In most environments, organisms compete for limited resources. The number and relative abundance of species that an ecosystem can host is referred to as ‘species diversity’. The competitive-exclusion principle is a hypothesis which proposes that, in an ecosystem, competition for resources results in decreased diversity: only species best equipped to consume the available resources thrive, while their less successful competitors die off. However, many natural ecosystems foster a wide array of species despite offering relatively few resources.

Researchers have proposed many competing theories to explain how this paradox can emerge, but they have mainly focused on ecosystems where nutrients are steadily supplied. By contrast, less is known about the way species diversity is maintained when nutrients are only intermittently available, for example in ecosystems that have seasons.

To address this question, Erez, Lopez et al. modeled communities of bacteria in which nutrients were repeatedly added and then used up. Depending on conditions, a variety of relationships between the amount of nutrient supplied and community diversity could emerge, suggesting that ecosystems do not follow a simple, universal rule that dictates species diversity. In particular, the resulting communities displayed a higher diversity of microbes than the limit imposed by the competitive-exclusion principle.

Further observations allowed Erez, Lopez et al. to suggest guiding principles for when diversity in ecosystems will be maintained or lost. In this framework, ‘early-bird’ species, which rapidly use a subset of the available nutrients, grow to dominate the ecosystem. Even though ‘late-bird’ species are more effective at consuming the remaining resources, they cannot compete with the increased sheer numbers of the ‘early-birds’, leading to a ‘rich-get-richer’ phenomenon.

Oceanic plankton, arctic permafrost and many other threatened, resource-poor ecosystems across the world can dramatically influence our daily lives. Closer to home, shifts in the microbe communities that live on the surface of the human body and in the digestive system are linked to poor health. Understanding how species diversity emerges and changes will help to protect our external and internal environments.

Microbial communities feature an immense diversity of organisms, with the typical human gut microbiota and a liter of seawater both containing hundreds of distinct microbial types (Lloyd-Price et al., 2016; Ladau et al., 2013; Weigel and Pfister, 2019). These observations appear to clash with a prediction of some resource-competition models, known as the competitive-exclusion principle – namely, that steady-state coexistence is possible for only as many species as resources (Levin, 1970; Armstrong and McGehee, 1980). This conundrum is familiarly known as the ‘paradox of the plankton’ (Hutchinson, 1961). Solving this paradox may provide one key to predicting and controlling outcomes ranging from ecosystem stability to successful cancer treatments in humans (Ptacnik et al., 2008; van Elsas et al., 2012; Taur et al., 2014; Stein et al., 2013). Chesson, 2000 classified mechanisms that purport to solve this paradox into two broad categories: stabilizing and equalizing. Stabilizing mechanisms prevent extinction by allowing species to recover from low populations, whereas equalizing mechanisms slow extinction by minimizing fitness differences between species.

Many possible solutions of the paradox that rely on stabilizing mechanisms have been offered: (i) interactions between microbes, such as cross-feeding or antibiotic production and degradation (Goyal and Maslov, 2018; Kelsic et al., 2015), (ii) spatial heterogeneity (Murrell and Law, 2003; Tilman, 1994), (iii) persistent non-steady-state dynamics (Hutchinson, 1961), and (iv) predation (Thingstad, 2000). Equalizing mechanisms have been studied through neutral theory, in which all species are assumed to have equal fitness (Hubbell, 2005), and recent work has proposed resource-competition models that self-organize to a neutral state (Posfai et al., 2017). Many proposed solutions for the paradox assume a chemostat framework wherein nutrients are continuously supplied and there is a continuous removal of biomass and unused nutrients (Palmer, 1994). However, in nature nutrients are rarely supplied in a constant and continuous fashion. In particular, seasonal variation is ubiquitous in ecology, influencing systems ranging from oceanic phytoplankton communities (Chang, 2003) to the microbiota of some human populations (Smits et al., 2017). How does a variable nutrient supply influence diversity?

Existing literature on seasonality focuses on stabilizing mechanisms and generally finds that seasonality either promotes or has little effect on coexistence (Chesson, 1994). But do these conclusions extend to equalizing mechanisms? To address this question, we consider a known resource-competition model that permits high diversity at steady state due to the equalizing effects of metabolic trade-offs, which assume that microbes have a limited enzyme production capacity they must apportion. Here, we investigate the equalizing effect of metabolic trade-offs in the context of serial dilution, to reflect a more realistic variable nutrient supply.

Serial dilution, in which cultures of bacteria are periodically diluted and supplied with fresh nutrients, is well-established as an experimental approach. For example, the bacterial populations in the Lenski long-term evolution experiment (Lenski and Travisano, 1994), experiments on community assembly (Goldford et al., 2018), and antibiotic cross-protection (Yurtsev et al., 2016) were all performed in serial dilution. While previous models of serial dilution have characterized competition between small numbers of species with trade-offs in their growth characteristics (Stewart and Levin, 1973; Smith, 2011), the theoretical understanding of diversity in serial dilution is much less developed than for chemostat-based steady-state growth.

