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Ohlmann M, Munoz F, Massol F, Thuiller W. Assessing mutualistic metacommunity capacity by integrating spatial and interaction networks. Theor Popul Biol 2024; 156:22-39. [PMID: 38219873 DOI: 10.1016/j.tpb.2024.01.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2023] [Revised: 12/26/2023] [Accepted: 01/10/2024] [Indexed: 01/16/2024]
Abstract
We develop a spatially realistic model of mutualistic metacommunities that exploits the joint structure of spatial and interaction networks. Assuming that all species have the same colonisation and extinction parameters, this model exhibits a sharp transition between stable non-null equilibrium states and a global extinction state. This behaviour allows defining a threshold on colonisation/extinction parameters for the long-term metacommunity persistence. This threshold, the 'metacommunity capacity', extends the metapopulation capacity concept and can be calculated from the spatial and interaction networks without needing to simulate the whole dynamics. In several applications we illustrate how the joint structure of the spatial and the interaction networks affects metacommunity capacity. It results that a weakly modular spatial network and a power-law degree distribution of the interaction network provide the most favourable configuration for the long-term persistence of a mutualistic metacommunity. Our model that encodes several explicit ecological assumptions should pave the way for a larger exploration of spatially realistic metacommunity models involving multiple interaction types.
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Affiliation(s)
- Marc Ohlmann
- Univ. Grenoble Alpes, CNRS, Univ. Savoie Mont-Blanc, LECA, Laboratoire d'Ecologie Alpine, F-38000 Grenoble, France
| | - François Munoz
- Univ. Grenoble Alpes, CNRS, Univ. Savoie Mont-Blanc, LECA, Laboratoire d'Ecologie Alpine, F-38000 Grenoble, France; Univ. Grenoble Alpes, CNRS, Liphy, Laboratoire Interdisciplinaire de Physique, F-38000 Grenoble, France
| | - François Massol
- Univ. Lille, CNRS, Inserm, CHU Lille, Institut Pasteur de Lille, U1019 - UMR 9017 - CIIL - Center for Infection and Immunity of Lille, F-59000 Lille, France
| | - Wilfried Thuiller
- Univ. Grenoble Alpes, CNRS, Univ. Savoie Mont-Blanc, LECA, Laboratoire d'Ecologie Alpine, F-38000 Grenoble, France.
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2
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Saulsbury JG, Parins-Fukuchi CT, Wilson CJ, Reitan T, Liow LH. Age-dependent extinction and the neutral theory of biodiversity. Proc Natl Acad Sci U S A 2024; 121:e2307629121. [PMID: 38150497 PMCID: PMC10769858 DOI: 10.1073/pnas.2307629121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/06/2023] [Accepted: 11/22/2023] [Indexed: 12/29/2023] Open
Abstract
Red Queen (RQ) theory states that adaptation does not protect species from extinction because their competitors are continually adapting alongside them. RQ was founded on the apparent independence of extinction risk and fossil taxon age, but analytical developments have since demonstrated that age-dependent extinction is widespread, usually most intense among young species. Here, we develop ecological neutral theory as a general framework for modeling fossil species survivorship under incomplete sampling. We show that it provides an excellent fit to a high-resolution dataset of species durations for Paleozoic zooplankton and more broadly can account for age-dependent extinction seen throughout the fossil record. Unlike widely used alternative models, the neutral model has parameters with biological meaning, thereby generating testable hypotheses on changes in ancient ecosystems. The success of this approach suggests reinterpretations of mass extinctions and of scaling in eco-evolutionary systems. Intense extinction among young species does not necessarily refute RQ or require a special explanation but can instead be parsimoniously explained by neutral dynamics operating across species regardless of age.
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Affiliation(s)
- James G. Saulsbury
- Natural History Museum, University of Oslo, Oslo0187, Norway
- Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence, KS66045
| | - C. Tomomi Parins-Fukuchi
- Department of Ecology and Evolutionary Biology, University of Toronto, Toronto, ONM5S 3B2, Canada
| | - Connor J. Wilson
- Natural History Museum, University of Oslo, Oslo0187, Norway
- School of Geography and the Environment, University of Oxford, OxfordOX1 3QY, United Kingdom
- Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ85721
| | - Trond Reitan
- Natural History Museum, University of Oslo, Oslo0187, Norway
- Center for Planetary Habitability, Department of Geosciences, University of Oslo, Oslo0371, Norway
| | - Lee Hsiang Liow
- Natural History Museum, University of Oslo, Oslo0187, Norway
- Center for Planetary Habitability, Department of Geosciences, University of Oslo, Oslo0371, Norway
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3
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An ergodic theorem for asymptotically periodic time-inhomogeneous Markov processes, with application to quasi-stationarity with moving boundaries. ADV APPL PROBAB 2023. [DOI: 10.1017/apr.2022.41] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/10/2023]
Abstract
Abstract
This paper deals with ergodic theorems for particular time-inhomogeneous Markov processes, whose time-inhomogeneity is asymptotically periodic. Under a Lyapunov/minorization condition, it is shown that, for any measurable bounded function f, the time average
$\frac{1}{t} \int_0^t f(X_s)ds$
converges in
$\mathbb{L}^2$
towards a limiting distribution, starting from any initial distribution for the process
$(X_t)_{t \geq 0}$
. This convergence can be improved to an almost sure convergence under an additional assumption on the initial measure. This result is then applied to show the existence of a quasi-ergodic distribution for processes absorbed by an asymptotically periodic moving boundary, satisfying a conditional Doeblin condition.
