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Data
All the data sets analyzed in this work have been previously published and were obtained from EBI Metagenomics69. Previous publications (Supplementart Table 1) report the original experiments and corresponding analysis. In order to test the robustness of the macroecological laws and the modeling framework presented in this work, we considered 7 data sets that differ not only for the biome considered, but also for the sequencing techniques and the pipeline used to process the data. Data sets were selected to represent a wide set of biomes. We considered only data sets with at least 50 samples with more than 104 reads. No data set was excluded a-posteriori.
Sampling and compositional data
In order to study how (relative) abundance varies across communities and species, one needs to remove the effect of sampling noise, as it is not a biologically informative source of variation. By explicitly modeling sampling (Supplementary Note 2), one finds that the probability of observing n reads of species i in a sample with N total number of reads, is given by

$${P}_{i}(n| N)=int_{0}^{1}dx,{rho }_{i}(x),left(begin{array}{*{20}{c}} {n} \ {N} end{array}right){x}^{n}{(1-x)}^{N-n},,$$
(6)

where ρi(x) is the AFD, i.e., the probability (over communities or times) that the relative abundance of i is equal to x. Note that this equation does not assume anything about independence across species or communities. It only assumes the sampling process is carried independently across communities.
Since the random variable xi, whose distribution is ρi(x), is a relative abundance, one has that ∑ixi = 1 (i.e., the data are compositional70). As discussed in Supplementary Note 2, given the range of variation of the empirical relative abundances, one can substitute Eq. (6) with

$${P}_{i}(n| N)=int_{0}^{infty }dx,{rho }_{i}(x)frac{{(xN)}^{n}}{n!}{e}^{-xN},,$$
(7)

and the condition (sum)ixi = 1 to ({sum }_{i}{bar{x}}_{i}=1), where ({bar{x}}_{i}=mathop{int}nolimits_{0}^{infty }dx,{rho }_{i}(x)x) is the mean value of xi. Under this assumption, one can also take the limits of the integration from 0 to ∞, instead of considering them from 0 to 1, as the contribution of the integrand from 1 to ∞ is negligible.
Note that, because of sampling, the average of a function f(x) over the pdf ρ(x) differs in general from the average of f(n/N) over P(n∣N)

$$int_{0}^{1}dx,rho (x)f(x), ne, sum _{n = 0}^{N}P(n| N)fleft(frac{n}{N}right)=int_{0}^{1}dx,rho (x)sum _{n = 0}^{N}fleft(frac{n}{N}right)frac{{(xN)}^{n}}{n!}{e}^{-xN},,$$
(8)

and the inequality becomes equality only if f(x) is linear. The important difference between the right- and the left-hand side is often neglected in the literature. In fact, the right-hand side is a good approximation of the left-hand size only in the limit xN ≫ 1, which is far from being realized in the data for most of the species. Supplementary Note 2 introduces a method to reconstruct the moments of ρ(x) from the moments of P(n∣N). More generally, I show that it is possible to infer the moment generating function of ρ(x) from the data, which allow to reconstruct the shape of the empirical ρ(x).
Excluding competitive exclusion
A Gamma-distributed AFD implies that all the species present in a community of a biome are present in all the communities from that biome. Therefore, when a species is not observed is because it is undetected due to sampling errors. I test this claim in two different ways. First, it is shown that one can in fact predict the occupancy of a species from its abundance fluctuations. Secondly, I show that a model without true absences is statistically more supported than a model where species are allowed to be absent.
The first way to test this hypothesis is to directly test its immediate prediction: if the absence is a consequence of sampling, one should be able to predict occupancy of a species (the probability that a species is present) simply from its average and variance of abundance (together with the total number of reads of each sample). In particular, assuming a Gamma AFD, the occupancy of species i is given by

$$langle {o}_{i}rangle =1-frac{1}{T}sum _{s}P(0| {N}_{s})=1-frac{1}{T}sum _{s = 1}^{T}{left(1+frac{{bar{x}}_{i}{N}_{s}}{{beta }_{i}}right)}^{-{beta }_{i}},,$$
(9)

where Ns is the total number of reads in sample s, T is the total number of samples, and ({beta }_{i}={bar{x}}_{i}^{2}/{sigma }_{{x}_{i}}^{2}). As shown in Fig. 2 and in Supplementary Fig. 3, this prediction well reproduces the observed occupancy across species. The prediction of Eq. (12) also matches the occupancy of temporal (longitudinal) data Supplementary Fig. 20.
The second, more rigorous, way to test the hypothesis that (most) species are always present is to use model selection. In this context we want to compare two (or more) models that aim at describing the observed number of reads of each species starting from alternative hypothesis. In particular I compare a purely Gamma AFD with a zero-inflated Gamma, which reads