Here, we show that under serial dilution, metabolic trade-offs can still support high diversity communities, but that this coexistence is now sensitive to environmental conditions. Interestingly, seasonality can both increase and decrease diversity in our model, contrasting what has been observed for stabilizing mechanisms. In particular, we find a surprising dependence between community diversity and the amount of nutrient provided to the community. These changes in diversity are driven by an ‘early-bird’ effect in which species that efficiently consume nutrients that are initially more abundant gain a population advantage early in the batch. To our knowledge, this is the first time this effect has been identified as a major influencer of diversity in seasonal ecosystems.

This dependence between community diversity and the supplied nutrient concentration allowed us to explore an unresolved question in ecology (Tilman, 1982; Abrams, 1995; Leibold, 1996): what is the relationship between the amount of nutrient supplied and the resulting diversity of the community? Experimental studies of this question have mainly been performed in macroecological contexts (Mittelbach et al., 2001; Waide et al., 1999; Adler et al., 2011), though recently there has been increased focus on microbial systems (Bienhold et al., 2012; Bernstein et al., 2017). In microbial experiments, some evidence has supported the ‘hump-shaped’ unimodal trend predicted by many theories (Kassen et al., 2000). However, a meta-analysis by Smith, 2007 found no consistent trend across microbial experiments. What we observe here is concordant with Smith’s result: even in our highly simplified model, there is no general relationship between nutrient supply and diversity. Among the factors we find that influence this relationship are cross-feeding, relative enzyme budgets, differences in enzyme affinities, and differences in nutrient yields. That so much variation appears in a simple model suggests that real ecosystems are not likely to display a single universal relationship between nutrient supply and diversity.

We employ the serial dilution model depicted in Figure 1A (see Appendix 1—table 1). At the beginning of each batch ($t=0$), we introduce the inoculum, defined as a collection of species $\{\sigma \}$ with initial biomass densities ${\rho }_{\sigma }(0)$ in the batch such that the total initial biomass density is ${\rho }_0={\sum }_{\sigma }{\rho }_{\sigma }(0)$. Together with the inoculum, we supply a nutrient bolus, defined as a mixture of p nutrients each with concentration in the batch $c_i(0)$, $i=1,\mathrm{\dots },p$ such that the total nutrient concentration is $c_0={\sum }_{i=1}^p{c}_i(0)$ (we also consider the case of cycles of single nutrient boluses that approach a mixture distribution, cf. Appendix 7—figure 1). It is assumed that all nutrients are substitutable (i.e. all nutrients are members of a single limiting class of nutrients, e.g. nitrogen sources). For simplicity, we assume ideal nutrient to biomass conversion, so that for a species to grow one unit of biomass density, it consumes one unit of nutrient concentration (we consider the case of nutrient-specific yields Y_i in a later section). During each batch, the species biomass densities ${\rho }_{\sigma }(t)$ increase with time, starting at $t=0$, and growth continues until the nutrients are fully depleted, ${\sum }_{i=1}^p{c}_i(\mathrm{\infty })\approx 0$ (we consider the case of incomplete depletion in Appendix 7—figure 2). Thus, at the end of a batch, the total biomass density of cells is ${\sum }_{\sigma }{\rho }_{\sigma }(\mathrm{\infty })={\rho }_0+c_0$. The next batch is then inoculated with a biomass density ${\rho }_0$ with a composition that reflects the relative abundance of each species in the total biomass at the end of the previous batch. This process is repeated until ‘steady state’ is reached, i.e. when the biomass composition at the beginning of each batch stops changing.

Figure 1

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In the model, a species σ is defined by its unique enzyme strategy ${\overrightarrow{\alpha }}_{\sigma }=({\alpha }_{\sigma ,i},\mathrm{\dots },{\alpha }_{\sigma ,p})$ which determines its ability to consume different nutrients. We assume that each species can consume multiple nutrients simultaneously, in line with the behavior of microbes at low nutrient concentrations (Kovárová-Kovar and Egli, 1998), though this assumption may not hold for all microbial species. Specifically, we assume that species σ consumes nutrient i at a rate $j_{\sigma ,i}$ (per unit biomass) that depends on nutrient availability c_i and on its enzyme-allocation strategy ${\alpha }_{\sigma ,i}$ according to 

For simplicity, we take all Monod constants to be equal, $K_i=K$ (a more general form of the nutrient model is considered in a later section). During each batch, the dynamics of nutrient concentrations and biomass densities then follow from the rates $j_{\sigma ,i}$ at which the species consume nutrients: 

Since the level of one enzyme inevitably comes at the expense of another, we model this trade-off via an approximately fixed total enzyme budget E. Formally, we take ${\sum }_i{\alpha }_{\sigma ,i}=E+\epsilon {\xi }_{\sigma }$, where ${\xi }_{\sigma }$ is a zero-mean and unit-variance Gaussian variable. Without loss of generality we take $E=1$; initially we set $\epsilon =0$, which we call exact trade-offs. This allows us to visualize the strategies ${\overrightarrow{\alpha }}_{\sigma }$ as points on a simplex, depicted as colored circles embedded in: (i) the interval $[0,1]$ for two nutrients (Figure 1B), or (ii) a triangle for three nutrients (Figure 1C), etc. One can plot the nutrient bolus composition $c_i/c_0$ on the same simplex, as depicted by the black diamonds in Figure 1B and C. In what follows, we focus on the case of two nutrients, though the main results extend to an arbitrarily large number of nutrients.

