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Lecture Notes

An Invitation to Physics for Biologists

Chuliang Song

Lecture notes for UCLA's EE BIOL 15, 370 pages, updated October 21, 2026

A bridge between physics and biology that is math-light yet quantitative: physical reasoning as a tool for answering fundamental questions about living systems. Four parts cover scaling and dimensional analysis, energy, information, and physical constraints, with problem sets.

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Submitted

Quantifying irreversibility of ecological systems

Junang Li, Stephan B. Munch, Tzer Han Tan, and Chuliang Song

bioRxiv preprint

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Irreversibility—the asymmetry of population dynamics when played forward versus backward in time—is a fundamental property of ecological dynamics. Despite its early recognition in ecology, irreversibility has remained a high-level and unquantifiable concept. Here, we introduce a quantitative framework rooted in non-equilibrium statistical physics to measure irreversibility in general ecological systems. Through theoretical analyses, we demonstrate that irreversibility quantifies the degree to which a system is out of equilibrium, a property not captured by traditional ecological metrics. We validate this prediction empirically across diverse ecological systems structured by different forces, such as rapid evolution, nutrient availability, and temperature. In sum, our study provides a rigorous formalism for quantifying irreversibility in ecological systems, with the potential to integrate dynamical, energetic, and informational perspectives in ecology.
Species interactions are internally constrained despite large climatic variability

Sergio Picó-Jordá, Chuliang Song†, and Oscar Godoy†

bioRxiv preprint

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Understanding how vital rates and species interactions vary over time is crucial for predicting community responses to environmental change. Considerable progress has been made in understanding the drivers of variation in vital rates. However, the question of whether interactions are highly plastic and context-dependent, or strongly constrained by internal (e.g., species traits and composition) and/or external factors (e.g., environmental conditions) remains unclear. We applied a theoretical approach based on the feasibility domain - the range of conditions allowing coexistence - to a nine-year dataset of time-changing interactions between annual plants under large variability in annual precipitation. Using subcommunities of three species, we found that species interactions are strongly constrained, forming a “core-periphery” structure of consistently feasible combinations across years. This main finding means that species sample repeatedly a restricted range of opportunities for coexistence. Similar findings were obtained for subcommunities of four species. Crucially, the constraints to variation in biotic interactions are determined by species identity (internal constraints) rather than precipitation or temporal autocorrelation (external environmental factors). Furthermore, we found a contrasting effect of precipitation on the feasibility of subcommunities. While wetter years increase similarity between subcommunities and reduce the overall feasible range, drier years increase dissimilarity between subcommunities and increase the probability of coexistence when the conditions seem harsher. These findings suggest that constraints to biotic interactions tend to be alike across species in wetter years, but more context dependency occurs across species in drier years. Our findings challenge the assumption of highly plastic species interactions even in a highly dynamic system of annual plants. Our results also highlight the critical importance of internal constraints generated by species identity in mediating community persistence and predicting community responses to environmental change.
Label invariance: a guiding principle for ecological models

Theo Gibbs*, Onofrio Mazzarisi*, Lorenzo Fant, Ruby An, Jacopo Grilli, György Barabás†, and Chuliang Song†

bioRxiv preprint

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Ecological models, though diverse in form, are strengthened when they obey guiding principles. We formalize and advocate for a foundational principle we call “label invariance”, which says that a model’s dynamics must remain the same when identical individuals are arbitrarily grouped into distinct sub-populations. This principle is a necessary consequence of trait continuity—the observation that ecological interactions change continuously as organisms become more similar. Violation of label invariance often implies a hidden, intrinsic niche differentiation between species, which may obscure the mechanisms of biodiversity maintenance. We provide a general framework for constructing both deterministic and stochastic models that follow label invariance. We further demonstrate its utility as a complementary, non-statistical tool for empirical model selection. In sum, label invariance provides an important test for evaluating existing ecological models and a guide for developing new ones, promoting clarity in model assumptions from the outset.

Published

2026

Toward a conceptual commons for theory in community ecology

Chuliang Song

Annual Review of Ecology, Evolution, and Systematics, 2026

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Community ecology has developed a rich and increasingly diverse body of theory, but its rapid expansion has outpaced efforts to compare and synthesize competing frameworks. As a result, semantic ambiguity often masks genuine scientific disagreement. Using theories of species interactions as a lens, I organize this landscape along two axes: formalization (how theories are built) and function (why they are used). Within formalization, I contrast the model-driven culture (which imposes structure) with the data-driven culture (which discovers it), highlighting the distinct forms of misspecification that plague each. Within function, I distinguish three goals: building predictive heuristics, developing definitional frameworks, and running experiments in model worlds. I use this mapping to identify where and why distinct traditions talk past one another, drawing on examples from modern coexistence theory, diversity–stability relationships, niche theory, and network ecology. This analysis provides a diagnostic roadmap for pinpointing why models disagree, justifying the choice of formalization, and aligning questions with appropriate theoretical tools. Beyond diagnosis, I survey current integration efforts and outline practical steps toward synthesis. Finally, I address the human dimension of theoretical practice, arguing that artificial intelligence offers a major opportunity to democratize access to theory if training pivots from derivation to verification. This review lays the groundwork for a conceptual commons where diverse theories can be rigorously compared and productively combined.
@article{song2026towards, author = {Song, Chuliang}, title = {Toward a conceptual commons for theory in community ecology}, year = {2026}, journal = {Annual Review of Ecology, Evolution, and Systematics}, volume = {57}, }
What have we learned from empirical applications of modern coexistence theory?

Peter B. Adler, Matteo Detto, Stephen P. Ellner, Theo L. Gibbs, Zachary J. Gold, Samuel J. Leonard, Jacob I. Levine, and 5 more authors

Ecology Letters, 2026

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Since the early 2000’s, Modern Coexistence Theory (MCT) has guided empirical investigations of coexistence. We reviewed empirical applications of MCT to answer two questions: (1) What have we learned about the strength of temporal, spatial and variation-independent mechanisms of coexistence in nature? (2) How have studies of niche and fitness differences advanced our understanding of coexistence? With respect to the first question, we now have evidence that the temporal storage effect and variation-independent mechanisms help stabilize coexistence in many systems. We have not learned more due to the low number of tests of these mechanisms (24 out of 84 MCT studies), the narrow focus on mechanisms involving temporal variation and grassland communities, and lack of rigorous model validation. With respect to our second question, studies of niche and fitness differences (61 of 84 studies) have answered fundamental questions about the role of phylogenetic relatedness, plant–soil feedbacks, and functional traits in coexistence. We recommend that future empirical MCT studies (1) match hypotheses to MCT quantities, (2) address gaps in empirical studies, (3) avoid known biases in estimating competition coefficients, and (4) validate models with independent data.
@article{adler2026empirical, author = {Adler, Peter B. and Detto, Matteo and Ellner, Stephen P. and Gibbs, Theo L. and Gold, Zachary J. and Leonard, Samuel J. and Levine, Jacob I. and Schiffer, Annie E. and Song, Chuliang and Stemkovski, Michael and Vahsen, Megan L. and Levine, Jonathan M.}, title = {What have we learned from empirical applications of modern coexistence theory?}, year = {2026}, journal = {Ecology Letters}, volume = {29}, number = {6}, pages = {e70404}, }
A geometric approach to beta diversity: Reply

Chuliang Song*, Muyang Lu*, Joseph R. Bennett, Benjamin Gilbert, Marie-Josée Fortin, and Andrew Gonzalez

