Introduction

Niche construction drives ecological dynamics[1], yet the “tragedy of the commons” predicts that non-contributing cheaters should undermine cooperative systems[2]. Microbial siderophores represent a classic model of this dilemma: microbes secrete costly iron-scavenging molecules as public goods, which are then uptaken via specific membrane receptors[3]. While functionally straightforward, siderophore systems display remarkable complexity stemming from two key features[4]: (1) extraordinary chemical diversity comprising at least a thousand structures[5]; and (2) high receptor specificity, creating “lock-and-key” patterns where receptors bind only specific siderophore subsets[6]. This specificity generates a rich ecological landscape where microbes possessing matching receptors can exploit “foreign” siderophores[7], forging intricate networks of competition, cooperation, and cheating[8].

This complexity makes siderophore-mediated interactions an ideal system for addressing a central puzzle in ecological theory: how does biodiversity persist in the face of cheating[9-11] ? Classical theory suggests cheaters should outcompete producers, yet producers and cheaters coexist broadly in nature [12]. Observations in the Pseudomonas genus further complicate this picture: while pathogenic strains often exhibit extreme strategies with isolated iron-interaction networks, more diverse environmental isolates likely to adopt “partial-producers” strategies that producing siderophores while cheating on others[6]. A fundamental challenge lies in bridging scales: how do fine-grained molecular attributes, including structural diversity and receptor specificity, give rise to macro-scale ecological patterns? While previous studies have explored siderophore dynamics in biofilms or pairwise competitions[13, 14], they often overlook the emergent properties of networks formed by multiple siderophore types [15-17]. Understanding how biodiversity persists in complex communities requires bridging the gap between molecular-level interaction networks and emergent ecosystem dynamics[18].

Here, we develop a general theoretical framework that integrates consumer-resource dynamics with graph-theoretic representation of benefit flows, to systematically address these questions. Through simulations and analytical derivations, we prove that cheating is actually required for stable and oscillatory coexistence in small communities. In large communities, we uncover a paradox: increasing the breadth of cheating heightens the risk of community-level extinction, yet fosters higher biodiversity within surviving communities. By mapping ecological dynamics onto a “maximal Benefit Transfer Graph,” we identify the graph’s core structure as a powerful predictor of ecological outcomes, explaining cheating’s paradoxical role: Increased cheating pervasiveness connects species through network percolation, simultaneously expanding the “terminator” core driving extinction and the cyclic core enabling coexistence. Thus, our findings reveal a universal topological rule: structured exploitation acts not as a destabilizing vulnerability but as a necessary architecture for diversity. This principle extends beyond iron competition to broadly explain persistence in systems governed by directed benefit transfers, from extracellular enzyme hydrolysis to complex metabolic cross-feeding networks.

Results

A Generalized Framework for Siderophore-mediated Interactions

To link molecular specificity with ecological dynamics, we developed an integrated framework coupling dynamic modeling with network topology (Fig. 1, SI Appendix, Section 1). Our dynamic model extends classical chemostat equations by incorporating diverse siderophore-receptor pairs, where growth depends on both iron uptake and primary metabolism (Fig. 1A). The model incorporates two key biological constraints: (1) a metabolic trade-off (parameter α), where species i allocate limited resources between growth (α_i0) and siderophore production (α_ij for siderophore type j, under the constrain ); and (2) a receptor profile (parameter v), which dictates the specificity of iron uptake (v_ij represents the fraction of receptors dedicated to siderophore type j in species i, under the constrain ).

Figure 1.

The dynamic model tracks three sets of variables: (1) Microbial biomass concentration M_i for each species i = 1,2, …, N_spe ; (2) concentrations of each siderophore type R_j (j = 1,2,…, N_sid); (3) Free iron concentration R_iron:

Varying parameters α and v generates a continuum of strategies, from “pure-producers” (exclusively utilizing their own siderophores) to “pure-cheaters” (no production) and “partial producers” (producing one siderophore while exploiting several other types) (Fig. 1B).

Crucially, we mapped these biochemical interactions onto a “Benefit Transfer Graph” (BTG) to quantify ecological dependencies (Fig. 1C). In this directed graph, nodes represent species, and edges represent the flow of growth benefits from a siderophore producer to its beneficiaries. An edge exists if a species possesses receptors matching another’s siderophore, with the weight b_m1,m2 quantifying the specific growth gain species m2 derives from species m1’s production:

This abstraction allows us to translate complex kinetic systems into topological structures, analyzing how benefit transfers drive community assembly.