Effect of total nutrient bolus on coexistence

In the chemostat limit, increasing the nutrient supply rate simply proportionally increases the steady-state population abundances. However, away from this limit we find that the magnitude of the nutrient bolus can qualitatively affect the steady-state outcome of serial dilutions. To understand this effect, we first consider a simple case of two nutrients and two species as depicted in Figure 2A. The two species will coexist if each species is invasible by the other. In our example, we first determine the invasibility of species R (strategy indicated by red circle) by species with strategies lying to its left. To this end, we choose a nutrient supply and perform model serial dilutions until steady state is reached. For a particular finite bolus size, we find that for all supplies within the hatched region an infinitesimal inoculum of any species lying to the left of R will increase more than R during a batch, and therefore can invade R. Similarly, we determine the invasibility of species B (strategy indicated by blue circle) by any species with a strategy lying to its right, and find the second hatched region. The intersection of these hatched regions for which (1) B can invade R and (2) R can invade B is the supply interval of mutual invasibility where these two species will stably coexist. The coexistence interval is bounded by the red and blue triangles, and each of these coexistence boundaries is a unique property of its corresponding species. We call these species-specific boundaries remapped because they generally lie at different locations on the simplex than the strategies they originated from, with the extent of remapping depending on the nutrient bolus size. At a more technical level, the remapped boundary for a given species and bolus size is the nutrient supply for which, over the course of a batch, all nutrients are equally valuable and so, a species with any strategy can neutrally invade and persist. This equality of nutrient value is defined in terms of the Monod function integrals for each nutrient (for details see Appendix 3).

Figure 2

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Since the remapped coexistence boundaries depend on the nutrient bolus size c₀, changing bolus size can qualitatively change the steady-state outcome of serial dilutions. Figure 2B–D depicts an example of how c₀ affects remapping, and the consequences for species coexistence. At low bolus size, $c_0\ll K$, corresponding to the chemostat limit, Figure 2B (left) shows that all species present achieve steady-state coexistence. This follows because the nutrient supply (black diamond) lies inside the convex hull. When c₀ is increased to $c_0\approx K$ (Figure 2C), the coexistence boundaries are remapped towards the center of the simplex (dashed arrow). In this example, the nutrient supply now lies outside the convex hull. This results in one winner species (the dark blue one nearest the supply), with all others decreasing exponentially from batch to batch. This loss of coexistence with increasing nutrient bolus size is reminiscent of Rosenzweig’s ‘paradox of enrichment’ in predator-prey systems (Rosenzweig, 1971). Strikingly, however, as bolus size is further increased to $c_0\gg K$, the coexistence boundaries are remapped back to their original positions, so that the nutrient supply once again lies within the convex hull, and so steady-state coexistence of all species is recovered.

What causes the remapping of the coexistence boundaries inwards as $c_0/K\to 1$? Let us consider a single species growing on two nutrients supplied in the same proportion as its strategy (i.e. the fraction of Nutrient 1 is equal to ${\alpha }_{\sigma ,1}$). At low $c_0/K$, this marks the remapped coexistence boundary and both nutrients will be equally valuable over the course of a batch. The balance is achieved because the nutrient with a higher initial abundance is more rapidly exhausted, while the nutrient with lower initial abundance is consumed more slowly and is therefore available for a longer span of time. At small $c_0/K$, the more rapid initial consumption of the more abundant nutrient does not influence the consumption rate of the less abundant nutrient because the bolus size is small relative to the initial population size, so the population does not grow substantially during the batch. This changes as $c_0/K$ increases: the rapid initial consumption of the more abundant nutrient leads to a substantial increase in the total population. The remaining low initial abundance nutrient is now consumed more quickly and is less available to an invader with a more balanced enzyme strategy. The nutrients are no longer equally valuable on average, and remedying this requires a more equally balanced nutrient bolus. Thus, the remapped coexistence boundary moves inwards (see Appendix 7—figure 3). In essence, as $c_0/K$ increases it is more difficult for the invader to grow because the resident gains an ‘early-bird’ advantage: its initial growth allows it to more effectively exhaust the nutrients.

Why does the coexistence boundary of a species map back to its original strategy in the limit of large bolus size, $c_0\gg K$? In this limit, the nutrient uptake functions in Equation 1 will be saturated during almost the entire period of a batch. Each species will therefore consume nutrients strictly in proportion to its strategy ${\alpha }_{\sigma ,i}$. For the case of two nutrients (e.g., as shown in Figure 1B), if there is only a single species present then if the supply lies anywhere to the left of its strategy, at some time during the batch there will be some of Nutrient 2 remaining after the bulk of Nutrient 1 has been consumed. Thus a single species can be invaded by any strategy to its left, provided the supply also lies to its left. Similarly, a species can be invaded by any strategy to its right if the supply lies to its right. This is exactly the condition for the coexistence boundary of a species to coincide with its actual strategy (details in Appendix 5).