Ecological Monographs, 2026

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In this reply, we address the formal comment by Baselga regarding our proposed geometric approach to beta diversity. While the critique argues that geometric beta diversity mathematically diverges from classic dissimilarity-based metrics, we counter that this divergence is a deliberate and mathematically rigorous feature of our framework, rather than a flaw. The classic paradigm prioritizes pairwise spatial isolation, whereas our geometric framework captures the structural complexity and combinatorial novelty of species assemblages. We clarify the ecological first principles underlying our geometric approach, specifically how it integrates spatial species associations and the structural fingerprint of metacommunities. Furthermore, we demonstrate that the distinct mathematical behaviors of geometric beta diversity are logically sound consequences of these principles. Ultimately, we emphasize that statistical ecology is best served by methodological pluralism, providing ecologists with distinct, mathematically rigorous tools for different analytical jobs.
@article{song2026reply, author = {Song, Chuliang and Lu, Muyang and Bennett, Joseph R. and Gilbert, Benjamin and Fortin, Marie-Josée and Gonzalez, Andrew}, title = {A geometric approach to beta diversity: Reply}, year = {2026}, journal = {Ecological Monographs}, volume = {96}, number = {2}, pages = {e70074}, }
Larger habitat islands maintain more persistent mutualistic assemblages

Chen Zhu, Peng Ren, Jordi Bascompte, Bo Dalsgaard, Chuliang Song, Wande Li, Fernando Gonçalves, and 5 more authors

Nature Communications, 2026

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Habitat loss and fragmentation threaten biodiversity by reducing species richness and disrupting ecological interactions. But it is poorly understood how reduced habitat area and increased isolation impact community persistence, as measured by its capacity to maintain multispecies coexistence. We draw on an intensive, multi-year field study of mutualistic interactions in plant-frugivore and plant-pollinator networks across 41 islands in an insular fragmented landscape formed by dam construction in 1959. We show that the loss of island area after inundation is the primary driver that reduces the persistence of mutualistic assemblages, beyond merely reducing species richness. We further identify structural mechanisms of persistence: on larger islands, decreasing network modularity enhances persistence in both plant-frugivore and plant-pollinator communities, whereas increasing nestedness contributes to persistence only in plant-pollinator communities. These findings represent a conceptual advance in understanding the impact of habitat loss on biodiversity by showing that species loss in small habitat fragments may result from the reduced capacity of their mutualistic communities to support species coexistence, mediated by changes in network structure. We urge to include evaluations of community persistence into the design of conservation and habitat restoration strategies to more effectively mitigate the long-term impacts of habitat loss and fragmentation on biodiversity.
@article{zhu2026larger, author = {Zhu, Chen and Ren, Peng and Bascompte, Jordi and Dalsgaard, Bo and Song, Chuliang and Li, Wande and Gonçalves, Fernando and Chang, Longxiao and Zhao, Yuhao and Zeng, Di and Ding, Ping and Si, Xingfeng}, title = {Larger habitat islands maintain more persistent mutualistic assemblages}, year = {2026}, journal = {Nature Communications}, volume = {17}, number = {1}, pages = {10105}, }

2025

Anthropogenic land consolidation intensifies zoonotic host diversity loss and disease transmission in human habitats

Shan Pei, Pengbo Yu, Jayna Raghwani, Yuxin Wang, Ziyan Liu, Yidan Li, Yanchao Cheng, and 19 more authors

Nature Ecology & Evolution, 2025

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Anthropogenic land use change is an important driver of global biodiversity loss and threatens public health through biological interactions. Understanding these landscape-ecological effects at local scales will help achieve the UN Sustainable Development Goals by balancing urbanization, biodiversity, and the spread of infectious diseases. Here, we address this knowledge gap by analyzing a 43-year-long monthly dataset (1980-2022) of synanthropic rodents in central China during intensive land use change. We observed a significant increase in the mean patch size, coinciding with a substantial change in rodent community composition and a marked decline in rodent diversity; eight of the nine local rodent species experienced near-extirpation. Our analysis reveals that these irregular species replacements can be attributed to the effect of land consolidation on species competition among rodents, favoring striped field mice, a critical reservoir host of Hantaan virus (HTNV). Consequently, land consolidation has facilitated the proliferation of striped field mice and increased the prevalence of HTNV among them. This study highlights the importance of considering both direct and indirect effects of anthropogenic activities in the management of biodiversity and public health.
@article{shan2024landscape, author = {Pei, Shan and Yu, Pengbo and Raghwani, Jayna and Wang, Yuxin and Liu, Ziyan and Li, Yidan and Cheng, Yanchao and Lin, Qiushi and Song, Chuliang and Dharmarajan, Guha and Faust, Christina L. and Tian, Yunyu and Xu, Yiting and Liang, Yilin and Qu, Jianhui and Wei, Jing and Li, Shen and Zhang, Tongjun and Ma, Chaofeng and Bharti, Nita and Cazelles, Bernard and Yang, Ruifu and Pybus, Oliver G. and Dobson, Andrew P. and Stenseth, Nils Chr. and Tian, Huaiyu}, title = {Anthropogenic land consolidation intensifies zoonotic host diversity loss and disease transmission in human habitats}, year = {2025}, journal = {Nature Ecology \& Evolution}, volume = {9}, number = {1}, pages = {99--110}, }
Rigorous validation of ecological models against empirical time series

Chuliang Song, and Jonathan M. Levine

Nature Ecology & Evolution, 2025

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The complexity of ecosystems poses a formidable challenge for validating ecological models. The prevailing inability to falsify models has resulted in an accumulation of models but not an accumulation of confidence. Here we introduce an approach rooted in queueing theory, termed the covariance criteria, that establishes a rigorous test for model validity based on covariance relationships between observable quantities. These criteria set a high bar for models to pass by specifying necessary conditions that must hold regardless of unobserved factors. We test our approach using observed time series data on three long-standing challenges in ecological theory: resolving competing models of predator–prey functional responses, disentangling ecological and evolutionary dynamics in systems with rapid evolution and detecting the often-elusive influence of higher-order species interactions. Across these diverse case studies, the covariance criteria consistently rule out inadequate models, while building confidence in those that provide strategically useful approximations. The covariance criteria approach is mathematically rigorous and computationally efficient, making it applicable to existing data and models.
@article{song2025covariance, author = {Song, Chuliang and Levine, Jonathan M.}, title = {Rigorous validation of ecological models against empirical time series}, year = {2025}, journal = {Nature Ecology \& Evolution}, volume = {9}, number = {12}, pages = {2265--2278}, }
A geometric approach to beta diversity

Chuliang Song*, Muyang Lu*, Joseph R. Bennett, Benjamin Gilbert, Marie-Josée Fortin, and Andrew Gonzalez

