Diverse communities behave like typical random ecosystems

In 1972, Robert May triggered a worldwide research program studying ecological communities using random matrix theory. Yet, it remains unclear if and when we can treat real communities as random ecosystems. Here, we draw on recent progress in random matrix theory and statistical physics to extend May's approach to generalized consumer-resource models. We show that in diverse ecosystems adding even modest amounts of noise to consumer preferences results in a transition to “typicality,” where macroscopic ecological properties of communities are indistinguishable from those of random ecosystems, even when resource preferences have prominent designed structures. We test these ideas using numerical simulations on a wide variety of ecological models. Our work offers an explanation for the success of random consumer resource models in reproducing experimentally observed ecological patterns in microbial communities and highlights the difficulty of scaling up bottom-up approaches in synthetic ecology to diverse communities.

Figure 1

Random interactions destabilize an ecosystem of specialist consumers. (a) Left: an ecosystem with system size $M=5$ starts with specialists consuming only one type of resource, resulting in a consumer preference matrix $\mathbf{B}=\mathbb{1}$. Right: off-target consumption coefficients $\mathbf{C}\sim \mathcal{N}(\frac{\mu }{M},\frac{{\sigma }_c}{\sqrt{M}})$ are sampled from a Gaussian distribution, resulting in an overall consumer preference matrix $\overline{\mathbf{C}}=\mathbf{B}+\mathbf{C}$. (b) Fraction of surviving species $S^{*}/M$ vs. ${\sigma }_c$, numerically computed using $M=100$ for an ecosystem described by Eq. (2), along with the corresponding results for a completely random ecosystem with $\mathbf{B}=0$. The error bar shows $\pm 1$ standard deviation from 10 000 independent realizations. Also shown are examples of the matrices $\overline{\mathbf{C}}$ employed in the simulations. (c) Heatmap for the identity matrix plus a Gaussian random matrix with ${\sigma }_c=1$ for two system sizes: $M=100$ and $M=500$.

Figure 2

Community properties for structured and random ecosystems. (a) Examples of designed interactions Top: the identity matrix; Middle: a Gaussian-type circulant matrix; Bottom: a block matrix (see Appendix pp1 for details). Simulations of designed and random ecosystems where the random component of the the consumer preferences $\mathbf{C}$ are sampled from a (b) Gaussian distribution $\mathcal{N}(0,\frac{{\sigma }_c}{\sqrt{M}})$, (c) Uniform Distribution: $\mathcal{U}(0,b)$, or a (d) Binomial distribution: $Bernoulli\left(p_c\right)$. The plots show the fraction of surviving species $S^{*}/M$, mean species abundance $\langle N\rangle$, and second moment of the species abundances $\langle N^2\rangle$ for designed and purely random ecosystems ($\mathbf{B}=0$) the number of nonspecific consumer preferences is increased.

Figure 3

Effect of random interactions on ecosystem sensitivity. (a) The bipartite interactions ${\overline{C}}_{i\alpha }$ in MacArthur's consumer-resource model can be mapped to pairwise competition coefficients $A_{ij}$ in generalized Lotka-Volterra equations through $A_{ij}={\sum }_{\alpha \in \mathbf{M}}{\overline{C}}_{i\alpha }{\overline{C}}_{\alpha j}^T$. (b) Spectra of $A_{ij}$ at different ${\sigma }_c$ for $\overline{\mathbf{C}}=\mathbb{1}+\mathbf{C}$, where $\mathbf{C}$ is a random matrix with i.i.d entries drawn from a normal distribution with mean zero and standard deviation ${\sigma }_c$. The red solid line is the Marchenko-Pastur distribution. (c) Comparison between numerical simulations and analytic results for the minimum eigenvalue of $\mathbf{A}$ at different ${\sigma }_c$. (d) Comparison between numerical simulations and analytic solutions for the mean sensitivity $\nu$ of steady-state population sizes to changes in species growth rates. See Appendix pp7 for details.