We have rationalized coexistence in our serial dilution model in terms of mutual invasibility, but have not explicitly stated the condition for an arbitrary number of species to coexist in steady state. In the chemostat, all species coexist when the concentrations of all nutrients are equal, implying the same growth rate for all strategies. However, for serial dilutions the nutrient concentrations are generally not equal and are not even constant in time. Instead, it is the integrated growth contribution of every nutrient that must be equal to allow for arbitrary coexistence. In the case of equal enzyme budgets ($\epsilon =0$), this condition occurs when the time integrals of the nutrient Monod functions within a batch are all equal, that is,

(7) $I_i={\int }_0^{\mathrm{\infty }}\frac{c_i}{K_i+c_i}𝑑t=\text{const}.$

To understand this condition for coexistence beyond competitive exclusion, note that the instantaneous rate of growth of a species σ is ${\sum }_i{\alpha }_{\sigma ,i}{c}_i/(K_i+c_i)$, so that the fold increase of a species during a batch is $\exp ({\overrightarrow{\alpha }}_{\sigma }\cdot \overrightarrow{I})$. This fold increase will be equal for all species if and only if Equation 7 holds. When there are two nutrients, Equation 7 holds at steady state whenever the supply is inside the convex hull of the coexistence boundaries of the species present (details in Appendix 3). For more nutrients, the corresponding condition is that the region of coexistence is bounded by contours that connect the outermost remapped nodes.

Given a fixed set of species and a choice of initial populations, repeating the growth-dilution batch procedure results in a steady state where the populations at the beginning of a batch do not change from batch to batch. The steady-state populations depend on the initial populations, with the set of all possible steady-state populations defining a coexistence manifold.

Steady-state diversity

As is apparent in Figure 2C, not all strategies are remapped to the same extent. In Figure 3A, we plot the remapping of coexistence boundaries as a function of nutrient bolus c₀. Note that: (i) the specialists $(0,1)$ and $(1,0)$ and the perfect generalist $(0.5,0.5)$ are not remapped at all; (ii) remapping is maximal for $c_0\approx K$; (iii) there is no remapping in both the $c_0\to 0$ and $c_0\to \mathrm{\infty }$ limits (see also Appendix 7—figure 4). The extent of remapping also depends on the inoculum size ${\rho }_0$ as shown in Figure 3B, which demonstrates that remapping is maximal for ${\rho }_0\ll K$ and vanishes for ${\rho }_0\gg K$.

Figure 3

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How does nutrient bolus size influence steady-state species diversity? A useful summary statistic for quantifying diversity (Jost, 2006) is the effective number of species $m_e=e^S$ with the Shannon diversity $S=-{\sum }_{\sigma }{P}_{\sigma }\ln P_{\sigma }$ and $P_{\sigma }={\rho }_{\sigma }^{*}(0)/{\rho }_0$, with ${\rho }_{\sigma }^{*}(0)$ the steady-state species abundances at the beginning of a batch. Diversity as measured by m_e is shown in Figure 3C for six choices of nutrient bolus composition. Notably, if the two nutrients are supplied equally (top curve, magenta), m_e is independent of c₀ and coincides with the maximal possible diversity (dashed black line), namely equal steady-state abundances of all species (Figure 3D, top). Conversely, if Nutrient 1 comprises only 5% of supplied nutrient (Figure 3C, bottom curve, cyan), the number of effective species m_e is lower than maximal even in the chemostat-limit of small bolus sizes $c_0\ll K$ and drops even further for $c_0\approx K$. This loss of diversity is due to the dramatically lowered steady-state abundances of strategies that favor Nutrient 1 (Figure 3D, bottom). Two different effects underlie this change in community structure. The first is the early-bird effect described above: species specializing in more abundant nutrients gain a population advantage that allows them to rapidly consume less abundant nutrients that would otherwise support species with different enzyme specializations. The second effect is a well-known property of single nutrient competition and can be viewed as a ‘single-nutrient’ early-bird effect. In this case, species that are superior competitors for a nutrient gain an exponential population advantage over inferior competitors, increasing their share of total nutrient beyond the ratio of initial consumption rates. Both of these effects increase in strength with larger bolus size because the early-bird advantage increases as growth proceeds. The combination of these effects results in the species specialized in consuming the most abundant nutrients consuming a larger fraction of all nutrients. However, for very large bolus sizes, saturation of nutrient uptake rates mitigates these two effects, leading to a lack of remapping for $c_0\gg K$ and diversity returning to its chemostat-limit. Though here we focused on the case of two nutrients, these results extend to more nutrients (for three nutrients see Appendix 7—figure 5).

Unequal enzyme affinities and nutrient yields

We have thus far made the simplifying assumption that all enzymes have the same substrate affinity, such that $K_i\equiv K$. However, in nature different nutrients may have drastically different values of K. For example, the methanogen Methanosarcina barkeri has K_i for the consumption of hydrogen and acetate that differ by approximately three orders of magnitude (Robinson and Tiedje, 1984; Wandrey and Aivasidis, 1983). How would such a large difference in the K_i values impact diversity in our serial dilution ecosystem? In Figure 4A we show diversity as a function of bolus size for a system with a large difference in K_i ($K_1={10}^{-3}$, $K_2=1$). Since the symmetry between nutrients is broken by the unequal K_i, we now show the entire range of nutrient proportions, not just the first half. In the chemostat limit, the diversity values are similar to those found in the system with equal K_i. This makes sense: in a chemostat the nutrients with higher K_i can accumulate to higher levels to compensate for their slow consumption, leaving the steady-state behavior unchanged. However, outside the chemostat regime, differences in the K_i have a drastic effect: when the nutrient with the lower K_i is scant in supply, diversity generally increases with increasing c₀, while the opposite occurs when the nutrient with the lower K_i is higher in supply.