Ecological Monographs, 2025

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Beta diversity—the variation among community compositions in a region—is a fundamental measure of biodiversity. Most classic measures have posited that beta diversity is maximized when each community has a distinct, non-overlapping set of species. However, this assumption overlooks the ecological significance of species interactions and non-additivity in ecological systems, where the function and behaviour of species depend on other species in a community. Here, we introduce a geometric approach to measure beta diversity as the hypervolume of the geometric embedding of a metacommunity. Besides considering compositional distinctiveness as in classic metrics, this geometric measure explicitly incorporates species associations and captures the idea that adding a unique, species-rich community to a metacommunity increases beta diversity. We show that our geometric measure is closely linked to and naturally extends previous information- and variation-based measures. Additionally, we provide a unifying geometric framework for widely adopted extensions of beta diversity. Applying our geometric measures to empirical data, we address two long-standing questions in beta diversity research—the latitudinal pattern of beta diversity and the effect of sampling effort—and present novel ecological insights that were previously obscured by the limitations of classic approaches. In sum, our geometric approach offers a new and complementary perspective on beta diversity, is immediately applicable to existing data, and holds promise for advancing our understanding of the complex relationships between species composition, ecosystem functioning, and stability.
@article{song2025beta, author = {Song, Chuliang and Lu, Muyang and Bennett, Joseph R. and Gilbert, Benjamin and Fortin, Marie-Josée and Gonzalez, Andrew}, title = {A geometric approach to beta diversity}, year = {2025}, journal = {Ecological Monographs}, volume = {95}, number = {1}, pages = {e70008}, }
Assembly graph as the Rosetta Stone of ecological assembly

Chuliang Song

Environmental Microbiology, 2025

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Ecological assembly—the process of ecological community formation through species introductions—has recently seen exciting theoretical advancements across dynamical, informational, and probabilistic approaches. However, these theories often remain inaccessible to non-theoreticians, and they lack a unifying lens. Here, I introduce the assembly graph as an integrative tool to connect these emerging theories. The assembly graph visually represents assembly dynamics, where nodes symbolize species combinations and edges represent transitions driven by species introductions. Through the lens of assembly graphs, I review how ecological processes reduce uncertainty in random species arrivals (informational approach), identify graphical properties that guarantee species coexistence and examine how the class of dynamical models constrain the topology of assembly graphs (dynamical approach), and quantify transition probabilities with incomplete information (probabilistic approach). To facilitate empirical testing, I also review methods to decompose complex assembly graphs into smaller, measurable components, as well as computational tools for deriving empirical assembly graphs. In sum, this math-light review of theoretical progress aims to catalyze empirical research towards a predictive understanding of ecological assembly.
@article{song2025review, author = {Song, Chuliang}, title = {Assembly graph as the Rosetta Stone of ecological assembly}, year = {2025}, journal = {Environmental Microbiology}, volume = {27}, number = {1}, pages = {e70030}, }
The equilibrium conundrum

Tess N. Grainger, Keila Stark, Chuliang Song, Matthew A. Barbour, and Rachel M. Germain

Ecology Letters, 2025

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The idea that natural systems tend to be in a state of balance, stability, or equilibrium dates back to the origin of the field of ecology and continues to underlie most ecological theory. However, empirical evidence for equilibrium dynamics in nature and in experiments is surprisingly elusive. Here, we address this conundrum by first exploring the deep history of equilibrium in ecological theory and contrasting this with the evidence for equilibrium dynamics in nature. We then search the literature to quantify how empiricists deal with equilibrium in their research, and find that fewer than half of top-cited studies testing major equilibrium-based theories mention or test for equilibrium, and that those that do employ a wide variety of approaches to assess equilibrium. Next, we break down perceived barriers to integrating equilibrium into empirical work by providing step-by-step instructions (with code) to determine whether a population is at equilibrium. And finally, we provide concrete options for what empiricists and theoreticians can each do to meet in the middle when it comes to equilibrium. We hope that this paper will stimulate new discussions from researchers from across the theory-empirical divide about this longstanding conundrum.
@article{grainger2025equilibrium, author = {Grainger, Tess N. and Stark, Keila and Song, Chuliang and Barbour, Matthew A. and Germain, Rachel M.}, title = {The equilibrium conundrum}, year = {2025}, journal = {Ecology Letters}, volume = {28}, number = {11}, pages = {e70232}, }

2024

Trophic tug-of-war: Coexistence mechanisms within and across trophic levels

Chuliang Song*, and Jurg W. Spaak*

Ecology Letters, 2024

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Ecological communities encompass rich diversity across multiple trophic levels. While modern coexistence theory has been widely applied to understand community assembly, its traditional formalism only allows assembly within a single trophic level. Here, using an expanded definition of niche and fitness differences applicable to multitrophic communities, we study how diversity within and across trophic levels affects species coexistence. If each trophic level is analyzed separately, both lower- and higher-trophic levels are governed by the same coexistence mechanisms. In contrast, if the multitrophic community is analyzed as a whole, different trophic levels are governed by different coexistence mechanisms: coexistence at lower trophic levels is predominantly limited by fitness differences, whereas coexistence at higher trophic levels is predominantly limited by niche differences. This dichotomy in coexistence mechanisms is supported by theoretical derivations, simulations of phenomenological and trait-based models, and a case study of a primeval forest ecosystem. Our work provides a general and testable prediction of coexistence mechanism operating in multitrophic communities.
@article{song2024multitrophic, author = {Song, Chuliang and Spaak, Jurg W.}, title = {Trophic tug-of-war: Coexistence mechanisms within and across trophic levels}, journal = {Ecology Letters}, volume = {27}, number = {4}, pages = {e14409}, keywords = {community assembly, fitness difference, food web, multitrophic, niche difference, predation}, year = {2024}, }
Soil microbial influences over coexistence potential in multispecies plant communities in a subtropical forest

Weitao Wang, Hangyu Wu, Tingting Wu, Zijing Luo, Wei Lin, Hanlun Liu, Junli Xiao, and 6 more authors

Ecology, 2024

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Soil microbes have long been recognized to substantially affect the coexistence of pairwise plant species across terrestrial ecosystems. However, projecting their impacts on the coexistence of multi-species plant systems remains a pressing challenge. To address this challenge, we conducted a greenhouse experiment with 540 seedlings of five tree species in a subtropical forest in China and evaluated microbial effects on multispecies coexistence using the structural method, which quantifies how the structure of species interactions influences the likelihood for multiple species to persist. Specifically, we grew seedlings alone or with competitors in different microbial contexts and fitted individual biomass to a population dynamic model to calculate intra- and inter-specific interaction strength with and without soil microbes. We then used these interaction structures to calculate two metrics of multispecies coexistence, structural niche differences (which promote coexistence) and structural fitness differences (which drive exclusion), for all possible communities comprising two to five plant species. We found that the soil microbes generally increased both the structural niche and fitness differences across all communities, with a much stronger effect on structural fitness differences. A further examination of functional traits between plant species pairs found that trait differences are stronger predictors of structural niche differences than of structural fitness differences, and that soil microbes have the potential to change trait-mediated plant interactions. Our findings underscore that soil microbes strongly influence coexistence of multispecies plant systems, and also add to the experimental evidence that the influence is more on fitness differences rather than niche differences.
@article{wu2024soil, author = {Wang, Weitao and Wu, Hangyu and Wu, Tingting and Luo, Zijing and Lin, Wei and Liu, Hanlun and Xiao, Junli and Luo, Wenqi and Li, Yuanzhi and Wang, Youshi and Song, Chuliang and Kandlikar, Gaurav and Chu, Chengjin}, title = {Soil microbial influences over coexistence potential in multispecies plant communities in a subtropical forest}, journal = {Ecology}, year = {2024}, data = {https://datadryad.org/stash/dataset/doi:10.5061/dryad.vt4b8gtx9#aff587}, volume = {105}, number = {11}, pages = {e4415}, }

