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Hofer M, Sneppen K. Diversity and ecotones in a model ecosystems of sessile species. Phys Rev E 2024; 109:024309. [PMID: 38491630 DOI: 10.1103/physreve.109.024309] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2023] [Accepted: 01/30/2024] [Indexed: 03/18/2024]
Abstract
Sessile species compete for space and accessible light, with directed interactions evident in one species overgrowing another and with multispecies systems characterized by nontransitive relationships. Such patterns are observed in coral reefs or lichens on rock surfaces. Open systems with episodic invasions of such species have been predicted to exhibit a stable high-diversity state when the interaction probability is below a certain critical threshold. Here, we explore this metastable high-diversity state and find that the diversity in the high-diversity state scales with the square root of the system area. When introducing two different environments, we predict a hugely increased diversity along mutual environment border. Further, the presence of spatially segregated environments is predicted to allow for increased robustness of the high-diversity state.
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Affiliation(s)
- Markus Hofer
- Medical University of Vienna, Center for Medical Data Science, Section for the Science of Complex Systems, Spitalgasse 23, 1090 Vienna, Austria and Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria
| | - Kim Sneppen
- Copenhagen University, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
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2
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Kirkegaard JB, Sneppen K. Emerging diversity in a population of evolving intransitive dice. Phys Rev E 2022; 106:054409. [PMID: 36559503 DOI: 10.1103/physreve.106.054409] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2022] [Accepted: 09/26/2022] [Indexed: 11/18/2022]
Abstract
Exploiting the mathematical curiosity of intransitive dice, we present a simple theoretical model for coevolution that captures scales ranging from the genome of the individual to the system-wide emergence of species diversity. We study a set of evolving agents that interact competitively in a closed system, in which both the dynamics of mutations and competitive advantage emerge directly from interpreting a genome as the sides of a die. The model demonstrates sympatric speciation where new species evolve from existing ones while in contact with the entire ecosystem. Allowing free mutations both in the genomes and the mutation rates, we find, in contrast to hierarchical models of fitness, the emergence of a metastable state of finite mutation rate and diversity.
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Affiliation(s)
| | - Kim Sneppen
- Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
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3
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Park HJ, Pichugin Y, Traulsen A. Why is cyclic dominance so rare? eLife 2020; 9:57857. [PMID: 32886604 PMCID: PMC7473768 DOI: 10.7554/elife.57857] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2020] [Accepted: 08/01/2020] [Indexed: 12/19/2022] Open
Abstract
Natural populations can contain multiple types of coexisting individuals. How does natural selection maintain such diversity within and across populations? A popular theoretical basis for the maintenance of diversity is cyclic dominance, illustrated by the rock-paper-scissor game. However, it appears difficult to find cyclic dominance in nature. Why is this the case? Focusing on continuously produced novel mutations, we theoretically addressed the rareness of cyclic dominance. We developed a model of an evolving population and studied the formation of cyclic dominance. Our results showed that the chance for cyclic dominance to emerge is lower when the newly introduced type is similar to existing types compared to the introduction of an unrelated type. This suggests that cyclic dominance is more likely to evolve through the assembly of unrelated types whereas it rarely evolves within a community of similar types.
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Affiliation(s)
- Hye Jin Park
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Plön, Germany.,Asia Pacific Center for Theoretical Physics, Pohang, Republic of Korea
| | - Yuriy Pichugin
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Plön, Germany
| | - Arne Traulsen
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Plön, Germany
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Abstract
Many ecosystems, from vegetation to biofilms, are composed of territorial populations that compete for both nutrients and physical space. What are the implications of such spatial organization for biodiversity? To address this question, we developed and analyzed a model of territorial resource competition. In the model, all species obey trade-offs inspired by biophysical constraints on metabolism; the species occupy nonoverlapping territories, while nutrients diffuse in space. We find that the nutrient diffusion time is an important control parameter for both biodiversity and the timescale of population dynamics. Interestingly, fast nutrient diffusion allows the populations of some species to fluctuate to zero, leading to extinctions. Moreover, territorial competition spontaneously gives rise to both multistability and the Allee effect (in which a minimum population is required for survival), so that small perturbations can have major ecological effects. While the assumption of trade-offs allows for the coexistence of more species than the number of nutrients-thus violating the principle of competitive exclusion-overall biodiversity is curbed by the domination of "oligotroph" species. Importantly, in contrast to well-mixed models, spatial structure renders diversity robust to inequalities in metabolic trade-offs. Our results suggest that territorial ecosystems can display high biodiversity and rich dynamics simply due to competition for resources in a spatial community.