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4
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Vezzani A, Muñoz MA, Burioni R. Anomalous finite-size scaling in higher-order processes with absorbing states. Phys Rev E 2023; 107:014105. [PMID: 36797930 DOI: 10.1103/physreve.107.014105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Accepted: 12/21/2022] [Indexed: 01/09/2023]
Abstract
Here we study standard and higher-order birth-death processes on fully connected networks, within the perspective of large-deviation theory [also referred to as the Wentzel-Kramers-Brillouin (WKB) method in some contexts]. We obtain a general expression for the leading and next-to-leading terms of the stationary probability distribution of the fraction of "active" sites as a function of parameters and network size N. We reproduce several results from the literature and, in particular, we derive all the moments of the stationary distribution for the q-susceptible-infected-susceptible (q-SIS) model, i.e., a high-order epidemic model requiring q active ("infected") sites to activate an additional one. We uncover a very rich scenario for the fluctuations of the fraction of active sites, with nontrivial finite-size-scaling properties. In particular, we show that the variance-to-mean ratio diverges at criticality for [1≤q≤3], with a maximal variability at q=2, confirming that complex-contagion processes can exhibit peculiar scaling features including wild variability. Moreover, the leading order in a large-deviation approach does not suffice to describe them: next-to-leading terms are essential to capture the intrinsic singularity at the origin of systems with absorbing states. Some possible extensions of this work are also discussed.
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Affiliation(s)
- Alessandro Vezzani
- Istituto dei Materiali per l'Elettronica ed il Magnetismo (IMEM-CNR), Parco Area delle Scienze, 37/A-43124 Parma, Italy; Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy; and INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy
| | - Miguel A Muñoz
- Departamento de Electromagnetismo y Física de la Materia and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada. E-18071 Granada, Spain
| | - Raffaella Burioni
- Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy and INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy
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5
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Sales de Queiroz A, Sales Santa Cruz G, Jean-Marie A, Mazauric D, Roux J, Cazals F. Gene prioritization based on random walks with restarts and absorbing states, to define gene sets regulating drug pharmacodynamics from single-cell analyses. PLoS One 2022; 17:e0268956. [PMID: 36342924 PMCID: PMC9639845 DOI: 10.1371/journal.pone.0268956] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/27/2021] [Accepted: 05/12/2022] [Indexed: 11/09/2022] Open
Abstract
Prioritizing genes for their role in drug sensitivity, is an important step in understanding drugs mechanisms of action and discovering new molecular targets for co-treatment. To formalize this problem, we consider two sets of genes X and P respectively composing the gene signature of cell sensitivity at the drug IC50 and the genes involved in its mechanism of action, as well as a protein interaction network (PPIN) containing the products of X and P as nodes. We introduce Genetrank, a method to prioritize the genes in X for their likelihood to regulate the genes in P. Genetrank uses asymmetric random walks with restarts, absorbing states, and a suitable renormalization scheme. Using novel so-called saturation indices, we show that the conjunction of absorbing states and renormalization yields an exploration of the PPIN which is much more progressive than that afforded by random walks with restarts only. Using MINT as underlying network, we apply Genetrank to a predictive gene signature of cancer cells sensitivity to tumor-necrosis-factor-related apoptosis-inducing ligand (TRAIL), performed in single-cells. Our ranking provides biological insights on drug sensitivity and a gene set considerably enriched in genes regulating TRAIL pharmacodynamics when compared to the most significant differentially expressed genes obtained from a statistical analysis framework alone. We also introduce gene expression radars, a visualization tool embedded in MA plots to assess all pairwise interactions at a glance on graphical representations of transcriptomics data. Genetrank is made available in the Structural Bioinformatics Library (https://sbl.inria.fr/doc/Genetrank-user-manual.html). It should prove useful for mining gene sets in conjunction with a signaling pathway, whenever other approaches yield relatively large sets of genes.
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Affiliation(s)
| | | | | | | | - Jérémie Roux
- CNRS UMR 7284, Inserm U 1081, Institut de Recherche sur le Cancer et le Vieillissement de Nice, Centre Antoine Lacassagne, Universite Côte d’Azur, Nice, France
- * E-mail: (FC); (JR)
| | - Frédéric Cazals
- Inria, Université Côte d’Azur, Nice, France
- * E-mail: (FC); (JR)
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6
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He G, Zhang H, Yang G. Exponential mixing property for absorbing Markov processes. Stat Probab Lett 2021. [DOI: 10.1016/j.spl.2021.109207] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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7
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Yaglom limit for stochastic fluid models. ADV APPL PROBAB 2021. [DOI: 10.1017/apr.2020.71] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
AbstractIn this paper we analyse the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of models in the theory of matrix-analytic methods. So far, only transient and stationary analyses of SFMs have been considered in the literature. The limiting conditional distribution gives useful insights into what happens when the process has been evolving for a long time, given that its busy period has not ended yet. We derive expressions for the Yaglom limit in terms of the singularity˜$s^*$ such that the key matrix of the SFM, ${\boldsymbol{\Psi}}(s)$, is finite (exists) for all $s\geq s^*$ and infinite for $s<s^*$. We show the uniqueness of the Yaglom limit and illustrate the application of the theory with simple examples.
