\approx 1$), while others will stably coexist with one or more of the residents by altering the resource availabilities in Eq. (1). The outcomes are particularly simple for $ℛ=2$ resources, since at most two strains can coexist at a time. The equilibrium frequency is given by

$$f^{\mathrm{*}}\approx \frac{\beta -{\alpha }_1}{\mathrm{\Delta }\alpha }+\frac{\beta (1-\beta )}{\mathrm{\Delta }{\alpha }^2}\cdot \mathrm{\Delta }X\equiv f_0^{\mathrm{*}}+\frac{\mathrm{\Delta }X}{s_c},$$ (3)

with coexistence occurring whenever $f^{\mathrm{*}}<1$ (34; SI Section 2). This leads to a new set of $g_i$ values, which can be substituted into Eq. (2) when the next mutation arises.

Equation (3) also defines two new parameters that will be important for much of our analysis below. The first is the baseline equilibrium frequency, $f_0^{\mathrm{*}}\equiv \left(\beta -{\alpha }_1\right)/\mathrm{\Delta }\alpha$, that is attained when the two strains share the same total fitness ($\mathrm{\Delta }X\approx 0$). The second is a characteristic fitness scale, $s_c\equiv \mathrm{\Delta }{\alpha }^2/\beta (1-\beta )$, which describes how constitutive fitness differences perturb this baseline equilibrium. This susceptibility parameter controls how the balance between two strains shifts when they acquire a constitutively beneficial mutation ($\mathrm{\Delta }X\to \mathrm{\Delta }X\pm s_b$), as well as the total fitness differences that are tolerated before one of the strains is driven to extinction (Fig. 1C).

While this existing theory applies to Fig. 1C, qualitatively different behavior emerges in larger populations when mutations occur more frequently ($NU\gtrsim 1$; Fig. 1D). The most striking difference is the absence of any apparent ecological equilibria: at any given timepoint, all of the resident strains exhibit nonzero relative growth rates, signaling that ecology and evolution are now operating on similar timescales. A consequence of this behavior is that the local strain diversity often exceeds the competitive exclusion bound set by the number of available resources (Fig. 1F), for the simple reason that new mutations are introduced into the population much faster than they can be purged (57–59).

Despite this high strain diversity, we find that when individuals are grouped together by their resource strategies ($\overrightarrow{\alpha }$), the population is usually dominated by only a few distinct strategies at a time (Fig. 1E,F), even when strategy mutations are common ($N{U}_{\alpha }\gg 1$). Moreover, these dominant strategies can persist at high frequencies for thousands of generations (Fig. 1E), while their underlying strains turn over at a much faster rate (Fig. 1D). This emergent simplicity highlights the utility of working in the resource strategy basis. It also suggests that further progress can be made by tracking the emergence and maintenance of a smaller number of distinct “ecotypes”, which share the same resource uptake strategies but contain strains with many different genotypes and fitnesses ($X_{\mu }$).

A dynamical priority effect hinders the invasion of new ecotypes

To understand the mechanisms driving the behavior in Fig. 1C–F, we start by considering the simplest case, where a new strategy mutation arises in a population with only a single resident ecotype. We initially assume that further strategy mutations can be neglected ($N{U}_{\alpha }\to 0$), so the fate of the new ecotype is shaped by fitness mutations alone. These “pairwise” invasions will provide the building blocks necessary for understanding the more general scenarios below.

In large populations ($N{U}_b\gg 1$), the initial stages of diversification are best understood by considering the population’s distribution across the strategy ($\alpha$) and fitness ($X$) dimensions of phenotype space (Fig. 2A; SI Section 3). When the new ecotype first arises, previous work has shown that the resident ecotype will have formed a “traveling fitness wave” along the $X$ axis, with a characteristic shape $\rho (x)$ that moves to the right a steady state rate $v$ (50–54). This distribution has a maximum width $x_c$, which roughly coincides with the location of the fittest individuals in the population: these individuals in the “nose” of the fitness distribution possess $q\approx x_c/s_b$ more beneficial mutations than the population average, and will grow to become the dominant fraction of the population in a characteristic “sweep time” $T_{\mathrm{s}\mathrm{w}}\approx x_c/v$ (Fig. 2A, left; SI Section 3.1). A new ecotype will arise on one of these extant genetic backgrounds, with a relative fitness drawn from $\rho (x)$. It will then go on to found its own fitness wave with resource strategy ${\alpha }_2={\alpha }_1+\mathrm{\Delta }\alpha$ (Fig. 2A, right).

Fig 2. Clonal interference creates a priority effect that favors the resident ecotype.