Figure 4

Effect of resource extinction on an ecosystem. A schematic for the consumer preference matrix with (a) and without (b) resource extinction for specialist consumers that each eat independent resources. The left schematic corresponds to the initial consumer matrix, and the right schematic to the consumer matrix after species and resource extinctions. Notice that resource extinctions can result in singular consumer matrices. (c) Spectra of $A_{ij}$ at ${\sigma }_c=0.3$ with consumer matrices chosen as in Fig. 3 with (left) and without resource extinction (right). The zero modes are marked with a red ellipse. (d) The mean sensitivity $\nu$ of steady-state at different ${\sigma }_c$. The dashed lines in panel (d) are cavity solutions. The scatter points are results from numerical simulations. See Appendix pp3 for detailed calculations.

Figure 5

The asymptotic spectrum of $A_{ij}$ for different values of ${\sigma }_c$ by solving Eq. (F10) numerically.

Figure 6

Species abundance $N$ in equilibrium at different ${\sigma }_c$. The simulation details can be found at Appendix pp7.

Figure 7

Reproduce Fig. 3 in the main text with model Eqs. (2). The parameters are the same as Fig. 9.

Figure 8

Reproduce Fig 1: the fraction of surviving species $S^{*}/M$ vs ${\sigma }_c$ for $M=25$ and $M=100$. It shows $M=25$ is enough to reproduce our result in the main text. Theoretically, numeric converges to our analytical result at the rate of $\frac{1}{M}$. And it is true that a smaller value of $M$ can result in a larger fluctuation, corresponding to a larger error bar. But the average converges to the same value which is the thermodynamic limit we care about.

Figure 9

Community properties for generalized consumer-resource models under Gaussian noise. (a) MacArthur's consumer resource model with resource extinction. (b) Linear resource dynamics: the resource dynamics is changed to $\frac{d{R}_{\alpha }}{dt}=K_{\alpha }-R_{\alpha }-{\sum }_i{N}_i{C}_{i\alpha }{R}_{\alpha }$. (c) With cross-feeding: the dynamics is described in Eq. (17) in the Supplemental Material of Ref. [19]. The noise is only applied on the consumption matrix and $D$ is kept the same at different ${\sigma }_c$. In both models, $\mathbf{B}=\mathbb{1}$. See Appendix pp7 for details.

Figure 10

Effects when $S\ne M$. (a) Community properties, (b) the minimum eigenvalue ${\lambda }_{\text{min}}$, (c) the mean sensitivity $\nu$. All simulations are the same as figures in the main text except $S=200,M=100$. See Appendix pp7 for details.

Figure 11

Spectra of $A_{ij}$ in different cases. (a) Uniform Noise: $\mathcal{U}(0,b)$ and (b) Binary Noise: $Bernoulli\left(p_c\right)$; the engineered matrix $\mathbf{B}$ is an identity matrix. (c) Gaussian noise and the engineered matrix $\mathbf{B}$ is a circulant matrix. (d) Gaussian noise and the engineered matrix $\mathbf{B}$ is a block matrix. Note that $A_{ij}$ are obtained from numerical simulations. See Appendix pp7 for details.

Figure 12

Comparison between numerical simulations (scatter points) and cavity solutions (solid lines) for $\chi$ at different ${\sigma }_c$ for different cases. (a) CRM without resource extinction, Eqs. (2). (b) CRM with resource extinction, Eqs. (1). Note $S^{*}$ and $M^{*}$ are obtained from the numerical simulations, although in principle they could be obtained by solving the cavity equations directly.

Figure 13

Comparison of the minimum eigenvalue ${\lambda }_{\text{min}}$ and the mean sensitivity $\nu$ between different distributions for the identity case at different ${\sigma }_c$. (a) CRM without resource extinction, Eqs. (2). (b) CRM with resource extinction, Eqs. (1). Note that the Bernoulli and uniform distribution are mapped to the corresponding Gaussian distribution $\mu =p_{c}M$, ${\sigma }_c=\sqrt{M{p}_c(1-p_c)}$ and $\mu =bM/2$, ${\sigma }_c=b\sqrt{M/12}$, respectively.

Figure 14

Comparison the minimum eigenvalue ${\lambda }_{\text{min}}$ and the mean sensitivity $\nu$ between different engineered matrices $\mathbf{B}$ at different ${\sigma }_c$. (a) CRM without resource extinction, Eqs. (2). (b) CRM with resource extinction, Eqs. (1).