$${varrho }_{i}(x| {vartheta }_{i},{beta }_{i},{bar{x}}_{i})={vartheta }_{i}delta (x)+(1-{vartheta }_{i})frac{1}{Gamma ({beta }_{i})}{left(frac{{beta }_{i}}{{bar{x}}_{i}}right)}^{{beta }_{i}}{x}^{{beta }_{i}-1}exp left(-{beta }_{i}frac{x}{{bar{x}}_{i}}right),,$$
(10)

where ϑi is the probability that a species is truly absent in a community and δ( ⋅ ) is the Dirac delta distribution. Our goal is to test whether the ϑis are significantly different from zero. Since the two models are nested, one can compare the maximum likelihood estimator in the case ϑi = 0 with the (maximum) likelihood marginalized over ϑ (which has prior μ(ϑ)). Given the number of reads ({n}_{i}^{s}) of species i in community s, with Ns total number of reads, one can compute the ratio (Supplementary Note 4)

$${ell }_{i}=frac{mathop{max }nolimits_{bar{x},beta }{prod }_{s}int dx{varrho }_{i}(x| 0,beta ,bar{x})frac{{(x{N}_{s})}^{{n}_{i}^{s}}}{{n}_{i}^{s}!}{e}^{-x{N}_{s}}}{int dvartheta ,mu (vartheta )left(mathop{max }nolimits_{bar{x},beta }{prod }_{s}int dx{varrho }_{i}(x| vartheta ,beta ,bar{x})frac{{(x{N}_{s})}^{{n}_{i}^{s}}}{{n}_{i}^{s}!}{e}^{-x{N}_{s}}right)},,$$
(11)

where μ(ϑ) is a prior over ϑ. If ℓi  > 1, the model with ϑi = 0 is more strongly supported than the model with ϑ ≠ 0. Under Beta prior with parameters 0.25 and 8, one obtains that ℓi  > 1 in 98.8% of the cases (averaged across data sets, ranging from 94.4 to 99.7%) and ℓi  > 100 in 97.5% cases (ranging from 92.8 to 99.2%). See Supplementary Note 4 for a more detailed description of the methodology and Supplementary Fig. 6 for results obtained with other priors.
Prediction of macroecological patterns
Given laws #1, #2, and #3, the probability to observe n reads of a randomly chosen species in a sample with N total reads is

$$P(n| N)=int_{-infty }^{infty }deta ,frac{Gamma (beta +n)}{n!Gamma (beta )}{left(frac{{e}^{eta }N}{beta +{e}^{eta }N}right)}^{n}{left(frac{beta }{beta +{e}^{eta }N}right)}^{beta }frac{exp left(-frac{{(eta -mu )}^{2}}{2{sigma }^{2}}right)}{sqrt{2pi {sigma }^{2}}},,$$
(12)

where (eta ={mathrm{log}},(bar{x})). All the properties of species are fully specified by its mean abundance (bar{x}={e}^{eta }). The probability of observing n reads of species with average abundance (bar{x}) in a sample with N total number of reads is therefore

$$P(n| N,bar{x})=frac{Gamma (beta +n)}{n!Gamma (beta )}{left(frac{bar{x}N}{beta +bar{x}N}right)}^{n}{left(frac{beta }{beta +bar{x}N}right)}^{beta },.$$
(13)

The predictions for the patterns shown in Fig. 3 are reported here. The full derivation of this and other patterns is presented in Supplementary Note 8.
The total number of observed species in a sample with N total number of reads can be easily calculated using Eq. (12). The probability of not observing a species is simply P(0∣N). The expected number of distinct species 〈s(N)〉 in a sample with N reads is therefore

$$langle s(N)rangle ={s}_{tot}left(1-P(0| N)right)={s}_{tot}left(1-int_{-infty }^{infty }deta ,frac{exp left(-frac{{(eta -mu )}^{2}}{2{sigma }^{2}}right)}{sqrt{2pi {sigma }^{2}}},{left(frac{beta }{beta +{e}^{eta }N}right)}^{beta }right),,$$
(14)

where stot is the total number of species in the biome (including unobserved ones, see Supplementary Note 7). Note that stot is (substantially) larger than sobs, the number of different species observed in the union of all the communities, which can instead be written as

$$langle {s}_{obs}rangle ={s}_{tot}left(1-int_{-infty }^{infty }deta ,frac{exp left(-frac{{(eta -mu )}^{2}}{2{sigma }^{2}}right)}{sqrt{2pi {sigma }^{2}}},{left(prod _{s = 1}^{T}frac{beta }{beta +{e}^{eta }{N}_{s}}right)}^{beta }right),.$$
(15)

