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

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.

💡 Research Summary

The paper presents a minimalist yet powerful dynamical model of a microbial community in which species interact through the leakage and uptake of metabolites – a process commonly referred to as cross‑feeding. Building on the classic MacArthur consumer‑resource framework, the authors introduce two bipartite adjacency matrices: τ (uptake) and λ (leakage). Each species μ can absorb a subset of m_take metabolites and release a subset of m_leak metabolites. The growth rate of a species depends on the equilibrium concentrations (\bar c_i) of the chemicals, which are themselves determined instantaneously by the total leakage (L_i) and total exchange (U_i) generated by the whole community. This “fast‑chemical‑relaxation” assumption collapses the full chemical dynamics into an algebraic expression for (\bar c_i), allowing the coupled system to be written as a set of ordinary differential equations for the species abundances x_μ.

Key parameters include the total number of potential species N (up to 300), the number of distinct chemicals C (up to 30), the numbers of uptake and leak chemicals (m_take, m_leak), and stochastic assignments of uptake/leak strengths and metabolite benefits b_i. Costs χ_0 and χ_i model basal metabolic expenses, while a saturation term with carrying capacity K_i limits the benefit from each metabolite.

Through extensive numerical simulations across randomly generated τ–λ networks, the authors identify four qualitatively distinct attractors:

1. Fixed points – both species abundances and chemical concentrations settle to steady values. The number of co‑existing species never exceeds C, fully consistent with Gause’s competitive exclusion principle.

2. Limit cycles – populations oscillate periodically, yet again the number of persistent species is bounded by C.

3. Low‑dimensional chaos – trajectories explore a chaotic attractor confined to 3–7 effective chemical dimensions (as measured by principal‑component analysis). Here the instantaneous species count N_inst can modestly exceed C (by roughly 10), and the cumulative set of species observed over time N_cum is larger but still far below the total pool N. A small fraction of species may persist permanently (N_perm > 0) but remains ≤ C.

4. High‑dimensional chaos – the chemical dynamics occupy most of the C‑dimensional space (d_c > 7). Species abundances display intermittent bursts and extinctions, with N_inst far surpassing C and N_cum approaching the total number of potential species N. In this regime N_perm typically collapses to zero, indicating that no species remains permanently; instead, the community continuously cycles through a large pool of species, each occupying quasi‑stationary states for limited periods.

A systematic exploration of the parameter space reveals that high‑dimensional chaos is most prevalent when the number of uptake chemicals modestly exceeds the number of leaked chemicals (e.g., m_take = 4–7, m_leak = 2). Under these conditions the system lacks a low‑dimensional invariant manifold and instead exhibits a high‑dimensional, weakly constrained flow that prevents the emergence of a single stable equilibrium.

The authors interpret these findings in ecological terms. In fixed‑point or limit‑cycle regimes, the classic competitive exclusion limit (species ≤ resources) holds because the chemical environment is static or periodically repeating, providing a fixed set of niches. In chaotic regimes, especially the high‑dimensional case, the chemical environment itself becomes a dynamic, high‑dimensional niche space. Species can temporarily exploit transiently favorable chemical configurations, leading to a “temporal niche partitioning” that effectively lifts the static resource bound. This mechanism offers a theoretical resolution to the “paradox of the plankton”: diverse microbial taxa coexist despite limited external resources by continuously reshaping the internal resource landscape through cross‑feeding.

The paper also discusses the shape of the rank‑abundance distribution. In the high‑dimensional chaotic regime the distribution is broader than exponential, matching empirical observations of microbial communities where many rare taxa coexist with a few dominant ones.

Overall, the study demonstrates that cross‑feeding networks can generate intrinsic high‑dimensional chaos, and that this chaotic dynamics serves as a generic stabilizing mechanism for biodiversity beyond the limits imposed by static resource competition. The authors suggest that experimental microcosms with controllable leak/uptake pathways could test these predictions, and that incorporating stochastic mutation or immigration would further enrich the dynamical landscape.