(a) Schematic of a strategy mutation arising in a population with a single resident ecotype. The mutation arises in an individual from the resident fitness distribution (grey), which has a maximum width $x_c$ and moves to the right at rate $v$. Its descendants found a new ecotype (green) that competes with the resident ecotype as they both acquire additional mutations. (b) The probability that a strategy mutation “establishes” (i.e. reaches 10% frequency) as a function of its initial invasion fitness $s_{\text{inv}}$. Green points show simulation results for $N={10}^{10},U_b={10}^{-6}$, and $s_b=0.01$; for each value of $s_{\text{inv}}$, the values of ${\alpha }_1,{\alpha }_2$, and $\beta$ were chosen so that $f^{\mathrm{*}}(0)=1/2$. Grey points show the establishment probabilities of constitutively beneficial mutations with the same values of $s_{\text{inv}}$. (c) The median frequencies of the mutations in panel $\mathrm{b}{T}_{\mathrm{s}\mathrm{w}}\approx 1000$ generations after first reaching 10% frequency. The dashed line denotes the expected frequency in the absence of clonal interference. (d) The long-term fixation probabilities of the mutations in panel c. Inset: the average length of time required for a mutation to transit from 10% frequency to either fixation (solid curves) or extinction (dashed).

When the frequency of the new ecotype is still small, the growth rate of its founding strain can be calculated from a generalization of Eq. (2),

$$x_{\mathrm{i}\mathrm{n}\mathrm{v}}\left(t\right)\approx s_{\mathrm{i}\mathrm{n}\mathrm{v}}+x-vt,$$ (4)

with $s_{\text{inv}}\approx \mathrm{\Delta }X+s_{\text{eco}}\approx \mathrm{\Delta }X+\mathrm{\Delta }\alpha \left(\beta -{\alpha }_1\right)/\beta (1-\beta )$. The two new terms in Eq. (4) account for (i) the initial relative fitness of the mutant’s previous genetic background, and (ii) the ongoing adaptation of the resident ecotype, which causes the relative fitness of any given strain to systematically decline with time. This declining growth rate implies that many initially adaptive mutants that survive genetic drift will eventually be outcompeted by the resident population before they reach appreciable frequencies (“clonal interference”). To reach a substantial fraction of the population, a new ecotype must either arise on such an anomalously fit genetic background that it “outruns” the resident population, or it must quickly acquire one or more additional beneficial mutations that accomplish the same effect. As we show below, these chance events all occur while the mutant lineage is still at a low frequency. This implies that both fitness and strategy mutations — and any other lineage with the same value of $x_{\text{inv}}(0)$ — will have the same probability of establishing.

We verify this prediction in Fig. 2B, by comparing the observed establishment probability of a new strategy mutation — operationally defined as the probability that it reaches 10% frequency — against a constitutively beneficial mutation with the same value of $s_{\text{inv}}$. For concreteness, we consider a simple class of strategy mutations with $\mathrm{\Delta }X=0$ and $\mathrm{\Delta }\alpha =2\left(\beta -{\alpha }_1\right)$, such that $s_{\text{inv}}\approx s_{\text{eco}}\approx \mathrm{\Delta }{\alpha }^2/2\beta (1-\beta )$. Fig. 2B shows that the establishment probabilities of these mutations closely track their non-ecological counterparts, and exhibit the classic hallmarks of clonal interference (53, 69): the establishment probability approaches the neutral limit ($p_{\text{est}}\sim 1/N$) when $s_{\text{inv}}\lesssim 1/T_{\text{sw}}$, and then increases sharply at higher values of $s_{\text{inv}}$ before returning to the SSWM expectation ($p_{\text{est}}\approx 2s_{\text{inv}}$) when $s_{\text{inv}}\gtrsim x_c$. This emergent neutrality at small values of $s_{\text{inv}}$ implies that sufficiently small changes in resource strategy $\left[\mathrm{\Delta }\alpha \lesssim \beta (1-\beta )/\left(\beta -{\alpha }_1\right)T_{\mathrm{s}\mathrm{w}}\right]$ will be effectively invisible to natural selection, even when their ecological selection pressures are nominally large ($N{s}_{\text{eco}}\gg 1$).

At higher frequencies, established strategy mutations start to diverge from their non-ecological counterparts, as environmental feedbacks from the invading ecotype eventually reduce the value of $s_{\text{eco}}(t)\equiv \sum _i\mathrm{\Delta }{\alpha }_i{g}_i(t)$. In the weak mutation regime, these feedbacks would normally cause the new ecotype to stabilize at the equilibrium frequency in Eq. (3), which is equal to $f^{\mathrm{*}}\approx 50\mathrm{\%}$ for all of the mutations that are simulated in Fig. 2B. Interestingly, however, Fig. 2C shows that the long-term frequency of the invading ecotype is consistently smaller than 50%, and reaches its lowest point for intermediate selection strengths where $s_{\mathrm{i}\mathrm{n}\mathrm{v}}\approx x_c$. On even longer timescales ($t\gg T_{\mathrm{s}\mathrm{w}}$), thelower initial frequency of the invading ecotype eventually impacts its ability to accumulate of future mutations. This implies that the invading ecotype is much more likely to be driven to extinction by the resident in the long run (Fig. 2D). We call this emergent bias in favor of the resident ecotype a priority effect (70), since it does not arise from any intrinsic differences between the two ecotypes, but rather from their order of arrival in the population.