Figure 3a shows that the prediction of Eq. (14) correctly matches the data (Supplementary Fig. 13).
The SAD, one of the most studied patterns in ecology and directly related to the Relative Species Abundance35, is defined as the fraction of species with a given abundance. According to our model, the expected SAD is given by

$$langle {Phi }_{n}(N)rangle := frac{langle {s}_{n}(N)rangle }{langle s(N)rangle }=frac{P(n| N)}{1-P(0,N)}=frac{int_{-infty }^{infty }deta ,frac{Gamma (beta +n)}{n!Gamma (beta )}{left(frac{{e}^{eta }N}{beta +{e}^{eta }N}right)}^{n}{left(frac{beta }{beta +{e}^{eta }N}right)}^{beta }frac{exp left(-frac{{(eta -mu )}^{2}}{2{sigma }^{2}}right)}{sqrt{2pi {sigma }^{2}}}}{1-int_{-infty }^{infty }deta ,{left(frac{beta }{beta +{e}^{eta }N}right)}^{beta },frac{exp left(-frac{{(eta -mu )}^{2}}{2{sigma }^{2}}right)}{sqrt{2pi {sigma }^{2}}}},,$$
(16)

where 〈sn(N)〉 is the number of species with n reads in a sample with N total number of reads. The cumulative SAD is defined as

$$langle {Phi }_{n}^{ ,{ > },}(N)rangle := sum _{m = n}^{infty }langle {Phi }_{m}(N)rangle =frac{int mathop{int}nolimits_{-infty }^{infty },{I}_{frac{{e}^{eta }N}{beta +{e}^{eta }N}}(n,beta )frac{exp left(-frac{{(eta -mu )}^{2}}{2{sigma }^{2}}right)}{sqrt{2pi {sigma }^{2}}}}{1-mathop{int}nolimits_{-infty }^{infty }eta ,{left(frac{beta }{beta +{e}^{eta }N}right)}^{beta },frac{exp left(-frac{{(eta -mu )}^{2}}{2{sigma }^{2}}right)}{sqrt{2pi {sigma }^{2}}}},,$$
(17)

where Ip(n, β) is the regularized incomplete Beta function. Figure 3b shows that the Eq. (17) captures the empirical cumulative SAD (Supplementary Fig. 17).
The occupancy probability is defined as the probability that a species is present in a given fraction of communities. This quantity has been extensively studied in a variety of contexts (from genomics71 to Lego sets and texts72) and has been more recently considered in microbial ecology37. The three macroecological laws predict (see derivation in Supplementary Note 8)

$${p}_{obs}(o)=frac{mathop{int}nolimits_{-infty }^{infty }deta ,sum_{t = 1}^{T}delta left(o-1+frac{1}{T}mathop{sum }nolimits_{s = 1}^{T}{left(frac{beta }{beta +{e}^{eta }{N}_{s}}right)}^{beta }right)frac{exp left(-frac{{(eta -mu )}^{2}}{2{sigma }^{2}}right)}{sqrt{2pi {sigma }^{2}}}mathop{prod }nolimits_{s = 1}^{T}left(1-{left(frac{beta }{beta +{e}^{eta }{N}_{s}}right)}^{beta }right)}{mathop{int}nolimits_{-infty }^{infty }deta ,frac{exp left(-frac{{(eta -mu )}^{2}}{2{sigma }^{2}}right)}{sqrt{2pi {sigma }^{2}}},mathop{prod }nolimits_{s = 1}^{T}left(1-{left(frac{beta }{beta +{e}^{eta }{N}_{s}}right)}^{beta }right)},,$$
(18)

where δ( ⋅ ) is a Dirac delta function. Figure 3c compares the prediction of Eq. (18) with the data (Supplementary Fig. 15).
Occupancy (the fraction of communities where a species is found) and abundance are not independent properties, and their relative dependence is often referred to as occupancy-abundance relationship21 Given an average (relative) abundance (bar{x}=exp (eta )), the expected occurrence is

$${langle orangle }_{eta }=1-frac{1}{T}sum _{s = 1}^{T}P(0| {N}_{s},bar{x})=1-frac{1}{T}sum _{s = 1}^{T}{left(frac{beta }{beta +bar{x}{N}_{s}}right)}^{beta },,$$
(19)