2023

Mean species responses predict effects of environmental change on coexistence

Frederik De Laender, Camille Carpentier, Timoteo Carletti, Chuliang Song, Samantha L. Rumschlag, Michael B. Mahon, Marie Simonin, and 2 more authors

Ecology Letters, 2023

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Environmental change research is plagued by the curse of dimensionality: the number of environmental drivers and the number of communities potentially at risk are both large. This raises the pressing question if a general understanding of ecological impact is achievable. Here, we show evidence that this is indeed possible. Theoretical results for both small and very large bi- and tritrophic communities, and simulation results for communities of intermediate size, show that linear combinations of mean species responses to environmental change predict the resulting effects on coexistence, quantified as the feasibility domain size. We next show that the same mean responses also predict effects on site occupancy in simple metacommunities. We then benchmark our findings using two relevant cases of environmental change (temperature change and pollution), showing that means of temperature optima and of species sensitivities to pollution predict concomitant effects on coexistence. Finally, we apply the concept to field data, finding support for effects of land use on coexistence in natural invertebrate communities.
@article{DeLaender2023Mean, author = {De Laender, Frederik and Carpentier, Camille and Carletti, Timoteo and Song, Chuliang and Rumschlag, Samantha L. and Mahon, Michael B. and Simonin, Marie and Meszéna, Géza and Barabás, György}, title = {Mean species responses predict effects of environmental change on coexistence}, journal = {Ecology Letters}, volume = {26}, number = {9}, pages = {1535--1547}, keywords = {climate change, community ecology, environmental ecology, environmental stressors, food chains, food webs, global change, pollution}, year = {2023}, }
Rapid monitoring of ecological persistence

Chuliang Song*, Benno I. Simmons*, Marie-Josée Fortin, Andrew Gonzalez, Christopher N. Kaiser-Bunbury, and Serguei Saavedra

Proceedings of the National Academy of Sciences, 2023

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Effective conservation of ecological communities requires accurate and up-to-date information about whether species are persisting or declining to extinction. The persistence of an ecological community is supported by its underlying network of species interactions. While the persistence of the network supporting the whole community is the most relevant scale for conservation, in practice, only small subsets of these networks can be monitored. There is therefore an urgent need to establish links between the small snapshots of data conservationists can collect, and the “big picture” conclusions about ecosystem health demanded by policymakers, scientists, and societies. Here, we show that the persistence of small subnetworks (motifs) in isolation—that is, their persistence when considered separately from the larger network of which they are a part—is a reliable probabilistic indicator of the persistence of the network as a whole. Our methods show that it is easier to detect if an ecological community is not persistent than if it is persistent, allowing for rapid detection of extinction risk in endangered systems. Our results also justify the common practice of predicting ecological persistence from incomplete surveys by simulating the population dynamics of sampled subnetworks. Empirically, we show that our theoretical predictions are supported by data on invaded networks in restored and unrestored areas, even in the presence of environmental variability. Our work suggests that coordinated action to aggregate information from incomplete sampling can provide a means to rapidly assess the persistence of entire ecological networks and the expected success of restoration strategies.
@article{song2023rapid, author = {Song, Chuliang and Simmons, Benno I. and Fortin, Marie-Josée and Gonzalez, Andrew and Kaiser-Bunbury, Christopher N. and Saavedra, Serguei}, title = {Rapid monitoring of ecological persistence}, journal = {Proceedings of the National Academy of Sciences}, volume = {120}, number = {20}, pages = {e2211288120}, year = {2023}, }

2022

Generalism drives abundance: A computational causal discovery approach

Chuliang Song, Benno I. Simmons, Marie-Josée Fortin, and Andrew Gonzalez

PLOS Computational Biology, 2022

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A ubiquitous pattern in ecological systems is that more abundant species tend to be more generalist; that is, they interact with more species or can occur in wider range of habitats. However, there is no consensus on whether generalism drives abundance (a selection process) or abundance drives generalism (a drift process). As it is difficult to conduct direct experiments to solve this chicken-and-egg dilemma, previous studies have used a causal discovery method based on formal logic and have found that abundance drives generalism. Here, we refine this method by correcting its bias regarding skewed distributions, and employ two other independent causal discovery methods based on nonparametric regression and on information theory, respectively. Contrary to previous work, all three independent methods strongly indicate that generalism drives abundance when applied to datasets on plant-hummingbird communities and reef fishes. Furthermore, we find that selection processes are more important than drift processes in structuring multispecies systems when the environment is variable. Our results showcase the power of the computational causal discovery approach to aid ecological research.
@article{song2022generalism, author = {Song, Chuliang and Simmons, Benno I. and Fortin, Marie-Josée and Gonzalez, Andrew}, journal = {PLOS Computational Biology}, publisher = {Public Library of Science}, title = {Generalism drives abundance: A computational causal discovery approach}, year = {2022}, month = sep, volume = {18}, pages = {e1010302}, number = {9}, }
Metapopulation persistence can be inferred from incomplete surveys

Chuliang Song, Marie-Josée Fortin, and Andrew Gonzalez

Proceedings of the Royal Society B: Biological Sciences, 2022

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Habitat destruction and fragmentation are principal causes of species loss. While a local population might go extinct, a metapopulation—populations inhabiting habitat patches connected by dispersal—can persist regionally by recolonizing empty patches. To assess metapopulation persistence, two widely adopted indicators in conservation management are metapopulation capacity and patch importance. However, we face a fundamental limitation in that assessing metapopulation persistence requires that we survey or sample all the patches in a landscape: often these surveys are logistically challenging to conduct and repeat, which raises the question whether we can learn enough about the metapopulation persistence from an incomplete survey. Here, we provide a robust statistical approach to infer metapopulation capacity and patch importance by sampling a portion of all patches. We provided analytic arguments on why the metapopulation capacity and patch importance can be well predicted from sub-samples of habitat patches. Full factorial simulations with more complex models corroborate our analytic predictions. We applied our model to an empirical metapopulation of mangrove hummingbirds (*Amazilia boucardi*). Based on our statistical framework, we provide some sampling suggestion for monitoring metapopulation persistence. Our approach allows for rapid and effective inference of metapopulation persistence from incomplete patch surveys.
@article{song2022metapopulation, title = {Metapopulation persistence can be inferred from incomplete surveys}, author = {Song, Chuliang and Fortin, Marie-Josée and Gonzalez, Andrew}, journal = {Proceedings of the Royal Society B: Biological Sciences}, volume = {289}, number = {1989}, pages = {20222029}, year = {2022}, publisher = {The Royal Society}, }
Synthesizing the effects of individual-level variation on coexistence

Simon Maccracken Stump, Chuliang Song, Serguei Saavedra, Jonathan M. Levine, and David A. Vasseur