Figure 4

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We can understand these K_i-driven shifts in the nutrient-diversity relationship as due to changes in the identity of the early bird. In a model with equal K_i, the identity of the early bird is determined by which nutrient is more abundant: if the two nutrients have equal K_i, a species can gain an early-bird advantage by preferentially consuming the more abundant nutrient. This changes if one nutrient has a much lower K_i than the other. In this case it may be advantageous to preferentially consume the nutrient with the lower K_i, even if it is the less abundant nutrient. If the nutrient with the lower K_i is also the more abundant nutrient, this will intensify the early-bird advantage. Why does this change in the early bird’s identity change the form of the nutrient-diversity relationship? This change arises from a clash between optimal feeding behavior in the chemostat and seasonal regimes. In the chemostat, it is advantageous to focus on the most abundant nutrient, regardless of the value of K_i. Thus, in the chemostat limit, species focusing on the more abundant nutrient have an advantage. In the case of equal K_i (or K_i favoring the more abundant nutrient), this advantage is intensified by the early-bird effect, increasing the biomass of already abundant species and lowering diversity. By contrast, if the low abundance nutrient has a low K_i, the early-bird effect will have the opposite effect on diversity. Now the early-bird effect benefits species that were disadvantaged in the chemostat limit, leading to more equal abundances and higher diversity. This shift in abundances is shown in Figure 4B. The change in the identity of the early bird can also explain more complex relationships between diversity and bolus size (see Appendix 7—figure 6).

In addition to unequal enzyme affinities, it is possible for different nutrients to have different yields, Y_i. In Figure 4C we show the relationship between bolus and diversity for a system with $Y_1=10$ and $Y_2=1$. As expected, at low c₀ the diversity is similar to that in the case of equal Y_i. As $c_0/K$ increases, the diversity decreases initially and the symmetry-related bolus-composition cases (e.g. [0.2,0.8] and [0.8,0.2]) eventually diverge, with one’s diversity rising and the other’s continuing to fall. This behavior is explainable by the same logic as in the variable K_i case: diversity rises or falls depending on whether the early-bird species was also favored in the chemostat limit. However, unlike the case of variable K_i, the diversity curves do not eventually return to the chemostat limit. Regardless of which nutrient the Y_i favor, the diversity eventually begins decreasing monotonically as $c_0/K$ increases. This difference between the variable K_i and variable Y_i cases can be understood by considering what occurs when both nutrients are saturating. In the variable K_i case, saturating nutrients are equal in value, implying a return to the chemostat limit as the early-bird effect weakens. In contrast, for variable Y_i, there remains a difference in the value of the two nutrients in the saturated regime, meaning that the early-bird effect will grow stronger and the early bird will take over the population. The beginning of this takeover can be seen at high bolus sizes in Figure 4D. Note that for both variable K_i and variable Y_i, these trends are also reflected in the remapping (see Appendix 7—figure 7).

Despite the large variation in relationships between diversity and bolus size, these phenomena can all be understood as consequences of the early-bird effect. As the model becomes more complex there are additional factors to consider in determining which nutrient will provide an early-bird advantage, but the fundamental mechanism of exploiting early growth advantages remains.

Cross-feeding

It is possible to extend Equations 2 and 3 beyond a single trophic layer, allowing for consumption of metabolic byproducts. This is a form of cross-feeding, which has generally been found to promote diversity (Goyal and Maslov, 2018) and stable community structure (Goldford et al., 2018). Here, cross-feeding is introduced through the byproduct matrix ${\mathrm{\Gamma }}_{i,i^{\prime }}^{\sigma }$, which converts the consumption of nutrient $i^{\prime }$ to production of nutrient i such that,

(8) $\frac{d{c}_i}{dt}=-\sum _{\sigma }{\rho }_{\sigma }\left(j_{\sigma ,i}-\sum _{i^{\prime }}{\mathrm{\Gamma }}_{i,i^{\prime }}^{\sigma }{j}_{\sigma ,i^{\prime }}\right).$

In this framework, nutrient $i^{\prime }$ is converted to nutrient i at no extra enzymatic cost, meaning that nutrient i is simply a byproduct whenever nutrient $i^{\prime }$ is consumed for growth (it would be straightforward to modify this framework so that nutrient conversion can be carried out independently from growth). We focus on the simplest case: initially supplying only Nutrient 2, with Nutrient 1 solely derived as a metabolic byproduct via ${\mathrm{\Gamma }}_{i,i^{\prime }}^{\sigma }=\left(\begin{array}{cc}\hfill 0\hfill & \hfill \mathrm{\Gamma }\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$ for all species. When $\mathrm{\Gamma }=1$, upon consumption Nutrient 2 is perfectly converted to Nutrient 1, leading to an equal total supply of the two nutrients. More generally, ${\int }_0^{\mathrm{\infty }}{\sum }_{\sigma }{\rho }_{\sigma }{j}_{\sigma ,1}dt=\mathrm{\Gamma }{c}_2(0)$ which allows a direct comparison between the unitrophic and bitrophic regimes: starting with $c_2(0)$ results in $(\mathrm{\Gamma }+1)c_2(0)$ total nutrient, and hence the Nutrient 1 fraction is $\frac{\mathrm{\Gamma }}{1+\mathrm{\Gamma }}$ of the total.