The origin of this priority effect can be understood by considering the genetic backgrounds of the invading ecotypes. The total establishment probability of a new mutation can be expressed as an average over genetic backgrounds,

$$p_{\mathrm{e}\mathrm{s}\mathrm{t}}\left(s_{\mathrm{i}\mathrm{n}\mathrm{v}}\right)=\int \rho \left(x\right)w\left(x+s_{\mathrm{i}\mathrm{n}\mathrm{v}}\right)dx,$$ (5)

where $w(x)$ represents the conditional establishment probability of a lineage with $x_{\mathrm{i}\mathrm{n}\mathrm{v}}(0)=x$ (SI Section 3.3). Previous work (53) has shown that when $s_{\text{inv}}\lesssim x_c$, the integral in Eq. (5) is strongly peaked around a characteristic value $x^{\mathrm{*}}\approx x_c-s_{\text{inv}}$, such that the initial growth rate of a successful mutation is likewise around $x_{\text{inv}}^{\mathrm{*}}(0)\approx x_c$. In other words, successful mutations appear to “tune” their genetic background so that their net growth rate coincides with the “nose” of $\rho (x)$, where the fittest individuals in the population currently reside.

When projected onto the $\alpha$ and $X$ axes in Fig. 2A, this tuning behavior implies that the $X$ coordinate of a successful strategy mutation will be strongly peaked around $X^{\mathrm{*}}\approx x_c-s_{\text{inv}}$, which lies below the nose of the resident ecotype by a characteristic amount $s_{\text{inv}}$. This constitutive fitness deficit only becomes important $T_{\mathrm{s}\mathrm{w}}\approx x_c/v$ generations later, when both sets of strains have grown to occupy the dominant fraction of the population. At this point, the invading ecotype experiences an effective fitness difference of $\mathrm{\Delta }{X}_{\text{eff}}\approx -s_{\text{inv}}$ compared to its competitors, which would lead to a lower equilibrium frequency according to Eq. (3). This heuristic argument helps explain the emergent priority effect observed above: successful strategy mutations tend to “mortgage” their ecological advantage to maximize their short-term probability of establishing. But they eventually pay a price at higher frequencies, when their rapid expansion causes their ecological advantage to suddenly dissipate.

Oscillatory invasions of new ecotypes

While this “fitness mortgaging” picture is essentially correct, quantitatively predicting the behavior in Fig. 2C,D requires a more careful analysis of an ecotype’s invasion dynamics over time. In rapidly evolving popluations, the frequency of a successful ecotype obeys a time-dependent generalization of Eq. (3),

$$\frac{\partial F}{\partial t}=\left[{\bar{X}}_2(t)-{\bar{X}}_1(t)+s_{\mathrm{e}\mathrm{c}\mathrm{o}}-s_{c}F\right]F(1-F),$$ (6)

where ${\bar{X}}_2(t)$ and ${\bar{X}}_1(t)$ denote the average $X$ coordinates of the invading and resident ecotypes, respectively (SI Section 3). Fig. 3 shows that these frequency trajectories can undergo large oscillations when the mutant ecotype first invades, reaching a peak frequency around $T_{\mathrm{s}\mathrm{w}}$ generations, before declining to a quasi-steady state level at later times. We will now show how these oscillatory dynamics emerge from a delayed feedback between two distinct subsets of the population: (i) the ecologically-relevant bulk, which shapes the current environment, and (ii) the evolutionarily-relevant nose, which will come to dominate the population at later times.

Fig 3. Oscillatory invasion dynamics of new ecotypes.

(a) Replicate frequency trajectories of an invading ecotype with $s_{\text{eco}}\approx 0.03$ and $f_0^{\mathrm{*}}\approx 0.8$ that arises on an initial genetic background with $x_{\text{inv}}(0)\approx x_c+s_b$. Simulated parameters are $N={10}^{12},U_b={10}^{-6}$, and $s_b=0.01$, for which $x_c\approx 4s_b$. Five random realizations are shown in solid lines, while the dashed line depicts an average over 50 independent replicates, which emphasizes the typical overshooting behavior between $T^{\mathrm{*}}$ (Eq. 7) and $T_{\mathrm{s}\mathrm{w}}\approx x_c/v$. Horizontal dashed lines show the original equilibrium frequency $f^{\mathrm{*}}$ (Eq. 3) and the “fitness mortgaging” term from Eq. (9). (b) The oscillations become more pronounced in the asymptotic limit, where $\mathrm{l}\mathrm{o}\mathrm{g}N=500,\mathrm{l}\mathrm{o}\mathrm{g}{U}_b=-50$, and $x_c\approx 21s_b$. (c) The initial growth rates of successive fitness classes (Fig. S1, SI Section 3.3) for the ecotype trajectories in panel b, illustrating how the runaway growth is eventually stabilized $\sim T_{\mathrm{s}\mathrm{w}}$ generations later.