Ecological Monographs, 2022

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Intraspecific trait variation (ITV) is a widespread feature of life, but it is an open question how ITV affects between-species coexistence. Recent theoretical studies have produced contradictory results, with ITV promoting coexistence in some models and undermining coexistence in others. Here we review recent work and propose a new conceptual framework to explain how ITV affects coexistence between two species. We propose that all traits belong to one of two categories—niche traits and hierarchical traits. Niche traits determine an individual’s location on a niche axis or trade-off axis, such that changing an individual’s trait makes it perform better in some circumstances and worse in others. Hierarchical traits represent cases where conspecifics with different traits have the same niche, but one performs better under all circumstances, such that there are winners and losers. Our framework makes predictions for how intraspecific variation in each type of trait affects coexistence by altering stabilizing mechanisms and fitness differences. For example, ITV in niche traits generally weakens the stabilizing mechanism, except when it generates a generalist-specialist trade-off. On the other hand, hierarchical traits tend to impact competitors differently, such that ITV in one species will strengthen the stabilizing mechanism while ITV in the other species will weaken the mechanism. We re-examine 10 studies on ITV and coexistence, along with four novel models, and show that our framework can explain why ITV promotes coexistence in some models and undermines coexistence in others. Overall, our framework reconciles what were previously considered to be contrasting results and provides both theoretical and empirical directions to study the effect of ITV on species coexistence.
@article{stump2022synthesizing, title = {Synthesizing the effects of individual-level variation on coexistence}, author = {Stump, Simon Maccracken and Song, Chuliang and Saavedra, Serguei and Levine, Jonathan M. and Vasseur, David A.}, journal = {Ecological Monographs}, volume = {92}, number = {1}, pages = {e01493}, year = {2022}, publisher = {Wiley Online Library}, }

2021

Coexistence holes characterize the assembly and disassembly of multispecies systems

Marco Tulio Angulo, Aaron Kelley, Luis Montejano, Chuliang Song†, and Serguei Saavedra†

Nature Ecology & Evolution, 2021

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A central goal of ecological research has been to understand the limits on the maximum number of species that can coexist under given constraints. However, we know little about the assembly and disassembly processes under which a community can reach such a maximum number, or whether this number is in fact attainable in practice. This limitation is partly due to the challenge of performing experimental work and partly due to the lack of a formalism under which one can systematically study such processes. Here, we introduce a formalism based on algebraic topology and homology theory to study the space of species coexistence formed by a given pool of species. We show that this space is characterized by ubiquitous discontinuities that we call coexistence holes (that is, empty spaces surrounded by filled space). Using theoretical and experimental systems, we provide direct evidence showing that these coexistence holes do not occur arbitrarily—their diversity is constrained by the internal structure of species interactions and their frequency can be explained by the external factors acting on these systems. Our work suggests that the assembly and disassembly of ecological systems is a discontinuous process that tends to obey regularities.
@article{angulo2021coexistence, title = {Coexistence holes characterize the assembly and disassembly of multispecies systems}, author = {Angulo, Marco Tulio and Kelley, Aaron and Montejano, Luis and Song, Chuliang and Saavedra, Serguei}, journal = {Nature Ecology \& Evolution}, volume = {5}, number = {8}, pages = {1091--1101}, year = {2021}, publisher = {Nature Publishing Group}, }
Bridging parametric and nonparametric measures of species interactions unveils new insights of non-equilibrium dynamics

Chuliang Song, and Serguei Saavedra

Oikos, 2021

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A central theme in ecological research is to understand how species interactions contribute to community dynamics. Species interactions are the basis of parametric (model-driven) and nonparametric (model-free) approaches in theoretical and empirical work. However, despite their different interpretations across these approaches, these measures have occasionally been used interchangeably, limiting our opportunity to use their differences to gain new insights about ecological systems. Here, we revisit two of the most used measures across these approaches— species interactions measured as constant direct effects (typically used in parametric approaches) and local aggregated effects (typically used in nonparametric approaches). We show two fundamental properties of species interactions that cannot be revealed without bridging these definitions. First, we show that the local aggregated intraspecific effect summarizes all potential pathways through which one species impacts itself, which are likely to be negative even without any constant direct self-regulation mechanism. This property has implications for the long-held debate on how communities can be stabilized when little evidence of self-regulation has been found among higher-trophic species. Second, we show that a local aggregated interspecific effect between two species is correlated with the constant direct interspecific effect if and only if the population dynamics do not have any higher-order direct effects. This other property provides a rigorous methodology to detect direct higher-order effects in the field and experimental data. Overall, our findings illustrate a practical route to gain further insights about non-equilibrium ecological dynamics and species interactions.
@article{song2021bridging, author = {Song, Chuliang and Saavedra, Serguei}, title = {Bridging parametric and nonparametric measures of species interactions unveils new insights of non-equilibrium dynamics}, year = {2021}, volume = {130}, number = {7}, pages = {1027--1034}, journal = {Oikos}, }
Understanding the emergence of contingent and deterministic exclusion in multispecies communities

Chuliang Song, Lawrence H. Uricchio, Erin A. Mordecai, and Serguei Saavedra

Ecology Letters, 2021

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Competitive exclusion can be classified as deterministic or as historically contingent. While competitive exclusion is common in nature, it has remained unclear when multispecies communities formed by more than two species should be dominated by deterministic or contingent exclusion. Here, we take a fully parameterised model of an empirical competitive system between invasive annual and native perennial plant species to explain both the emergence and sources of competitive exclusion in multispecies communities. Using a structural approach to understand the range of parameters promoting deterministic and contingent exclusions, we then find heuristic theoretical support for the following three general conclusions. First, we find that the life-history of perennial species increases the probability of observing contingent exclusion by increasing their effective intrinsic growth rates. Second, we find that the probability of observing contingent exclusion increases with weaker intraspecific competition, and not with the level of hierarchical competition. Third, we find a shift from contingent exclusion to deterministic exclusion with increasing numbers of competing species. Our work provides a heuristic framework to increase our understanding about the predictability of species persistence within multispecies communities.
@article{song2021understanding, title = {Understanding the emergence of contingent and deterministic exclusion in multispecies communities}, author = {Song, Chuliang and Uricchio, Lawrence H. and Mordecai, Erin A. and Saavedra, Serguei}, journal = {Ecology Letters}, volume = {24}, number = {10}, pages = {2155--2168}, year = {2021}, publisher = {Wiley Online Library}, }
Untangling the complexity of priority effects in multispecies communities

Chuliang Song, Tadashi Fukami, and Serguei Saavedra

Ecology Letters, 2021

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The history of species immigration can dictate how species interact in local communities, thereby causing historical contingency in community assembly. Since immigration history is rarely known, these historical influences, or priority effects, pose a major challenge in predicting community assembly. Here, we provide a graph-based, non-parametric, theoretical framework for understanding the predictability of community assembly as affected by priority effects. To develop this framework, we first show that the diversity of possible priority effects increases super-exponentially with the number of species. We then point out that, despite this diversity, the consequences of priority effects for multispecies communities can be classified into four basic types, each of which reduces community predictability: alternative stable states, alternative transient paths, compositional cycles, and the lack of escapes from compositional cycles to stable states. Using a neural network, we show that this classification of priority effects enables accurate explanation of community predictability, particularly when each species immigrates repeatedly. We also demonstrate the empirical utility of our theoretical framework by applying it to two experimentally derived assembly graphs of algal and ciliate communities. Based on these analyses, we discuss how the framework proposed here can help guide experimental investigation of the predictability of history-dependent community assembly.
@article{song2021untangling, title = {Untangling the complexity of priority effects in multispecies communities}, author = {Song, Chuliang and Fukami, Tadashi and Saavedra, Serguei}, journal = {Ecology Letters}, volume = {24}, number = {11}, pages = {2301--2313}, year = {2021}, publisher = {Wiley Online Library}, }
Merging dynamical and structural indicators to measure resilience in multispecies systems