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8
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Affiliation(s)
- Sven Knoth
- Department of Mathematics and Statistics, Helmut Schmidt University, Hamburg, Germany
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9
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Wang AQ, Pollock M, Roberts GO, Steinsaltz D. Regeneration-enriched Markov processes with application to Monte Carlo. ANN APPL PROBAB 2021. [DOI: 10.1214/20-aap1602] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Andi Q. Wang
- Department of Mathematics, University of Bristol
| | - Murray Pollock
- School of Mathematics, Statistics and Physics, Newcastle University
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Alberti F, Baake E, Letter I, Martínez S. Solving the migration-recombination equation from a genealogical point of view. J Math Biol 2021; 82:41. [PMID: 33774735 PMCID: PMC8004498 DOI: 10.1007/s00285-021-01584-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2020] [Revised: 10/08/2020] [Accepted: 02/14/2021] [Indexed: 11/29/2022]
Abstract
We consider the discrete-time migration–recombination equation, a deterministic, nonlinear dynamical system that describes the evolution of the genetic type distribution of a population evolving under migration and recombination in a law of large numbers setting. We relate this dynamics (forward in time) to a Markov chain, namely a labelled partitioning process, backward in time. This way, we obtain a stochastic representation of the solution of the migration–recombination equation. As a consequence, one obtains an explicit solution of the nonlinear dynamics, simply in terms of powers of the transition matrix of the Markov chain. The limiting and quasi-limiting behaviour of the Markov chain are investigated, which gives immediate access to the asymptotic behaviour of the dynamical system. We finally sketch the analogous situation in continuous time.
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Affiliation(s)
- F Alberti
- Faculty of Mathematics, Bielefeld University, Postbox 100131, 33501, Bielefeld, Germany
| | - E Baake
- Faculty of Mathematics, Bielefeld University, Postbox 100131, 33501, Bielefeld, Germany.
| | - I Letter
- Statistics Department, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB, UK
| | - S Martínez
- Department of Mathematical Engineering and Center of Mathematical Modeling, UMI 2807 UCHILE-CNRS, Universidad de Chile, Santiago, Chile
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11
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Entropy of killed-resurrected stationary Markov chains. J Appl Probab 2021. [DOI: 10.1017/jpr.2020.81] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
AbstractWe consider a strictly substochastic matrix or a stochastic matrix with absorbing states. By using quasi-stationary distributions we show that there is an associated canonical Markov chain that is built from the resurrected chain, the absorbing states, and the hitting times, together with a random walk on the absorbing states, which is necessary for achieving time stationarity. Based upon the 2-stringing representation of the resurrected chain, we supply a stationary representation of the killed and the absorbed chains. The entropies of these representations have a clear meaning when one identifies the probability measure of natural factors. The balance between the entropies of these representations and the entropy of the canonical chain serves to check the correctness of the whole construction.
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12
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A probabilistic proof of Cooper and Frieze's First Visit Time Lemma. LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS 2021. [DOI: 10.30757/alea.v18-64] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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13
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Denisov S, Vershinina O, Thingna J, Hänggi P, Ivanchenko M. Quasi-stationary states of game-driven systems: A dynamical approach. CHAOS (WOODBURY, N.Y.) 2020; 30:123145. [PMID: 33380033 DOI: 10.1063/5.0019736] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2020] [Accepted: 12/02/2020] [Indexed: 06/12/2023]
Abstract
Evolutionary game theory is a framework to formalize the evolution of collectives ("populations") of competing agents that are playing a game and, after every round, update their strategies to maximize individual payoffs. There are two complementary approaches to modeling evolution of player populations. The first addresses essentially finite populations by implementing the apparatus of Markov chains. The second assumes that the populations are infinite and operates with a system of mean-field deterministic differential equations. By using a model of two antagonistic populations, which are playing a game with stationary or periodically varying payoffs, we demonstrate that it exhibits metastable dynamics that is reducible neither to an immediate transition to a fixation (extinction of all but one strategy in a finite-size population) nor to the mean-field picture. In the case of stationary payoffs, this dynamics can be captured with a system of stochastic differential equations and interpreted as a stochastic Hopf bifurcation. In the case of varying payoffs, the metastable dynamics is much more complex than the dynamics of the means.