How does cross-feeding influence diversity in our serial dilution model? In Figure 5A we compare bitrophic diversity for six values of $\mathrm{\Gamma }$ to their unitrophic equivalents (in Figure 3C). We note that: (i) bitrophy still supports diversity greater than the competitive-exclusion limit; (ii) in the chemostat regime, $c_0\ll K$, the unitrophic and bitrophic schemes have identical values of m_e, and these drop as $c_0\to K$; (iii) but for bitrophy the m_e does not recover for $c_0\gg K$; (iv) even when the total supply of both nutrients is equal ($\mathrm{\Gamma }=1$), bitrophy leads to lower than maximal m_e outside the chemostat limit. These features are clarified in Figure 5B, which shows steady-state species abundances for $\mathrm{\Gamma }$ values leading to a total Nutrient 1 supply fraction of 0.5 and 0.05, and highlights the lower diversity for bitrophy compared to unitrophy for large nutrient bolus size. This difference is due to an early-bird effect: the species consuming supplied nutrient early in the batch can build a sizable population before the competing species that rely on its byproduct. The early-bird population then outcompetes the others for byproduct consumption. As such, this effect increases with $c_0/{\rho }_0$. The effect also becomes stronger at low $c_0/K$ (with constant $c_0/{\rho }_0$), since this allows the early-bird species more time to grow before the byproduct accumulates to high enough levels to be significantly consumed (Appendix 7—figure 8). Note that this effect is dependent on metabolite byproducts being also consumed by their producer. If the species in each trophic layer are single-nutrient specialists, then changes in $c_0/K$ have no impact on community diversity.

Figure 5

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The behavior of the model with cross-feeding shows that the early-bird effect extends beyond simple metabolic trade-offs. More broadly, when species compete for multiple resources that are supplied in batches, a species’ survival depends on more than its ability to efficiently consume nutrients. An early-bird species, being more specialized in consuming the nutrients that are initially more abundant, gains a population advantage early in the batch. This population advantage may allow the early-bird species to out-compete other species even when consuming nutrients it is not specialized to consume. Despite its consumption inefficiencies, through sheer numbers the early-bird species can consume more of the remaining nutrients than its more specialized competitors.

Population bottlenecks

So far we have considered deterministic dynamics, which is appropriate for large populations. In natural settings, however, there are often small semi-isolated communities. For these communities, fluctuations can play an important role. In particular, population bottlenecks can lead to large demographic changes (Abel et al., 2015). In our model, how does the nutrient supply affect diversity in such communities? To address this question, we applied discrete sampling of a finite population when diluting from one batch to the next (see Appendix 1). With this protocol, an ‘extinction’ occurs when sampling yields zero individuals of a species. For a long enough series of dilutions such extinctions would ultimately lead to near-complete loss of diversity. For small real-world populations, however, diversity may be maintained by migration. To model such migration we augmented the population at each dilution with a ‘spike-in’ from a global pool of species, in the spirit of MacArthur’s theory of island biogeography (MacArthur and Wilson, 2001). Specifically, in the spike-in procedure, to prevent extinctions caused by sampling fluctuations, every new batch is inoculated with a small number of the original, founder species.

In Figure 6A we show results of spike-in serial dilutions for a population bottleneck of 1008 cells. 95% of these cells are sampled from the previous batch, while 5% are sampled from a global pool, with equal abundances of 21 equally spaced strategies (cf. Figure 3A). The resulting m_e vs. c₀ curves have maximal m_e for all six nutrient fractions in the regime $c_0\ll K$ where the 5% spike-in dominates sampling noise. As expected, for a balanced nutrient supply at any c₀, all species have the same average abundance (Figure 6B top). By contrast, when Nutrient 1’s fraction is low (Figure 6A cyan and 6B bottom), increasing c₀ increases the abundance gaps between the species, reflecting the uneven competition for Nutrient 2. Overall, the spike-in protocol leads to higher diversity at low c₀ than the deterministic case (starting from equal species abundances but with no spike-in, Figure 3C). For large c₀, the m_e vs. c₀ curves for these two protocols are indistinguishable. The only noticeable difference is that the spike-in maintains a higher level of the least competitive strains, but since these abundances are still low, this difference in not reflected in the m_e values.