Lucas P. Medeiros*, Chuliang Song*, and Serguei Saavedra

Journal of Animal Ecology, 2021

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Resilience is broadly understood as the ability of a system to recover and resist perturbations coming in abundance from the environment. However, one of the main problems in assessing resilience has been linked to its measurement and the interconnection among its components, which may not be complementary (i.e., respond to the same types of perturbations) or even possible to quantify. While recovery has a strong tradition under the mathematical analysis of asymptotic dynamical stability (i.e., return to a reference equilibrium state after infinitesimal perturbations acting on state variables), it is unclear whether this same formalism can be used to measure resistance and whether it is independent from recovery. Importantly, resilience and, in particular, resistance can also be linked to structural stability (i.e., the response of a system to structural perturbations). Formally, this structure can be represented by a model describing the governing laws of a system and its parameters. Furthermore, it has already been shown that stochastic perturbations of state variables in the vicinity of the equilibrium are equivalent to fluctuations of model parameters within the same infinitesimal border. Here, we extend the link between dynamical and structural stability beyond the vicinity of an equilibrium point to provide a framework to measure the resilience of the species composition of an ecological system to perturbations in species abundances and model parameters. We show that the return rate of the slowest-recovered species (what we call full recovery) is negatively associated with the largest random parameter perturbation that a system can withstand before losing any species (what we call full resistance). Next, we show that the return rate of the second slowest-recovered species (what we call partial recovery) is negatively associated with the largest random parameter perturbation that a system can withstand before losing all but one species (what we call partial resistance). Then, using random and experimental systems with different types of ecological interactions, we show that full and partial indicators of resilience are complimentary measurements. Because it is expected that non-infinitesimal perturbations in abundances and model parameters happen simultaneously in nature, our findings reveal that recovery (dynamical indicator) and resistance (structural indicator) are interdependent. Therefore, our results suggest that merging full and partial indicators of resilience, whether dynamical or structural, can allow a more complimentary assessment of risk in ecological systems under model-driven and data-driven applications.
@article{medeiros2021merging, title = {Merging dynamical and structural indicators to measure resilience in multispecies systems}, author = {Medeiros, Lucas P. and Song, Chuliang and Saavedra, Serguei}, journal = {Journal of Animal Ecology}, volume = {90}, number = {9}, pages = {2027--2040}, year = {2021}, publisher = {Wiley Online Library}, }

2020

Disentangling the effects of external perturbations on coexistence and priority effects

Chuliang Song, Rudolf P. Rohr, David Vasseur, and Serguei Saavedra

Journal of Ecology, 2020

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A major challenge in ecological research is to identify the tolerance of ecological communities to external perturbations. Modern coexistence theory (MCT) has been widely adopted as a framework to investigate the tolerance to perturbations in relative reductions of per capita growth rates, often using metrics that explicitly eliminate the independent role of intrinsic growth rates. More recently, the structural approach (SA) was introduced to investigate the tolerance of communities to perturbations in intrinsic growth rates as a function of the strength of intraspecific and interspecific competition. Because the external perturbations are likely to happen in both intrinsic growth rates and competition strengths, no framework alone can fully disentangle the effects of external perturbations. Here we combine MCT and SA to disentangle the tolerance in coexistence and priority effects of a pair of competing species when subject to perturbations in intrinsic growth rates and competition strengths. Through this combination, we reveal the emergence of a key trade-off: increasing the tolerance to perturbations in intrinsic growth rates typically decreases the tolerance in competition strengths, and vice versa. Furthermore, this trade-off is stronger under coexistence than under priority effects. We test this combined framework on competing pairs of 18 California annual plant species. For both coexistence and priority effects, we find that the tolerance to perturbations in intrinsic growth rates is maximized instead of that to perturbations in competition strengths in the studied annual plant communities. Synthesis. Our combined framework of modern coexistence theory and structural approach illustrates that it is possible to disentangle the impact of different external perturbations on the persistence of species. Importantly, our findings show that species interactions may reveal whether communities are dominated either by changes in intrinsic growth rates or by competition strengths. Overall, this combined framework can open a new perspective to understand and predict the response of populations to changing environmental conditions.
@article{song2020disentangling, author = {Song, Chuliang and Rohr, Rudolf P. and Vasseur, David and Saavedra, Serguei}, title = {Disentangling the effects of external perturbations on coexistence and priority effects}, journal = {Journal of Ecology}, volume = {108}, number = {4}, pages = {1677--1689}, keywords = {coexistence, competition strengths, intrinsic growth rates, modern coexistence theory, priority effects, structural approach, tolerance to perturbations}, year = {2020}, }
Telling ecological networks apart by their structure: An environment-dependent approach

Chuliang Song, and Serguei Saavedra

PLOS Computational Biology, 2020

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The network architecture of an ecological community describes the structure of species interactions established in a given place and time. It has been suggested that this architecture presents unique features for each type of ecological interaction e.g., nested and modular architectures would correspond to mutualistic and antagonistic interactions, respectively. Recently, Michalska-Smith and Allesina (2019) proposed a computational challenge to test whether it is indeed possible to differentiate ecological interactions based on network architecture. Contrary to the expectation, they found that this differentiation is practically impossible, moving the question to why it is not possible to differentiate ecological interactions based on their network architecture alone. Here, we show that this differentiation becomes possible by adding the local environmental information where the networks were sampled. We show that this can be explained by the fact that environmental conditions are a confounder of ecological interactions and network architecture. That is, the lack of association between network architecture and type of ecological interactions changes by conditioning on the local environmental conditions. Additionally, we find that environmental conditions are linked to the stability of ecological networks, but the direction of this effect depends on the type of interaction network. This suggests that the association between ecological interactions and network architectures exists, but cannot be fully understood without attention to the environmental conditions acting upon them.
@article{song2020telling, author = {Song, Chuliang and Saavedra, Serguei}, journal = {PLOS Computational Biology}, publisher = {Public Library of Science}, title = {Telling ecological networks apart by their structure: An environment-dependent approach}, year = {2020}, month = apr, volume = {16}, pages = {e1007787}, number = {4}, }
Structural stability: concepts, methods, and applications (in Chinese)

Chuliang Song

Biodiversity Science, 2020

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Ecological networks—how species interactions are organized within ecological communities—are highly structured, which has motivated generations of ecologists to elucidate how these structures affect species coexistence. Unfortunately, we still do not have a clear and consistent answer about the link between network structure and species coexistence. A possible explanation is that most of the studies do not take into account that the environment affects both network structure and species coexistence due to the multidimensional and changing nature of environmental factors. In this context, the structural stability approach provides a theoretical framework grounded on biological realism to quantitatively link network structure, species coexistence, and environmental factors. I begin by an overview of the heated debates in the study of ecological networks. Then I introduce the theoretical framework and computational tools of the structural stability approach in a nutshell. Then I show the empirical applications in different ecological questions across a broad range of ecological systems. Overall, the structural stability approach provides a new perspective to understand how biodiversity is generated and maintained in ecological communities.
@article{song2020review, author = {Song, Chuliang}, journal = {Biodiversity Science}, title = {Structural stability: concepts, methods, and applications (in Chinese)}, year = {2020}, volume = {28}, pages = {1345--1361}, number = {11}, }
Towards a probabilistic understanding about the context-dependency of species interactions