Closed Benefit Loops Drive Transitions from Exclusion to Coexistence and Chaos

We first analyzed minimal community motifs to dissect the topological rules of coexistence. In single-species systems with only one producer (Fig. 2A), the Benefit Transfer Graph only consists of self-loops. Dynamic modeling reveals that siderophore-mediated positive feedback creates an Allee effect [19], leading to bistability where survival depends on initial biomass thresholds (Fig. 2B; SI Appendix, Section 2). Consequently, when two pure-producers compete, they are driven to competitive exclusion: either one species becomes completely dominant, or the system exhibits priority effects in which the final winner is determined by who is initially more abundant (SI Appendix, Section 3)

Figure 2.

Stable coexistence emerges only when species adopt “partial-producer” strategies, forming a closed mutual cheating loop in the BTG (Fig. 2C–D). Analytically, we derived that each species must deliver greater growth benefits to its competitor than to itself (b_2,1 > b_2,2, b_1,2 > b_1,1, SI Appendix, Section 4). Strikingly, this mutual exploitation generates a synergistic rescue effect: in parameter regimes where single species would collapse, the combined siderophore pool allows the mutualistic pair to survive (Fig. 2E–G).

Expanding these loops to multi-species motifs identifies key topologies that enable dynamic coexistence. In three-species systems, an intransitive “rock–paper–scissors” BTG loop (Fig. 2H) generates sustained population oscillations (Fig. 2I). Bifurcation analysis demonstrates that increasing self-reliance drives a transition from oscillatory coexistence to competitive exclusion via Heteroclinic bifurcation[20], while balanced benefit flows favor stable coexistence over limit cycles (SI Appendix, Section 5).

As the interaction network grows more complex, such as in five-species motifs with overlapping loops (Fig. 2J), coupled feedback can drive the system into deterministic chaos (Fig. 2K). Similarly, increasing self-reliance progressively drives transition from chaotic dynamics to stable periodic cycles and, eventually, to exclusion. Collectively, these results demonstrate that closed benefit-transfer loops serve as the structural scaffold for diversity: short mutual loops support stable coexistence, while longer cycles enable dynamic fluctuations.

The Paradox of Cheating in Large Communities

We extended our framework to complex ecosystems by simulating 1.5 · 10⁶ communities (N_spe = 50) with varying “cheating breadth” (the average number of foreign siderophores a species can exploit) and fractions of “pure-cheaters” (Fig. 3A). Simulation outcomes reveal an unintuitive paradox. On one hand, broad cheating and high pure-cheater ratio act as destabilizing forces, monotonically increasing the probability of community-wide extinction (Fig. 3B).

Figure 3.

On the other hand, within non-extinct communities (having at least one surviving species), cheating promotes biodiversity. Species richness increases monotonically with cheating breadth (Fig. 3C), and these diverse communities are more likely to exhibit dynamic behaviors (oscillations or chaos) (SI Appendix, Section 6). Notably, dynamic communities consistently support larger populations than stable ones, suggesting that temporal fluctuations create niches for species maintenance (Fig. 3D).

The paradox of cheating breadth implies that surviving communities must harbor non-random interaction structures that enable persistence. To identify these structures, we analyzed the Benefit Transfer Graph within each non-extinct community. For every siderophore producer i, we ranked all its outgoing benefit edges (b_ij) by magnitude, designating the single strongest flow as the “Rank-1 edge” (Fig. 3E). Notably, we found that the subgraph formed by surviving species is overwhelmingly dominated by these Rank-1 incoming edges (Fig. 3F), far exceeding their frequency in randomly assembled subgraphs (SI Appendix, Section 6). This finding suggests that community fate is not determined by average interaction strengths, but by a specific topological backbone formed by maximal benefit flows.

The Maximal Benefit Transfer Graph Resolves the Cheating Paradox

To decode the structural basis of the cheating paradox, we formalized the “Rank-1” backbone as a maximal Benefit Transfer Graph (mBTG). By retaining only the strongest outgoing benefit edge for each producer, the mBTG assumes the topology of a directed pseudoforest, a graph class where every node has at most one outgoing edge (Fig. 4A). A central feature of such graphs is their decomposition into Weakly Connected Components (WCCs), which are groups of nodes connected by paths regardless of direction. In directed pseudoforests, benefit flows within each WCC inevitably converge to a unique “Core”: the minimal subset of nodes with no outgoing edges to the rest of the graph (Fig. 4B).

Figure 4.

