Cross-feeding yields high-dimensional chaos and coexistence of species beyond exclusion principle

TL;DR

This work studies microbial communities where species exchange metabolites through leakage and uptake by coupling population sizes and chemical concentrations in a minimal leak–uptake model. By numerically exploring networks with varying numbers of uptakes $m_{take}$ and leaks $m_{ leak}$, the authors identify four attractor classes: fixed points, limit cycles, low-dimensional chaos, and high-dimensional chaos. While fixed-point and limit-cycle regimes respect Gause’s competitive-exclusion bound with coexisting species $N_{inst}\le C$, high-dimensional chaos supports coexistence of many more species and exhibits intermittent switching and a broad, fat-tailed rank-abundance distribution; chemical dynamics occupy a high-dimensional space with dimension $d_c$ often exceeding 7. This mechanism provides a generic dynamical route to sustaining diversity in cross-feeding microbial ecosystems under resource limitations, with potential implications for understanding the maintenance of microbial diversity in soils, guts, and other ecosystems.

Abstract

Species interactions through cross-feeding via leakage and uptake of chemicals are important in microbial communities, and play an essential role in the coexistence of diverse species. Here, we study a simple dynamical model of a microbial community in which species interact by competing for the uptake of common metabolites that are leaked by other species. The model includes coupled dynamics of species populations and chemical concentrations in the medium, allowing for a variety of uptake and leakage networks among species. Depending on the structure of these networks, the system exhibits different attractors, including fixed points, limit cycles, low-dimensional chaos, and high-dimensional chaos. In the fixed-point and limit-cycle cases, the number of coexisting species is bounded by the number of exchangeable chemicals, consistent with the well-known competitive exclusion principle. In contrast, in the low-dimensional chaotic regime, the number of coexisting species exhibits noticeable but limited excess over this limit. Remarkably, in the high-dimensional chaotic regime, a much larger number of species beyond this limit coexist persistently over time. In this case, the rank-abundance distribution is broader than exponential, as often observed in real ecosystems. The population dynamics displays intermittent switching among quasi-stationary states, while the chemical dynamics explore most of the high dimensions. We find that such high-dimensional chaos is ubiquitous when the number of uptake chemicals is moderately larger than the number of leaked chemicals. Our results identify high-dimensional chaos with intermittent switching as a generic dynamical mechanism that stabilizes coexistence in interacting systems. We discuss its relevance to sustaining diverse microbial communities with leak-uptake cross-feeding.

Figures (15)

• Figure 1: The schematic of the model ecosystem interacting through leak and uptake of chemicals, defined by Eqs. [\ref{['TSKKmodel_minimal']}]. The filled symbols represent the chemicals those are leaked and taken by at least one of the species, and the open symbols are for the chemicals those are involved in the metabolic network inside the cells but only taken or leaked by them (i.e. common resource and waste, respectively).
• Figure 2: Typical behaviors of the system classified as fixed point, limit cycle, low-dimensional chaos, and high-dimensional chaos, respectively from top to bottom. Panels in each row show the following information of each sample: (a) Time series of the species population in a stacked plot, (b) Relative species abundance based on the cumulative populations observed over the sampling period, the species, (c) Trajectory of chemical dynamics in the plane defined by its first and second principal components, and (d) Cumulative explained-variance ratio plot from PCA of chemical dynamics.
• Figure 3: Transitional change in $N_{\rm inst}$ (the temporal average of instant number of coexisting species at each moment) and $N_{\rm cum}$ (cumulative number of species which appears in the observation time) to the change in $m_{\rm take}$ (the lower main panel). In the upper sub panel, the corresponding relation between $N_{\rm inst}$ and $N_{\rm perm}$ (number of species those are stably observed in the whole observation time) is shown. System parameters are fixed at $N = 300, C = 30, m_{\rm leak} = 2$. In the dynamically coexisting regime ($N_{\rm cum} \gg C \cap N_{\rm inst} > C$), only a small minority of species show stable presence ($N_{\rm perm} \approx 0$) indicating that the set of species above the observation threshold keep alternating.
• Figure 4: Distributions of cumulative number of species ($N_{\rm cum}$, top) and instant number of species ($N_{\rm inst}$, bottom), to the effective dimension of the chemical dynamics of corresponding timeseries, $d_c$. While $N_{\rm cum}$ starts to exceed the Gause's limit $C=30$ at around $d_c=3$, $d_c = 6$ or higher dimension is needed to overcome the Gause's limit in typical $N_{\rm inst}$.
• Figure 5: A typical example of the dynamics classified as (iv) high-dimensional chaos. (Top) The timeseries of stacked species populations, which shows an intermittent dynamics among different community structures. (Bottom left) Relative species abundance (RSA) curves of cumulative populations (bold black line) and instantaneous populations at different moments (colored thin lines). In this regime, instantaneous RSA also becomes broader beyond Gause's limit ($C=30$, shown by the vertical dotted line). (Bottom center) Distribution of the instantaneous populations of species. Colors from red through yellow and green to blue represents the rank of that species in the cumulative population. It shows that top major species often take large population size but also sometimes take very small population, and the species with following majority in average take large population less frequently. (Bottom right) Temporal decay of the community similarity. In consistent with the intermittent feature of the population timeseries, community similarity shows a power-law decay in time as $\sim t^{-0.25}$.