Chuliang Song, Sarah Von Ahn, Rudolf P. Rohr, and Serguei Saavedra

Trends in Ecology & Evolution, 2020

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Observational and experimental studies have shown that an interaction class between two species (be it mutualistic, competitive, antagonistic, or neutral) may switch to a different class, depending on the biotic and abiotic factors within which species are observed. This complexity arising from the evidence of context-dependencies has underscored a difficulty in establishing a systematic analysis about the extent to which species interactions are expected to switch in nature and experiments. Here, we propose an overarching theoretical framework, by integrating probabilistic and structural approaches, to establish null expectations about switches of interaction classes across environmental contexts. This integration provides a systematic platform upon which it is possible to establish new hypotheses, clear predictions, and quantifiable expectations about the context-dependency of species interactions.
@article{song2020towards, title = {Towards a probabilistic understanding about the context-dependency of species interactions}, author = {Song, Chuliang and Von Ahn, Sarah and Rohr, Rudolf P. and Saavedra, Serguei}, journal = {Trends in Ecology \& Evolution}, volume = {35}, number = {5}, pages = {384--396}, year = {2020}, publisher = {Elsevier}, }
Accelerating the emergence of order in swarming systems

Yandong Xiao, Chuliang Song, Liang Tian, and Yang-Yu Liu

Advances in Complex Systems, 2020

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Our ability to understand and control the emergence of order in swarming systems is a fundamental challenge in contemporary science. The standard Vicsek model (SVM) — a minimal model for swarming systems of self-propelled particles — describes a large population of agents reaching global alignment without the need of central control. Yet, the emergence of order in this model takes time and is not robust to noise. In many real-world scenarios, we need a decentralized protocol to guide a swarming system (e.g., unmanned vehicles or nanorobots) to reach an ordered state in a prompt and noise-robust manner. Here, we find that introducing a simple adaptive rule based on the heading differences of neighboring particles in the Vicsek model can effectively speed up their global alignment, mitigate the disturbance of noise to alignment, and maintain a robust alignment under predation. This simple adaptive model of swarming systems could offer new insights in understanding the prompt and flexible formation of animals and help us design better protocols to achieve fast and robust alignment for multi-agent systems.
@article{xiao2020accelerating, title = {Accelerating the emergence of order in swarming systems}, author = {Xiao, Yandong and Song, Chuliang and Tian, Liang and Liu, Yang-Yu}, journal = {Advances in Complex Systems}, volume = {23}, number = {01}, pages = {1950015}, year = {2020}, publisher = {World Scientific}, }

2019

On the consequences of the interdependence of stabilizing and equalizing mechanisms

Chuliang Song*, György Barabás*, and Serguei Saavedra

The American Naturalist, 2019

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We present an overlooked but important property of modern coexistence theory (MCT), along with two key new results and their consequences. The overlooked property is that stabilizing mechanisms (increasing species niche differences) and equalizing mechanisms (reducing species fitness differences) have two distinct sets of meanings within MCT; one in a two-species context and another in a general multispecies context. We demonstrate that the two-species framework is not a special case of the multispecies one, and therefore these two parallel frameworks must be studied independently. Our first result is that, using the two-species framework and mechanistic consumer-resource models, stabilizing and equalizing mechanisms exhibit complex interdependence, such that changing one will simultaneously change the other. Furthermore, the nature and direction of this simultaneous change sensitively depend on model parameters. The second result states that while MCT is often seen as bridging niche and neutral modes of coexistence by building a niche-neutrality continuum, the interdependence between stabilizing and equalizing mechanisms acts to break this continuum under almost any biologically relevant circumstance. We conclude that the complex entanglement of stabilizing and equalizing terms makes their impact on coexistence difficult to understand, but by seeing them as aggregated effects (rather than underlying causes) of coexistence, we may increase our understanding of ecological dynamics.
@article{song2019consequences, title = {On the consequences of the interdependence of stabilizing and equalizing mechanisms}, author = {Song, Chuliang and Barabás, György and Saavedra, Serguei}, journal = {The American Naturalist}, volume = {194}, number = {5}, pages = {627--639}, year = {2019}, publisher = {The University of Chicago Press Chicago, IL}, }
Beware z-scores

Chuliang Song, Rudolf P. Rohr, and Serguei Saavedra

Journal of Animal Ecology, 2019

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Song, Rohr, and Saavedra (2017) have proposed a methodology to compare network properties across systems with different sizes and constraints, in response to the fact that z‐scores cannot be used for such purposes. Simmons, Hoeppke, and Sutherland (2019) have shown that part of the methodology can be improved. Here, we show that all previous results hold and are strengthened by the new methodology.
@article{song2019beware, title = {Beware z-scores}, author = {Song, Chuliang and Rohr, Rudolf P. and Saavedra, Serguei}, journal = {Journal of Animal Ecology}, volume = {88}, number = {5}, pages = {808--809}, year = {2019}, publisher = {Wiley Online Library}, }

2018

Rethinking the importance of the structure of ecological networks under an environment-dependent framework

Simone Cenci*, Chuliang Song*, and Serguei Saavedra

Ecology and Evolution, 2018

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A major quest in network and community ecology has been centered on understanding the importance of structural patterns in species interaction networks—the synthesis of who interacts with whom in a given location and time. In the past decades, much effort has been devoted to infer the importance of a particular structure by its capacity to tolerate an external perturbation on its structure or dynamics. Here, we demonstrate that such a perspective leads to inconsistent conclusions. That is, the importance of a network structure changes as a function of the external perturbations acting on a community at any given point in time. Thus, we discuss a research agenda to investigate the relative importance of the structure of ecological networks under an environment‐dependent framework. We hypothesize that only by studying systematically the link between network structure and community dynamics under an environment‐dependent framework, we can uncover the limits at which communities can tolerate environmental changes.
@article{cenci2018rethinking, title = {Rethinking the importance of the structure of ecological networks under an environment-dependent framework}, author = {Cenci, Simone and Song, Chuliang and Saavedra, Serguei}, journal = {Ecology and Evolution}, volume = {8}, number = {14}, pages = {6852--6859}, year = {2018}, publisher = {Wiley Online Library}, }
Will a small randomly assembled community be feasible and stable?

Chuliang Song, and Serguei Saavedra

Ecology, 2018

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How likely is it that few species can randomly assemble into a feasible and stable community? Some studies have answered that as long as the community is feasible, it will nearly always be stable. In contrast, other studies have answered that the likelihood is almost null. Here, we show that the origin of this debate has been the underestimation of the association of the parameter space of intrinsic growth rates with the feasibility and stability properties of small randomly‐assembled communities. In particular, we demonstrate that not all parameterizations and sampling distributions of intrinsic growth rates lead to the same probabilities of stability and feasibility, which could mistakenly lead to under‐ or overestimate the stability properties of feasible communities. Additionally, we find that stability imposes a filtering of species abundances “towards” more even distributions in small feasible randomly‐assembled communities. This indicates that the stability of feasible communities is inherently linked to the starting distribution of species abundances, a characteristic that many times has been ignored, but should be incorporated in manageable lab and field experiments. Overall, the return to this debate is a central reminder that a more systematic exploration of the feasible parameter space is necessary to derive general conclusions about the stability properties of ecological communities.
@article{song2018will, title = {Will a small randomly assembled community be feasible and stable?}, author = {Song, Chuliang and Saavedra, Serguei}, journal = {Ecology}, volume = {99}, number = {3}, pages = {743--751}, year = {2018}, publisher = {Wiley Online Library}, }
Structural changes within trophic levels are constrained by within-family assembly rules at lower trophic levels