Figure 3d shows the comparison between data and predictions (Supplementary Fig. 16). These predictions are also tested for temporal (longitudinal) data in Supplementary Figs. 22–24.
Transition probabilities in longitudinal data
For longitudinal data, in addition to the stationary AFD, one can study the probability ({rho }_{i}(x^{prime} ,t+Delta t| x,t)) that a species i has abundance (x^{prime}) at time t + Δt conditioned on having abundance x at time t. Instead of focusing on the full distribution, we study its first two (conditional) central moments, i.e. the average and variance of the abundance at t + Δt conditioned to abundance x at time t. In the analysis of the data stationarity is assumed (the distribution ({rho }_{i}(x^{prime} ,t+Delta t| x,t)) depends on Δt but not on t). I also assume that the dynamics of different species are governed by similar equations that only differ in their parameters. One would like therefore to average over species, by properly rescaling their abundances. The average over species is potentially problematic, as it could add a spurious effect to the conditional averages. For instance, only species with larger fluctuations would appear for extreme values of the initial abundance. In order to avoid these problems, instead of consider the actual abundance, its cumulative probability distribution value (calculated using the empirical AFD of each species) was used, that is referred as “quantile abundance”. This is equivalent to rank the abundances of each species over communities and use the (relative) ranking of each community instead of the abundance. A value equal to 0 corresponds to the lowest observed abundance, and a value equal to 1 to the highest. By definition, the quantile abundance is always uniformly distributed.
Ruling out demographic stochasticity
Demographic stochasticity can reproduce a Gamma AFD. A birth, death, and immigration process has a Gamma as stationary distribution35. In the limit of large populations sizes, it corresponds to the following equation35

$$frac{dx}{dt}=m-(d-b)x+sqrt{(b+d)x}xi (t),,$$
(20)

where m is the migration rate, while b and d are the per-capita birth and death rate. The Gaussian white noise term ξ(t) has mean zero and time-correlation (langle xi (t)xi (t^{prime} )rangle =delta (t-t^{prime} )). The stationary distribution of this process turns out to be

$$rho (x)=frac{1}{Gamma left(2frac{m}{b+d}right)}{left(frac{b+d}{2(d-b)}right)}^{-2frac{m}{b+d}}{x}^{2frac{m}{b+d}-1}exp left(-2frac{d-b}{b+d}xright),.$$
(21)

The average abundance is equal to (bar{x}=m/(d-b)), while the variance turns out to be ({sigma }_{x}^{2}=(m/2)(b+d)/{(b-d)}^{2}). The square of the coefficient of variation would therefore be equal to (b + d)/(2m).
More generally, one can assume that all the parameters are species dependent, and the population of species i is described by

$$frac{d{x}_{i}}{dt}={m}_{i}-({d}_{i}-{b}_{i}){x}_{i}+sqrt{({b}_{i}+{d}_{i}){x}_{i}}{xi }_{i}(t),,$$
(22)

where (langle {xi }_{i}(t){xi }_{j}(t^{prime} )rangle ={delta }_{ij}delta (t-t^{prime} )) was assumed.
Taylor’s Law and the wide variation of average abundance together imply that mi/(bi + di) is constant while mi/(di − bi) varies across species on several orders of magnitudes. This imposes a constraint on the variation of parameter values across species.
For instance, one can consider the scenario where species migrate to local communities from a common species pool (metacommunity). As abundance in the metacommunity varies across species the migration rate is a species-dependent quantity. Under neutrality, the per-capita birth and death rates in the local communities are constant and independent of the identity of the species. In this case mi depends on the species, while b and d do not. One could recover the Lognormal MAD by imposing that mi is Lognormally distributed. On the other hand, this model would fail in reproducing Taylor’s law with exponent 2, as it would predict and exponent 1.
More in general, the condition imposed on the parameters corresponds to an unnatural fine-tuned relationship between migration, birth, and death rates. Variation of the average abundance is observed across, at least, 7 orders of magnitudes. In order to reproduce this variation across species and Taylor’s law with exponent 2, the range of variability of (bi − di)/(bi +  di) should be of the same order. It is unrealistic that the relative difference between birth and death rates, which have strong and direct connection to fundamental biological processes, vary so much across bacterial species. It is important to underline however, that the model of Eq. (22) can, in fact, for a proper parameterization, explain the observed variation of the data. But the choice of parameters explaining the empirical variation require for achieving this goal requires careful and unrealistic fine-tuning of the microscopic parameters.
Stochastic logistic model
The SLM is defined as