We quantify these dynamics by considering the trajectories of successive fitness classes, $f_k(t)$, which denote the collection of mutant individuals that have acquired $k$ additional mutations. We focus on the limiting behavior of these cohorts in the empirically-relevant scenario where $s_b\gg U_b\gg 1/N$, where the key fitness scales in Fig. 2A satisfy $\sqrt{v}\ll s_b\ll x_c$ (52). In this regime, the fitness of the invading ecotype is usually dominated by a single value of $k$, corresponding to the largest fitness class at that instant in time. We can combine this approximation with Eq. (6) to obtain a piecewise description of the ecotype’s frequency trajectory over time (SI Section 3.4).

In the initial phase of each trajectory ($t<T_{\mathrm{s}\mathrm{w}}$), the invading ecotype is dominated by the founding $k=0$ class, and the first three terms in Eq. (6) are well-approximated by the initial growth rate $x_{\text{inv}}(t)$ in Eq. (4). The resulting dynamics strongly depend on how $x_{\text{inv}}(0)$ compares to $x_c$. When $x_{\text{inv}}(0)>x_c$, the invading ecotype will eventually reach a frequency where the ecological feedbacks become important. Before this point, the invading ecotype is well-approximated by the linear model in Eq. (4), and increases exponentially with a steadily declining growth rate $x_{\text{inv}}(t)$. After a critical time

$$T^{\mathrm{*}}\approx \frac{x_{\mathrm{i}\mathrm{n}\mathrm{v}}\left(0\right)}{v}-\sqrt{\frac{x_{\mathrm{i}\mathrm{n}\mathrm{v}}(0{)}^2-x_c^2}{v^2}},$$ (7)

the ecological feedbacks in Eq. (6) lead to a rapid equilibration of the ecotype frequency around a quasi-steady-state value,

$$F\left(t\right)\approx \frac{x_{\mathrm{i}\mathrm{n}\mathrm{v}}\left(0\right)-vt}{s_c},$$ (8)

which applies when $s_{c}F(t)\gtrsim \sqrt{v}$. The decreasing form of this expression implies that the invading ecotype will attain a maximum value $F(T^{\mathrm{*}})\approx s_c^{-1}\sqrt{x_{\mathrm{i}\mathrm{n}\mathrm{v}}(0{)}^2-x_c^2}$ and will then begin to decline over time, as the resident ecotype continues to adapt relative to the initial founding clone.

On longer timescales, the $k=0$ class will eventually be replaced by mutant lineages that have acquired additional beneficial mutations. In our parameter regime of interest ($s_b\gg U_b\gg 1/N$), these replacement dynamics take on a particularly simple form: the $k$ th class starts to dominate the new ecotype at a characteristic time $t\approx T_{\mathrm{s}\mathrm{w}}+T_k$, where $T_k$ denotes the corresponding establishment time where $f_k\left(T_k\right)\approx 1/2N{x}_{\text{inv},\mathrm{k}}\left(T_k\right)$ (Fig. S1; SI Section 3.4). After the $k$th class takes over, the fitness of the invading ecotype will have advanced by a total amount $k{s}_b$, while the fitness of the resident ecotype will have increased by $v\left(T_{\mathrm{s}\mathrm{w}}+T_k\right)$ over the same time interval. This leads to a modified version of Eq. (8) that applies for $t\gtrsim T_{\mathrm{s}\mathrm{w}}$:

$$F\left(t\approx T_{\mathrm{s}\mathrm{w}}+T_k\right)\approx \frac{{\bar{X}}_2\left(t\right)-{\bar{X}}_1\left(t\right)+s_{\mathrm{e}\mathrm{c}\mathrm{o}}}{s_c}\approx \frac{x_{\mathrm{i}\mathrm{n}\mathrm{v}}\left(0\right)-x_c}{s_c}+\frac{k{s}_b-v{T}_k}{s_c}.$$ (9)

This expression constitutes the rapid-evolution analogue of the ecological equilibrium in Eq. (3). The first term matches the naive expectation from our “fitness mortgaging” argument above, while the second term accounts for differences in the rates of establishment between the invading and resident ecotypes.

The statistics of the first few $T_k$ values can be calculated by generalizing the calculations in Ref. (52) (SI Section 3.3). The $k=1$ class establishes slightly earlier than the resident nose, since its feeding class grows at a higher initial rate ($x_{\mathrm{i}\mathrm{n}\mathrm{v}}(0)>x_c$; Fig. 3C). This difference in establishment times causes the invading ecotype frequency in Eq. (9) to be slightly larger than expected from the fitness mortgaging term alone. These early establishments continue for several additional fitness classes, as each successive establishment widens the gap in the initial growth rate (Fig. 3C), leading to an increase in the overall frequency. However, the trend eventually reverses for $T_k\gtrsim T^{\mathrm{*}}$, when the ecological feedbacks cause the effective growth rate to drop to $k{s}_b<x_c$ (SI Section 3.4). These damped oscillations can persist for several additional multiples of $T_{\mathrm{s}\mathrm{w}}$, as observed in Fig. 3. However, the corrections are often modest in practice, such that the first term in Eq. (9) provides a reasonable approximation to the ecotype frequency when $t\propto T_{\mathrm{s}\mathrm{w}}$.