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Affiliation(s)
| | - Anna Posfai
- Simons Center for Quantitative Biology, Cold Spring Harbor Laboratory, Cold Spring Harbor, NY 11724
| | - Ned S Wingreen
- Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544
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Farahpour F, Saeedghalati M, Brauer VS, Hoffmann D. Trade-off shapes diversity in eco-evolutionary dynamics. eLife 2018; 7:e36273. [PMID: 30117415 PMCID: PMC6126925 DOI: 10.7554/elife.36273] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2018] [Accepted: 08/03/2018] [Indexed: 12/22/2022] Open
Abstract
We introduce an Interaction- and Trade-off-based Eco-Evolutionary Model (ITEEM), in which species are competing in a well-mixed system, and their evolution in interaction trait space is subject to a life-history trade-off between replication rate and competitive ability. We demonstrate that the shape of the trade-off has a fundamental impact on eco-evolutionary dynamics, as it imposes four phases of diversity, including a sharp phase transition. Despite its minimalism, ITEEM produces a remarkable range of patterns of eco-evolutionary dynamics that are observed in experimental and natural systems. Most notably we find self-organization towards structured communities with high and sustained diversity, in which competing species form interaction cycles similar to rock-paper-scissors games.
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Affiliation(s)
- Farnoush Farahpour
- Bioinformatics and Computational BiophysicsUniversity of Duisburg-EssenEssenGermany
| | | | | | - Daniel Hoffmann
- Bioinformatics and Computational BiophysicsUniversity of Duisburg-EssenEssenGermany
- Center for Computational Sciences and SimulationUniversity of Duisburg-EssenEssenGermany
- Center for Medical BiotechnologyUniversity of Duisburg-EssenEssenGermany
- Center for Water and Environmental ResearchUniversity of Duisburg-EssenEssenGermany
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Sneppen K. Models of life: epigenetics, diversity and cycles. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:042601. [PMID: 28106010 DOI: 10.1088/1361-6633/aa5aeb] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
This review emphasizes aspects of biology that can be understood through repeated applications of simple causal rules. The selected topics include perspectives on gene regulation, phage lambda development, epigenetics, microbial ecology, as well as model approaches to diversity and to punctuated equilibrium in evolution. Two outstanding features are repeatedly described. One is the minimal number of rules to sustain specific states of complex systems for a long time. The other is the collapse of such states and the subsequent dynamical cycle of situations that restitute the system to a potentially new metastable state.
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Affiliation(s)
- Kim Sneppen
- Center for Models of Life, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
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Uekermann F, Mathiesen J, Mitarai N. Characterization of phase transitions in a model ecosystem of sessile species. Phys Rev E 2017; 95:032409. [PMID: 28415313 DOI: 10.1103/physreve.95.032409] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2016] [Indexed: 06/07/2023]
Abstract
We consider a model ecosystem of sessile species competing for space. In particular, we consider the system introduced by Mathiesen et al. [J. Mathiesen, N. Mitarai, K. Sneppen, and A. Trusina, Phys. Rev. Lett. 107, 188101 (2011)PRLTAO0031-900710.1103/PhysRevLett.107.188101] where species compete according to a fixed interaction network with links determined by a Bernoulli process. In the limit of a small introduction rate of new species, the model exhibits a discontinuous transition from a high-diversity state to a low-diversity state as the interaction probability between species, γ, is increased from zero. Here we explore the effects of finite introduction rates and system size on the phase transition by utilizing efficient parallel computing. We find that the low state appears for γ>γ_{c}. As γ is increased further, the high state approaches the low state, suggesting the possibility that the two states merge at a high γ. We find that the fraction of time spent in the high state becomes longer with higher introduction rates, but the availability of the two states is rather insensitive to the value of the introduction rate. Furthermore, we establish a relation between the introduction rate and the system size, which preserves the probability for the system to remain in the high-diversity state.