Figure 6

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Unequal enzyme budgets

While we have assumed exact trade-offs to achieve diversity within a resource-competition model, the trade-offs present among real microorganisms will not be exact. For the serial dilution protocol with spike-ins, diversity is maintained by migration and so it is possible to relax the constraint of exact trade-offs. How does diversity depend on the nutrient supply if we allow species to have different enzyme budgets? We implemented random differences in species enzyme budgets by setting $\epsilon =0.1$, that is, a standard deviation of 10%, and plotted effective number of species m_e in Figure 6C. As in the $\epsilon =0$ limit (Figure 6A), at sufficiently small c₀ the spike-in procedure dominates both sampling noise and differential growth rates due to unequal enzyme budgets. Raising c₀ leads to a drop in m_e (albeit still above the competitive-exclusion limit). Examining the species abundances in Figure 6D, we note that differences in enzyme budget establish a fitness hierarchy even when nutrient fractions are equal (top), with those species with the highest budgets increasing in relative abundance as c₀ increases. The asterisk (*) marks the species with the highest total enzyme budget, which becomes the most abundant for ${\displaystyle c_0\gg K}$. Reducing Nutrient 1’s fraction to 0.05 results in a shifting abundance hierarchy (Figure 6D, bottom): at low c₀ the highest abundance species is the one that consumes only Nutrient 2, as in the equivalent $\epsilon =0$ case. However, increasing c₀ results in increased abundance for the species with the highest enzyme budget – which would ultimately lead to its domination for sufficiently large c₀. This increasing dominance of the species with the highest enzyme budget is another manifestation of the early-bird effect: as the amount of growth in a batch increases, the advantage of a larger enzyme budget further compounds. In short, for spike-in serial dilutions the influence of unequal enzyme budgets depends on the nutrient supply, such that the species with the largest budgets dominate for large, unbiased supplies.

Natural ecosystems experience variations in the timing and magnitude of nutrient supply, and the impact of these variations on species diversity is not fully understood (Smith, 2011; Smith, 2007). To explore the impact of variable nutrient supply, we modeled resource competition in a serial dilution framework and analyzed the model’s steady states. We found that variable nutrient supply still allows for the high diversity seen in the continuous supply (‘chemostat’) version of the model. Indeed, the serial dilution steady state mimics that of a chemostat when the amount of nutrients supplied in each batch is small. Surprisingly, however, supplying the nutrients as a bolus led to a dependence of diversity on the amount of supplied nutrients.

In contrast to existing literature on seasonality, we find that environmental fluctuations can both weaken and strengthen coexistence in this model. This occurs as the result of an ‘early-bird’ effect associated with supplying nutrients as large seasonal boluses instead of continuously. Some species can capitalize on rapid initial growth on an abundant nutrient to reach a large population size, which then allows them to deplete the remaining nutrients at the expense of their competitors. This early-bird effect can both restrict and expand the range of environments in which communities can self-organize to a neutral state. We show that even when metabolic trade-offs are combined with stabilizing mechanisms, the impact of the early-bird effect remains. For example, in the case of cross-feeding, the community diversity falls as a function of $c_0/K$ due to the early-bird advantages gained by species at higher trophic levels.

While the idea of species gaining early advantages has been explored, such as in the literature on founder effects and speciation (Barton and Charlesworth, 1984; Brown,, 1957), to the best of our knowledge this is the first demonstration of the influence of the early-bird effect on the diversity of seasonal ecosystems. We believe that this effect will occur in a variety of such ecosystems, as its only fundamental requirement is competition for multiple nutrients that are supplied in a time-dependent manner. Interestingly, while the early-bird effect plays a large role in our model, it is not the only bolus-dependent effect that influences diversity. We also observe another effect that can be viewed as a ‘single-nutrient’ version of the early-bird effect. This effect arises from a well-studied property of competition: as growth proceeds, a superior competitor for a nutrient gains an exponential advantage over inferior competitors for that nutrient. Like the early-bird effect, this shifts the system’s biomass towards species more specialized in initially abundant nutrients, particularly for large but non-saturating nutrient bolus sizes. This single-nutrient effect can co-occur with the early-bird effect, for example in competition for an abundant nutrient between two early-bird species.

The form of seasonality we explore in this manuscript, where mixed boluses are supplied periodically, is only one possible form of seasonal nutrient supply. The impact of the early-bird effect and single-nutrient competition will likely differ between different forms of seasonality. For example, we show in Appendix 7—figure 1 that supplying cycles of single nutrient boluses that approach an equal distribution of nutrients results in lower diversity than supplying mixed equal nutrient boluses. While this form of seasonality differs from the one we characterized, we can still understand the loss of diversity as arising from the single-nutrient competition effect initially observed in our mixed-bolus models. We expect the principles gleaned from our models to be of use in understanding diversity in a variety of seasonal ecosystems.

Finding a general relation between the amount of nutrient supplied to a community and its diversity is a long-standing goal of theoretical ecology (Tilman, 1982; Abrams, 1995; Leibold, 1996). We found that in our model the form of the nutrient-diversity relation (NDR) can change based on model details. The model has two regimes: a low diversity and a high diversity regime. The former satisfies competitive exclusion (no more species coexisting than resources), whereas the latter exceeds competitive exclusion and occurs when the nutrient supply lies within the convex hull of the remapped metabolic strategies present (Posfai et al., 2017). At the bifurcation point between the two regimes, we observe critical slowing down in that the number of dilutions required to reach steady state diverges.

In the high diversity regime, the NDR can take several forms, resulting from the interplay of the early-bird effect and other mechanisms. Even with a single trophic layer, the NDR can be U-shaped, hump-shaped, monotonically decreasing, or have multiple peaks. These trends can then be further modified by the addition of more trophic layers, differences in enzyme budgets, etc.