These results show that the frequency trajectory of the invading ecotype strongly depends on the critical quantity $\delta \left(s_{\text{inv}}\right)\equiv x_{\text{inv}}(0)-x_c$, which represents the growth advantage of mutant over the resident nose, and implicitly depends on $s_{\text{inv}}$ through the distribution of successful background fitnesses $x$ in Eq. (5). A more careful analysis in SI Sections 3.2 and 3.3 reveals that the distribution of positive $\delta$ values can be approximated by a truncated Gaussian distribution,

$$p(\delta \mid \delta >0)\propto e^{-\frac{{\left(\delta +x_c-s_{\mathrm{i}\mathrm{n}\mathrm{v}}\right)}^2}{2v}},$$ (10)

which is peaked near $\delta \approx s_{\text{inv}}-x_c$ for $s_{\text{inv}}\gtrsim x_c$ and $\delta \sim v/x_c$ otherwise. These results imply that mutants with larger values of $s_c\equiv \mathrm{\Delta }{\alpha }^2/\beta (1-\beta )$ must sample proportionally larger values of $\delta$ to reach a given post-invasion frequency $F(t)$. The fitness mortgaging effect in Eq. (10) makes this increasingly unlikely until $s_{\text{inv}}\gtrsim x_c$, which explains the non-monotonic behavior in Fig. 2C. At the same time, Eq. (7) shows that populations with stronger clonal interference ($x_c\gg s_b$) can amplify smaller values of $\delta$ to equivalent peak frequencies $F(T^{\mathrm{*}})$ during the initial transient phase of the invasion. This highlights the counter-intuitive dynamical effects that emerge in rapidly evolving populations.

Long-term dynamics of coexisting ecotypes

On longer timescales ($t\gg T_{\mathrm{s}\mathrm{w}}$), the cumulative differences in the rates of establishment will start to dominate the local equilibrium in Eq. (8), and the ecotype frequencies will gradually diverge from their initial post-invasion levels. In the absence of additional strategy mutations, these fitness differences will eventually grow so large that they will drive one of the two ecotypes to extinction (Fig. 2D).

Since these long-term differences reflect the contributions of many accumulated mutations ($k\gg 1$), it becomes natural to model their dynamics as a generalized random walk. Building on previous results derived in non-ecological settings (52, 54), we show that the long-term fitness of each ecotype can be modeled as an effective diffusion process,

$$\frac{\partial {\bar{X}}_i}{\partial t}\approx v\left(N{F}_i\left(t\right)\right)+\sqrt{\frac{2{\pi }^2}{3T_{\mathrm{s}\mathrm{w}}^3}}\eta \left(t\right),$$ (11)

where $v(N)$ denotes the rate of adaptation as a function of the population size $N$ (SI Section 3.5). Substituting this expression into Eq. (9) yields a corresponding model for the ecotype frequency,

$$\frac{\partial F}{\partial t}\approx \frac{2}{s_c{T}_{\mathrm{s}\mathrm{w}}^2}\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{F}{1-F}\right)+\sqrt{\frac{4{\pi }^2}{3s_c^2{T}_{\mathrm{s}\mathrm{w}}^3}}\eta (t),$$ (12)

which applies for $t\gg T_{\mathrm{s}\mathrm{w}}\gg 1/s_c$. The first term in this model reflects the deterministic difference in the rates of adaptation of the two ecotypes due to their unequal population sizes. The second term arises from the stochasticity in the timing of individual establishment events (52, 54), which are correlated on intermediate timescales ($\mathrm{\Delta }t\sim T_{\mathrm{s}\mathrm{w}}$) but behave diffusively when $\mathrm{\Delta }t\gg T_{\mathrm{s}\mathrm{w}}$.

Analysis of Eq. (12) reveals that the deterministic bias dominates over the stochastic term when the difference in ecotype frequencies is larger than $\sim 1/\sqrt{s_c{T}_{\mathrm{s}\mathrm{w}}}$ (SI Section 3.5). Beyond this critical point, the biased rates of adaptation will deterministically drive the minority strain to extinction on a timescale $T_{\text{ext}}\sim s_c{T}_{\mathrm{s}\mathrm{w}}^2\gg T_{\mathrm{s}\mathrm{w}}$. The gap between these timescales justifies the original diffusion approximation in Eq. (12), and illustrates why established ecotypes can coexist for much longer than the lifetime of a typical lineage (Fig. 2D inset). At the same time, these results also show that the window of deterministic extinction becomes increasingly large when $s_c{T}_{\mathrm{s}\mathrm{w}}\gg 1$. This implies that even small initial frequency differences (Fig. 2C) can generate strong priority effects in the long-term composition of the population (Fig. 2D).