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Affiliation(s)
- Sergey Denisov
- Department of Computer Science, Oslo Metropolitan University, N-0130 Oslo, Norway
| | - Olga Vershinina
- Department of Applied Mathematics, Lobachevsky University, 603950 Nizhny Novgorod, Russia
| | - Juzar Thingna
- Center for Theoretical Physics of Complex Systems (IBS), Daejeon 34126, South Korea
| | - Peter Hänggi
- Institut für Physik, Universität Augsburg, D-86135 Augsburg, Germany
| | - Mikhail Ivanchenko
- Department of Applied Mathematics, Lobachevsky University, 603950 Nizhny Novgorod, Russia
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14
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Zhou K, Dobson I, Wang Z, Roitershtein A, Ghosh AP. A Markovian influence graph formed from utility line outage data to mitigate large cascades. IEEE TRANSACTIONS ON POWER SYSTEMS : A PUBLICATION OF THE POWER ENGINEERING SOCIETY 2020; 35:3224-3235. [PMID: 32565614 PMCID: PMC7304557 DOI: 10.1109/tpwrs.2020.2970406] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
We use observed transmission line outage data to make a Markovian influence graph that describes the probabili- ties of transitions between generations of cascading line outages. Each generation of a cascade consists of a single line outage or multiple line outages. The new influence graph defines a Markov chain and generalizes previous influence graphs by including multiple line outages as Markov chain states. The generalized influence graph can reproduce the distribution of cascade size in the utility data. In particular, it can estimate the probabilities of small, medium and large cascades. The influence graph has the key advantage of allowing the effect of mitigations to be analyzed and readily tested, which is not available from the observed data. We exploit the asymptotic properties of the Markov chain to find the lines most involved in large cascades and show how upgrades to these critical lines can reduce the probability of large cascades.
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Affiliation(s)
- Kai Zhou
- KZ, ID, ZW are with Electrical and Computer Engineering dept., Iowa State University, Ames IA USA; AR is with Statistics dept., Texas A&M University, College Station TX USA; APG is with Statistics dept., Iowa State University
| | - Ian Dobson
- KZ, ID, ZW are with Electrical and Computer Engineering dept., Iowa State University, Ames IA USA; AR is with Statistics dept., Texas A&M University, College Station TX USA; APG is with Statistics dept., Iowa State University
| | - Zhaoyu Wang
- KZ, ID, ZW are with Electrical and Computer Engineering dept., Iowa State University, Ames IA USA; AR is with Statistics dept., Texas A&M University, College Station TX USA; APG is with Statistics dept., Iowa State University
| | - Alexander Roitershtein
- KZ, ID, ZW are with Electrical and Computer Engineering dept., Iowa State University, Ames IA USA; AR is with Statistics dept., Texas A&M University, College Station TX USA; APG is with Statistics dept., Iowa State University
| | - Arka P Ghosh
- KZ, ID, ZW are with Electrical and Computer Engineering dept., Iowa State University, Ames IA USA; AR is with Statistics dept., Texas A&M University, College Station TX USA; APG is with Statistics dept., Iowa State University
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15
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On the Structure of the World Economy: An Absorbing Markov Chain Approach. ENTROPY 2020; 22:e22040482. [PMID: 33286256 PMCID: PMC7516966 DOI: 10.3390/e22040482] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/10/2020] [Revised: 04/13/2020] [Accepted: 04/21/2020] [Indexed: 11/24/2022]
Abstract
The expansion of global production networks has raised many important questions about the interdependence among countries and how future changes in the world economy are likely to affect the countries’ positioning in global value chains. We are approaching the structure and lengths of value chains from a completely different perspective than has been available so far. By assigning a random endogenous variable to a network linkage representing the number of intermediate sales/purchases before absorption (final use or value added), the discrete-time absorbing Markov chains proposed here shed new light on the world input/output networks. The variance of this variable can help assess the risk when shaping the chain length and optimize the level of production. Contrary to what might be expected simply on the basis of comparative advantage, the results reveal that both the input and output chains exhibit the same quasi-stationary product distribution. Put differently, the expected proportion of time spent in a state before absorption is invariant to changes of the network type. Finally, the several global metrics proposed here, including the probability distribution of global value added/final output, provide guidance for policy makers when estimating the resilience of world trading system and forecasting the macroeconomic developments.