Experimental studies that characterize the NDRs of microbial ecosystems have reached similarly variable conclusions. For example, one work studying bacterial communities in Arctic deep-sea sediments found an increasing trend between energy input and richness (Bienhold et al., 2012), while a study on photosynthetic microbial mats found a negative relationship between energy input and richness (Bernstein et al., 2017). A meta-analysis of aquatic microbial ecosystems found examples of both monotonic and non-monotonic NDRs, with no single form dominating (Smith, 2007). Our theoretical results, together with these experimental findings, indicate that there may be no single universal NDR in microbial ecosystems. This conclusion suggests that the best approach for characterizing the NDR of a given ecosystem is not to apply a one-size-fits-all theory, but to analyze the role of different factors such as cross-feeding, trade-offs, and immigration in determining that particular ecosystem’s NDR. While we have focused on microbial systems, the absence of a universal NDR is consistent with results from recent work in plants (Adler et al., 2011).

We found that the stringency of metabolic trade-offs has a large impact on community diversity. We imposed a metabolic enzyme budget on each species to reflect the reality that microbial cells have a finite capacity to synthesize proteins and must carefully apportion their proteome (Basan et al., 2015). However, while it is true that microbes have limited biosynthetic capacity, it is unclear how strict are the resulting trade-offs. For this reason, we characterized versions of the model with both exact and inexact trade-offs. Our results show that the form of an ecosystem’s NDR can depend on the stringency of metabolic trade-offs. This finding is not exclusive to the serial dilution model. The stringency of trade-offs was also important in the original chemostat setting: in a birth-death-immigration framework, small violations of the enzyme budget still allowed for high levels of coexistence, but large violations disrupted coexistence (Posfai et al., 2017). These results suggest that an experimental characterization of the stringency of metabolic trade-offs among microbes would provide a valuable ecological parameter. Note that metabolic trade-offs are only one of the many types of trade-offs microbes are subject to; other types of trade-offs, such as constraints between biomass yield and growth rate (Wortel et al., 2018), may also shape a community’s NDR.

In constructing a model, we made a number of assumptions about the way in which microbes consume and utilize nutrients. Some of these assumptions do not apply to all microbial communities, and the impact of relaxing these assumptions can affect the NDR. For example, we mostly focused on communities where all nutrients are equally valuable (i.e. $Y_i=Y_j\forall i,j$). However, biomass yields can vary between nutrients and between species, which we explored in Figure 4C–D. Notably, unequal yields create differences between nutrients even in the saturating regime ($c_0\gg K$), leading to a departure from the chemostat limit at large nutrient boluses. Coexistence in the serial dilution model is robust to varying yield, as long as all species have the same yield on a given nutrient. The scenario where species have different biomass yields on the same nutrient is conceptually similar to the case of inexact trade-offs, since some species will have a strict advantage over others. Thus, it is likely that these unequal yields between species will lead to a reduction in community diversity. However, varying the yield in this manner also allows for the inclusion of new trade-offs that may impact diversity, such as the aforementioned trade-off between yield and growth rate (Wortel et al., 2018). We also explored the effects of unequal Monod constants for different nutrients (cf. Figure 4A–B). We found that if a low-abundance nutrient also has a low K_i, the early-bird effect favors species that were disadvantaged in the chemostat limit, thus reversing the equal-K_i NDR and leading to hump-shaped NDR curves. Indeed, large differences in K_i values can lead to a multi-peaked NDR as shown in Appendix 7—figure 6.

Our model assumes that all nutrients are substitutable (i.e. only one of the multiple nutrients is required for growth). In real ecosystems, microbes can require multiple complementary nutrients to grow, e.g. sources of carbon, nitrogen, and phosphorus. In cases where one class of complementary nutrient is strongly limiting, a model with both complementary and substitutable resources would essentially reduce to the current model of only substitutable resources. This case is likely the more common one, e.g. as many soils are carbon limited (Aldén et al., 2001; Demoling et al., 2007). However, in cases where no single nutrient is strongly limiting, the presence of complementary nutrients would possibly lead to different NDRs, which will be an interesting direction for future study.

Our modeling predictions, e.g. the convex hull condition and the changes in diversity due to the early-bird effect, are in principle testable. To connect our modeling assumptions to real microbial systems, we compare our growth model of substitutable and simultaneous nutrient consumption to previously published experimental data from Escherichia coli growing in batch and chemostat conditions. We find that our modeling assumptions are consistent with both datasets and outline potential future experiments to test the model’s multispecies predictions, detailed in Appendix 6. As is apparent in Appendix 6—figure 1, the growth dynamics of E. coli at low nutrient levels is well described by our modeling framework. The experiments we compared were performed with the same strain of E. coli, meaning that inclusion of different microbes would be needed to test the multispecies predictions. To determine the strategies of other microbes, including other strains of E. coli, the most practical approach would likely be batch culturing. Once strains with different strategies have been identified, nutrient-diversity relationships could then be obtained by competing strains in serial dilution culture and measuring the community diversity (e.g. via fluorescent tags or by 16S rRNA sequencing) as a function of the total concentration of multiple, substitutable nutrients provided at the start of each batch.

No datasets were generated in this work. All code and data used in this manuscript available at: https://github.com/AmirErez/SeasonalEcosystem (copy archived at https://github.com/elifesciences-publications/SeasonalEcosystem).

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