Rediversification from recurrent mutations

For finite mutation rates ($N{U}_{\alpha }>0$), additional strategy mutations can arise before one of the resident ecotypes goes extinct, which alters the simple pairwise picture above. To build intuition for this more realistic case, we start by assuming that there are still just two possible ecotypes, while allowing mutations (and reversions) to occur between the two strategies at a symmetric rate $U_{\alpha }$ (Fig. 4A). Simulations of this model recapitulate some of the behaviors observed in the multi-ecotype setting in Fig. 1. In both cases, we observe a priority effect similar to Fig. 2, where one ecotype persists at a higher frequency for extended “epochs” of length $T_{\text{epoch}}\gg T_{\mathrm{s}\mathrm{w}}$ (Fig. 4B, top), with rare transitions between successive epochs where the majority ecotypes flip. However, within the minority ecotype, there is constant genetic turnover, as new strategy mutations from the majority background continually arise and replace the previous lineages (Fig. 4B, bottom). These asymmetric cross-invasions stabilize the frequency of the minority ecotype, but lead to even stronger priority effects at the genetic level, since lineages in the minority background will have a vanishingly small probability of contributing to the future genetic composition of the population.

Fig 4. Recurrent strategy mutations stabilize coexistence by replacing the lagging ecotype.

(a) Recurrent strategy mutations between established ecotypes can replace the resident lineage if the fitness differences grow sufficiently large. (b) Example simulation showing bistable ecotype frequencies on long timescales, with continuous lineage turnover on shorter timescales due to successful cross-invasions. Thick shaded curves represent overall ecotype frequencies, while thinner lines denote lineages founded by distinct strategy mutations. Ecotype parameters are $s_c=0.1,\mathrm{\Delta }X=0$, and $f_0^{\mathrm{*}}=0.5$, with $U_{\alpha }={10}^{-11}$; the remaining parameters are the same as Fig. 3A. (c,d) Lineage turnover time (c) and overall frequency (d) of the trailing ecotype for different values of $s_c$ and $s_b$. Symbols denote the median across simulations with the same $N,U_b$ parameters as panel b. Dashed lines denote theoretical predictions (SI Section 3.6).

We can understand these new dynamics by combining several of the results derived above. In this case, a strategy mutation arising in the majority ecotype at time $t_0$ will have an initial growth rate similar to Eq. (4), but with an altered ecological component,

$$s_{\mathrm{e}\mathrm{c}\mathrm{o}}\left(t_0\right)\approx {\bar{X}}_1\left(t_0\right)-{\bar{X}}_2\left(t_0\right),$$ (13)

which reflects the fact that the second niche is now already occupied (SI Section 3.6). Since we have seen that the fitness difference between the two ecotypes changes on timescales much longer than $T_{\mathrm{s}\mathrm{w}}$, the establishment probability of this mutation will still be captured by Eq. (5), with an effective invasion fitness $s_{\text{inv}}\left(t_0\right)\approx \mathrm{\Delta }X+s_{\text{eco}}\left(t_0\right)$. The average time to the next successful cross-invasion ($T_{\text{cross}}$) can then be defined from the standard condition that approximately one mutation establishes within that interval,

$${\int }_0^{T_{\text{cross}}}N{U}_{\alpha }{p}_{\text{est}}\left(s_{\text{inv}}\left(t_0\right)\right)d{t}_0\sim 1,$$ (14)

which implicitly depends on the initial fitness difference between the two ecotypes. At steady-state, the change in fitness during $T_{\text{cross}}$ must balance the net fitness increase from a successful cross-invasion, which is proportional to the fitness difference $\delta \left(s_{\text{inv}}\right)$ in Eq. (10). Combining this criterion with Eq. (14) yields a self-consistent solution for $T_{\text{cross}}$.

These steady-state dynamics are particularly simple in the empirically-relevant regime where $N{U}_{\alpha }\gg 1$ and $U_{\alpha }\ll s_b$ (SI Section 3.6). Provided that the direct cost of the mutation is not too large, the fitness difference between the ecotypes remains approximately constant between invasions, with $s_{\text{inv}}(t)\ll x_c$. The self-consistency condition then reduces to $T_{\text{cross}}\sim T_{\text{sw}}$, which is independent of the underlying resource strategies or their mutation rate $U_{\alpha }$ (Figs. 4C and S2). This result shows that rapid evolution pushes the ecosystem to a point where the effective fitness benefits and mutation rates of strategy mutations balance each other, so that successful cross invasions occur approximately once every $T_{\mathrm{s}\mathrm{w}}$ generations. The emergent fitness benefits of these mutations depend only logarithmically on $U_{\alpha }$,

$$s_{\mathrm{i}\mathrm{n}\mathrm{v}}\left(T_{\text{cross}}\right)\approx \frac{1}{T_{\mathrm{s}\mathrm{w}}}\log \left(\frac{1}{U_{\alpha }{T}_{\mathrm{s}\mathrm{w}}}\right),$$ (15)

and fall in the exponential portion of $p_{\text{est}}\left(s_{\text{inv}}\right)$ in Fig. 2B. This steep dependence on $s_{\text{inv}}$ implies that even small changes in the relative fitnesses of the two ecotypes will generate large shifts in $T_{\text{cross}}$, which effectively stabilizes the relative fitnesses – and the corresponding ecotype frequencies in Eq. (9) – close to their average steady-state values (Figs. 4D and S2).