We discovered that the topology of this Core serves as a potent predictor of community fate, achieving arround 80% classification accuracy. Irrespective of initial conditions, asymptotic biomass consistently concentrates within a single WCC, and specifically within its core (SI Appendix, Section 7). This indicates that each WCC represents a distinct basin of attraction, with the core acting as the structural and dynamic nucleus. Characterizing these cores by their “loop length” (the number of edges forming the core) reveals three distinct architectures that dictate ecological outcomes (Fig. 4B):

1. Sink Core (Loop length 0, W0 in Fig. 4B): A core consisting of a single pure-cheater acts as a resource “black hole,” absorbing benefits without reciprocation. This topology leads to community extinction with 97% probability.

2. Self-loop Core (Loop length 1, W1 in Fig. 4B): A core formed by a producer benefiting maximally from itself leads to a steady state (Fig. 4C) where the core species survives alone, driving all others to exclusion (Fig. 4D).

3. Cyclic Core (Loop length ≥: 2, W2 and W3 in Fig.4B): A closed loop of two or more species supports coexistence. Notably, as the loop length reaches three, the probability of entering oscillatory or chaotic attractors increases sharply (Fig. 4C). While species richness in steady-state systems plateaus at short loop lengths, dynamic systems continuously support higher diversity (Fig. 4D). These dynamic communities often harbor twice as many total survivors (biomass >10⁻⁶) as high-abundance species (biomass > 10⁻³), suggesting that oscillatory dynamics enable rare species to persist by intermittently accessing benefits from abundant ones (Fig. 4D).

This topological framework resolves the paradox of cheating breadth. The mBTG is composed of four functional node types, whose abundance changes by cheating breadth (Fig. 4E, SI Appendix, Section 8):

1. “Self-loop” Maximal Beneficiaries (MBs): Producers that gain the highest benefit from their own siderophores. These form Self-loop Cores, driving exclusion. However, increasing cheating breadth expands the pool of potential recipients, “diluting” a producer’s probability of retaining its own siderophore as the maximal benefit (Fig. 4F). This causes a monotonic decline in exclusion-driving Self-loops.

2. “Connector” MBs: Partial-producers possessing both incoming edges and an outgoing edge. This type benefits most from foregin siderophores while producing a siderophore whose maximal beneficiary is not itself. They are the essential “glue” for Cyclic Cores, linking species into loops that support stable or dynamic coexistence.

3. “Terminator” MBs: Nodes with incoming edges but no outgoing edges, corresponding exclusively to pure-cheaters. They form Sink Cores. Selection of maximal beneficiaries probabilistically favors species with higher growth allocation (like pure-cheaters with α₀ = 1), and broader cheating amplifies this bias (Fig. 4G), causing a surge in Terminator MBs.

4. “Leaf” Nodes: Species with no incoming edges. The number of Leaf nodes increases as cheating breadth expands, indicating that incoming edges concentrate disproportionately on a small subset of species. This trend can also be explained by the increased bias towards high-α₀ species, which leads to a heavy-tailed in-degree distribution in which a small set of “super-beneficiaries” capture maximal benefits from many sources whereas most species get none.

Taken together, as cheating breadth increases, the mBTG undergoes a percolation-like transition. The suppression of Self-loops releases outgoing edges, while the concentration of edges toward “super-beneficiaries” promotes the coalescence of fragmented components into a giant connected cluster (Fig. 4H). This reduces multi-stability, forcing the community into few dominant attractors. Thus, while broad cheating eliminates the self-loops that trigger exclusion, it forces the community to “gamble”: it either collapses into a Terminator Sink or stabilizes into a complex, high-diversity Cyclic Core. Cheating therefore does not always act as a destabilizing vulnerability, but can also provide the necessary architectural scaffold for biodiversity.

Discussion

In this study, we introduce an integrated framework that bridges molecular specificity, ecological dynamics, and network topology to resolve the “cheating paradox” in microbial communities. By mapping siderophore-mediated interaction dynamics to a maximal Benefit Transfer Graph (mBTG), we demonstrate that community fate, whether extinction, exclusion, or coexistence, is encoded in the core topology of benefit flows. Our findings challenge the classical view that cheating is purely detrimental. While unchecked cheating (Sink Cores) indeed drives collapse, structured exploitation (Cyclic Cores) acts as a necessary architectural scaffold for biodiversity.

The concept of niche construction traditionally emphasizes how organisms modify environments to their benefit, but it also creates vulnerabilities to exploitation[21]. While mechanisms such as spatial segregation[22], metabolic cross-feeding[23], and kin selection[24] have been proposed to resolve this dilemma, our model demonstrates that cheating itself can be transformed into an organizing force. This can be analogous to the central tenet of Modern Coexistence Theory, which posits that stable coexistence arise from either niche differentiation or fitness equivalence [25, 26]. Similarly, when microbes create multiple “chemical niches” by diverse siderophore types, partial-producers emerge as essential “loop connectors,” linking distinct niches through their dual capacity to produce and exploit. This aligns with empirical observations in Pseudomonas, where partial-producers are associated with diverse, non-pathogenic communities, whereas pure strategies are linked to low diversity and pathogenicity[6].