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Affiliation(s)
- Florian Uekermann
- Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen, Denmark
| | - Joachim Mathiesen
- Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen, Denmark
| | - Namiko Mitarai
- Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen, Denmark
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Mitarai N, Gunnarson I, Pedersen BN, Rosiek CA, Sneppen K. Three is much more than two in coarsening dynamics of cyclic competitions. Phys Rev E 2016; 93:042408. [PMID: 27176330 DOI: 10.1103/physreve.93.042408] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2015] [Indexed: 06/05/2023]
Abstract
The classical game of rock-paper-scissors has inspired experiments and spatial model systems that address the robustness of biological diversity. In particular, the game nicely illustrates that cyclic interactions allow multiple strategies to coexist for long-time intervals. When formulated in terms of a one-dimensional cellular automata, the spatial distribution of strategies exhibits coarsening with algebraically growing domain size over time, while the two-dimensional version allows domains to break and thereby opens the possibility for long-time coexistence. We consider a quasi-one-dimensional implementation of the cyclic competition, and study the long-term dynamics as a function of rare invasions between parallel linear ecosystems. We find that increasing the complexity from two to three parallel subsystems allows a transition from complete coarsening to an active steady state where the domain size stays finite. We further find that this transition happens irrespective of whether the update is done in parallel for all sites simultaneously or done randomly in sequential order. In both cases, the active state is characterized by localized bursts of dislocations, followed by longer periods of coarsening. In the case of the parallel dynamics, we find that there is another phase transition between the active steady state and the coarsening state within the three-line system when the invasion rate between the subsystems is varied. We identify the critical parameter for this transition and show that the density of active boundaries has critical exponents that are consistent with the directed percolation universality class. On the other hand, numerical simulations with the random sequential dynamics suggest that the system may exhibit an active steady state as long as the invasion rate is finite.
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Affiliation(s)
- Namiko Mitarai
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
| | - Ivar Gunnarson
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
| | - Buster Niels Pedersen
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
| | - Christian Anker Rosiek
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
| | - Kim Sneppen
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
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Mitarai N, Heinsalu E, Sneppen K. Speciation, diversification, and coexistence of sessile species that compete for space. PLoS One 2014; 9:e96665. [PMID: 24819515 PMCID: PMC4018333 DOI: 10.1371/journal.pone.0096665] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2013] [Accepted: 04/10/2014] [Indexed: 11/18/2022] Open
Abstract
Speciation, diversification, and competition between species challenge the stability of complex ecosystems. Laboratory experiments often focus on one or two species competing under conditions where they may grow exponentially. Field studies, in contrast, emphasize multi-species communities characterized by many types of ecological interactions. A general problem is to understand conditions that support a dynamically maintained coexistence of many species in an ecosystem over a long time span. In the present paper we propose a lattice model of multiple competing and evolving sessile species. When allowing the interspecies interactions to mutate, we obtain coexistence of many species in a complex ecosystem, provided that there is a cost for each interaction. The diversity reached by the model incorporating speciation is found to be substantially higher than in the case when entirely new species appear due to immigration from outside of the considered ecosystem. The species self-organize their spatial distribution through competitive interactions to create many patches, implicitly protecting each other from competitively superior species, and speciation in each patch leads the system to high diversity. We also show that species that exist a long time tend to have a relatively small population, as this allows them to avoid encounter with competitive invaders.