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16
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Nguyen QT, Tran KP, Castagliola P, Celano G, Lardjane S. One-sided synthetic control charts for monitoring the multivariate coefficient of variation. J STAT COMPUT SIM 2019. [DOI: 10.1080/00949655.2019.1600694] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
| | - Kim Phuc Tran
- GEMTEX Laboratory, Ecole Nationale Supérieure des Arts et Industries Textiles, Roubaix Cedex 1, France
| | | | | | - Salim Lardjane
- LMBA UMR CNRS 6205, Université de Bretagne-Sud, Vannes, France
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17
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Knoth S. The steady-state behavior of multivariate exponentially weighted moving average control charts. Seq Anal 2019. [DOI: 10.1080/07474946.2018.1554890] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- Sven Knoth
- Department of Mathematics and Statistics, Helmut Schmidt University, Hamburg, Germany
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18
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Liu X, Mubayi A, Reinhold D, Zhu L. Approximation methods for analyzing multiscale stochastic vector-borne epidemic models. Math Biosci 2019; 309:42-65. [PMID: 30658089 DOI: 10.1016/j.mbs.2019.01.003] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/21/2018] [Revised: 01/08/2019] [Accepted: 01/11/2019] [Indexed: 11/29/2022]
Abstract
Stochastic epidemic models, generally more realistic than deterministic counterparts, have often been seen too complex for rigorous mathematical analysis because of level of details it requires to comprehensively capture the dynamics of diseases. This problem further becomes intense when complexity of diseases increases as in the case of vector-borne diseases (VBD). The VBDs are human illnesses caused by pathogens transmitted among humans by intermediate species, which are primarily arthropods. In this study, a stochastic VBD model is developed and novel mathematical methods are described and evaluated to systematically analyze the model and understand its complex dynamics. The VBD model incorporates some relevant features of the VBD transmission process including demographical, ecological and social mechanisms, and different host and vector dynamic scales. The analysis is based on dimensional reductions and model simplifications via scaling limit theorems. The results suggest that the dynamics of the stochastic VBD depends on a threshold quantity R0, the initial size of infectives, and the type of scaling in terms of host population size. The quantity R0 for deterministic counterpart of the model is interpreted as a threshold condition for infection persistence as is mentioned in the literature for many infectious disease models. Different scalings yield different approximations of the model, and in particular, if vectors have much faster dynamics, the effect of the vector dynamics on the host population averages out, which largely reduces the dimension of the model. Specific scenarios are also studied using simulations for some fixed sets of parameters to draw conclusions on dynamics.
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Affiliation(s)
- Xin Liu
- Department of Mathematical Sciences, Clemson University, South Carolina, United States.
| | - Anuj Mubayi
- School of Human Evolution and Social Change; Simon A. Levin Mathematical Computational and Modeling Science Center, Arizona State University, Tempe, Arizona, United States.
| | - Dominik Reinhold
- Department of Biostatistics and Informatics, University of Colorado, Denver, Colorado, United States.
| | - Liu Zhu
- Department of Mathematical Sciences, Clemson University, South Carolina, United States.
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19
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Girardin V, Lhote L, Regnault P. Different Closed-Form Expressions for Generalized Entropy Rates of Markov Chains. Methodol Comput Appl Probab 2018. [DOI: 10.1007/s11009-018-9679-3] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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20
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Singer A, Frank K. Viability of cyclic populations. Ecology 2016; 97:3143-3153. [PMID: 27870022 DOI: 10.1002/ecy.1564] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 03/22/2016] [Revised: 06/13/2016] [Accepted: 08/05/2016] [Indexed: 11/05/2022]
Abstract
Theory on viability of small populations is well developed and has led to the standard methodology of population viability analysis (PVA) to assess vulnerability of single species. However, more complex situations involving community dynamics or environmental change violate theoretical assumptions. Synthesizing concepts from population, community, and conservation ecology, we develop a generic theory on the viability of cyclic populations. The interplay of periodic population decline and demography causes varying risk patterns that aggregate during cycles and modify the temporal structure of viability. This variability is visualized and quantitatively assessed. For two standard viability metrics that summarize immediate extinction risk and the general long-term conditions of populations, we mathematically describe the impact of population cycles. Finally, we suggest and demonstrate PVA for cyclic populations that respond to, e.g., seasonality, interannual variation, or trophic interactions. Our theoretical and methodological advancement opens a route to viability analysis in food webs and trophic meta-communities and equips biodiversity conservation with a long-missing tool.
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Affiliation(s)
- Alexander Singer
- Department of Ecological Modelling, Helmholtz-Centre for Environmental Research-UFZ, Permoserstrasse 15, D-04318, Leipzig, Germany.,German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Deutscher Platz 5e, D-04103, Leipzig, Germany
| | - Karin Frank
- Department of Ecological Modelling, Helmholtz-Centre for Environmental Research-UFZ, Permoserstrasse 15, D-04318, Leipzig, Germany.,German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Deutscher Platz 5e, D-04103, Leipzig, Germany.,Institute for Environmental Systems Research, University of Osnabrück, Barbarastrasse 12, D-49076, Osnabrück, Germany
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21
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Zhao L, Lascoux M, Waxman D. An informational transition in conditioned Markov chains: Applied to genetics and evolution. J Theor Biol 2016; 402:158-70. [PMID: 27105672 DOI: 10.1016/j.jtbi.2016.04.021] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2015] [Revised: 02/19/2016] [Accepted: 04/17/2016] [Indexed: 11/18/2022]
Abstract
In this work we assume that we have some knowledge about the state of a population at two known times, when the dynamics is governed by a Markov chain such as a Wright-Fisher model. Such knowledge could be obtained, for example, from observations made on ancient and contemporary DNA, or during laboratory experiments involving long term evolution. A natural assumption is that the behaviour of the population, between observations, is related to (or constrained by) what was actually observed. The present work shows that this assumption has limited validity. When the time interval between observations is larger than a characteristic value, which is a property of the population under consideration, there is a range of intermediate times where the behaviour of the population has reduced or no dependence on what was observed and an equilibrium-like distribution applies. Thus, for example, if the frequency of an allele is observed at two different times, then for a large enough time interval between observations, the population has reduced or no dependence on the two observed frequencies for a range of intermediate times. Given observations of a population at two times, we provide a general theoretical analysis of the behaviour of the population at all intermediate times, and determine an expression for the characteristic time interval, beyond which the observations do not constrain the population's behaviour over a range of intermediate times. The findings of this work relate to what can be meaningfully inferred about a population at intermediate times, given knowledge of terminal states.