Chuliang Song, Florian Altermatt, Ian Pearse, and Serguei Saavedra

Ecology Letters, 2018

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Historical contingency broadly refers to the proposition that even random historical events can constrain the ecological and evolutionary pathways of organisms and that of entire communities. Focusing on communities, these pathways can be reflected into specific structural changes within and across trophic levels – how species interact with and affect each other – which has important consequences for species coexistence. Using the registry of the last 2000 years of plant introductions and their novel herbivores encountered in Central Europe, we find that the order of arrival of closely related (but not of distantly related) plant species constrained the structural changes within the trophic level formed by herbivore species across the observation period. Because it is difficult for field and lab experiments to be conducted over hundreds of years to record and replay the assembly history of a community, our study provides an alternative to understand how structural changes have occurred across extensive periods of time.
@article{song2018structural, title = {Structural changes within trophic levels are constrained by within-family assembly rules at lower trophic levels}, author = {Song, Chuliang and Altermatt, Florian and Pearse, Ian and Saavedra, Serguei}, journal = {Ecology Letters}, volume = {21}, number = {8}, pages = {1221--1228}, year = {2018}, publisher = {Wiley Online Library}, }
A guideline to study the feasibility domain of multi-trophic and changing ecological communities

Chuliang Song, Rudolf P. Rohr, and Serguei Saavedra

Journal of Theoretical Biology, 2018

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The feasibility domain of an ecological community can be described by the set of environmental abiotic and biotic conditions under which all co-occurring and interacting species in a given site and time can have positive abundances. Mathematically, the feasibility domain corresponds to the parameter space compatible with positive (feasible) solutions at equilibrium for all the state variables in a system under a given model of population dynamics. Under specific dynamics, the existence of a feasible equilibrium is a necessary condition for species persistence regardless of whether the feasible equilibrium is dynamically stable or not. Thus, the size of the feasibility domain can also be used as an indicator of the tolerance of a community to random environmental variations. This has motivated a rich research agenda to estimate the feasibility domain of ecological communities. However, these methodologies typically assume that species interactions are static, or that input and output energy flows on each trophic level are unconstrained. Yet, this is different to how communities behave in nature. Here, we present a step-by-step quantitative guideline providing illustrative examples, computational code, and mathematical proofs to study systematically the feasibility domain of ecological communities under changes of interspecific interactions and subject to different constraints on the trophic energy flows. This guideline covers multi-trophic communities that can be formed by any type of interspecific interactions. Importantly, we show that the relative size of the feasibility domain can significantly change as a function of the biological information taken into consideration. We believe that the availability of these methods can allow us to increase our understanding about the limits at which ecological communities may no longer tolerate further environmental perturbations, and can facilitate a stronger integration of theoretical and empirical research.
@article{song2018guideline, title = {A guideline to study the feasibility domain of multi-trophic and changing ecological communities}, author = {Song, Chuliang and Rohr, Rudolf P. and Saavedra, Serguei}, journal = {Journal of Theoretical Biology}, volume = {450}, pages = {30--36}, year = {2018}, publisher = {Elsevier}, }
Structural stability as a consistent predictor of phenological events

Chuliang Song, and Serguei Saavedra

Proceedings of the Royal Society B: Biological Sciences, 2018

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The timing of the first and last seasonal appearance of a species in a community typically follows a pattern that is governed by temporal factors. While it has been shown that changes in the environment are linked to phenological changes, the direction of this link appears elusive and context-dependent. Thus, finding consistent predictors of phenological events is of central importance for a better assessment of expected changes in the temporal dynamics of ecological communities. Here we introduce a measure of structural stability derived from species interaction networks as an estimator of the expected range of environmental conditions compatible with the existence of a community. We test this measure as a predictor of changes in species richness recorded on a daily basis in a high-arctic plant–pollinator community during two spring seasons. We find that our measure of structural stability is the only consistent predictor of changes in species richness among different ecological and environmental variables. Our findings suggest that measures based on the notion of structural stability can synthesize the expected variation of environmental conditions tolerated by a community, and explain more consistently the phenological changes observed in ecological communities.
@article{song2018phenology, title = {Structural stability as a consistent predictor of phenological events}, author = {Song, Chuliang and Saavedra, Serguei}, journal = {Proceedings of the Royal Society B: Biological Sciences}, volume = {285}, number = {1880}, pages = {20180767}, year = {2018}, publisher = {The Royal Society}, }

2017

Existence of positive solutions for an approximation of stationary mean-field games

Nojood Almayouf, Elena Bachini, Andreia Chapouto, Rita Ferreira, Diogo Gomes, Daniela Jordão, David Evangelista Junior, and 18 more authors

Involve, a Journal of Mathematics, 2017

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Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

@article{almayouf2016existence,
  title = {Existence of positive solutions for an approximation of stationary mean-field games},
  author = {Almayouf, Nojood and Bachini, Elena and Chapouto, Andreia and Ferreira, Rita and Gomes, Diogo and Jordão, Daniela and Evangelista Junior, David and Karagulyan, Avetik and Monasterio, Juan and Nurbekyan, Levon and Pagliar, Giorgia and Piccirilli, Marco and Pratapsi, Sagar and Prazeres, Mariana and Reis, João and Rodrigues, André and Romero, Orlando and Sargsyan, Maria and Seneci, Tommaso and Song, Chuliang and Terai, Kengo and Tomisaki, Ryota and Velasco-Perez, Hector and Voskanyan, Vardan and Yang, Xianjin},
  journal = {Involve, a Journal of Mathematics},
  volume = {10},
  number = {3},
  pages = {473--493},
  year = {2017},
  publisher = {Mathematical Sciences Publishers},
}

Why are some plant–pollinator networks more nested than others?

Chuliang Song, Rudolf P. Rohr, and Serguei Saavedra

Journal of Animal Ecology, 2017

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Empirical studies have found that the mutualistic interactions forming the structure of plant–pollinator networks are typically more nested than expected by chance alone. Additionally, theoretical studies have shown a positive association between the nested structure of mutualistic networks and community persistence. Yet, it has been shown that some plant–pollinator networks may be more nested than others, raising the interesting question of which factors are responsible for such enhanced nested structure. It has been argued that ordered network structures may increase the persistence of ecological communities under less predictable environments. This suggests that nested structures of plant–pollinator networks could be more advantageous under highly seasonal environments. While several studies have investigated the link between nestedness and various environmental variables, unfortunately, there has been no unified answer to validate these predictions. Here, we move from the problem of describing network structures to the problem of comparing network structures. We develop comparative statistics, and apply them to investigate the association between the nested structure of 59 plant–pollinator networks and the temperature seasonality present in their locations. We demonstrate that higher levels of nestedness are associated with a higher temperature seasonality. We show that the previous lack of agreement came from an extended practice of using standardized measures of nestedness that cannot be compared across different networks. Importantly, our observations complement theory showing that more nested network structures can increase the range of environmental conditions compatible with species coexistence in mutualistic systems, also known as structural stability. This increase in nestedness should be more advantageous and occur more often in locations subject to random environmental perturbations, which could be driven by highly changing or seasonal environments. This synthesis of theory and observations could prove relevant for a better understanding of the ecological processes driving the assembly and persistence of ecological communities.
@article{song2017some, title = {Why are some plant--pollinator networks more nested than others?}, author = {Song, Chuliang and Rohr, Rudolf P. and Saavedra, Serguei}, journal = {Journal of Animal Ecology}, volume = {86}, number = {6}, pages = {1417--1424}, year = {2017}, publisher = {Wiley Online Library}, }