$$frac{d{x}_{i}}{dt}=frac{{x}_{i}}{{tau }_{i}}left(1-frac{{x}_{i}}{{K}_{i}}right)+sqrt{frac{{sigma }_{i}}{{tau }_{i}}}{x}_{i}{xi }_{i}(t),,$$
(23)

where ξ(t) is a Gaussian white noise term with mean zero and correlation (langle {xi }_{i}(t){xi }_{j}(t^{prime} )rangle ={delta }_{ij}delta (t-t^{prime} )). Taylor’s Law and the observed Lognormal MAD constraints the parameter value. The parameters 1/τi, Ki and σi are the intrinsic growth rate, the carrying capacity and the coefficient of variation of the growth-rate fluctuations. Taylor’s Law requires σi = σ (independently of i). Since the average abundance of the SLM is ({bar{x}}_{i}={K}_{i}(1-{sigma }_{i}/2)), if σi = σ, the average abundance and the carrying capacity turn out to be proportional to each other. The lognormal MAD implies therefore that the Kis are lognormally distributed. The stationary distribution corresponding to Eq. (23) reads

$${rho }_{i}(x)=frac{1}{Gamma (2{sigma }_{i}^{-1}-1)}{left(frac{2}{{K}_{i}{sigma }_{i}}right)}^{2{sigma }_{i}^{-1}-1}exp left(-frac{2}{{K}_{i}{sigma }_{i}}xright){x}^{2{sigma }_{i}^{-1}-2},.$$
(24)

The parameter τi does not affect stationary properties, but determines the timescale of relaxation to the stationary distribution. For small deviation of abundance from the average and for large times, the conditional expected abundance behaves as

$${langle {x}_{i}(t+Delta t)rangle }_{{x}_{i}(t)}={bar{x}}_{i}+left({x}_{i}(t)-{bar{x}}_{i}right){e}^{-frac{Delta t}{{tau }_{i}}},.$$
(25)

From the slopes of Fig. 4g one can then determine the timescales τi, which turn out to be approximately equal to 19 h. In Fig. 4 it was assumed τi = 19 h for all species.
Equation (23) can emerge as effective description of more complicated coupled equations. For instance, it is possible to show that a Lotka-Volterra system of equation with random interactions reduces to Eq. (23) (with colored noise to be self-consistently determined)42. If the coefficient of variation of the interaction coefficient does not increase with the number of species (e.g., if it is constant) then the Lotka-Volterra equations can be effectively approximated with Eq. (23).
The noise term in Eq. (23) can be interpreted as corresponding to environmental fluctuations. These fluctuations are typically known to have a characteristic timescale and are not white40,41. Supplementary Note 13 and Supplementary Fig. 25 show that colored noise in Eq. (23) does not affect significantly the predictions obtained with the SLM with white noise.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. More

Construction of the five-compartment model
Here we explain the construction of the five-compartment model (Fig. 1a). This is step 1 among the four steps described in “Results” section (subsection “Ecosystem dynamics at microsite scale”). We use upper bars in our initial notations to indicate parameters prior to rescaling.
The five-compartment model captures the stochastic processes acting at the level of C, D, M, Z, X entities (molecules, cells) (Fig. 1a) within a microsite. Dynamics of C, D, M, Z, X occur in continuous time. Mt is the number of cells at time t. Ct, Dt, Zt are the numbers of SOC molecules, DOC molecules, and enzyme molecules respectively. Xt is the number of complexes formed by an enzyme molecule binding a SOC molecule. There are constant external sources of SOC and DOC. When a cell dies, a fraction p of the molecules released are recycled into SOC, while the rest is recycled into DOC. A fraction l of dead microbes and deactivated enzymes may be lost due to leaching.
We denote by α the structural cost of a cell, which is the equivalent in number of DOC molecules of one cell (without storage). We denote by (alpha ^{prime}) the energetic cost of a cell, which is the number of DOC molecules consumed to produce the energy needed for the synthesis reactions involved in the production of one cell. We denote by β the equivalent in number of DOC molecules of one SOC molecule, and the structural and energetic cost of producing one molecule of enzyme by ρ and (rho ^{prime}), respectively. We assume that the energetic costs are carbon released by cells as CO2 (cell respiration) that diffuses out of the system instantly. We define the biomass production fraction and enzyme allocation fraction as

$${bar{gamma }}_{M}:=frac{1}{alpha +alpha ^{prime} },quad {bar{gamma }}_{Z}:=frac{1}{rho +rho ^{prime} }.$$
(1)