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Affiliation(s)
- Lei Zhao
- Centre for Computational Systems Biology, Fudan University, 220 Handan Road, Shanghai 200433, PR China
| | - Martin Lascoux
- Centre for Computational Systems Biology, Fudan University, 220 Handan Road, Shanghai 200433, PR China; Evolutionary Biology Center, Department of Ecology and Genetics, Uppsala University, Uppsala 75236, Sweden
| | - David Waxman
- Centre for Computational Systems Biology, Fudan University, 220 Handan Road, Shanghai 200433, PR China.
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Abstract
The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.
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23
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Abstract
We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.
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24
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Abstract
The quasi-stationary behaviour of a Markov chain which is φ-irreducible when restricted to a subspace of a general state space is investigated. It is shown that previous work on the case where the subspace is finite or countably infinite can be extended to general chains, and the existence of certain quasi-stationary limits as honest distributions is equivalent to the restricted chain being R-positive with the unique R-invariant measure satisfying a certain finiteness condition.
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25
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Abstract
Quasi-stationary distributions, as discussed in Darroch and Seneta (1965), have been used in biology to describe the steady state behaviour of population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. These distributions have some drawbacks: they need not exist, nor be unique, and their calculation can present problems. In this paper, we give biologically plausible conditions under which the quasi-stationary distribution is unique, and can be closely approximated by distributions that are simple to compute.
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26
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Ledoux J, Rubino G, Sericola B. Exact aggregation of absorbing Markov processes using the quasi-stationary distribution. J Appl Probab 2016. [DOI: 10.2307/3215143] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
We characterize the conditions under which an absorbing Markovian finite process (in discrete or continuous time) can be transformed into a new aggregated process conserving the Markovian property, whose states are elements of a given partition of the original state space. To obtain this characterization, a key tool is the quasi-stationary distribution associated with absorbing processes. It allows the absorbing case to be related to the irreducible one. We are able to calculate the set of all initial distributions of the starting process leading to an aggregated homogeneous Markov process by means of a finite algorithm. Finally, it is shown that the continuous-time case can always be reduced to the discrete one using the uniformization technique.
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27
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Abstract
We characterize the conditions under which an absorbing Markovian finite process (in discrete or continuous time) can be transformed into a new aggregated process conserving the Markovian property, whose states are elements of a given partition of the original state space. To obtain this characterization, a key tool is the quasi-stationary distribution associated with absorbing processes. It allows the absorbing case to be related to the irreducible one. We are able to calculate the set of all initial distributions of the starting process leading to an aggregated homogeneous Markov process by means of a finite algorithm. Finally, it is shown that the continuous-time case can always be reduced to the discrete one using the uniformization technique.
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28
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Abstract
In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.
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29
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Abstract
The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.
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30
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On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals. J Appl Probab 2016. [DOI: 10.1017/s0021900200035580] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable
denote the first busy period initiated by a waiting time u at time t = 0.
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31
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Abstract
The general (non-spatial) stochastic epidemic is extended to allow infective individuals to move forward through a system of spatially connected locations · ··, L
1, L
2, · ·· (on the line) each containing susceptible individuals and the outcome of the epidemic in each of these locations is then considered. In the deterministic case, a (spatial) equilibrium solution and threshold behaviour are discussed. In the stochastic case, a (spatial) quasi-equilibrium behaviour (conditional on sufficient numbers of infectives present) is discussed; numerical results suggest some correspondence between this stochastic quasi-equilibrium and the deterministic equilibrium.
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32
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Tweedie RL. Quasi-stationary distributions for Markov chains on a general state space. J Appl Probab 2016. [DOI: 10.2307/3212556] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
The quasi-stationary behaviour of a Markov chain which is φ-irreducible when restricted to a subspace of a general state space is investigated. It is shown that previous work on the case where the subspace is finite or countably infinite can be extended to general chains, and the existence of certain quasi-stationary limits as honest distributions is equivalent to the restricted chain being R-positive with the unique R-invariant measure satisfying a certain finiteness condition.
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33
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Abstract
Our main concern in this paper is the convergence, as t → ∞, of the quantities
i, j ∈ E; where Pij
(t) is the transition probability of a semi-Markov process whose state space E is irreducible but not closed (i.e., escape from E is possible), and rj
is the probability of eventual escape from E conditional on the initial state being i. The theorems proved here generalize some results of Seneta and Vere-Jones ([8] and [11]) for Markov processes.