The weak dependence of the ecosystem on the strategy mutation rate strongly deviates from the traditional SSWM regime (34). Instead, it echoes the key lessons derived from non-ecological models of clonal interference: since most strategy mutations are wasted on insufficiently fit genetic backgrounds, ecosystem dynamics are only logarithmically dependent on the underlying mutation supply rate.

Dynamics of multiple competing ecotypes

We can also extend the above analysis to the more general case where multiple types of strategy mutations are available. In this case, an arbitrary strategy mutation arising in the majority ecotype, $\mathrm{\Delta }{\alpha }^{\prime }={\alpha }_3-{\alpha }_1$, will satisfy a generalization of Eq. (13),

$$s_{\mathrm{e}\mathrm{c}\mathrm{o}}\left(t_0\right)\approx \left(\frac{\mathrm{\Delta }{\alpha }^{\prime }}{\mathrm{\Delta }\alpha }\right)\cdot \left[{\bar{X}}_1\left(t_0\right)-{\bar{X}}_2\left(t_0\right)\right]\equiv \mathrm{\Delta }{\alpha }^{\prime }\cdot \mathrm{\Gamma }\left(t_0\right),$$ (16)

where the quantity $\mathrm{\Gamma }\left(t_0\right)\equiv \left({\bar{X}}_1-{\bar{X}}_2\right)/\left({\alpha }_2-{\alpha }_1\right)$ now depends on both the fitness difference and resource strategies of the resident ecotypes. These new ecotypes will continue to establish with probability $p_{\text{est}}\left(\mathrm{\Delta }{X}^{\prime }+\mathrm{\Delta }{\alpha }^{\prime }\cdot \mathrm{\Gamma }\left(t_0\right)\right)$ in Eq. (5), and will replace either the majority or minority ecotype (or both) depending on the relative values of $\mathrm{\Delta }{\alpha }^{\prime }$ and $\mathrm{\Delta }{X}^{\prime }$.

While a wide range of dynamics are now possible, we find that many populations reach a steady-state similar to Fig. 4, where a single majority ecotype seeds periodic cross-invasions into the opposing niche, producing a stochastic sequence of minority ecotypes. In this case, the steady-state population has a constant value of $\mathrm{\Gamma }\left(T_{\text{cross}}\right)$, which satisfies a generalization of Eq. (14):

$${\int }_0^{T_{\text{cross}}}\sum _{i}N{U}_i{p}_{\text{est}}\left(\mathrm{\Delta }{X}_i+\mathrm{\Delta }{\alpha }_i\cdot \mathrm{\Gamma }\left(t_0\right)\right)\sim 1,$$ (17)

with $U_i$ denoting the corresponding mutation rate for strategy mutations with effect ($\mathrm{\Delta }{\alpha }_i,\mathrm{\Delta }{X}_i$).

The presence of multiple mutation classes produces new effects that were not present in the pairwise case above. In particular, the exponential growth of $p_{\text{est}}\left(s_{\text{inv}}\right)$ implies that natural selection will strongly favor mutations with larger values of $\mathrm{\Delta }{\alpha }_i$, corresponding to ecotypes with more specialist metabolic strategies. This bias is much larger than the linear scaling expected in the weak mutation limit (34), and is often enough to outweigh much larger-fold differences in the underlying mutation rates (Fig. 5). As a consequence of this rapid dependence on $s_{\text{inv}}$, the sum in Eq. (17) will often be dominated by a narrow range of $\mathrm{\Delta }\alpha$ and $\mathrm{\Delta }X$ values, so that only a small fraction of the available ecotypes are likely to be observed at high frequencies.

Fig 5. Clonal interference biases the metabolic structure of the population.

(a) Schematic of a mutational landscape with three possible resource strategies. (b) Ecotype frequency trajectories from three example populations with $U_{1\to 2}/U_{1\to 3}=3,30$, and 300, with $U_{1\to 3}={10}^{-10}$ held fixed. The resource strategies satisfy ${\alpha }_1=0.43,{\alpha }_2=0.61$, and ${\alpha }_3=0.69$, with $\beta =0.5$, while the remaining parameters are $N={10}^{12},U_b={10}^{-6}$, and $s_b={10}^{-2}$. (c) The fraction of time ($\mathrm{\Phi }$) spent in the (${\alpha }_1,{\alpha }_2$)-dominated state, for different values of ${\alpha }_3$ (color-scale) and $U_{1\to 2}$ (x axis); all other parameters are the same as panel b. Inset: the relative mutation rate where the two lagging ecotypes are equally likely ($\mathrm{\Phi }\approx 1/2$) for different values of ${\alpha }_3$. Clonal interference amplifies selection on small differences in resource consumption, which can overwhelm much larger differences in the relative mutation rates of different strategies.