The event times are given by independent exponential random variables whose parameters are defined by event rates (Supplementary Tables 2–4). These event rates give an approximation of the average frequency of each event. The rates of cell growth and enzyme production depend on the trait φ. Once a cell has doubled its initial size, reproduction occurs by releasing the mother cell at its initial size, and the daughter cell at its same size. The cell must therefore take up and store its structural and energetic cost, ((alpha +alpha ^{prime} )), in DOC molecules in order to reproduce. We denote N the number of uptake events before reproduction. The number of DOC molecules taken up at each uptake event is then ((alpha +alpha ^{prime} )/N), hence the notation ({{bf{1}}}_{{Dge (alpha +alpha ^{prime} )/N}}) which equals 1 if (Dge (alpha +alpha ^{prime} )/N), and 0 otherwise. The same notation, ({{bf{1}}}_{{Dge rho +rho ^{prime} }}), is used for the production event of an enzyme molecule. Uptake is stochastic, but reproduction is deterministic, which means that when a cell has performed N uptake events, it reproduces with probability 1. A larger N means a larger number of uptake events between 2 reproduction events, which also means less DOC molecules taken up at each uptake event. The model tracks the dynamics of the number of cells, SOC, DOC, enzyme molecules, and also of the DOC stored in each cell.
Enzyme–substrate complexes form at rate ({bar{lambda }}^{k}) as one enzyme molecule (e.g. cellulase) bind one SOC molecule (e.g. cellulose). A complex may either dissociate (with no decomposition) at rate ({bar{lambda }}_{-1}^{varepsilon }), or react at rate ({bar{mu }}^{varepsilon }) and convert the molecule of SOC into β molecule of DOC while the enzyme is released and free again to react with new molecules of SOC (Supplementary Table 3).
System size k does not appear in this system of equations, yet it enters the volume-dependent parameters of the model, IC, ID, θ, and Km. We denote V the unit volume of soil that contains on average one microbial cell and the corresponding equilibrium of carbon mass of SOC, DOC and enzymes. The system size k is the number of well-mixed unit volumes in one microsite, which determines the number of cells sharing SOC, DOC, and enzymes. The volume of one microsite is therefore k × V. Increasing k amounts to increasing microsite volume, the number of cells sharing resources in one microsite, the amount of resources per microsite, and the volume-dependent parameters, such as the amount of SOC entering a microsite per unit of time, IC. In our analysis, the unit volume V is fixed, and we vary k to investigate the effect of microsite volume on the system’s eco-evolutionary dynamics. With very large k, the hybrid model can be approximated by a fully deterministic model which takes the form of a system of four ordinary differential equations (see Supplementary Information 3.3 and Supplementary Fig. 1), similar to the microbial decomposition model first introduced by Allison et al.53. However, empirical data suggest that k is of the order of 10–10054. When k = 1, there is only one cell in the microsite, which volume is V defined as the unit soil volume expected to contain a single cell. A value of k greater than 1 means that each microsite contains k cells and k times the amount of SOC, DOC and enzyme molecules of 1 cell; thus, the microsite volume is k × V, and volume-dependent parameters are rescaled by k. Specifically, there are four volume-dependent parameters: the external input of C, ({bar{I}}_{C}^{k}), the external input of D, ({bar{I}}_{D}^{k}), the half-saturation constant of DOC uptake, ({bar{K}}_{m}^{k}), and the encounter intensity of two given SOC and enzyme molecules, ({bar{lambda }}^{k}). The external inputs increase proportionally with the volume of the microsite, while the encounter intensity of two given molecules in a microsite decreases as its volume increases. The half-saturation is inversely proportional to the affinity between a given cell M and a given DOC molecule d, which decreases with increasing microsite volume. We thus obtain the following scaling relationships:

$${bar{I}}_{C}^{k}=k{bar{I}}_{C}, {bar{I}}_{D}^{k}=k{bar{I}}_{D}, {bar{lambda }}^{k}=frac{lambda }{k} {rm{and}} {bar{K}}_{m}^{k}=k{bar{K}}_{m}.$$
(2)