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Affiliation(s)
- Namiko Mitarai
- Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
- * E-mail:
| | - Els Heinsalu
- Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
- National Institute of Chemical Physics and Biophysics, Tallinn, Estonia
| | - Kim Sneppen
- Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
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Giavazzi F, Vailati A. Geometry for a penguin-albatross rookery. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052706. [PMID: 25353828 DOI: 10.1103/physreve.89.052706] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/22/2013] [Indexed: 06/04/2023]
Abstract
We introduce a simple ecological model describing the spatial organization of two interacting populations whose individuals are indifferent to conspecifics and avoid the proximity to heterospecifics. At small population densities Φ a nontrivial structure is observed where clusters of individuals arrange into a rhomboidal bipartite network with an average degree of 4. For Φ → 0 the length scale, order parameter, and susceptibility of the network exhibit power-law divergences compatible with hyperscaling, suggesting the existence of a zero-density nontrivial critical point. At larger densities a critical threshold Φ(c) is identified above which the evolution toward a partially ordered configuration is prevented and the system becomes jammed in a fully mixed state.
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Affiliation(s)
- Fabio Giavazzi
- Dipartimento di Biotecnologie Mediche e Medicina Traslazionale, Università degli Studi di Milano, Milano, Italy
| | - Alberto Vailati
- Dipartimento di Fisica, Università degli Studi di Milano, I-20133 Milano, Italy
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Botta F, Mitarai N. Disturbance accelerates the transition from low- to high-diversity state in a model ecosystem. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022704. [PMID: 25353506 DOI: 10.1103/physreve.89.022704] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2013] [Indexed: 06/04/2023]
Abstract
The effect of disturbance on a model ecosystem of sessile and mutually competitive species [Mathiesen et al., Phys. Rev. Lett. 107, 188101 (2011); Mitarai et al., Phys. Rev. E 86, 011929 (2012)] is studied. The disturbance stochastically removes individuals from the system, and the created empty sites are recolonized by neighboring species. We show that the stable high-diversity state, maintained by occasional cyclic species interactions that create isolated patches of metapopulations, is robust against small disturbance. We further demonstrate that finite disturbance can accelerate the transition from the low- to high-diversity state by helping the creation of small patches through diffusion of boundaries between species with standoff relations.
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Affiliation(s)
- Filippo Botta
- Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
| | - Namiko Mitarai
- Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
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12
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Walker SI, Callahan BJ, Arya G, Barry JD, Bhattacharya T, Grigoryev S, Pellegrini M, Rippe K, Rosenberg SM. Evolutionary dynamics and information hierarchies in biological systems. Ann N Y Acad Sci 2013; 1305:1-17. [DOI: 10.1111/nyas.12140] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Affiliation(s)
- Sara Imari Walker
- BEYOND: Center for Fundamental Concepts in Science Arizona State University Tempe Arizona
- Blue Marble Space Institute of Science Seattle Washington
| | | | - Gaurav Arya
- Department of NanoEngineering University of California, San Diego La Jolla California
| | - J. David Barry
- Wellcome Trust Centre for Molecular Parasitology Institute of Infection Immunity and Inflammation University of Glasgow Glasgow United Kingdom
| | - Tanmoy Bhattacharya
- Sante Fe Institute Sante Fe New Mexico
- Grp T‐2, MSB285, Los Alamos National Laboratory Los Alamos New Mexico
| | - Sergei Grigoryev
- Penn State University College of Medicine Department Biochemistry and Molecular Biology Pennsylvania State University Hershey Pennsylvania
| | - Matteo Pellegrini
- Department of Molecular, Cell, and Developmental Biology University of California Los Angeles Los Angeles California
| | - Karsten Rippe
- Deutsches Krebsforschungszentrum (DKFZ) and BioQuant Research Group Genome Organization & Function Heidelberg Germany
| | - Susan M. Rosenberg
- Departments of Molecular and Human Genetics Biochemistry and Molecular Biology Molecular Virology and Microbiology, and Dan L. Duncan Cancer Center Baylor College of Medicine Houston Texas
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Rulands S, Zielinski A, Frey E. Global attractors and extinction dynamics of cyclically competing species. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052710. [PMID: 23767569 DOI: 10.1103/physreve.87.052710] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2013] [Indexed: 06/02/2023]
Abstract
Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases.
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Affiliation(s)
- Steffen Rulands
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Physics Department, Ludwig-Maximilians-Universität München, Theresienstrasse 33, D-80333 München, Germany
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