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34
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Abstract
Quasi-stationary distributions are considered in their own right, and from the standpoint of finite approximations, for absorbing birth-death processes. Results on convergence of finite quasi-stationary distributions and a stochastic bound for an infinite quasi-stationary distribution are obtained. These results are akin to those of Keilson and Ramaswamy (1984). The methodology is a synthesis of Good (1968) and Cavender (1978).
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35
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An Efficient Procedure for Computing Quasi-Stationary Distributions of Markov Chains by Sparse Transition Structure. ADV APPL PROBAB 2016. [DOI: 10.1017/s000186780002601x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
We describe a computational procedure for evaluating the quasi-stationary distributions of a continuous-time Markov chain. Our method, which is an ‘iterative version' of Arnoldi's algorithm, is appropriate for dealing with cases where the matrix of transition rates is large and sparse, but does not exhibit a banded structure which might otherwise be usefully exploited. We illustrate the method with reference to an epidemic model and we compare the computed quasi-stationary distribution with an appropriate diffusion approximation.
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36
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Abstract
We consider an absorbing semi-Markov chain for which each time absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The new process is a semi-Markov replacement chain and we study its properties in terms of those of the imbedded Markov replacement chain. A time-dependent version of the model is also defined and analysed asymptotically for two types of environmental behaviour, i.e. either convergent or cyclic. The results contribute to the control theory of semi-Markov chains and extend in a natural manner a wide variety of applied probability models. An application to the modelling of populations with semi-Markovian replacements is also presented.
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37
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38
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Numerically Stable Methods for the Computation of Exit Rates in Markov Chains. Methodol Comput Appl Probab 2016. [DOI: 10.1007/s11009-014-9417-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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39
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Abstract
In this paper we use a simple transient Markov process with an absorbing point to investigate the qualitative behavior of a large-scale storage network of nonreliable file servers across which files can be duplicated. When the size of the system goes to ∞, we show that there is a critical value for the maximum number of files per server such that, below this quantity, most files have a maximum number of copies. Above this value, the network loses a significant number of files until some equilibrium is reached. When the network is stable, we show that, with convenient time scales, the evolution of the network towards the absorbing state can be described via a stochastic averaging principle.
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40
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The influence of genetic drift on the formation and stability of polymorphisms arising from negative frequency-dependent selection. J Theor Biol 2016; 391:51-64. [DOI: 10.1016/j.jtbi.2015.11.011] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2015] [Revised: 11/13/2015] [Accepted: 11/17/2015] [Indexed: 11/20/2022]
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41
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Keller P, Rœlly S, Valleriani A. On Time Duality for Markov Chains. STOCH MODELS 2015. [DOI: 10.1080/15326349.2014.969736] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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42
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Exponential convergence to quasi-stationary distribution and $$Q$$ Q -process. Probab Theory Relat Fields 2015. [DOI: 10.1007/s00440-014-0611-7] [Citation(s) in RCA: 62] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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43
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Benaïm M, Cloez B. A stochastic approximation approach to quasi-stationary
distributions on finite spaces. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2015. [DOI: 10.1214/ecp.v20-3956] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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44
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Barbillon P, Thomas M, Goldringer I, Hospital F, Robin S. Network impact on persistence in a finite population dynamic diffusion model: application to an emergent seed exchange network. J Theor Biol 2014; 365:365-76. [PMID: 25451529 DOI: 10.1016/j.jtbi.2014.10.032] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2014] [Revised: 09/16/2014] [Accepted: 10/24/2014] [Indexed: 11/26/2022]
Abstract
Dynamic extinction colonisation models (also called contact processes) are widely studied in epidemiology and in metapopulation theory. Contacts are usually assumed to be possible only through a network of connected patches. This network accounts for a spatial landscape or a social organization of interactions. Thanks to social network literature, heterogeneous networks of contacts can be considered. A major issue is to assess the influence of the network in the dynamic model. Most work with this common purpose uses deterministic models or an approximation of a stochastic Extinction-Colonisation model (sEC) which are relevant only for large networks. When working with a limited size network, the induced stochasticity is essential and has to be taken into account in the conclusions. Here, a rigorous framework is proposed for limited size networks and the limitations of the deterministic approximation are exhibited. This framework allows exact computations when the number of patches is small. Otherwise, simulations are used and enhanced by adapted simulation techniques when necessary. A sensitivity analysis was conducted to compare four main topologies of networks in contrasting settings to determine the role of the network. A challenging case was studied in this context: seed exchange of crop species in the Réseau Semences Paysannes (RSP), an emergent French farmers׳ organisation. A stochastic Extinction-Colonisation model was used to characterize the consequences of substantial changes in terms of RSP׳s social organization on the ability of the system to maintain crop varieties.