Despite these differences, the timescale of cross-invasion remains similar to the pairwise case above. In both cases, the population self-organizes to a state where the ecological selection pressures are low enough to ensure that only ~1 new ecotype establishes every $T_{\text{cross}}\gtrsim T_{\mathrm{s}\mathrm{w}}$ generations, even when $N{U}_{\alpha }\gg 1$. Moreover, since the window for successful establishment is comparable to the $T^{\mathrm{*}}\sim T_{\mathrm{s}\mathrm{w}}$ timescale in Fig. 3, the first such ecotype to reach high frequency will rapidly exclude its competitors, ensuring that only $R\approx 2$ ecotypes will dominate the population at any given time. This creates an emergent priority effect that favors the resident ecotypes, sustaining long periods of ecotype stability in the face of rapid within-ecotype evolution (Fig. 1).

Discussion

In large microbial populations, ecological and evolutionary dynamics often unfold on similar timescales. Our study introduces a theoretical framework for modeling these coupled dynamics in a widely used class of resource competition models, where individual cells acquire mutations that alter their resource consumption phenotypes. By focusing on the simplest case of two substitutable resources, we showed how the stochastic competition between large numbers of linked mutations impacts the emergence of new ecotypes and the structure of the resulting ecosystem.

Accounting for these realistically large mutation rates led to qualitative departures from existing models of ecological diversification (33–40), which assume that evolution is much slower than ecology. However, the direction of this effect often conflicted with the simple view that a higher mutation rate will tend to enhance ecological diversity (34, 57, 58). Instead, we found that clonal interference creates a series of “priority effects” that favor the resident lineages, and cause the population as a whole to cluster into a handful of discrete metabolic strategies (or “ecotypes”). Our analysis shows that these priority effects arise from the delayed feedback between the ecologically relevant bulk of the population – which shapes the current environment – and the high fitness “nose” that comes to dominate the population at later times. These delays combine to give the resident strains an additional opportunity to encroach on an invading mutant’s niche. Since these delayed feedbacks are a generic feature of evolution in large asexual populations (54, 71, 72), we expect that similar priority effects will likely emerge in other ecological settings (19, 73) beyond the resource competition models studied here.

Our results also provide a framework for interpreting data from microbial evolution experiments, where ecological diversification is often observed. For example, we have seen that the delayed feedbacks from clonal interference can cause invading ecotypes to undergo substantial oscillations in frequency when they first establish (Fig. 3). Similar dynamics have previously been observed experimentally, e.g. during the emergence of the crossfeeding interactions in Lenski’s long-term evolution experiment in E. coli (LTEE) (3, 28). Our results suggest that these dramatic reversals might be a natural consequence of ecological diversification in large populations, even in the absence of additional factors like epistasis or environmental fluctuations. On the other hand, our results also highlight cases where theory and data appear to diverge. For example, the coexisting lineages in the LTEE persist for much longer (and with larger fluctuations) than expected under our simple model, which predicts that the lagging ecotype should be replaced by cross-invaders every $\sim T_{\mathrm{s}\mathrm{w}}$ generations. These examples illustrate how a quantitative understanding of diversification can uncover qualitative features of the data that require further theoretical explanation.

Our mathematical analysis relied on several simplifying assumptions that could be relaxed in future work. One important simplification was the assumption that constitutively beneficial mutations confer the same characteristic fitness benefit $s_b$. While this caricature is unlikely to be realized in nature, previous work has shown that more general distributions of fitness effects can often be mapped to an equivalent single-$s$ model with appropriately chosen values of $s_b$ and $U_b$ (52–54, 56). We therefore expect that most of our conclusions will be robust to relaxations of this particular assumption. Simulations with a more realistic exponential distribution of fitness effects support this intuition (Fig. S3), exhibiting qualitatively similar priority effects as Fig. 2.

Another critical assumption was the absence of epistasis, which could influence our current results in a number of important ways. Previous work has shown that global epistasis can cause the fitness effects of new mutations to systematically decline over time as the population fitness increases (74–82). Our analysis shows that this effect would tend to stabilize an existing pair of ecotypes by increasing $T_{\mathrm{s}\mathrm{w}}$ and slowing the rate of cross-invasion — an effect that was previously observed in simulations in Ref. (58) (SI Section 3.5). On the other hand, it is possible that strategy mutations could also alter the supply of new mutations by rewiring an organism’s metabolism, creating new opportunities for compensatory mutation (81). In this case, the different rates of adaptation of the ecotypes could overwhelm the dynamical priority effects we identified above (56). Accounting for this interplay between diversification and evolvability — as well as additional effects like recombination (83–85) — would be a valuable direction for future work.

Finally, many natural microbial communities are considerably more diverse than the two-resource environments we have focused on here. While most of our results continue to hold for larger numbers of supplied resources (SI Section 3), accounting for larger numbers of coexisting ecotypes remains a pressing open challenge (16, 34, 35). We expect that the analytical tools developed in this work — particularly the dynamical establishment picture in SI Section 3.3 — may be useful for extending our current results to the many-ecotype limit as well. However, in this case, the genealogical relationships between ecotypes can be considerably more complex than Fig. 5, and the resulting network of diversification can play an important role in shaping the long-term structure of the ecosystem (34). Future efforts to account for these effects will be critical for understanding how large microbial communities evolve.

Code availability:

Scripts to generate all simulations and plots are avaiable at https://github.com/DanielWongPGH/consumer_resource_clonal_interference.