In our simulations, we generally assume that k is equal to 10, to match the empirical observation that (cells) in soil habitat tend to interact with 10 to 100 other cells at all time54.
Derivation of the hybrid model
In Supplementary Information 3, we present the next two steps (2 and 3 in “Ecosystem dynamics at microsite scale” of “Results” section) to derive the hybrid model on which all our results are based. In Supplementary Information 3.1, we explain how the dynamics generated by the five-compartment model can be captured with a reduced model with four-state variables (step 2). In Supplementary Information 3.2, we explain how the stochastic-deterministic PDMP model can be derived from the stochastic four-state variable model (step 3). In the hybrid model (step 4), only cell death remains stochastic, and cell dynamics is measured in unit of number of individuals (M), while other entities are now in carbon mass unit. The rescaled SOC, DOC and enzyme abundances are denoted with lower case letters c, d, and z.
Simulation algorithm for the hybrid spatial model
One technical benefit of the hybrid model is its much greater computational tractability. Here we describe the algorithm used to perform simulations of the hybrid model. We ran the model on single microsite to produce the simulations reported in Fig. 2. We ran the model on a 10 × 10 lattice of microsites for the subsequent figures. The algorithm is based on the Gillepsie algorithm55 as used in Champagnat et al.37, Fournier and Méléard56, which straightforwardly extends to the simulation of PDMPs.
To couple PDMP models across microsites, we account for the DOC and dispersal of cells between adjacent microsites. The DOC diffusion between microsites is modeled by approximating a continuous diffusion with a Euler scheme in which time is discretized with a fixed time step interval, τdiff. τdiff is chosen sufficiently small to provide a fine enough discretization of the DOC diffusion.
A simulation starts with a given amount of M, z, c, and d in each microsite at time t = 0, while the initial amount of DOC stored within each cell is determined uniformly at random. Two stochastic events (death of a cell) may not occur at the same time. Assume that the process has been computed until time ti; to continue the computation to time ti+1, we proceed as follows.
First, we simulate T, an exponential random variable with parameter r(ti) = dMM(ti), which corresponds to the death rate of the total cell population at time ti (M(ti) being the total number of cells on the entire lattice). We then compute

$${t}_{i+1}:={t}_{i}+min left(T,{tau }_{{rm{diff}}}right).$$

To obtain c(ti+1), d(ti+1), and z(ti+1) in each microsite at time ti+1, and the variation in amount of DOC stored within a cell in the corresponding microsite, we use a Euler scheme that solves the dynamical system

$$left{begin{array}{l}hskip -136pt dot{c}(t)={I}_{C}-{l}_{C}c-theta zc,\ dot{d}(t)={I}_{D}-{l}_{D}d+theta zc+(1-l){d}_{Z}z-{V}_{max }frac{d}{{K}_{m}{,}+{,}d}{omega }_{M}M,\ hskip -84pt dot{z}(t)=varphi {gamma }_{Z}{V}_{max }frac{d}{{K}_{m}{,}+{,}d}{omega }_{M}M-{d}_{Z}z,\ hskip -93pt dot{Delta }(t)=(1-varphi ){gamma }_{M}{V}_{max }frac{d}{{K}_{m}{,}+{,}d}{omega }_{M},end{array}right.$$

in each microsite between ti and ti+1, where M is the number of cells in the microsite at time ti, Δ gives the amount variation of DOC stored within a cell, Δ(ti) = 0 and the other initial conditions are the biomass of c, d, z in the corresponding microsite at time ti.
Note that, within a microsite, the variation of stored DOC is the same for all cells and corresponds to Δ(ti+1). Hence, this amount is added to the amount of DOC stored within each cell living in the corresponding microsite. If, for a cell j, the resulting amount Sj(Ti) + Δ(ti+1) is over ωM, a new cell appears. The amount of stored DOC within the new cell and the mother cell is then updated: Sj(ti+1) = (Sj(Ti) + Δ(ti+1) − ωM)/2. Otherwise, Sj(ti+1) = Sj(Ti) + Δ(ti+1). To determine the position of the new cell, the following steps are taken:

A uniform random variable ϑ1 in [0, 1] is simulated.

If ϑ1  More

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