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Affiliation(s)
- Pierre Barbillon
- AgroParisTech / UMR INRA MIA, F-75005 Paris, France; INRA, UMR 518, F-75005 Paris, France.
| | - Mathieu Thomas
- AgroParisTech / UMR INRA MIA, F-75005 Paris, France; INRA, UMR 518, F-75005 Paris, France; INRA, UMR 0320 / UMR 8120 Génétique Végétale, F-91190 Gif-sur-Yvette, France
| | - Isabelle Goldringer
- INRA, UMR 0320 / UMR 8120 Génétique Végétale, F-91190 Gif-sur-Yvette, France
| | - Frédéric Hospital
- INRA, UMR 1313 Génétique Animale et Biologie Intégrative, F-78352 Jouy-en-Josas, France
| | - Stéphane Robin
- AgroParisTech / UMR INRA MIA, F-75005 Paris, France; INRA, UMR 518, F-75005 Paris, France
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45
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Hössjer O. On the eigenvalue effective size of structured populations. J Math Biol 2014; 71:595-646. [PMID: 25230676 PMCID: PMC4532751 DOI: 10.1007/s00285-014-0832-5] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2013] [Revised: 08/16/2014] [Indexed: 11/26/2022]
Abstract
A general theory is developed for the eigenvalue effective size (N(e)E) of structured populations in which a gene with two alleles segregates in discrete time. Generalizing results of Ewens (Theor Popul Biol 21:373-378, 1982), we characterize N(e)E in terms of the largest non-unit eigenvalue of the transition matrix of a Markov chain of allele frequencies. We use Perron-Frobenius Theorem to prove that the same eigenvalue appears in a linear recursion of predicted gene diversities between all pairs of subpopulations. Coalescence theory is employed in order to characterize this recursion, so that explicit novel expressions for N(e)E can be derived. We then study N(e)E asymptotically, when either the inverse size and/or the overall migration rate between subpopulations tend to zero. It is demonstrated that several previously known results can be deduced as special cases. In particular when the coalescence effective size N(e)C exists, it is an asymptotic version of N(e)E in the limit of large populations.
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Affiliation(s)
- Ola Hössjer
- Divsion of Mathematical Statistics, Department of Mathematics, Stockholm University, Stockholm, Sweden,
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46
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Waxman D, Weinert LA, Welch JJ. Inferring host range dynamics from comparative data: the protozoan parasites of new world monkeys. Am Nat 2014; 184:65-74. [PMID: 24921601 DOI: 10.1086/676589] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Abstract
Uncovering the ecological determinants of parasite host range is a central goal of comparative parasitology and infectious disease ecology. But while parasites are often distributed nonrandomly across the host phylogeny, such patterns are difficult to interpret without a genealogy for the parasite samples and without knowing what sorts of ecological dynamics might lead to what sorts of nonrandomness. We investigated inferences from comparative data, using presence/absence records from protozoan parasites of the New World monkeys. We first demonstrate several distinct types of phylogenetic signal in these data, showing, for example, that parasite species are clustered on the host tree and that closely related host species harbor similar numbers of parasite species. We then show that all of these patterns can be generated by a single, simple dynamical model, in which parasite host range changes more rapidly than host speciation/extinction and parasites preferentially colonize uninfected host species that are closely related to their existing hosts. Fitting this model to data, we then estimate its parameters. Finally, we caution that quite different ecological processes can lead to similar signatures but show how phylogenetic variation in host susceptibility can be distinguished from a tendency for parasites to colonize closely related hosts. Our new process-based analyses, which estimate meaningful parameters, should be useful for inferring the determinants of parasite host range and transmission success.
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Affiliation(s)
- David Waxman
- Centre for Computational Systems Biology, Fudan University, Shanghai 200433, People's Republic of China
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47
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Faure M, Schreiber SJ. Quasi-stationary distributions for randomly perturbed dynamical systems. ANN APPL PROBAB 2014. [DOI: 10.1214/13-aap923] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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48
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Olsson F, Hössjer O, Laikre L, Ryman N. Characteristics of the variance effective population size over time using an age structured model with variable size. Theor Popul Biol 2013; 90:91-103. [DOI: 10.1016/j.tpb.2013.09.014] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2013] [Revised: 09/23/2013] [Accepted: 09/26/2013] [Indexed: 10/26/2022]
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49
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Abstract
We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth-death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.
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50
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Gerlee P, Lundh T. Effect of space in the game "war of attrition". PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:041115. [PMID: 22680427 DOI: 10.1103/physreve.85.041115] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2011] [Revised: 08/31/2011] [Indexed: 06/01/2023]
Abstract
Spatial dynamics has in many cases been invoked as a mechanism that can promote the evolution of coexistence and cooperation, although the precise conditions for this to occur have not yet been characterised. In an effort to address this question we have analyzed an alternative version of the theoretical game "war of attrition," which exhibits unusual behavior: The well-mixed system exhibits quasistationary coexistence and a relaxation time that scales exponentially with the system size, while the spatial system shows a relaxation time that is considerably smaller and scales with a power α≈1.4 of the system size. Further, the spatial system exhibits a first-order phase transition in the strategy distribution at a consolation prize of k≈1/3. Close to this point the relaxation time diverges with an exponent γ≈1.2. This analysis shows that the effect of space is highly dependent on the type of game considered.
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Affiliation(s)
- Philip Gerlee
- Mathematical Sciences, Chalmers University of Technology, S-412 96 Gothenburg, Sweden.
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