This graph-theoretic framework bridges experiments, bioinformatics, and ecological theories. While kinetic parameters are often elusive in wild communities, the structure of the Benefit Transfer Graph can be inferred from genomic analysis[4] or cross-feeding assays[27]. By coarse-graining molecular specificity into topological motifs (WCCs and Cores), the mBTG approach can forecast community fate not only in systems governed by siderophore-mediated interactions, but also in other systems driven by shared chemical resources, such as extracellular enzyme hydrolysis[28], antibiotic degradation[29], or metabolic cross-feeding networks[30].

Percolation theory describes abrupt transitions from local connectivity to global connectivity [31]. Previously applied to physical connectivity like fragmented habitats [32, 33], percolation concepts now extend to abstract interaction networks [34]. Our Maximal Benefit Transfer Graph (mBTG) exhibits distinctive percolation behavior shaped by its pseudoforest topology: increasing cheating breadth rapidly merges Weakly Connected Components into a giant component, while Strongly Connected Components (formed the cores) approach but never reach the percolation threshold. Under evolutionary dynamics, this structure self-organizes further. Pure-cheaters face extinction, driving the network toward self-sustaining structures composed by either Self-loops or Connector-based cycles. This mirrors autocatalytic set emergence, where mutually reinforcing cycles evolve spontaneously from random networks [35]. Biologically, this suggests that ecosystems operate near a critical point. Bacteria generally possess multiple siderophore receptors, ranging from 1–4 in E. coli to 20–30 in Pseudomonas and other environmental strains [7, 36-38]. This range corresponds to the “near-critical spot” in our simulations—high enough to ensure global connectivity and diversity, yet structured enough to maintain stability.

Our work provides a mechanistic foundation that grounds abstract ecological theory in molecular specificity. We show that siderophore-mediated interactions generate rich phenomena, ranging from synergistic rescue and priority effects to heteroclinic bifurcations. While phenomenological models highlight the role of intransitive interactions in maintaining diversity[39], our work identifies a concrete chemical basis for such dynamics. Furthermore, the emergence of oscillation and chaos supports theoretical predictions that non-equilibrium dynamics promote diversity through temporal niche partitioning[40].

Our current framework relies on simplified assumptions to ensure analytical tractability, such as well-mixed environments and single-siderophore production. These simplifications point toward fruitful avenues for future exploration. For instance, incorporating spatial structure could reveal how biofilm stabilize or fragment large interaction loops [41], and integrating more interactions like nutrient competition and antibiotic antagonism would offer a more holistic ecological picture [42]. Additionally, incorporating evolutionary dynamics will be crucial to understand the selective origins of these topological motifs [8, 43]. Ultimately, these future complexities will build upon the fundamental principle established here: molecular specificity transforms the social dilemma of public goods, recasting cheating from a destabilizing threat into the structural scaffold of diverse ecosystems.

Data availability

All computer code and scripts used for the simulations, analysis, and figure generation in this study have been deposited in the Zenodo repository: (https://zenodo.org/records/18158439). This study is a theoretical work and did not generate new biological materials or empirical datasets; all data supporting the findings are derived from the model simulations and are available within the article and its Supplementary Information.

Acknowledgements

This work is supported by National Natural Science Foundation of China (No. T2321001 to ZL, 32588202 and 32425036 to SW, and T2422010 and 62172170 to XL), Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (JYB2025XDXM502 to ZL), National Key Research and Development Programme of China (2022YFF0802103 to SW), the Peking-Tsinghua Center for Life Sciences, and the Fundamental Research Funds for Central Universities.

Additional files

SI Appendix

Additional information

Funding

National Natural Science Foundation of China (NSFC) (T2321001)

• Zhiyuan Li

National Natural Science Foundation of China (NSFC) (32588202)

• Shaopeng Wang

National Natural Science Foundation of China (NSFC) (32425036)

• Shaopeng Wang

National Natural Science Foundation of China (NSFC) (2422010)

• Xueming Liu

National Natural Science Foundation of China (NSFC) (62172170)

• Xueming Liu

Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (JYB2025XDXM502)

• Zhiyuan Li

National Key Research and Development Program of China (NKPs) (2022YFF0802103)

• Shaopeng Wang